pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which...

408
ISSN 1361 - 6161 Computer Science University of Manchester AiML-2004: Advances in Modal Logic International Conference on Advances in Modal Logic Manchester, UK, September 9–11, 2004, Preliminary Conference Proceedings Renate Schmidt, Ian Pratt-Hartmann, Mark Reynolds, Heinrich Wansing (eds.) Department of Computer Science University of Manchester Technical Report Series UMCS-04-9-1

Transcript of pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which...

Page 1: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

ISSN 1361 - 6161

Computer ScienceUnivers i ty o f Manchester

AiML-2004: Advances in Modal LogicInternational Conference on Advances in Modal LogicManchester, UK, September 9–11, 2004,Preliminary Conference Proceedings

Renate Schmidt, Ian Pratt-Hartmann, Mark Reynolds,

Heinrich Wansing (eds.)

Department of Computer Science

University of Manchester

Technical Report Series

UMCS-04-9-1

Page 2: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative
Page 3: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

AiML-2004: Advances in Modal Logic

International Conference on Advances in Modal LogicManchester, UK, September 9–11, 2004Preliminary Conference Proceedings

Renate A. Schmidt, Ian Pratt-Hartmann

Department of Computer ScienceUniversity of Manchester, UK

{schmidt,ipratt}@cs.man.ac.uk

Mark Reynolds

School of Computer Science & Software EngineeringUniversity of Western Australia, Perth, Australia

[email protected]

Heinrich Wansing

Institute of PhilosophyDresden University of Technology, Germany

[email protected]

September 2004

Copyright c©2004, University of Manchester. All rights reserved. Reproduction (electron-ically or by other means) of all or part of this work is permitted for educational or researchpurposes only, on condition that (1) this copyright notice is included, (2) proper attributionto the author or authors is made, (3) no commercial gain is involved, and (4) the document isreproduced without any alteration whatsoever.

Recent technical reports issued by the Department of Computer Science, Manchester Uni-

versity, can be obtained via http://www.cs.man.ac.uk/cstechrep/index.html.

Page 4: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative
Page 5: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Preface

Advances in Modal Logic is the main international forum at which research on all aspectsof modal logic is presented. The Advances in Modal Logic Initiative was founded in 1995and the first AiML Meeting was held in 1996 in Berlin, Germany. Since then the AiMLMeeting has been organised on a bi-annual basis with previous meetings being held in1998 in Uppsala, Sweden, in 2000 in Leipzig, Germany (jointly with ICTL-2000), and in2002 in Toulouse, France. AiML-2004 is the fifth event organized as part of this initiativeand held September 9–11, 2004 in Manchester, UK. As the first meeting to have thestatus of an international conference it reflects the overall technical quality of the event,the increasing impact research in modal logic has and the growing recognition the eventreceives among researchers in the modal logic community and those working in areasrelated to modal logic.

This report contains the preliminary versions of papers presented at AiML-2004, in-cluding 6 contributions by invited speakers and 28 accepted regular papers selected from61 submissions in total. The contributed papers cover a wide range of topics in modallogic, extending from foundational papers on the mathematics of modal logic, prooftheory, computational aspects, philosophical issues, to more ‘applied’ research where ex-tended modal logic formalisms and modelling of knowledge, belief, time and actions arecrucial. After another round of refereeing, the revised full papers will be collected andsubmitted to King’s College Publications for publication as Volume 5 of the Advances in

Modal Logic book series.We thank all authors who submitted papers, all participants of the conference as

well as the distinguished invited speakers for their contributions. We are grateful tothe members of the programme committee and the additional referees for reviewing thesubmitted papers and helping with the selection process. We are especially grateful to themembers of the local organization team for all their help: the staff in the ACSO office andthe finance office, in particular, Helen Spragg who handled the registration of participants,Zhen Li for collecting the papers and compiling this report, and Michael Ebert for thecomputer support. Finally, we thank the sponsors for their generous support.

September 2004 Renate SchmidtIan Pratt-Hartmann

Mark ReynoldsHeinrich Wansing

ii

Page 6: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Organization

AiML-2004 was hosted by the Department of Computer Science at the Chancellors Res-idential Conference Centre of the University of Manchester. It was co-located with theTARSKI Workshop of EU COST Action 274.

Programme Co-Chairs

Mark Reynolds (University of Western Australia, Australia)Heinrich Wansing (Dresden University of Technology, Germany)

Programme Committee

Patrick Blackburn (INRIA Lorraine, Nancy, France)Alexander Chagrov (Tver University, Russia)Vincent Hendricks (Roskilde University, Denmark)Ian Pratt-Hartmann (University of Manchester, UK)Mark Reynolds (University of Western Australia, Australia)Maarten de Rijke (University of Amsterdam, The Netherlands)Ulrike Sattler (University of Manchester, UK)Holger Schlingloff (Berlin, Germany)Renate Schmidt (University of Manchester, UK)Nobu-Yuki Suzuki (Shizuoka University, Japan)Heinrich Wansing (Dresden University of Technology, Germany)Frank Wolter (University of Liverpool, UK)Michael Zakharyaschev (King’s College London, UK)

Local Organizing Committee

Renate Schmidt (Chair) (University of Manchester, UK)Ian Pratt-Hartmann (University of Manchester, UK)

AiML Steering Committee

Mark Reynolds (University of Western Australia, Australia)Maarten de Rijke (University of Amsterdam, The Netherlands)Renate Schmidt (University of Manchester, UK)Nobu-Yuki Suzuki (Shizuoka University, Japan)Heinrich Wansing (Dresden University of Technology, Germany)Frank Wolter (University of Liverpool, UK)Michael Zakharyaschev (King’s College London, UK)

iii

Page 7: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Sponsors

– TARSKI EU COST Action 274 on the ‘Theory and Applications of RelationalStructures as Knowledge Instruments’

– CologNet, Work areas ‘Logic and multi-agent system’ and ‘Automated reasoning,deduction, theorem proving, and model-checking’

– School of Information Technology at Murdoch University

– British Logic Colloquium

– Language and Inference Technology Group at the University Amsterdam

– Department of Computer Science at the University Manchester

– Advances in Modal Logic Initiative

Additional Reviewers

Adrianna AlexanderPhilippe BalbianiHoward BarringerJochen BurghardtMelvin FittingTim FrenchPaul GochetIan HodkinsonJohn HortyUllrich HustadtRosalie IemhoffManfred JaegerRyo KashimaOliver KutzGerhard LakemeyerChristoph Luth

Carsten LutzMaarten MarxTill MossakowskiLarry MossMarco RagniMarkus RoggenbachTakafumi SakuraiKatsumi SasakiKen SatohKarsten SchmidtLutz SchroederYoshihito TanakaDmitry TishkovskyStefan Wolflplus other anonymous reviewers

iv

Page 8: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Table of Contents

I Invited Talks 1Variants of PDL with Intersection of Programs

Philippe Balbiani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2The Logic of Jon Barwise (1942–2000)

Keith Devlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Elementary Canonical Formulae: Syntactic, Model-Theoretic, and Algorithmic

AspectsValentin Goranko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Model Checking Epistemic PropertiesWiebe van der Hoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Modal Logic, Xpath and XMLMaarten Marx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

On what there isn’t (but might have been)Robert Stalnaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

II Regular Papers 15Pure Extensions, Proof Rules and Hybrid Axiomatics

Patrick Blackburn, Balder ten Cate . . . . . . . . . . . . . . . . . . . . . 16The Complexity of Strict Implication Logics

Felix Bou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Sahlqvist Theory and Transfer Results for Hybrid Logic

Balder ten Cate, Maarten Marx, Petrucio Viana, Nick Bezhanishvilli . . 44Public Announcements and Belief Revision

Hans van Ditmarsch, Wiebe van der Hoek, Barteld Kooi . . . . . . . . . 62Model Checking, Preprocessing, and BDD Size

Andrea Ferrara, Paolo Liberatore, Marco Schaerf . . . . . . . . . . . . . 74Complete Axiomatizations for Logics of Knowledge and Past Time

Tim French, Ron van der Meyden, Mark Reynolds . . . . . . . . . . . . . 89Products of ‘Transitive’ Modal Logics without the (Abstract) Finite Model

PropertyD. Gabelaia, A. Kurucz, M. Zakharyaschev . . . . . . . . . . . . . . . . . 104

Interpolation and the interpretability logic of PAEvan Goris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

On the Axiomatization of Elgesem’s Logic of AgencyGuido Governatori, Antonino Rotolo . . . . . . . . . . . . . . . . . . . . 130

A Two-Sorted Hybrid Logic with Guarded JumpsBernhard Heinemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

On Modularity of TheoriesAndreas Herzig, Ivan Varzinczak . . . . . . . . . . . . . . . . . . . . . . 158

v

Page 9: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Decidability of IF Modal Logic of Perfect RecallTapani Hyttinen, Tero Tulenheimo . . . . . . . . . . . . . . . . . . . . . 167

On Dynamic Topological and Metric LogicsBoris Konev, Roman Kontchakov, Frank Wolter, Michael Zakharyaschev 182

Reduction Axioms for Epistemic ActionsBarteld Kooi, Johan van Benthem . . . . . . . . . . . . . . . . . . . . . . 197

Strong Completeness for Non-Compact Hybrid LogicsBarteld Kooi, Gerard Renardel de Lavalette, Rineke Verbrugge . . . . . . 212

A Lower Complexity Bound for Propositional Dynamic Logic with IntersectionMartin Lange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

On Notions of Completeness Weaker than Kripke CompletenessTadeusz Litak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Modal Logics of Topological RelationsCarsten Lutz, Frank Wolter . . . . . . . . . . . . . . . . . . . . . . . . . 249

Normal Modal Logics Containing KTB with some Finiteness ConditionsYutaka Miyazaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

On the Formal Structure of Continuous ActionThomas Muller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Utilitarian Deontic LogicYuko Murakami . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Resolution for Synchrony and No LearningClaudia Nalon, Clare Dixon, Michael Fisher . . . . . . . . . . . . . . . . 303

On the Complexity of Fragments of Modal LogicsLinh Anh Nguyen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

PSPACE Decision Procedure for some Transitive Modal LogicsIlya Shapirovsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

A New Version of the Filtration MethodValentin Shehtman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

A Systematic Proof Theory for Several Modal LogicsCharles Stewart, Phiniki Stouppa . . . . . . . . . . . . . . . . . . . . . . 357

Consistency proofs for systems of multi-agent only knowingArild Waaler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

Connexive Modal LogicHeinrich Wansing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

vi

Page 10: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Part I

Invited Talks

1

Page 11: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Variants of PDL with intersection of programs

Philippe Balbiani

Institut de recherche en informatique de Toulouse

Dynamic logics are logics arising from the combination of relation algebraswith the modality of necessity. Together with their variations and close relatives,they constitute a part of the field of modal logic that is concerned in actions andprograms. The propositional versions of dynamic logics are collectively calledPDL. In PDL, a modal connective [α] is associated with each program α of aprogramming language formed from atomic programs by induction using oper-ations on binary relations. The chief operations on binary relations consideredsince the first development of PDL have been the operations ;, ∪, and ?. The op-eration ; is the sequential composition operation, with programs like α; β beingread “do α, then do β”. ∪ is the operation of nondeterministic choice. Programsof the form α∪ β mean “nondeterministically choose one of α or β and executeit”. As for ?, the operation of iteration, the intended meaning of a programlike α? is “execute α some nondeterministically chosen finite number of times”.The semantics of PDL comes from the semantics of modal logic. A frame forthe language sketchily described above should be a Kripke structure of the form(W, {Rα: α is a program}) with Rα a binary relation on W for each programα. Intuitively, we can think of the binary relations Rα as sets of input / outputpairs of states. Thus: Rα;β should be equal to the relational composition of Rα

and Rβ , the input / output relation associated to α ∪ β should correspond tothe union of the relations Rα, Rβ, and α? should be interpreted by the reflexivetransitive closure of Rα. From the beginning, a great deal of attention has beenfocused on the complexity of PDL. Papers [5, 14] are but a sample of the workthat has been done: the exponential-time lower bound for PDL being estab-lished by Fischer and Ladner [5] and deterministic exponential-time algorithmsbeing given by Pratt [14]. But perhaps the key results on PDL come fromSegerberg [15] who formulated a Hilbert-style deductive system for PDL andfrom Gabbay [6] and Parikh [12] who showed independently its completeness.

Subsequently, a variety of different extensions of PDL have been proposedfor increasing the expressive power of the logic: PDL with nonregular pro-grams [8], PDL with converse operation [12], PDL with predicates for well-foundedness [16], etc. We shall concentrate in this paper on the extension ofPDL studied in Danecki [3] and Harel et al. [9], PDL with intersection ofprograms, because this extension is linked to a number of PDL-like systemsinvestigated in Farinas del Cerro and Or lowska [4], Gargov and Passy [7], andMirkowska [11] under the various names DAL, BML, and PAL. PDL with

1

2

Page 12: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

intersection is called PDL∩. The intersection operation ∩ is a program oper-ation that allows programs to be run in parallel: programs like α ∩ β beingread “do α and β in parallel” for each α, β. The essential feature of PDL∩ isthat Rα ∩Rβ , the input / output relation associated to α∩β in Kripke frames,is not modally definable in the language of PDL. As a result, less is knownabout the complexity and the proof theory of PDL∩. Regarding the complex-ity of PDL∩, an exponential-space lower bound for PDL∩ has been establishedonly this year [10] whereas it is still unknown at present whether the doublyexponential-time algorithm given by Danecki [3] is near-optimal. As regardsthe proof theory of PDL∩, a sound and complete Hilbert-style deductive sys-tem for an extension of PDL∩ with data constants, i.e. special atomic formulasinterpreted in Kripke frames by singletons, has been presented by Passy andTinchev [13] but technical difficulties have made impracticable the project offinding a sound and complete axiomatization for PDL∩ in its ordinary language.We shall present such an axiom system for PDL∩. Based on a step-by-step con-struction, its completeness proof brings in the new concept of maximal programinitiated by [2] and furthered by [1]. Like maximal theories, which are specialsets of formulas, maximal programs are special sets of programs with propertiesto be thoroughly studied in this paper.

References

[1] Balbiani, P.: Eliminating unorthodox derivation rules in an axiom systemfor iteration-free PDL with intersection. Fundamenta Informaticæ56 (2003)211–242.

[2] Balbiani, P., Vakarelov, D.: Iteration-free PDL with intersection: a com-plete axiomatization. Fundamenta Informaticæ45 (2001) 173–194.

[3] Danecki, R.: Non-deterministic propositional dynamic logic is decidable. InSkowron, A. (Editor): Computation Theory. Springer-Verlag, Lecture Notesin Computer Science 208 (1985) 34–53.

[4] Farinas del Cerro, L., Orlowska, E.: DAL — a logic for data analysis. The-oretical Computer Science 36 (1985) 251–264.

[5] Fisher, M., Ladner, R.: Propositional dynamic logic of regular programs.Journal of Computer and System Sciences 18 (1979) 194–211.

[6] Gabbay, D.: Axiomatizations of logics of programs. Unpublished (1977).

[7] Gargov, G., Passy, S.: A note on Boolean modal logic. In Petkov, P. (Editor):Mathematical Logic. Plenum Press (1990) 299–309.

[8] Harel, D., Pnueli, A., Stavi, J.: Propositional dynamic logic of nonregularprograms. Journal of Computer and System Sciences 26 (1983) 222–243.

2

3

Page 13: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[9] Harel, D., Pnueli, A., Vardi, M.: Two dimensional temporal logic and PDL

with intersection. Unpublished (1982).

[10] Lange, M.: A lower complexity bound for propositional dynamic logic withintersection. This volume.

[11] Mirkowska, G.: PAL — propositional algorithmic logic. Fundamenta In-formaticæ4 (1981) 675–760.

[12] Parikh, R.: The completeness of propositional dynamic logic. InWinkowski, J. (Editor): Mathematical Foundations of Computer Science1978. Springer-Verlag, Lecture Notes in Computer Science 64 (1978) 403–415.

[13] Passy, S., Tinchev, T.: An essay in combinatory dynamic logic. Informationand Computation 93 (1991) 263–332.

[14] Pratt, V.: A near-optimal method for reasoning about actions. Journal ofComputer and System Sciences 20 (1980) 231–254.

[15] Segerberg, K.: A completeness theorem in the modal logic of programs. InTraczyk, T. (Editor): Universal Algebra and Applications. Polish ScientificPublishers, Banach Center Publications 9 (1982) 31–36.

[16] Streett, R.: Propositional dynamic logic of of looping and converse is ele-mentarily decidable. Information and Control 54 (1982) 121–141.

3

4

Page 14: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The Logic of Jon Barwise (1942-2000) Keith Devlin

ABSTRACT

My good friend and colleague Kenneth Jon Barwise died of cancer in 2000 at the early age of 58. In the March 2004 issue of the Bulletin of Symbolic Logic, I published an overview of his work on the applications of methods of logic to natural language (JSL Volume 10, Number 1, pp.54–85). Although Barwise never, to my knowledge, did any work in formal modal logic, much of his work in logic was very much in the spirit that led to modal logic, and indeed I suspect that some of his work could provide fertile hunting ground for new developments in modal logic. In my talk, I will provide a general summary of my JSL paper, ending with a few brief remarks about some recent research of my own that finds its inspiration in Barwise’s work. The main lesson for us logicians to learn from Barwise is to continually step back from all the formalisms and the theorems of logic and ask the fundamental questions that motivate a study of logic in the first place: how do people reason and communicate? Barwise never ceased to do this. His landmark 1983 book Situations and Attitudes, co-authored with Stanford’s John Perry, introduced the world to the new subject of situation semantics that Perry and he had developed in the early 1980s. Situation semantics arose from asking how much of Tarski’s theory of mathematical truth could be applied to truth in the real world. In his 1997 monograph Information Flow: The Logic of Distributed Systems (1997), coauthored with Indiana’ University’s Jerry Seligman, Barwise extended some of the central ideas in Situations and Attitudes to propose a theory of how information flows through complex systems such as computers and natural languages. Central to this theory is the notion of an information channel, capable of preserving information as it is transmitted through a complex, causally interacting system. Fundamental to much of his work was Barwise’s strong belief that human reasoning is a situated activity, and that logic should seek to take that into account. In his collected work The Situation in Logic (CSLI Lecture Notes 17, 1989), he wrote [pp.xv–xvi]:

Back in the days before I became interested in the situated aspects of logic, I sometimes used to wonder how logicians felt in the first quarter of this century. Did they feel confused. Reading the literature of that period, one senses the extent to which they were groping toward the view of logic that eventually emerged, but also the extent to which they were still in the dark about what was central and what was peripheral. One also realizes that they were just missing certain key distinctions. In other words, they were confused. It was only with the pioneering work of Gödel, Church, Turing, Tarski, and Kleene in the 1930's that the modern conception of logic really took hold. I now feel I have some idea of how logicians must have felt in that period before the really seminal work, since I feel we are in an analogous stage now … As we try to let go of some of the simplifying idealizations made in standard logic, we too are groping for the key notions, and probably missing some key distinctions. In giving up these simplifying assumptions, there are many things to be rethought, many choices to be made, and many things to be tried. It is an exciting time, if you have

5

Page 15: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

the patience for that sort of thing, and a taste for the basic task of conceptual clarification. But it is also frustrating … …There is only one point about which I am really certain. That is that the view of language and logic as situated activities is an important one, and that situating logic is a task that must be carried out if we are to come to grips with some of the problems that currently vex the field.

Barwise received his B.A. in Mathematics and Philosophy from Yale University in 1963 and a Ph.D. in Mathematics in 1967 from Stanford University under Professor Solomon Feferman. His first book, Admissible Sets and Structures (1975), developed the theory of admissible sets, and applied it to definability theory. For the latter part of his career, starting in 1990, Barwise was College Professor of Philosophy, Computer Science, and Mathematics at Indiana University. Between 1983 and 1990, he was a professor of philosophy at Stanford, where he was a co-founder and first director of the Center for the Study of Language and Information, and the first director of the Symbolic Systems Program. With his Stanford colleague John Etchemendy, Barwise developed numerous pieces of innovative courseware to help convey abstract concepts in logic to students. These were published with a series of textbooks, including The Language of First-order Logic (1990), Tarski’s World (1991), Turing’s World (1993), Hyperproof (1994), and Language, Proof and Logic (2000). Keith Devlin CSLI, Stanford University Stanford, CA 94305-4115

6

Page 16: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Elementary canonical formulae:

syntactic, model-theoretic, and algorithmic aspects

Valentin Goranko1

Joint work with Dimiter Vakarelov2 and Willem Conradie1

1Department of Mathematics, Rand Afrikaans University

PO Box 524, Auckland Park 2006, Johannesburg, South Africa

2Department of Mathematical Logic with Laboratory for Applied Logic,

Faculty of Mathematics and Computer Science, Sofia University

blvd James Bouchier 5, 1126 Sofia, Bulgaria

e-mails: {wec,vfg}@rau.ac.za, [email protected]

This is a survey of a study of modal formulae which are elementary i.e. determine first-order definable frame conditions, and canonical in an appropriate sense which impliescompleteness of the logics axiomatized with such formulae. These formulae, which axioma-tize many important modal logics, are of particular practical interest as they lend themselvesto the computational tools developed for first-order logic.

The class of elementary canonical fomulae is not decidable, hence the problem arises to es-tablish useful characterizations and to identify rich natural classes of effectively recognizablesuch formulae.

The best-known class of elementary canonical fomulae, which was also the starting pointof this study, is the class of Sahlqvist formulae introduced in [8]. While bearing a clearsemantic motivation, these formulae are defined purely syntactically, and the syntacticdefinition is only a lower approximation of their underlying semantic idea (the method ofminimal substitutions, see [1]). That syntactic definition is extremely fragile, as it doesnot survive even simple boolean transformations or substitutions changing the polarity ofpropositional variables. It has, therefore, become customary to tacitly consider the Sahlqvistformulae closed under such simple transformations. On the other hand, it is known (see[2]) that axiomatic equivalence to a Sahlqvist formula is not decidable, and hence it wouldbe unreasonable to close the class of Sahlqvist formulae under such equivalences. Thus,the notion of Sahlqvist formulae has become fuzzy, and the question ‘What is a Sahlqvist

formula?’ has gained an increasing importance.

In [3] we have extended the Sahlqvist formulae to the class of inductive formulae in arbitrarypolyadic languages, also generalized for hybrid modal languages in [4]. These are stillsyntactically defined elementary canonical fomulae, and their first-order equivalents are stillcomputed by means of minimal, first-order definable substitutions which enable eliminationof the second-order predicate variables. These minimal substitutions are defined inductively,

1

7

Page 17: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

in order determined by certain syntactic dependancies between the propositional variableswithin the formula. Like Sahlqvist formulae, the syntactic shape of inductive formulae israther vulnerable to otherwise inessential transformations, and thus the question ‘What is

an inductive formula?’ remains actual.

The study follows three main threads which will be discussed in the talk:

1. Syntactic. In this line of research we analyse the possibilities to extend further theclass of syntactically determined elementary canonical fomulae:

(a) directly, by extending the explicit syntactic definition of inductive formulae;

(b) by adding and refining a pre-processing phase, in attempt to transform the for-mula into an inductive formula, while preserving its frame condition and itspersistence.

(c) by closing the class of inductive formulae under a suitable equivalence preservingthe elementary canonical formulae, to a larger, effectively recognizable class.

Also, we investigate and develop purely syntactic procedures for computation of thefirst-order equivalents of effectively defined classes of elementary canonical formulae.For instance, in [4] we have have presented such method for inductive formulae intemporal (more generally, reversive) languages with nominals.

2. Algorithmic. This is a natural extension of the syntactic approach, aiming at devel-opment of algorithms which identify elementary canonical formulae, and thus produceeffectively enumerable classes of such formulae. Such algorithmically definable classesneed not be decidable, but they are much less tied-up with the syntactic shape of theformulae, and penetrate deeper into their semantic nature.

To establish first-order definability of a modal formula amounts to elimination of themonadic second-order quantifiers occurring in its standard translation. Therefore,prime candidates for algorithms producing elementary canonical formulae are the twocurrently available and implemented algorithms for second-order quantifier elimina-tion, viz. SCAN and DLS (see [7]). Both are provably correct and incomplete, andnone of them is stronger than the other for that task. While seemingly based on dif-ferent ideas and with quite distinct computational behaviour, they converge in theirlogical core.

The main directions of this line of research are:

(a) To outline, in syntactic and model-theoretic terms, the scope of applicability ofboth algorithms.

For instance, it has been proved in [6] that SCAN succeeds for all Sahlqvistformulae. Moreover, this holds for all inductive formulae, and the same appliesto DLS.

(b) To establish canonicity of the formulae for which the algortihms succeed.

(c) To analyse and compare the strengths and weaknesses of the theoretical funda-mentals and implementational solutions of both algorithms.

(d) To develop their optimizations and new, stronger algorithms for second-orderquantifier elimination combining their best features.

2

8

Page 18: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(e) To develop strong, purely modal, algorithms for identification of elementarycanonical formulae and computation of their first-order equivalents.

3. Model-theoretic. This direction of research aims at characterizing the elementarycanonical formulae of a given modal language in practically useful model-theoreticterms. A typical such characterization (as a sufficient condition) is a suitable notionof persistence, e.g.:

(a) d-persistence (with respect to all descriptive frames), in the case of standardpolyadic modal languages;

(b) di-persistence (with respect to all discrete frames), in the case of hybrid modallanguages with nominals;

(c) r-persistence (with respect to all refined frames), in the case of logics with addi-tional non-orthodox rules of inference.

For instance, in [5] we have introduced a new notion of persistence, which separatesthe Sahlqvist formulae from the inductive ones. Also, we have proved that, up tolocal equivalence in all descrete general frames in reversive languages with nominals,the inductive, pure (not containing propositional variables) and locally di-persistentformulae coincide, thus neatly delineating a very large and natural class of elementarycanonical formulae in such languages.

In general, various persistence properties have emerged as useful tools for model the-oretic analysis and classification of elementary canonical and related formulae. Forinstance, as noted by van Benthem in [1], the modal formulae amenable to the methodof substitutions turn out to be precisely those persistent with respect to the generalframes containing as admissible all parametrically first-order definable sets.

Still in this line of research, following results from [5], we pursue characterization ofelementary canonical formulae in terms of the topological properties of their associatedset-theoretic operators on general frames.

Ultimately, this study intends to cross the border of first-order definability, and to investi-gate the classes of canonical formulae at all, as well as classes of well-behaved modal formulaedefinable in other natural fragments of monadic second-order logic, such as first-order logicwith least fixpoints.

References

[1] van Benthem J.F.A.K., Modal Logic and Classical Logic , Bibleapolis, Napoli, 1985.

[2] Chagrov, A. and M Zakharyaschev, Sahlqvist formulae are not so elementary, Logic

Colloquium’92, L. Csirmaz, D. Gabbay and M. de Rijke (eds.), CSLI Publications,Stanford, 1995, 61-73.

[3] Goranko, V. and D. Vakarelov, Sahlqvist formulae Unleashed in Polyadic Modal Lan-guages, Advances in Modal Logic, vol. 3, World Scientific, Singapore, 2002, pp.221-240.

3

9

Page 19: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[4] Goranko, V. and D. Vakarelov, Sahlqvist formulae in Hybrid Polyadic Modal Languages,J. of Logic and Computation, vol. 11(5), 2001, 737-754.

[5] Goranko, V. and D. Vakarelov, Elementary Canonical Formulae, I. Extending Sahlqvisttheorem, 2003, submitted.

[6] Goranko, V., U. Hustadt, R. Schmidt, and D. Vakarelov. SCAN is complete for allSahlqvist formulae, Proc. of RelMiCS’03, to appear.

[7] Nonnengart, A., H. J. Ohlbach, and A. Szalas. Quantifier elimination for second-orderpredicate logic, in: Logic, Language and Reasoning: Essays in honour of Dov

Gabbay, Part I, H. J. Ohlbach and U. Reyle (eds.), Kluwer, 1997.

[8] Sahlqvist, H., Correspondence and completeness in the first and second-order semanticsfor modal logic, Proc. of the 3rd Scandinavial Logic Symposium, Uppsala

1973, S. Kanger (ed.), North-Holland, Amsterdam, 1975, 110-143.

4

10

Page 20: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Model Checking Epistemic Properties

Wiebe van der Hoek

Department of Computer Science

University of Liverpool

[email protected]

Introduction

In my talk for AiML I will report on work that has been done in the Agent art group in Liverpoolfor the last two years on model checking epistemic properties. In this abstract, I will do no justice toother work in this area, and I will only refer to such work as far as it relates to ours.

Although model checking is a dominant verification technique for Multi Agent Systems, the em-phasis has been on temporal, rather than for instance epistemic properties. To the best of ourknowledge, there are no model checkers available to which one can directly feed epistemic propertiesfor verification.

Knowledge and Linear Time: Local Propositions

In [6] we use the concept of local propositions, as introduced in [2], to propose a way to modelcheck knowledge properties, with an underlying interpreted systems semantics ([3]) in Linear Timesystems. Let us use ltl for the language of Linear Time Logic, and ckl for the propositional temporallogic with operators Ki (i an agent) and C for Common Knowledge.

Roughly, the idea is as follows. Call a propositional formula ϕ to be i-local if agent i knows itsvalue, i.e., if in all states u and v with u ∼i v, formula ϕ has the same truth-value. For any ckl

formula ψ, and an interpreted system I = 〈R, π〉, we are now interested whether ψ holds initially inevery run r in I, or more formally, whether (〈R, π〉, (r, 0)) |=ckl ψ, for all r ∈ R. Let us abbreviatethis to mcckl(I, ψ). Our aim is now to reduce this problem to an ltl model checking problemmcltl(I, ψ′).

The idea to do this is simple: rather than the agent’s knowledge, we use his (propositional)evidence: Suppose ψ is an epistemic formula Kiβ, and that there is an i-local property ϕ. Then,verify that mcltl(I,3ϕ ∧ 2(ϕ → β ′), where β ′ is obtained from β using the same procedure. Weshow that this procedure is ‘correct’, and will discuss an example. Also, we will argue that in anadversarial setting, i.e. one in which absence of knowledge is to be proven, this technique is rathercumbersome.

Alternating-time Temporal Epistemic Logic

The framework atl extends ctl in that it generalises the path quantifiers A (“on all paths”) andits dual E to coalition modalities 〈〈Σ〉〉 for every set of agents Σ, with intended meaning of 〈〈Σ〉〉ϕ, thatthe coalition Σ has a strategy, so that, no matter what the agents outside Σ do, ϕ will be true. Inatl, ϕ’s main operator must be a temporal one.

In [7] we proposed to enrich atl with an epistemic component, giving rise to a language atel. Inthis language a whole range of properties are expressible, referring to (secret) communication, likeKaφ∧¬Kbφ∧¬Kcφ∧〈〈a, b〉〉♦(Kbφ∧¬Kcφ), or to knowledge pre-conditions, as in (¬〈〈a〉〉 gφ)U Kaψ

or as in the atl∗ formula Kb(c = s) → 〈〈b〉〉(〈〈b〉〉 go)U ¬(c = s) (“If Bob knows the code c of the

safe s, he is able to open it until the code changes”). In the simplest setting (no interaction between

11

Page 21: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

knowledge and abilities assumed), the setting that is the subject of [7], the model checking problemfor atel is ptime-complete.

Agents that Know how to Play

Once a basic framework for atl with epistemics is in place, it becomes interesting to look atproperties one can impose on agents with incomplete information. Some first steps in this directionare taken in [4], although perhaps the main message of that paper is that matters can become prettycomplicated. One main source of complication in atl is the way a strategy is defined, which assumesperfect information and perfect memory by the agents. Analogous to an insight by Moore ([5]), itis noted in [4] that the distinction between de dicto and de re when referring to the knowledge ofhaving a strategy is helpfuld. [4] then proposes various ways to restrict the set of allowed strategies,but especially for the multi-agent abilities, it remains unclear to how exactly interpret a formula likeCΣ〈〈Σ〉〉3winΣ; even if it is common knowledge within Σ that they have a strategy for winning, thisdoes not immediately imply that this strategy will be played (e.g., it may not be unique!).

There are some other unexpected logical problems with expressing intuıtively simple atel prop-erties. As an example, take the scheme 〈〈i〉〉 gϕ → Ki〈〈i〉〉 gϕ: agents are aware of what they canachieve. The most obvious requirement to impose this seems to be that agents have the same capa-bilities in indistinguishable states (formally: q ∼i q

′ ⇒ δ(q, i) = δ(q, i), where δ(q, i) represents theset of possible sets of next states that i can bring about, in q). However, as recently demonstratedby Agotnes ([1]) things don’t work out that way. The key problem here is that, even if the choices ofagent i are ‘the same’ in two states q and q ′, it is possible that he can achieve gp in state q becauseof the other agents’ limited abilities there, while the other agents’ abilities are not so much restrictedin q′.

Acknowledgements

The work reported here is based on joint work with Wojciech Jamroga, Sieuwert van Otterlooand Michael Wooldridge. During the past years, our progress in this benefited a lot from work anddiscussions with Valentin Goranko, Alessio Lomuscio, Carsten Lutz, Ron van der Meyden, WojciechPenczek, Dirk Walther, and Frank Wolter.

References

[1] Agotnes, T., A Note on Syntactic Characterization of Incomplete Information in atel, in: S. van Otterloo,P. McBurney, W. van der Hoek and M. Wooldridge, editors, Knowledge and Games workshop, Liverpool,

UK, 2004, pp. 34 – 42, informal proceedings.

[2] Engelhardt, K., R. van der Meyden and Y. Moses, Knowledge and the logic of local propositions, in:Proceedings of the 1998 Conference on Theoretical Aspects of Reasoning about Knowledge (TARK98), 1998,pp. 29–41.

[3] Fagin, R., J. Y. Halpern, Y. Moses and M. Y. Vardi, “Reasoning About Knowledge,” The MIT Press,Cambridge, MA, 1995.

[4] Jamroga, W. and W. van der Hoek, Agents that know how to play (2004), accepted for Fundamenta

Informaticae.

[5] Moore, R., A formal theory of knowledge and action, in: A. J. F., H. J. and Tate, editors, Readings in

Planning, Morgan Kaufmann Publishers, San Mateo, CA, 1990 pp. 480–519.

[6] van der Hoek, W. and M. Wooldridge, Model Checking Knowledge and Time, in: D. Bosnacki and S. Leue,editors, Model Checking Software (Procs. SPIN 2002), number 2318 in LNCS, 2002, pp. 95–111.

[7] van der Hoek, W. and M. Wooldridge, Time, Knowledge, and Cooperation: Alternating-time Temporal

Epistemic Logic and its Applications, Studia Logica 75 (2003), pp. 125–157.

2

12

Page 22: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Modal logic, XPath and XML

Maarten MarxInstitute for Logic, Language and Computation

University of Amsterdam

Reasoning about finite ordered trees is becoming very important with theadvent of the new data storage format XML. XPath is the most widely usedlanguage to retrieve data from XML documents. We show the close connec-tion between XPath and various well investigated variable free formalismsinterpreted on trees: Kleene algebras with Tests, Propositional Dynamiclogic, since/until temporal logic, and the Blackburn, Meyer-Viol, de Rijkelogic of finite trees.

(1) We present an extension of XPath which can express all first orderdefinable queries over ordered trees.

(2) We give a linear time algorithm for query evaluation for a languagewhich extends XPath with all regular expressions and tests over the fourbasic axes (regular XPath).

(3) We give an EXPTIME decision algorithm which can be used to decideequivalence of regular XPath expressions given a set of constraints (eg, anXML Schema Definition or a Document Type Definition (DTD).

All these results follow more or less directly from well-known resultsin modal logic. In particular (1) simply extends Kamp’s theorem to or-dered trees, (2) follows from the linear time model checking result for PDL(Alechina–Immerman), and (3) from the EXPTIME algorithm for determin-istic PDL with inverse by Vardi and Wolper.

We will use these results as a counterexample to the following line of”reasoning”:

> The results obtained in this paper are interesting.

> My problem is that the proofs are almost trivial. [...]

> That‘s the reason why I don‘t think that the contribution is

> sufficiently original nor deep to be presented [...].

13

Page 23: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

ON WHAT THERE ISN’T (BUT MIGHTHAVE BEEN)

Robert StalnakerDepartment of Linguistics and Philosophy, MIT

Actualism claims that nothing exists that does not actually exist. How isactualism to be reconciled with possible worlds semantics, which seems torequire a commitment to worlds that are merely possible? According toan actualist interpretation of possible worlds semantics, possible worlds, orpossible states of the world, are properties—ways a world might be. So amerely possible world is a property that a world might have had, but doesnot. The state of the world actually exists, but is uninstantiated. But thepossible states postulated by possible worlds semantics are usually taken tobe properties that are complete, or maximally specific, and this gives rise toa problem about merely possible individuals. A possible worlds semanticsfor a quantified language associates with each possible state of the worlda domain—all the individuals that exist in that world (that would exist ifthat state of the world were realized). It might be that some or all of theindividuals who exist in some counterfactual situation are actually existingindividuals, but it seems reasonable to believe that there might have existedthings that do not in fact exist, which seems to require that there be domainsthat include nonactual, and so nonexistent things. Some philosophers, in re-sponse to this problem, have argued that nothing could exist except whatdoes exist, and that what does exist exists necessarily. Others have arguedthat there must exist surrogates for merely possible individuals—individualessences that for each potential individual that exist necessarily. But I willargue that actualism can be reconciled with the possible existence of non-actual things, and that it can do this without commitments to individualessences. And I will argue that while actualism may require a reinterpreta-tion of orthodox possible worlds semantics, it does not require modifying thatsemantics. Specifically, I will respond to an argument of Timothy Williamsonthat it is not possible that there exist anything other than what does exist,and that what does exist exists necessarily.

14

Page 24: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Part II

Regular Papers

15

Page 25: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Pure Extensions, Proof Rules, and Hybrid Axiomatics

Patrick Blackburn¤and Balder ten Cate†

Keywords: hybrid logic, completeness, proof rules

Abstract

We examine the role played by proof rules in general axiomatisationsfor hybrid logic. We prove three main results. First, all known axioma-tisations for the basic hybrid language HL(@) that are strong enoughto yield completeness for any pure extension of the minimal logic makeuse of unorthodox proof rules. We show that this is not accidental: anyaxiomatisation of the minimal logic that uses only a finite collection oforthodox proof rules will be incomplete for some pure extension. Second,we introduce what we call existential saturation rules and show how theygive rise to completeness results for frame classes not definable using pureformulas. Third, we provide an axiomatisation for the stronger languageHL(@, ↓) which does not make use of unorthodox rules, but which doesyield completeness for any pure extension of the minimal logic, and indeed,for any extension by existential saturation rules.

1 Introduction

The basic hybrid language is the result of enriching ordinary modal logic withnominals, a second sort of atomic formula, typically written i, j, k,. . . , andthe @ operator. More precisely, given a countable set of ordinary propositionletters prop and a countable set of nominals nom, we define the formulas ofthe basic hybrid language HL(@) to be

φ ::= p | i | ¬φ | φ ∧ ψ | 3φ | @iφ

where p ∈ prop and i ∈ nom. Frames are defined just as in orthodox modallogic, but there is an extra constraint on what counts as a model: in any model,every nominal must be true at a unique world. That is, a model for the basichybrid language is a pair (F , V ) where F is a frame and V is a valuation(defined for all symbols in prop ∪ nom) such that |V (i)| = 1 for all i ∈ nom.

∗INRIA Lorraine, 615, rue du Jardin Botanique 54602 Villers les Nancy Cedex, France.Email: [email protected]

†ILLC/Dept. of Philosophy, University of Amsterdam, Nieuwe Doelenstraat 15, 1012CPAmsterdam. Email: [email protected]

1

16

Page 26: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Because each nominal is true at a unique world in any model, nominals in effectname worlds. The idea of using ‘formulas as terms’ in this way is due to ArthurPrior [10, 11]. Moreover, when we have nominals at our disposal, it becomesnatural to have a device for evaluating a formula at a named world, and thatis the role of the @ operator:

(F , V ), w |= @iφ iff (F , V ), u |= φ, where u is the unique world in V (i).

The nominals and @-operators make it possible to define classes of framesthat are not definable in the basic modal language. For example, no formulaof the basic modal language can define the class of irreflexive frames, but it iseasy to see that

@i¬3i

is valid on a frame iff the frame is irreflexive (important: valid here means trueat all worlds in the frame under all hybrid valuations — that is, all valuationswhich make nominals true at a unique world).

The formula @i¬3i is a simple example of a pure formula: it contains noordinary propositional letters. As is well known, modal formulas containingordinary proposition letters may well define non-elementary frame classes (theMcKinsey formula 23p → 32p is the simplest example). But any class offrames definable by pure formulas will be an elementary class, for if we extendthe standard translation (of modal logic to first-order logic) to hybrid logic(see Chapter 7 of [3] for details) nominals correspond to first-order constants.It is therefore reasonable to look for minimal proof systems that are not onlycomplete for the class of all frames, but have the following additional property:whenever pure formulas are added as extra axioms, completeness with respectto the (elementary) class of frames defined by these axioms is guaranteed. Thatis, researchers working on hybrid logic have long been interested in axiomati-sations that are automatically complete for pure extensions, and a number ofsuch results are known for various hybrid languages (see for example [7, 9, 5]).

This paper takes a closer look at such axiomatisations. We shall prove threemain results. First, all known axiomatisations for the basic hybrid languageHL(@) that are complete for pure extensions, make use of unorthodox proofrules (i.e., rules involving syntactic side-conditions). We shall show that thisis not accidental: any axiomatisation of the minimal logic (in the languageHL(@)) that uses only a finite collection of orthodox proof rules will be incom-plete for some pure extension. Second, we introduce what we call existentialsaturation rules and show how they give rise to completeness results that coverframe classes not definable by pure formulas. Third, we examine what happenswhen we move from the basic hybrid language HL(@) to a stronger hybridlanguage, HL(@, ↓), that has been the focus of much recent work. We shallshow that there is an axiomatisation in this language which does not make useof unorthodox rules, but which is complete for pure extensions, and indeed,complete for any extension with existential saturation rules.

2

17

Page 27: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 Unorthodox rules and pure extensions

The following axiomatisation is a simplification of the minimal axiomatisationfor the basic hybrid language given in Chapter 7 of [3]:

HL(@)

Axioms:CT All classical tautologiesK2 ` 2(p→ q)→ 2p→ 2q

K@ ` @i(p→ q)→ @ip→ @iq

Selfdual@ ` @ip↔ ¬@i¬pRef@ ` @ii

Agree ` @i@jp↔ @jp

Intro ` i→ (p↔ @ip)Back ` 3@iφ→ @iφ

Rules:MP If ` φ and ` φ→ ψ then ` ψSubst If ` φ then ` φσ

Gen@ If ` φ then ` @iφ

Gen2 If ` φ then ` 2φ

Name If ` @iφ and i does not occur in φ, then ` φBG If ` @i3j → @jφ and j 6= i does not occur in φ, then ` @i2φ

The validity of the axioms should be clear, but two comments on the rulesshould be made. First, the substitution rule allows us to uniformly replacenominals by nominals and atomic propositions by arbitrary formulas. Second,and more importantly, note that the Name and BG rules are unorthodox: theyinvolve syntactic side conditions of the form X does not occur in Y . Let’s takea closer look.

First the BG rule. This stands for Bounded Generalisation, and can bethought of as a modal analog of the UG (Universal Generalisation) rule of first-order logic. Because j is a nominal distinct from i that does not occur in φ, wecan read @i3j as asserting the existence of a successor (arbitrarily labeled j)of the world labeled i. Accordingly, the antecedent condition of the rule can beread as follows: suppose we can prove that if i has an arbitrary successor j thenφ holds at j. But then, since j was an arbitrary successor of i, the consequentcondition of the rule tells us that φ must hold at all successors of i (that is,@i2φ). The analogy with the first-order rule of universal generalisation shouldbe clear. It’s also worth noting that our informal explanation of the BG rulehas a natural deduction flavour, and in fact the Box-introduction rule in thenatural deduction system for hybrid logic given in [6] is closely related to BG.

What of the Name rule? This tells us that if it is provable that φ holds atan arbitrary world i (the world is arbitrary because i does not occur in φ) thenwe can prove φ. This rule plays a simple, but crucial, role in the completenessproof given in the following section.

3

18

Page 28: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

This axiomatisation is sound and complete with respect to the class of allframes. But it is also complete for all pure extensions: any extension withpure axioms is complete with respect to the class of frames defined by thoseaxioms (recall that a formula of the basic hybrid language is pure iff it containsno ordinary propositional variables). More precisely, given any set of pureformulas Λ, let HL(@)+Λ denote the above axiomatisation, extended with theaxioms in Λ. Then we have:

Theorem 1 (Completeness). Let Λ be any set of pure axioms. A set of

formulas Σ is HL(@) + Λ consistent iff Σ is satisfiable in a model satisfying

the frame properties defined by Λ.

For example, if we choose Λ to be empty set (that is, if we add no additionalaxioms) then this theorem tells us HL(@) is complete with respect to the classof all frames. If we choose Λ to be

{@i¬3i,@i3j ∧@j3k → @i3k}

then we have an axiomatisation that is complete with respect to strict partialorders, for the first axiom defines irreflexivity, and the second defines transitiv-ity.

We won’t prove Theorem 1 here — a similar result is proved in Chapter 7 of[3], and we shall prove something more general in the following section. Insteadwe shall address the following question. Theorem 1 covers all pure extensions,but it is based on an axiomatisation that makes use of unorthodox rules (namelyName and BG). Is this necessary, or is the use of such rules avoidable? From[8], we know that if Λ consists of canonical modal formulas, Name and BG areadmissible. However, we shall now show that if completeness for arbitrary pureextensions is required, the use of unorthodox rules can only be avoided at thecost of introducing infinitely many rules.

By an orthodox modal rule we mean a rule of the form

` φ1(α1, . . . , αn) & · · · & ` φk(α1, . . . , αn)

` ψ(α1, . . . , αn)

Here, α1, . . . , αn range over arbitrary formulas, and are implicitly universallyquantified. In the presence of a modus ponens rule (together with enoughpropositional axioms), we can assume without loss of generality that there isonly a single antecedent (a big conjunction), hence all orthodox rules can beassumed to be of the form

` φ(α1, . . . , αn)

` ψ(α1, . . . , αn)

In fact, we may assume that φ and ψ do not contain any proposition letters,i.e., they are built up from α1, . . . , αn using the Boolean connectives and modaloperators. The rank of such a rule will be n. For example, the rank of the Gen2

4

19

Page 29: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

rule is 1. A rule preserves validity on a class of frames F, if for all formulasα1, . . . , αn, F |= φ(α1, . . . , αn) implies F |= ψ(α1, . . . , αn). We can now provethe desired result: no finite collection of orthodox rules can be complete for allpure extensions, even if we take as axioms all validities of HL(@).

Theorem 2. Let Kh be the set of all formulas in the basic hybrid language

that are valid on all frames, let P be a finite set set of orthodox rules, and

let L be the axiomatic system formed by taking as axioms Kh, and taking as

rules modus ponens, substitution, and all the rules in P . Then there is a pure

extension L + Λ that is not sound and complete with respect to the class of

frames defined by Λ.

Proof. Let n be the maximal rank of the rules in P — this information is allwe need to construct a pure extension that is incomplete with respect to theframe class it defines. Define Λ be the set containing only the the followingpure formula:

1≤l≤2n+2

3il →∨

1≤k<l≤2n+2

3(ik ∧ il).

L + Λ is the axiomatic system L enriched by this single pure axiom (closedunder modus ponens, substitution and the rules in P ). Let F be the class offrames defined by Λ, that is, the class of all frames in which each world has atmost 2n + 1 successors. Either the rules in P preserve validity on F or they donot. If they do not, soundness is lost and there is nothing to prove, so assumethat the rules P do preserve validity on F. We shall now show that L + Λ isnot complete for F.

Define M to be the class of models based on frames in F. Further-more, define F = (W,R) to be the frame where W = {1, . . . , 2n + 2}and R = W 2; clearly, F 6∈ F. Finally, let M

′ = M ∪ {(F , V ) |V is a valuation for F such that V (i) = V (j) for all nominals i, j}. Now forthe heart of the proof: we shall show that L + Λ is sound for the class ofmodels M

′.

Claim 1.

• All formulas in Kh are valid on M′.

• Validity on M′ is closed under uniform substitution of formulas for propo-

sitional variables and nominals for nominals.

• Validity on M′ is closed under modus ponens

The proof of Claim 1 is straightforward and is left to the reader.

Claim 2. All formulas valid on F with at most n propositional variables are

valid on M′. Hence M

′ |= Λ.

5

20

Page 30: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Let φ be a formula with at most n propositional variables, and suppose forthe sake of contradiction that F |= φ and M

′ 6|= φ. Then there is a valuationV and a world w such that F , V, w ° ¬φ, and such that V assigns the sameworld to each nominal. Since only n propositional variables occur in φ, and allnominals are true at the same world, it follows that the bisimulation contractionof M (over this restricted vocabulary) has at most 2n + 1 worlds; hence, thisbisimulation contraction is in F. It follows that F 6|= φ, which contradicts ourinitial assumption.

Claim 3. All rules in P preserve validity on M′.

Let ρ ∈ P be a rule` φ(α1, . . . , αm)

` ψ(α1, . . . , αm)

with m ≤ n, and suppose that M′ |= φ(α1, . . . , αm) for particular formulas

α1, . . . , αm. Uniformly substitute > for each of the propositional variablesoccurring in α1, . . . , αm. We then obtain pure formulas β1, . . . , βm, and byClaim 1 it follows that M

′ |= φ(β1, . . . , βm). Let p1, . . . , pm be new, distinctpropositional variables. Then it follows that

M′ |= φ((p1 ¢ φ(p1, . . . , pm) ¤ β1), . . . , (pm ¢ φ(p1, . . . , pm) ¤ βm))

where (φ¢ ψ ¤ χ) is shorthand for (ψ ∧ φ) ∨ (¬ψ ∧ χ). Hence

F |= φ((p1 ¢ φ(p1, . . . , pm) ¤ β1), . . . , (pm ¢ φ(p1, . . . , pm) ¤ βm))

Since F is a purely definable frame class, ρ preserves validity. Hence,

F |= ψ((p1 ¢ φ(p1, . . . , pm) ¤ β1), . . . , (pm ¢ φ(p1, . . . , pm) ¤ βm))

Since this formula contains at most n propositional variables, it follows byClaim 2 that

M′ |= ψ((p1 ¢ φ(p1, . . . , pm) ¤ β1), . . . , (pm ¢ φ(p1, . . . , pm) ¤ βm))

By closure under uniform substitution (Claim 1), it follows that

M′ |= ψ((α1 ¢ φ(α1, . . . , αm) ¤ β1), . . . , (αm ¢ φ(α1, . . . , αm) ¤ βm))

Recall that M′ |= φ(α1, . . . , αm). It follows that M

′ |= (αi ¢ φ(α1, . . . , αm) ¤

βi)↔ αi. Hence,M

′ |= ψ(α1, . . . , αm)

This completes the proof of the third claim, and hence we have shown thatL+ Λ is sound with respect to M

But now the incompleteness result follows. Consider the following formula

η =∧

1≤i≤2n+2

3pi →∨

1≤i<j≤2n+2

3(pi ∧ pj)

6

21

Page 31: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Notice that M′ 6|= η. By Claim 1–3, it follows that L+ Λ 6` η. However F |= η,

so it follows that L+ Λ is not complete for F.

In short, if we want completeness results that cover arbitrary pure exten-sions, such as Theorem 1, and we don’t want to make use of infinitely manyrules, then unorthodox rules (for example, BG and Name) are unavoidable.

3 Existential Saturation Rules

While Theorem 1 is quite general, there are many interesting frame classesthat are not definable using pure axioms. One example is the class of framesin which every world has a predecessor, another is the class of Church-Rosserframes. How can we axiomatise such frame classes?

First a simple observation: while neither of these frame classes can bedefined in the basic hybrid language, they can be defined in a much strongerhybrid language, namely Prior’s original hybrid language which allowed fullclassical quantification over nominals (see [10, 11]). For example, the class offrames in which every world has a predecessor can be defined in such a languageby the formula ∀s∃t@t3s, and the class of Church-Rosser frames can be definedby ∀stu∃v(@s3t ∧@s3u→ @t3v ∧@u3v).

How does this help? After all, for many purposes we don’t want to work insuch a strong language (which is essentially a notational variant of the full first-order correspondence language). We want to work in the basic hybrid language.The answer is this: any Priorian formula ξ of the form ∀s1 . . . sn∃t1 . . . tm.φ,where φ is a formula of HL(@) that contains no proposition letters and nonominals besides s1, . . . , sn, t1, . . . , tm gives rise to what we call an existential

saturation rule for ξ, namely the following (unorthodox) proof rule:

If ` φ(i1, . . . , in, j1, . . . , jm)→ ψ then ` ψ,provided that i1, . . . , in, j1, . . . , jm are distinct and j1, . . . , jm do not occur in

φ.

For example, the existential saturation rule for ∀stu∃v(@s3t∧@s3u→ @t3v∧@u3v) (the Church-Rosser property) is:

If ` @i3j ∧@i3k → @j3l ∧@k3l→ ψ then ` ψ,

provided i, j, k, l are distinct and l does not occur in ψ.

Incidentally, notice the similarity between these existential saturation rulesand the proof rules used by [12].

Let ρ be the existential saturation rule for some ξ. We say that a frameadmits ρ if the set of formulas valid on that frame is closed under ρ. The class

of frames defined by ρ will be the class of frames defined by ξ.

7

22

Page 32: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Lemma 1. Let ρ be the existential saturation rule for some formula ξ of the

form ∀~s∃~t.φ, where φ is quantifier-free, pure and nominal free. Every frame

satisfying ξ admits ρ.

Proof. Suppose F |= ∀s1 . . . sn∃t1 . . . tn.φ, and suppose that the antecedent of ρis valid on F, i.e., suppose F |= φ(i1, . . . , in, j1, . . . , jm)→ ψ(i1, . . . , in) , wherei1, . . . , in, j1, . . . , jm are distinct. Then φ(i1, . . . , in) is valid on F. For, pickand world w and any valuation V . Since F |= ∀~s∃~t.φ and the truth of ψ onlydepends on the value of the nominals i1, . . . , jn, we can assume without loss ofgenerality that V assigns worlds to the nominals j1, . . . , jm in such a way thatφ is true. It follows that under this valuation, at the given world, ψ is true.

This tells us that in order to axiomatise (in the basic hybrid language) frameclasses involving properties such as the Church-Rosser property, we can addthe relevant existential saturation rules to the axiomatisation while retainingsoundness. In fact, we will see that the resulting axiomatisation is complete aswell.

Given a set of pure axioms Λ and a set of existential saturation rules P , wewill use HL(@) + Λ + P to denote the HL(@) axiomatisation extended withthe axioms in Λ and the rules in P .

Theorem 3 (Extended completeness). Let Λ be a set of pure axioms and let

P be a set of existential saturation rules. A set of formulas Σ is HL(@)+Λ+Pconsistent iff Σ is satisfiable in a model satisfying the frame properties defined

by Λ and P .

Note that Theorem 1 is a special case of this result, namely when P isempty. The remainder of this section is dedicated to the proof of Theorem 3.It closely resembles a Henkin-style completeness proof for first-order logic, withnominals playing the role of first-order constants.

First, we remark that the following validities and rules are derivable:

Lemma 2. The following are derivable

K−1

@` (@iφ→ @iψ)→ @i(φ→ ψ)

Nom ` @ij → (@iφ↔ @jφ)Sym ` @ij → @ji

Bridge ` @i3j ∧@jφ→ @i3φ

Name′ If ` i→ φ then ` φ where i does not occur in φ

Paste3 If ` @i3j ∧@jφ→ ψ and j 6= i does not occur in φ or ψ, then ` @i3φ→ ψ

Proof. Left to the reader.

Definition 1. Let Σ be a set of HL(@) formulas.

• Σ is named if one of its elements is a nominal.

8

23

Page 33: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

• Σ is 3-saturated if for all @i3φ ∈ Σ, there is a nominal j such that

@i3j ∈ Σ and @jφ ∈ Σ.

• Let ρ be the existential saturation rule corresponding to the strong Pri-

orean formula ∀s1 · · · sn∃t1 · · · tk.Λ(s1, . . . , sn, t1, . . . , tm). Then Σ is ρ-

saturated, if for all nominals i1 . . . in there are nominals j1 . . . jm such

that Λ(i1, . . . , im, j1, . . . , jm) ∈ Σ.

Lemma 3 (Lindenbaum Lemma). Every HL(@) + Λ+ P consistent set of

formulas can be extended to a named, 3-saturated HL(@) + Λ + P MCS, by

adding countably many new nominals to the language.

Proof. Suppose Σ is HL(@)+Λ+P consistent. Let (in)n∈N be an enumerationof a countably infinite set of new nominals, and let (φn)n∈N be an enumerationof the formulas of the extended language. Let Σ0 denote Σ∪{i0}. The Name

rule guarantees that Σ0 is consistent. For all n ∈ N, Σn+1 is defined as follows.If Σn ∪ {φn} is HL(@) + Λ + P inconsistent, then Σn+1 = Σn. Otherwise:

1. Σn+1 = Σn ∪ {φn} if φn is not of the form @i3ψ.

2. Σn+1 = Σn ∪ {φn} ∪ {@i3im,@imψ} if φn is of the form @i3ψ,where im is the first new nominal that does not occur in Σn or φn.

Let Σω =⋃

n≥0Σn. Then Σ ⊆ Σω and Σω is named, 3-saturated, maximal

and consistent. The only non-trivial step is in 2., and that this step preservesconsistency is guaranteed by the Paste3 rule.

Lemma 4 (Rule Saturation Lemma). Every HL(@)+Λ+P consistent set

of formulas can be extended to a named, 3-saturated, P -saturated HL(@) +Λ + P MCS, by adding countably many new nominals to the language.

Proof. The proof proceeds in two steps. First, we show that every HL(@)+Λ+P consistent set of formulas Σ can be extended to a set of formulas Σ+, whichis still HL(@) + Λ + P consistent, such that Σ+ provides witnesses for Σ, inthe following sense: for each existential saturation rule ρ ∈ P corresponding toa strong Priorean formula ∀s1 · · · sk∃t1 · · · tm.Λ, and for all nominals i1, . . . , ikoccurring in Σ, there are nominals j1, . . . , jk such that Λ(i1, . . . , ik, j1, . . . , jm) ∈Σ+. Such Σ+ can be constructed as follows.

Let (in)n∈N be an enumeration of a countably infinite set of new nomi-nals, and let (ρn,~in)n∈N be an enumeration of all pairs (ρn, in1 . . . ink) whereρn ∈ P is an existential saturation rule for a strong Priorean formula∀s1 · · · sk∃t1 · · · tm.Λ(s1, . . . , sk, t1, . . . , tm), and in1 . . . ink are nominals occur-ring in Σ (note that there are at most countably many such pairs). Let Σ0 = Σ,and for each n ∈ N, let Σn+1 = Σ∪{Λ(in1, . . . , ink, j1, . . . , jm)}, where ρn is theexistential saturation rule for the strong Priorean formula ∀s1 · · · sk∃t1 · · · tm.Λand j1, . . . , jm are the first m distinct nominals in the enumeration not occur-ring in Σn. Let Σ+ =

nΣn. Then Σ ⊆ Σ+, Σ+ is HL(@) + Λ + P consistent

and Σ+ provides witnesses for Σ in the sense described above.

9

24

Page 34: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The main argument now runs as follows. Consider any HL(@) + Λ + P

consistent set of formulas Γ. Let Γ0 = Γ and for all n ∈ N, let Γn+1 be a 3-saturated named MCS extending (Γn)+ (the Lindenbaum Lemma guaranteesthere is one). This gives rise to the following chain of consistent sets of formulas:

Γ = Γ0 ⊆ (Γ0)+ ⊆ Γ1 ⊆ (Γ1)+ ⊆ · · ·

Let Γω =⋃

nΓn. Then Γω is a 3-saturated, named P -saturated MCS. Inciden-

tally, during the entire process we expanded the language with only countablymany new nominals, and therefore Γω is a countable set.

Definition 2 (Henkin model obtained from anMCS). Let Γ be a maximal

consistent set of HL(@) formulas. For all nominals i, let |i| be {j | @ij ∈ Γ}.Then MΓ = (W,R, V ) is given by

W = {|i| | i is a nominal}

|i|R|j| iff @i3j ∈ Γ

V (p) = {|i| ∈W | @ip ∈ Γ}

V (i) = {|i|}

That MΓ is well-defined follows from Ref, Sym and Nom (note that transi-

tivity is just a special case of Nom).

Lemma 5 (Truth Lemma). For all 3-saturated MCSs Γ, nominals i and

formulas φ, MΓ, |i| |= φ iff @iφ ∈ Γ

Proof. By induction on the length of φ.

Lemma 6 (Frame Lemma). If Γ is a 3-saturated, P -saturated HL(@)+Λ+PMCS, then the underlying frame of MΓ satisfies the frame properties defined

by Λ and P .

Proof. Since MΓ is a named model and Γ contains all instances of elementsof Λ, it follows that the underlying frame of MΓ validates Λ. Since MΓ is anamed model and Γ is P -saturated, it follows that the underlying frame of MΓ

satisfies (the strong Priorean formulas corresponding to) P .

At this point, we have all the required apparatus in place, and we can finishoff the proof by the usual kind of argument.

Proof of Theorem 3. Suppose Σ is HL(@) + Λ + P consistent. By Lemma 4,Σ can be extended to a named, 3-saturated, P -saturated MCS Γ. Let i ∈ Γ.By Lemma 5, MΓ, |i| |= Σ. By Lemma 6, MΓ satisfies all required frameproperties.

10

25

Page 35: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A final remark. Existential saturation rules are essentially reflections in thebasic hybrid language of certain type of formulas in the stronger Priorean lan-guage, namely sentences of the form ∀s1 . . . sn∃t1 . . . tm.φ(s1, . . . , sn, t1, . . . , tm),where φ is quantifier-free, pure and nominal free. A natural question for furtherwork is whether the effects of more complex formulas in the richer languagecan be captured in a simple way.

4 Axiomatisations for HL(@, ↓)

The completeness result just proved can be straightforwardly extended to covera number of other hybrid languages. Here we sketch how to adapt it to thericher HL(@, ↓). This language, which allows us to bind a nominal to theworld of evaluation using the ↓ binder, has been one of the most extensivelyexplored in contemporary hybrid logic: it corresponds to exactly the generatedsubmodel invariant fragment of first-order logic [1], and once we have ↓ in ourlanguage, it becomes possible to prove very general interpolation results [1, 4].The language HL(@, ↓) was first axiomatised in [5]. Here we shall improveon this axiomatisation in two ways. First, we will show how to axiomatiseHL(@, ↓) by adding a single axiom schema to our axiomatisation to HL(@).Second, we show how to eliminate the Name and BG rules.

Here is the first axiomatisation. We remark that since we’re now dealingwith a language with variables and binding, we need to adjust the substitutionrule to allow variables and nominals to be substituted for each other, and weneed to take the standard precautions to prevent accidental binding of variables.Bearing this in mind, here is the axiomatisation HL(@, ↓)-I:

HL(@, ↓)IAxioms:All axioms for HL(@)DA ` @i(↓s.φ↔ φ[s := i])Rules:All rules for HL(@)

As promised, our first axiomatisation extendsH(@) with a single axiom schemawhich states the semantics of the ↓ operator at some arbitrary world named i(the notation φ[s := i] means substitute the nominal i for all free occurrencesof the variable s).

Theorem 4 (Completeness). Let Λ be a set of pure axioms and let P be a

set of existential saturation rules. A set of sentences Σ is HL(@, ↓)I + Λ + P

consistent iff Σ is satisfiable in a model satisfying the frame properties defined

by Λ and P .

Proof. Almost no changes to our completeness proof for HL(@) are required.Once again we build the model out of (equivalence classes of) nominals, and

11

26

Page 36: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

the Lindenbaum Lemma and the Rule Saturation Lemma are unchanged. Infact all we need to do is add an extra clause to our proof of the Truth Lemmafor formulas of the form ↓s.ψ, and this is where DA is used. With the TruthLemma established, completeness follows in the expected way.

Before going further, a remark on existential saturation rules in HL(@, ↓).First note that for some frame conditions they are no longer needed. For exam-ple, the Church-Rosser property, which could not be defined by any pureHL(@)formula, is defined by the pure HL(@, ↓) formula 3i ∧ 3j → @i(3↓s.@j3s).However the class of frames in which every world has a predecessor is not de-finable by means of any pure HL(@, ↓) axiom (since it is not closed undergenerated subframes) so to axiomatise this frame class we still make use of anexistential saturation rule.

Let’s turn to our second axiomatisation. While HL(@, ↓)I is a particularlysimple extension of HL(@), it inherits from HL(@) the use of the unorthodoxName and BG rules. As we shall now show, if we add a Gen↓ rule, the Nameand BG rules can be replaced by axiom schemas:

HL(@, ↓)IIAxioms:All axioms for HL(@)DA ` @i(↓s.φ↔ φ[s := i])Name↓ ` ↓s.(s→ φ)→ φ provided that s does not occur in φBG↓ ` @i2↓s.@i3s

Rules:MP If ` φ and ` φ→ ψ then ` ψSubst If ` φ then ` φσ

Gen@ If ` φ then ` @iφ

Gen↓ If ` φ then ` ↓s.φGen2 If ` φ then ` 2φ

Theorem 5 (Completeness). Let Λ be a set of pure axioms and let P be a

set of existential saturation rules. A set of sentences Σ is HL(@, ↓)II + Λ+ P

consistent iff Σ is satisfiable in a model satisfying the frame properties defined

by Λ and P .

Proof. By Theorem 4, it suffices to show that the Name and BG rules areHL(@, ↓)II derivable. Here’s how to derive BG (we leave the derivation ofName to the reader):

@i2↓s.@i3sBG↓

@i3j → @jÁ

@j(@i3j → Á)Gen@,K@, Selfdual

↓s.@i3s→ ÁDA,Name

@i2(↓s.@i3s→ Á)Gen2, Gen@

@i2ÁK2, Gen@,K@

BG

12

27

Page 37: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Thus HL(@, ↓)II can derive all axioms and rules of HL(@, ↓)I, and henceits completeness follows by Theorem 4.

5 Conclusion

In this paper we examined the use of proof rules in hybrid axiomatics, and inparticular the role they play in establishing completeness results for elementaryframe classes. We first showed (Theorem 2) that finite axiomatisations forthe basic hybrid language using only orthodox rules cannot be complete forall pure extensions. We then showed (Theorem 3) that existential saturationrules, reminiscent of Venema [12]’s SD rule, allow us to prove completenessresults for many elementary frame classes not definable by pure formulas ofthe basic hybrid language. Finally, we proved a completeness result for thestronger hybrid language HL(@, ↓) which covered all pure extensions, and allextensions by means of existential saturation rules, but which did not involveunorthodox rules. To conclude this paper, let us make some brief remarksabout the relation of the work reported here to other recent work on hybridaxiomatics.

Ten Cate, Marx and Viana [8] give an axiomatization of the basic hybridlanguage that does not make use of unorthodox rules, and that is complete forarbitrary extension with canonical modal axioms. In particular, their resultcovers the class of Church-Rosser frames, since it is defined by the canonicalmodal formula 32p → 23p. One might ask whether completeness is alsoobtained when pure and canonical modal formulas are mixed, for if so, the ex-istential saturation rule for Church-Rosser can always be avoided. The answeris No [8]: there is a pure formula φ and a canonical modal formula φ (in fact,a Sahlqvist formula) such that HL(@)+ {φ, ψ} is not complete. This shows anincompatibility between the Henkin model construction used in this paper andthe canonical model method used in [8].

Another recent line of work should also be mentioned. Both the pure exten-sion and the Sahlqvist based results have an obvious limitation: they only applyto elementary, or at least canonical, frame classes. In recent work Bezhanishviliand ten Cate [2] have shown how completeness results for modal logics thatadmit filtration can be lifted to completeness results for their hybrid versions(the ‘hybrid companion logics’). This approach allows completeness results fornon-elementary non-canonical logics, such as hybrid versions of GL and Grz,to be established straightforwardly, without the use of unorthodox rules.

Summing up, at present that are three general techniques for establishingcompleteness of hybrid logics: via pure extensions, via canonicity, and viatransfer from orthodox modal logic. Each handles certain logics the otherscan’t, and mapping the trade-offs between the methods more precisely is animportant topic for further research.

13

28

Page 38: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

References

[1] Carlos Areces, Patrick Blackburn, and Maarten Marx. Hybrid logics:Characterization, interpolation and complexity. Journal of Symbolic Logic,66(3):977–1010, 2001.

[2] Nick Bezhanishvili and Balder ten Cate. Transfer results for hybrid logic.Part II of Bezhanishvili et al., Sahlqvist theory and transfer results for

hybrid logic. This volume., 2004.

[3] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal logic.Cambridge University Press, Cambridge, UK, 2001.

[4] Patrick Blackburn and Maarten Marx. Constructive interpolation in hy-brid logic. Journal of Symbolic Logic, 68(2):463–480, 2003.

[5] Patrick Blackburn and Miroslava Tzakova. Hybrid languages and temporallogic. Logic Journal of the IGPL, 7(1):27–54, 1999.

[6] Torben Brauner. Natural deduction for hybrid logic. Journal of Logic and

Computation, To appear.

[7] R. Bull. An approach to tense logic. Theoria, 36:282–300, 1970.

[8] Balder ten Cate, Maarten Marx, and Petrucio Viana. Sahlqvist theory forhybrid logic. Part I of Bezhanishvili et al., Sahlqvist theory and transfer

results for hybrid logic. This volume., 2004.

[9] G. Gargov and V. Goranko. Modal logic with names. Journal of Philo-

sophical Logic, 22:607–636, 1993.

[10] A.N. Prior. Past, Present and Future. Oxford University Press, 1967.

[11] A.N. Prior. Papers on Time and Tense. Oxford University Press, Newedition, 2003. Edited by Hasle, Øhrstrom, Brauner, and Copeland.

[12] Yde Venema. Derivation rules as anti-axioms in modal logic. Journal of

Symbolic Logic, 58(3):1003–1034, 1993.

14

29

Page 39: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Title: The complexity of strict implication logics

Author: Felix Bou

Postal Address:

Faculty of MathematicsUniversity of Barcelona (UB)Gran Via 585E-08007 Barcelona, Spain

E-mail: [email protected], [email protected].

Keywords: modal logic, strict implication, strict impli-cation logics, complexity of modal logics, PSPACE-complete, basic propositional logic, formal propositionallogic, embeddings.

30

Page 40: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The complexity of strict implication logics

Felix Bou∗

Abstract

The aim of the present paper is to analyze the complexity of strictimplication logics. We prove that Ladner’s Theorem remains validwhen we restrict us to strict implication fragments. As a consequencewe have that the validity problem of most normal modal logics is in thesame complexity class than the validity problem of its strict implicationfragment. We also obtain that the validity problems for basic propo-sitional logic and formal propositional logic are PSPACE complete.Finally, for some normal modal logics we give a polynomial reductionfrom it into its strict implication fragment.

Ladner proved in [13] that all modal subsystems of S4 have a validityproblem that is PSPACE hard. The theorem is stated in the framework ofnormal modal logics, where we have classical negation ∼. For certain logicsformulated in classical negation free languages it is also known its complexityclass [19]. However, for classical negation free languages there are no generalstatements like Ladner’s Theorem solving the complexity problem for a lotof logics. We present a classical negation free language, the strict implicationlanguage, where it is possible to prove a version of Ladner’s Theorem. Thislanguage is a very natural fragment of modal language where it is possibleto formulate a lot of examples from the literature. Since strict implicationlanguage is a fragment of modal language, our version of Ladner’s Theoremcan be considered as an improvement of it.

1 Preliminaries

We devote the section to fix our notation and to get certain acquaintancewith the modal case.

Throughout the paper we will consider two languages. One is the well-known modal language and the other is its strict implication fragment (withconjunction, disjunction and falsity). Let Prop be an infinite set of proposi-tions. The modal formulas and the strict implication formulas are defined,

∗I would like to thank Nick Bezhanishvili for some discussions. The author is partiallysupported by Catalan grant 2001SGR-00017 and by Spanish grant BFM2001-3329.

1

31

Page 41: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

respectively, by

ϕ ::= p | ⊥ | ϕ0 ∧ ϕ1 | ϕ0 ∨ ϕ1 | ϕ0 ⊃ ϕ1 | �ϕand

ϕ ::= p | ⊥ | ϕ0 ∧ ϕ1 | ϕ0 ∨ ϕ1 | ϕ0 → ϕ1,

where p ranges over elements of Prop. The set of modal formulas willbe denoted by Lmod(Prop), and the set of strict implication formulas byLs(Prop). The set of subformulas of a formula ϕ is denoted by Sub(ϕ).

These formulas are to be interpreted in Krikpe models. Given a Kripkemodel M = 〈M,R, V 〉, a world m ∈ M and a formula ϕ the satisfiabilityrelation M,m � ϕ is defined as follows:

M,m � p iff m ∈ V (p)M,m � ⊥M,m � ϕ0 ∧ ϕ1 iff M,m � ϕ0 and M,m � ϕ1

M,m � ϕ0 ∨ ϕ1 iff M,m � ϕ0 or M,m � ϕ1

M,m � ϕ0 ⊃ ϕ1 iff M,m � ϕ0 or M,m � ϕ1

M,m � �ϕ iff ∀m′(mRm′ ⇒ M,m′ � ϕ)M,m � ϕ0 → ϕ1 iff ∀m′(mRm′&M,m′ � ϕ0 ⇒ M,m′ � ϕ1).

If M,m � ϕ we say that ϕ is satisfied in M at a. It is said that ϕ is validin M (notation: M � ϕ) if M,m � ϕ for every world m ∈ M . Giventwo formulas ϕ0 and ϕ1 we will write ϕ0 ≡ ϕ1 whenever both formulasare satisfied in the same worlds for every Kripke model. We will use thefollowing abbreviations in the strict implication language: � := ⊥ → ⊥,¬ϕ := ϕ → ⊥ and �ϕ := � → ϕ. In the modal case we use ∼ϕ := ϕ ⊃ ⊥ϕ0 ⊃⊂ ϕ1 := (ϕ0 ⊃ ϕ1) ∧ (ϕ1 ⊃ ϕ0) and �ϕ := ∼�∼ϕ. Given n ∈ ω wedenote by �nϕ and �(n)ϕ the formulas

n times︷ ︸︸ ︷�. . .�ϕ and ϕ ∧ �ϕ ∧ . . . ∧ �nϕ.

It is clear that the semantics of the non-primitive connectives is the expectedone1. In particular,

M,m � ∼ϕ iff M,m � ϕ.We notice that if we add the classical negation ∼ to Ls(Prop) then thelanguage that we obtain have the same expressive power than Lmod(Prop).We also single out that semantics on strict implication language is preciselythe common semantics given for intuitionistic propositional logic except forthe fact that we range over arbitrary Kripke models (and not only overintuitionistic models2).

1We notice that the semantics of � for the modal case coincides with its semantics asa defined connective for the strict implication case. Therefore, there is no ambiguity.

2For this sake we don’t call our language the intuitionistic propositional language. Wethink it is better to use a different name to point out this fact.

2

32

Page 42: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Our interest in the strict implication language comes from the fact thatthere is a wide range of well-known logics that can be formalized in it.As we have already pointed out the more famous example is intuitionis-tic propositional logic IPL, which is the set of strict implication formulasthat are valid in all reflexive and transitive frames under persistent valua-tions [12, 4]. We recall that a valuation V is persistent when it satisfies thatfor every p ∈ Prop and every worlds m and m′, if mRm′ and m ∈ V (p)then m′ ∈ V (p). Besides IPL it also holds that classical propositional logicCPL can be obtained in this way. It is the set consisting of all strict im-plication formulas that are valid in all reflexive, transitive and symmetricframes under persistent valuations (or simply the set of all strict implica-tion formulas that are valid in the reflexive frame that has a single point).Other examples in the literature are all superintuitionistic logics [4], somesubintuitionistic logics [6, 7, 15, 22, 3], formal propositional logic FPL [21]and basic propositional logic BPL [21, 16, 17]. Let us explain which logicsare the last two. We define the map T : Ls(Prop) −→ Lmod(Prop) usingthe following clauses:

i)T(p) = �p, ii)T(⊥) = ⊥, iii)T(ϕ0 ∧ ϕ1) = T(ϕ0) ∧ T(ϕ1),iv)T(ϕ0 ∨ ϕ1) = T(ϕ) ∨ T(ϕ1), v) T(ϕ0 → ϕ1) = �(T(ϕ0) ⊃ T(ϕ1)).

It is well-known that T is an embedding of IPL into both the normal modallogic S4 and the normal modal logic Grz, i.e.,

ϕ ∈ IPL iff T(ϕ) ∈ S4 iff T(ϕ) ∈ Grz.

The same map is also an embedding of CPL into S5. The logics FPLand BPL were introduced by Visser in [21] as the logics such that T is anembedding of them into GL and K4, respectively. That is,

FPL = {ϕ ∈ Ls(Prop) : T(ϕ) ∈ GL},and

BPL = {ϕ ∈ Ls(Prop) : T(ϕ) ∈ K4}.It is known that FPL is precisely the set of strict implication formulas thatare valid in all frames that are Noetherian strict orders under persistentvaluations3, and that BPL is the set of strict implication formulas that arevalid in all transitive frames under persistent valuations. Visser introducedBPL for technical reasons, but nowadays BPL has acquired an interest ofits own. Ruitenburg has argued its philosophical interest as a constructivelogic (see [16, 17]).

There is a general method to obtain a strict implication logic from anormal modal logic. The idea is to consider its strict implication fragment.

3If we replace ‘Noetherian’ with ‘finite’ we also obtain FPL.

3

33

Page 43: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Given a normal modal logic L we define its strict implication fragment Ls

as the set {ϕ ∈ Ls(Prop) : σ(ϕ) ∈ L} where σ is the translation defined by:

σ(p) := pσ(⊥) := ⊥σ(ϕ0 ∧ ϕ1) := σ(ϕ0) ∧ σ(ϕ1)σ(ϕ0 ∨ ϕ1) := σ(ϕ0) ∨ σ(ϕ1)σ(ϕ0 → ϕ1) := �(σ(ϕ0) ⊃ σ(ϕ1)).

The map σ has been considered several times in the literature. Its firstappearance under this name is in [7], and since then it has been also usedby other authors, e.g., Celani and Jansana in [3].

The aim of the present paper is to analyze the complexity of the strictimplication logics obtained in this way, i.e., to analyze the complexity ofstrict implication fragments. We will prove that in most cases we obtainthe same complexity than in the modal case4. The following easy exampleshows that this does not seems to be true for all normal modal logics.

Example 1. Let Verum be the normal modal logic given by the frame F

that is a single irreflexive point. Using that the problem “ϕ ∈ CPL?” isco-NP complete [5] it is clear that “ϕ ∈ Verum?” is also co-NP complete.On the other hand, it is not hard to prove that “ϕ ∈ Verums?” is in P. Itfollows from the fact that ϕ ∈ Verums iff ϕ holds in the Kripke model overF such that all propositions hold in the unique state.

In the rest of the section we recall two of the main results concerningcomplexity of normal modal logics. The first one is due to Hemaspaandra(nee Spaan) [18, 11].

Theorem 2 (Hemaspaandra). Let L be a normal modal logic extendingS4.3. Then, the problem “ϕ ∈ L?” is co-NP complete.

The other result is due to Ladner [13] and gives a lower bound of thecomplexity of normal modal logics that are subsystems of S4.

Theorem 3 (Ladner). Let L be a normal modal logic such that K ⊆ L ⊆S4. Then, the problem “ϕ ∈ L?” is PSPACE hard.

Using that for the validity problem of normal modal logics like K, K4and S4 it is known the existence of algorithms running in PSPACE, it fol-lows that the validity problem for these three examples is PSPACE com-plete.

4Besides the complexity results there are a lot of other results suggesting that theexpressive power of the strict implication fragment is very close to the one of the modallanguage. A thorough study can be found in [2].

4

34

Page 44: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

• •

•p1=0

���������p1=1

���������

p0=0��

• •

•p1=0

���������p1=1

���������

p0=1��

(a) The binary trees generated by β

• •

p0,∼ p1,q0,q1,q2,∼ q3

���������

p0,p1,q0,q1,q2,∼ q3

���������

p0,∼ p1,q0,q1,∼ q2,∼ q3

∼ p0,∼ p1,q0,∼ q1,∼ q2,∼ q3

��

(b) A Kripke model witnessing that β is true

Figure 1: Let β be the quantified Boolean formula ∃p0∀p1(p0 ∨ p1)

Now we review the lower bound proofs of Ladner’s Theorem (see [13,10]), since we plan to follow the same strategy in the next section. Theproof shows a (polynomial time) reduction from the membership problemof the logic QBF of quantified Boolean formulas. It is enough because it iswell known that this last problem is PSPACE complete [20]. A quantifiedBoolean formula β is a formula of the form

Q0p0Q1p1 . . . Qn−1pn−1ϕ

where Qi ∈ {∀,∃} and ϕ is a propositional formula (i.e., a Lmod(Prop)-formula without any occurrence of �) called the matrix of β whose onlyprimitive propositions are among p0, . . . , pn−1. The quantifiers range overthe truth values 1 (true) and 0 (false), and a quantified Boolean formulawithout free variables is true if and only if it evaluates to 1, i.e., the sub-formulas ∀p ϕ(p) and ∃p ϕ(p) are regarded to be true iff ϕ(�) ∧ ϕ(⊥) andϕ(�) ∨ ϕ(⊥) are true, respectively. The set QBF is defined as the set ofquantified Boolean formulas that are true.

It is clear that if we restrict5 to quantified Boolean formulas such thatQ0 = ∃ and n ≥ 2 then the membership problem is also PSPACE complete.Let β be a quantified Boolean formula Q0p0Q1p1 . . . Qn−1pn−1ϕ such thatQ0 = ∃ and n ≥ 2. Ladner’s idea is that when we are evaluating β we areessentially generating binary trees of height n such that each one of their

5There is no reason to adopt this restriction in the modal case, but in the strict impli-cation case this restriction simplifies the argument.

5

35

Page 45: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

branches gives us a propositional valuation (we illustrate it with an examplein Figure 1(a)). Then, β is true iff it can generate a certain binary treeof height n such that all propositional valuations it records ensure that thematrix evaluates to 1 (in our example the second tree witnesses that theconsidered quantified Boolean formula is true). The connection with Kripkemodels comes from the fact that we can associate with each one of thebinary trees of height n generated by β a Kripke model over the propositions{p0, . . . , pn−1, q0, . . . , qn+1} with a persistent valuation. Let us see how todefine this Kripke model. The accessibility relation of this associated Kripkemodel is the one given by the tree6, and the behaviour of its valuation at anode of height i (≤ n) is the following one:

• the propositions in {q0, . . . , qi} are true.

• the propositions in {pi, . . . , pn, qi+1, . . . , qn+1} are false.

• the propositions in {p0, . . . , pi−1} behaves in the same way than in thepropositional valuation given by a branch containing the node.

In our previous example the Kripke model associated with the second treein Figure 1(a) is the one depicted in Figure 1(b).

Having these ideas in mind Ladner succeed to define a modal formulag(β) with propositions in {p0, . . . , pn−1, q0, . . . , qn+1} such that the followingstatements are equivalent:

1. β is true.

2. g(β) is satisfiable.

3. g(β) is satisfiable in a finite strict order with a persistent valuation.

4. g(β) is satisfiable in a finite partial order with a persistent valuation.

In particular it follows that

β ∈ QBF iff ∼ g(β) ∈ K iff ∼ g(β) ∈ S4.

This g gives us the polynomial reduction that we need for proving Ladner’sTheorem. Let us present now the definition of g(β)7. It is the conjunctionof the following formulas:

(i) q06Indeed, here we have some choice: we can also consider its transitive closure or its

reflexive-transitive closure.7Our definition is a bit different of the one given in [13, 10], but it is also clear that it

satisfies the equivalences stated above.

6

36

Page 46: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(ii) ∼(q1 ∨ q2 ∨ . . . ∨ qn ∨ qn+1 ∨ p0 ∨ p1 ∨ . . . ∨ pn−1)

(iii) �(q1 ∧ ∼ q2)

(iv) �1γ1n ∧ �2γ2n ∧ . . . ∧ �nγnn

(v) �1θ1 ∧ �2θ2 ∧ . . . ∧ �n−1θn−1

(vi)∧

i∈{j<n:Qj=∀} �iδi

(vii) �1ψ1∧�2(ψ1∧ψ2)∧�3(ψ1∧ψ2∧ψ3)∧. . .∧�n−1(ψ1∧ψ2∧ψ3∧. . .∧ψn−1)

(viii) �n(qn ⊃ ϕ)

where

γin := (qi∧∼ qi+1) ⊃((q0∧. . .∧qi−1)∧(∼ qi+2∧. . .∧∼ qn+1)∧(∼ pi∧. . .∧∼ pn−1)

),

θi := (qi ∧ ∼ qi+1) ⊃ �(qi+1 ∧ ∼ qi+2),

δi := (qi ∧ ∼ qi+1) ⊃(�(qi+1 ∧ ∼ qi+2 ∧ pi) ∧ �(qi+1 ∧ ∼ qi+2 ∧ ∼ pi)

),

andψi :=

((qi ∧ pi−1) ⊃ �pi−1

) ∧ ((qi ∧ ∼ pi−1) ⊃ �∼ pi−1

).

2 Complexity Results

In this section we prove that Hemaspaandra’s Theorem and Ladner’s The-orem are also true when we restrict to the strict implication language.

Theorem 4. Let L be a normal modal logic extending S4.3. Then, theproblems “ϕ ∈ L?” and “ϕ ∈ Ls?” are both co-NP complete.

Proof. By Hemaspaandra’s Theorem it is enough to show that “ϕ ∈ Ls?”is co-NP hard. It is known that the equational logic of distributive latticesis co-NP hard (see [8]). Given two formulas ϕ0 and ϕ1 using only theconnectives ∧ and ∨, it is clear that ϕ0 ≈ ϕ1 holds in all distributive latticesiff (ϕ0 → ϕ1) ∧ (ϕ0 → ϕ0) ∈ Ls. Hence, we have obtained a polynomialreduction that allow us to conclude that “ϕ ∈ Ls?” is co-NP hard.

In order to prove our strengthening of Ladner’s Theorem we are goingto consider the truth problem for quantified Boolean formulas such that itsmatrix is in conjunctive normal form. It is also known that this problemis PSPACE hard (see [1, Corollary 1.36]). We recall that a matrix ϕ is inconjunctive normal form when ϕ is of the form

(ν0 ⊃ π0) ∧ . . . ∧ (νk−1 ⊃ πk−1)

where each νi is a conjunction of propositions and each πi is a disjunctionof propositions.

7

37

Page 47: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Theorem 5. Let X be a set of strict implication formulas such that Ks ⊆X ⊆ IPL or Ks ⊆ X ⊆ FPL. Then, the problem “ϕ ∈ X?” is PSPACEhard.

Proof. Let β be a quantified Boolean formula Q0p0Q1p1 . . . Qn−1pn−1ϕ suchthat Q0 = ∃, n ≥ 2 and ϕ is in conjunctive normal form. We suppose that ϕis (ν0 ⊃ π0)∧ . . .∧ (νk−1 ⊃ πk−1). We define the following strict implicationformulas:

γ′in :=

⎛⎝qi → (qi+1 ∨

∧j∈{0,...,i−1}

qj)

⎞⎠ ∧

∧j∈{i+2,...,n+1}

((qi ∧ qj) → qi+1) ∧

∧∧

j∈{i,...,n−1}((qi ∧ pj) → qi+1),

θ′i := (qi ∧ (qi+1 → qi+2)) → qi+1,

δ′i :=((

qi ∧((qi+1 ∧ pi) → qi+2

)) → qi+1

)∧

∧(

(qi ∧(qi+1 → (pi ∨ qi+2)

)) → qi+1

),

andψ′

i :=(qi ∧ pi−1) → �pi−1

) ∧ (qi → (pi−1 ∨ ¬pi−1)

).

It is easy to check that γ′in ≡ �γin, θ′i ≡ �θi, δ′i ≡ �δi and ψ′i ≡ �ψi. Let

us consider the following list of strict implication formulas:8

(i) q0

(ii) q1 ∨ q2 ∨ . . . ∨ qn ∨ qn+1 ∨ p0 ∨ p1 ∨ . . . ∨ pn−1

(iii) q1 → q2

(iv) γ′1n ∧ �1γ′2n ∧ . . . ∧ �n−1γ′nn

(v) θ′1 ∧ �1θ′2 ∧ . . . ∧ �n−2θ′n−19

(vi)∧

i∈{j<n:Qj=∀} �i−1δ′i10

(vii) ψ′1∧�1(ψ′

1∧ψ′2)∧�2(ψ′

1∧ψ′2∧ψ′

3)∧ . . .∧�n−2(ψ′1∧ψ′

2∧ψ′3∧. . .∧ψ′

n−1)

8We notice that each one of them is either equivalent to the one considered in theprevious section or equivalent to the classical negation ∼ of the formula considered in theprevious section.

9The assumption n ≥ 2 guarantees that it is a strict implication formula.10Since Q0 = ∃ we know that 0 �∈ {j < n : Qj = ∀}. Hence, i − 1 ≥ 0 whenever

i ∈ {j < n : Qj = ∀}.

8

38

Page 48: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(viii) �n−1((

(qn∧ν0) → π0

)∧((qn∧ν1) → π1

)∧. . .∧((qn∧νk−1) → πk−1

)).

We define f0(β) as the conjunction of (i), (iv), (v), (vi), (vii), (viii); and wedefine f1(β) as the disjunction of (ii), (iii). Finally, let f(β) be the strictimplication formula f0(β) → f1(β). A moment of reflection shows that�g(β) ≡ ∼ f(β). Using this and the equivalences stated in page 2 it easilyfollows that the following statements are equivalent:

1. β is true.

2. ∼ f(β) is satisfiable, i.e., f(β) ∈ Ks.

3. ∼ f(β) is satisfiable in a finite strict order with a persistent valuation,i.e., f(β) ∈ FPL.

4. ∼ f(β) is satisfiable in a finite partial order with a persistent valuation,i.e., f(β) ∈ IPL.

Therefore,

β ∈ QBF iff f(β) ∈ Ks iff f(β) ∈ FPL iff f(β) ∈ IPL.

Thus, for any set X such that Ks ⊆ X ⊆ IPL or Ks ⊆ X ⊆ FPL it holdsthat

β ∈ QBF iff f(β) ∈ X.

It is not hard to prove that the length of f(β) increases polynomially in thelength of β. Therefore, we have showed that f is a polynomial reductionfrom the truth problem for quantified Boolean formulas such that its matrixis in conjunctive normal form, which is a PSPACE hard problem. Thisconcludes the proof.

It follows that the membership problem for the logics Ks, Ts, K4s,S4s, Grzs and GLs is PSPACE complete: it is in PSPACE because thevalidity problem for the corresponding normal modal logic is in PSPACE.Another easy consequence of our theorem is that the logics BPL and FPLintroduced by Visser are PSPACE complete: we know that they are inPSPACE thanks to the embedding T.

3 A Polynomial Reduction from Normal ModalLogics

Up to now we have proved that most common normal modal logics are inthe same complexity class than its strict implication fragment, but we have

9

39

Page 49: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

not given any polynomial reduction from the normal modal logic into itsstrict implication fragment. In this section we present (without any proof)a reduction of this type that works well under certain (general) assumptions.

Let Prop be the set {pn : n ∈ ω}, and let Prop′ be {pn : n ∈ ω} ∪ {qn :n ∈ ω} ∪ {rϕ : ϕ ∈ Lmod(Prop)}. Then, we simultaneously define twotranslations + and − from Lmod(Prop) into Ls(Prop′):

p+n := pn p−n := qn

⊥+ := ⊥ ⊥− := �(ϕ0 ∧ ϕ1)+ := ϕ+

0 ∧ ϕ+1 (ϕ0 ∧ ϕ1)− := ϕ−

0 ∨ ϕ−1

(ϕ0 ∨ ϕ1)+ := ϕ+0 ∨ ϕ+

1 (ϕ0 ∨ ϕ1)− := ϕ−0 ∧ ϕ−

1

(ϕ0 ⊃ ϕ1)+ := ϕ−0 ∨ ϕ+

1 (ϕ0 ⊃ ϕ1)− := ϕ+0 ∧ ϕ−

1

(�ϕ)+ := �ϕ+ (�ϕ)− := rϕ.

Let ϕ be a modal formula such that its propositions are among p0, . . . , pn−1

and its modal degree is k (i.e., the number of nested modalities). A simpleinduction shows that⎛⎝�(k)

( ∧0≤i<n

(pi ⊃⊂ ∼ qi) ∧∧

ψ∈Sub(ϕ)

(rψ ⊃⊂ ∼�σ(ψ+)))⎞⎠ ⊃ (ϕ ⊃⊂ σ(ϕ+)) ∈ K

and⎛⎝�(k)

( ∧0≤i<n

(pi ⊃⊂ ∼ qi) ∧∧

ψ∈Sub(ϕ)

(rψ ⊃⊂ ∼�σ(ψ+)))⎞⎠ ⊃ (∼ϕ ⊃⊂ σ(ϕ−)) ∈ K.

It suggests us to introduce the following definition.

Definition 6. Let ϕ be a modal formula such that its propositions are inp0, . . . , pn−1 and its modal degree is k. Then, we define h(ϕ) as the strictimplication formula over Prop′ obtained as (�(k)δ) → �ϕ+ where δ is∧0≤i≤n

(((pi ∧ qi) → ⊥) ∧ �(pi ∨ qi))∧∧

ψ∈Sub(ϕ)

(((rψ ∧ �ψ+) → ⊥) ∧ �(rψ ∨ �ψ+)

).

Now it is possible to prove the following theorem.

Theorem 7. Let L be a normal modal logic that is characterized by a certainclass of frames F where it holds that

for every F ∈ F and every world m in the frame F there is anotherframe F′ ∈ F with a distinguished world m′ such that i)m′ isnot an initial point (i.e., R−1

F′ [{m′}] = ∅), and ii) the subframegenerated by m and the subframe generated by m′ are isomorphic.

Then, h is a polynomial reduction from L into Ls, i.e., for every modalformula ϕ it holds that

ϕ ∈ L iff h(ϕ) ∈ Ls.

10

40

Page 50: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

We single out that most common normal modal logics satisfies the re-quirement of the previous theorem, e.g., K, T, K4, S4, Grz and GL. Andalso all frame complete extensions of T.

Corollary 8. Let L be a frame complete extension of T. Then, the problems“ϕ ∈ L?” and “ϕ ∈ Ls?” are in the same complexity classes.

4 Further Research

Halpern proved in [9] that Ladner’s Theorem for normal modal logics holdseven in the case that there is a single proposition. In this paper we havedemonstrated that strict implication language satisfies a version of Ladner’sTheorem. So, a natural question is whether in the case that there is a singleproposition strict implication language also has a certain version of Ladner’sTheorem. Of course we cannot give the same version than in Theorem 5because it is known that intuitionistic propositional logic with one variableis decidable in linear time [14]. But perhaps it is possible to give the sameversion when there are only two propositions. Finally, we single out thatin [2, Chapter 5] it is proved that Ks is PSPACE hard even when Prop isempty.

References

[1] J. L. Balcazar, J. Dıaz, and J. Gabarro. Structural complexity. I. Textsin Theoretical Computer Science. An EATCS Series. Springer-Verlag,Berlin, second edition, 1995.

[2] F. Bou. Strict-weak languages. Ph. D. Dissertation, University ofBarcelona, 2004. In preparation.

[3] S. Celani and R. Jansana. A closer look at some subintuitionistic logics.Notre Dame Journal of Formal Logic, 42(4):225–255, 2001.

[4] A. Chagrov and M. Zakharyaschev. Modal Logic, volume 35 of OxfordLogic Guides. Oxford University Press, 1997.

[5] S. A. Cook. The complexity of theorem-proving procedures. In ACM,Proc. 3rd ann. ACM Sympos. Theory Computing, Shaker Heights, Ohio1971, 151-158. 1971.

[6] G. Corsi. Weak logics with strict implication. Zeitschrift fur Mathema-tische Logik und Grundlagen der Mathematik, 33:389–406, 1987.

[7] K. Dosen. Modal translations in K and D. In M. de Rijke, editor,Diamond and Defaults, pages 103–127. Kluwer Academic Publishers,1993.

11

41

Page 51: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[8] R. Freese. Algorithms in finite, finitely presented and free lattices.Preprint, July 1999.

[9] J. Y. Halpern. The effect of bounding the number of primitive propo-sitions and the depth of nesting on the complexity of modal logic. Ar-tificial Intelligence, 75(2):361–372, 1995.

[10] J. Y. Halpern and Y. Moses. A guide to completeness and complexityfor modal logics of knowledge and belief. Artificial Intelligence, 54:319–379, 1992.

[11] E. Hemaspaandra. The price of universality. Notre Dame Journal ofFormal Logic, 37:174–203, 1996.

[12] S. Kripke. Semantical analysis of intuitionistic logic I. In J. M. Crossleyand M. A. E. Dummet, editors, Formal Systems and Recursive Func-tions, pages 92–130. North-Holland, Amsterdam, 1965.

[13] R. E. Ladner. The computational complexity of provability in systemsof modal propositional logic. SIAM J. Comput., 6(3):467–480, 1977.

[14] I. Nishimura. On formulas of one variable in intuitionistic propositionalcalculus. The Journal of Symbolic Logic, 25:327–331 (1962), 1960.

[15] G. Restall. Subintuitionistic logics. Notre Dame Journal of FormalLogic, 35(1):116–129, 1994.

[16] W. Ruitenburg. Constructive logic and the paradoxes. Modern Logic,1:207–301, 1991.

[17] W. Ruitenburg. Basic logic and Fregean set theory. In H. Barendregt,M. Bezem, and J. W. Klop, editors, Dirk van Dalen Festschrift, Ques-tiones Infinitae, volume 5, pages 121–142. Department of Philosophy,Utrecht University, 1993.

[18] E. Spaan. Complexity of modal logics. Ph. D. Dissertation, Institutefor Logic, Language and Computation, 1993.

[19] R. Statman. Intuitionistic propositional logic is polynomial-space com-plete. Theoretical Computer Science, 9(1):67–72, 1979.

[20] L. J. Stockmeyer and A. R. Meyer. Word problems requiring expo-nential time: preliminary report. In Fifth Annual ACM Symposium onTheory of Computing (Austin, Tex., 1973), pages 1–9. Assoc. Comput.Mach., New York, 1973.

[21] A. Visser. A propositional logic with explicit fixed points. Studia Logica,40:155–175, 1981.

12

42

Page 52: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[22] H. Wansing. Displaying as temporalizing. Sequent systems for subintu-itionistic logic. In S. Akama, editor, Logic, Language and Computation,pages 159–178. Kluwer, 1997.

Felix Bou. Departament de Filosofia.Autonomous University of Barcelona (UAB).Edifici B. Campus de la UAB. 08193 Bellaterra. Barcelona, Spain.E-mail: [email protected], [email protected]

13

43

Page 53: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��������� ���� � � ��� ��� ������ �� ������ �����

���� ������������ ������ �� ����� ������ ������� �������� �����

��������

�� ��� ���� ��� �� ��� � ������������ ��� ���� ��� ���������� �� ��� ���� ����������� �� �� ��� � ��� ��� ���� ������ ����� �� ��� ���� ��������� �� ��� ������� ���������� �� ��� ���� ������ ����� ��� ������ ���� ��� ������� � ���� � ����������� �� �� ������ ��� ���������

�� ��� ������ ��� �� ����� ��� � ��� ������ �������� ���� ����� � ��� �������������� �� �� �� ������� ����� ���������� ������� ���� � �� �� � �� ����������� �� ������ �� ������� �����������!������ ��� ��������� ���� �� �� �� �� ��� ���� ��� ���� ���� �� ������ ���������� "�������# ������������ �� ����� ������$����������� ���� � �� �� � �� ��� ��� ��� ������� ������� ��� �� ������ ���� ������������ ����������� �� ��� ����� ����� �������� ��������

���� �� ������� �� ��� ��� ������ ����������� �� ���� ������ ���� �� �������� ����

� �����������

������ ���� ��� ��� � ������� ���������� ������� ����� ��������� ��� ���� ������� ��� ���� ������ ���� ��� ����� ��� ���� ����� �� ������ ��� ��� � ���� ���� �� ������� ��������� ��� ������� ����� ���� ��� ��������� ��������� ����������� �������� ��� ����� ��������� �� !��� ����� ���� �������� ��� ��� "���� ���������� ���� #���������� �� ��� ��!��� ��� ���� �� ����$����� ������

%� �� ����� ���� ������� ������ ��� &���"����'� ������ ��� ���� ����� � ������������� ���������� ������� &������ "�������� �� �� ���� (� ����� &���"���� ���� ��)��������� �� � ���� ����* +�� ��� ,����-.����� ���� ���� ��� �� � �������������/0�� 1� ��� ����� ����� �� ��� ������� �� ��� ����� ���������� ��� ��� �� �� ��� �/2�� 3�������� �� � � �� "���������

/� (� ��� ��������� �� ��� ���� ������ ���� ��� � ��� �� &���"���� ����� �� ��� ������*3��� ��� ���� &���"����'� ������ �� ������� ��� ������ ����*

0� ,�� � ����� ��� � � ������� ���������� �������* 3��� ��� �� ����� ��������� �� ������� ������ ���� ��� � ��� 4 �� &���"���� ����� ��� � ��� 5 �� ���� ����� ������ ��� ������ �� ��� ���� �� ����� ��!��� �� 4 ��� 5 ��������*

#���� ���� �� ��� ����� ��� ��� ���� "��������� %� ��� ���� ����� ��������� �� ��� ���������� ���� ��� ������� ���� ����� 6��� ����� &���"���� �����7 �� ������ ���� ������ ��� ����� ��� ���� ������ ��� � ���� � &���"���� ������ � ��� � ���� ������ ���� ���� ��� ���� ������ ���� �������� ��� ��� ����� � ��� � �� �������� ���� �� ���������� �� ��� ����� ��� ��� ���� ������

���� ������� � ��� ��� ���� ��� � ��� �� ����� ������� � ������� �� ���� ����������� ���� ������������ ������ ��� ������� � ��� ������ ���� ��� � ��� �� ����� �� ����� ������� ��� ����� ��� ��������

������ ��������! � "��������� ������������������ ������ ��������! � "��������� �� �������������� ����������� ���������� ��������! � "��������� �������������� ���������� �� #����������� $����� $ �������� ��������!� ����������������

/

44

Page 54: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� � ��������� � ����� �� ����������� ���� ������ �� ������ ����� ������ 8���'���!�������� �������� ����� 6�� �9� ��� � ��������� �� ���� ���� ������ ��� ��� ��������������7� ������� �������� �� ��� �������� �� ���� ����� ��� ���� ��� �����$���� ��� ������� ��� ���� ������ ������ ��� ���� ��� ����� &���"���� ���������� 3��� �� � ��������� ����������� ����� ����� ��� ���)�������� �� ��� ����� ���� ���� ������� ������� ����)������������ ��:� ��� ;�����'� ����$������� �����

3��� ���� �� ��� ����� �� ������<�� �� ����� �� 3��� ������ ����$� ������ ������ �����&����� 0 ��� � &���"����'� ������ ��� ������ ����� (� &����� = � ������ ���������������� 8���'� ��������� &����� � ��� � ���� � ��������� �� &���"���� ��� ���� ����� �� ������������� �� �� ������� %� ������ �� &����� ��

�� �� ����

%��� ����� � �� � ����� �������:)����� ������������ �� ������ ����� ����� ��� ���� ������ ������ ��� ������ �� ��������� ��� ���� ���� �������� ��� � ����� ���� �� ���� ����������������� �������� ��� ��� ����������� ���������� 3�� ������� ������ ������ �� ������������������� �������� ����� ���� ����� �������������� �� ����� �� ��������� �� ��������� �����(� ����� ����� ������� �� �� ���� ��� ������ &���������� ��������� ���� ��� �� ����������� � ������ ����� �� ��� ���� ���� �� ������� #�� ������� >�� ��������� ���� � ������� ��� ���� ���� �� ��� ������ ��

#������� ��� ���� �� � �������� ��!���� ��� �� ����������� ������� ��� ��� � ����������!���� ��� �� ��������� 3�� ������� �� ��� ���� ������ ���� ��� ����� �� ����� ��

� ��? � � � � � � �� � � � � � �� � >��

���� � � ���� ��� � � ���� 3�� ����� ��!������ ��� ��� ������� �� ��� ��� �� ������ ����������� �������� ��� ����� ��� �� ��� ��� � ? 6�� � 7� ���� � �� � ���� ���� � ��������� ������� ��� ��� ����������� ������� ��� �������� 3�� ����� ��!������ ������ ������� �� ��� ��� �� ��� ��� ����������� �������� �� �? � �@ � � 6�7� 3�� ������@����� �� �� ��� ��������� ����������� ���� ��������� �������� ��� ���� �� ���� ������ ����� ������ � ��������� ���� 3�� �������������� �� ��� ����������� ��������� �� �� ���� ���������� �� �? >�� �@ �� �? �� ���� � 6�7 ? ���

+���� ��� �� ���� �� �������<������ ��� ���� ��������� A�� B �� ��� ��� �� ����� ������� �������� ���:�� �������������� ������ ��� �����)������ ��� ��� �� � � ����

������� >�>��� >��

���:� �>��� >��

������������� � � �� >��

����� >��

�����)����� >��� �>����

A�� ����� �� ��� ������� ��� �� ������� ��������� ��� ������������ ����� �6� � 7 �6��� � 7� >�6�� 7� 6>��� >� 7 ��� � � ���� ��� ��� ����� �� B� ����� ����� ������������ ����� ������������ � ��� �� � ���������� ����� ��� ������ � � ������ ������� ������� 6(� � ���� ��7� ��� >������ ������� 6(� � ���� >��7� ;���� � ��� 4�� ������ �������� ��� ���� �����4 �� �������� �� ������ ��� ������� �� 4 �� ����� ������� ������ ��� ������ ����� ��� ����� ������

������� �� ����� �� ����� � ����� ��� ���� ��� � ��� � ��� � � �����

� �:��� �� ��� ����� �� 3����� 0�/ �� �� ����� �� ���� 3����� 0�/ ���� ����� � ��� ���������� �� &����� =�

���� ���� �� ��������� �� ���� ����� ��� ����� ����� � �� ��� ! � ��� �� �����

0

45

Page 55: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ������� ���������� ������ ����� ��� ���������� �� ��� ���� ���� ��� ���� �������������� � � ����� ���������� ����� ��� ����� �� ��� ������� � ���� ���� �� �� ���� ����������� �� ����������� �������� ���� ��������

����� (� >�� ��� � ���� ��� ��� �� �� ���� ����� (� >��� � >�� ���� � ? � ��� � ���� ��� ��� �� �� ���� >���

&������ �������� �� ����� ����� ��� �� ����������� ����� ���� ��� �� ��� �/0� ��� ������� ����� ���� (� ��� ����� ������ ��� ����� !��� ������ �� ����

A�� ������ �� ��� ���� �������� �� ������ ����� � � �������� ����� �� ������ ;���� �

��� 4 �� ������ �������� ��� ���� ������4 �� �������� �� ������ ��� ������� �� 4 �� ��

����

�� ����� ������ ��� ������ ����� ��� ������ �������� ��� � � ����� ������

������� � ����� ��� 4 �� ��� ��� � ��� ��� ��� ���� ������4 �� ����� � ����� ���

���� ��� � ��� ��� � ��� ������ �� 4�

1�� "������� ���� ���� �� 3����� 0�0 �� ���� �� ��� � �� ������� %���� ��� ����� ��3����� 0�0 �� ����� �� � ���:�� ���� ���������� ���� ������� ������� �� ��� ��������� ��� ������ ���� ���� ��� ������ ��� ����������� �� ���������� ������ ������ �������"������� ���� ������ �� �� ����� ��������� �� ��

���� ��� &���"���� ����� �� �������

.���� ��� ��� ����������� ���� ��� ����� &���"���� ���� ���������� �� � ���� �����

� ������ � ���!������ "�� �� �� ����

,������� ����� �� ��� ��� � ? ����� 6��7������ 6��7������� ���� ��� �� �� � �������������� �� � ��� ��� �� �� � ������ �� � � A�� �� ��� ��� ����� ���������� ������� ����������� ��� ��� ��� �������� ������ A�� �� ��� ���� ��� � ���)�������� ���� ������ �� ��� ��� � � ���� �� �� � ��������� ��� ��6���� � ���7� � ���)�������� ���� �� ����� 6�� � 7 ���� � �� � ���)�������� ���� ��� ��� ��������� � ���������� ��� ������������������ 6��� �������������� �� ��� ������� �� ������� ����� �� �7�

C�� ��� ��� ����������� ��������� �� ��������� ��� ��� ������� �� ���� ��������� ��� �������� ������ ������� �� ���)�������� ������ � ����� ������ ��� ����� �������������!������ ��� ���� ���� ��� ��� ����� ��� �������

�� � � �@ � ���

�� � >�� �@ ��6��� ��� �� � � �7�

8� ���� ����� �� �������� ��� ������� �� B ��!�� ���������� �� ���)�������� ������ #���������� �����)����� ���� ���� ��� ��������� �� ��� ���������� �� � ����� �� ���� ��� ����� ������ ����� �� &���"���� ���� �� ������� ��� ��� � !���)����� ������������� ��������� �

������� ���6���� ����� � ����7

���:� ���6��� ����� � ����7

������������� �6���� ����7

����� ���6���� � ���7

�����)����� ���6���� ����� � � ? �7�

(� �� ��� ���� �� ��� ��� ����� !���)����� ������������� ���� ��� ����� ��� ������

#���� ��� � �������������� ����������� ��� � �� ��� �� � �? B�

=

46

Page 56: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

8� ��������� ��� ����� ��� ���������� ������ ����� ���

$�������� �� ��� 4 �� � ��� � �������� ���� ��� ��� ���� �����4 �� ����� � �������� ���� ��� � ��� � ��� � ���� ����������� ���� �� ������� 4�

%��� ��� ��� ���)�������� ���� � ? ����� 6��7������ 6��7������ � �� � �� ��������� �������� ������ ���� �� ? ����� � � �6�� ��7 � � � ����7� (� ���� ���� ��������� ������� �� ��D������ �� ��� ����� ���� ��� ����� �������� ������ ������ ����� �� ������ ������ ���)�������� ���� � ��� ����� ? ��� � ����������� ��� �������� ������� ��� ����� ��� ��������� 6�7� ��������� ���� ����� �� �������� ��� ��� ������ ������� ���� �? ��@ ��� �? �� E�������� �� � ������� �� ������� �� ����������� ���������� ���� � �� ������� ��� ���������� ���� �� �� �@ � �� ����� �� ��� ���������� 6���)��������7 ���� �� �������� � ������ ��� ����� ����

������� ��� ��� 4 �� � ��� � �������� ���� ��� �� ��� ���������� ������ � � ������������ �������� ���� �����4 �� ����� � ����� ��� ���� ��� � ��� � ��� � ���������� �� 4�

$�������� ��% ��� �!������� � ����� ���� ���� "�� #���� �!���� ��� ���������� ������� � � ����������� ������� �� ����� � ����� ��� ���� ��� � ��� � ��� � ���� �������� ��� �!�����

;����� ��� ;����:� �/0� ������ � ������ ������ ��� ��� ������ �������� ��� ��� �������������� ��� � �������� ��@����� ������

% �����!������ ��� &���' !��!����

�� ���� ������ �� ������ ���� �� ������ ��� ���� ������ ���� ��� 8���'� ��!���������������� �9�� (� �� :�� � ���� ������������� ����� �/�� ��� 8���'� �������� �� � ��� ��:��� #��8���'� �������� �� ����� � � ���� D��� � ��������� ������� �� �������������� ��� ������������� ���� ������ ������ ����������� ������� ��� �� �� ������ ������� ������� ���� ����� ��������� �� �� ��� ����"����� �� �� ������� �������� �� 3����� / � ������ ������� �� ������������� ��� ���� 8���'� ���������

� ���� ��� ������ ����� ��� ���������� ����� �� ������� � � � �� ��������� ����������� �� ����������� �� ��� ���� � � � ��� � � � ��� ��������� ��� ��� ����������� �������������� �� � ��� ���� �� � ��� �� ��

E��� �0F� ,�������� 8���/� ��� �� ���� ����� ������� ���� ���� ��� �� ������ ��� ������ �� � ��������� ���� ��!����� ���� �� ����� ��� ������������� ���� ������������������� %��� ��� �������� �� ������ ��� !���)����� ������������� �� ����� �� B ��� ������������� ��������� 3�� ����� �� ������ � ������ ������ ����������� �������� 8� �00�� ��������� ����� ���� ��� ������������� ���� ����������� ������� ��� ������� ������ ����������� ������������� ����� � ���� ��� ������������� ���� ����������� �������� ����� � ������ ��� ����� ����

������� %�� ��� 4 �� � ��� � �������� ���� ��� �� ���� ������� $�� �������������� ���� �����4 ��� ������ ����� ��� ���������� ������

� ���� ���� �� ���� �� ���� 8���'� ��!�������� �������� �� ����� ������ ��!������ ���� ��� �������� E��� ��������� ��� G6�7 �� � ��� �� ������� ��������� ��� ����������������� � ��� �������� ����� ����������� ������� ��� �������� G6�7 ��!��� � �� �� ��� �����

�"� �������� �������� ���� ��% ��� �&������'����� � ����� ��� ���� ��� � ����( � �&��� �� �������������� ���������% � )�*� � �+ � )�� � )��� ���� �&��� �� ��� �� �� �������������� � �������������������� �� ��&��� ,� ��� ����� ����� �� ��� ������� �������� �� ����� ����������� ��������� �� ���������-����� ������ ! �������(� �� ���� ��� ��� �.�&���% � )�*� � �+ � )�� � )��� ���� ���� �� �� �� !��������� ����� ��� ����� �� ������� � ��� ��� ����� �� ��� �������� � /�� .��� 0 ��� /��0�

47

Page 57: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� ��� ���� G6�7 ��� G6��7 ��� ���� �� ����� ������ ���� � � �� �� ���� �� ����� ������� (������ ����� G6�7 ��!��� � �� G6�7�G6��7 �?��� �� ��� ���� �?��� ������� ������ ����������8���'� �������� ������ ���� ������� ���� �������� ����� ������ � ������ � �� ��� � ������� ���� ��� ���� G6�7 �?��� �� �� ,������� � �� �� ������� ��!������ �� �� �������� �� ��������� G6�7�

8���'� �������� �� � ���������� ������ ��� ��!�������� �� ������ ���� ����� ��������!������ ���������� �� �� �������� �������� ��!������� � �������� ������� ������� �������� ��� ��� 3����� ��/�

������� %� ��� 4 �� � ��� � �������� ���� ��� �� ���� ������� $�� �������������� ���� �����4 ��� %���&� ������� ��� �������

� $�� ��� !��� ��� ������ � �(��

�� � �������� �� ��� ������������ ��� ����� &���"���� ���� ���������� �� � ���� �����(� �� ������� �� ��: �� ���������� ������� ��� � ������ ��� ���� ������ ���� ��

���� ���

� ��������� �� ���� ��� ������� ������ 3�� ��� �� �� ���������

������� ��� ���� �� � ��� �!��� � ��� � "�� #���� �!��� � ���� ���� ��� ����� ������

������� �� �� ��� ���� ��� � ��� ��� � ��� ������ �� � � ��

)���"* ,������� ��� ����� ��� ����� 6��� !���)����� ���� ��������� ���� ��!�� ��� ������� ���7�

��� ���� ��� ���6��� ���� � ��6��� ����77�������� �6� ���7� �6�� � �7 ����6��� ���� ���� ����� � ? �7����� ��� �� ���6��� ���� � � ? �7

��� �� � &���"���� ������ ��� �������� �� ����� �� �� �� ������ ���� ��� ��� !���)������������������� ����� ���� ���������� ��� ��� ����������������� ������ �� ����� � ������� ����� �� ��� ��������� ��� ��� ��� ��������6��� ���� ����� ��� ����� ��� ���� �����7�3� ��� ����� ������� ��� ����� ��� �������� �������� ���� �� ��!���� ������ A�� � �� ���������� ���� ��� ���� �������� �� ��� ��� ��������� ���� ��� ������ ��� !���� ��� �)!�������� ����

1��������� ����� ���� �� ������ ���� �� ����� �� �� #���������� ��� ��� �� ������������ �� � �� ����� ����� ��� ���������� ����� �� ������ ����� �������� ����� ��� ���� 6��������� ����� � ��� ��� ��� ���� ����� ��������� ��� �� ���������7� �� � ��� ����� �� ��� �� �? ���� � �? �������� ��� � �? ������ .���� ���� ��

��������� ��������� �� ��!��� �� ���

��� ��� ��� ��� �� �������� ����� ��� ������� ��� �� ������� ��������� ��� ����� �� ����������� ������ ��������� ���� �� ����� ����� ��� �������� ����� 6�������� ����� ��� ����7� (��� ������ ����� � ���� ����� � ��

��������� ����������

3� ��� ���� � �? ���� ������� �� �� � ���� &��� � 6�7 ���������� �� ��� �������� !���� �� �)!����� &��� �����!�� ���� ����� ��� �� � ����� ��� ���� ������������������ �� &��� ����� ����� �� �� ��� ��!������ ��� ��������� �� ����� � ���� � 6�7��� �� ��!����� ���� �)!����� (� ����� � ���� ����� ���� ��� � ������� ���������� ����� ���������� �� �� �? ���� 3��� ����������� ��� �������� �� ���� 3��� � �? �������� ���� �? ����� �� ����� �

�-���� �� �� � ����������� ����� ��� ��������( �� 1� ��� 1*��+ �� ��� ���� � � ��� ����( � ����������� � ��! �� �� 1*�+�

48

Page 58: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

+ $����� ��

(� ������ ���� � ���� � � ������� ���������� �������� &���"����'� ������ ��� ��� ��������� ���� ������ %� ��� �� ���� ���� ����� �� ������� �� ����� ��� �� ��� ������� ���3�� ��������� �� �������� ��@����� �� ����� ������ ����� ���� ��� ��������� ������ �� ������������ �� ����� &���"���� ���� �� ����������� �� � ���� ����� (� ���� ���� ��� ���������� � ������ �������� ����� �� ��� ���� ����� ��� ��� ��@����� ��������� (� ���� ���������� �� �� ������� ���������� ������ ��:� C����'� &H ������ �0�� ����� �� ��� ��� ������� ������ C���� ��������� ���� �� ��� ���)����� ���� ��� �� �� ��� ��� ������������ ������ �������� ������ ��� ����� �� �� ������� ����� ��������� ��� ������ 3�� �������� ����� �� �� ����� �� ��� ��� �� 3����� ��/ �� ������� ���������� �� ���� �� !��� (����� ����� ������ �����

� �������� ��@����� �������������� �� 3����� =�= �� �� �����<�� �� ����� ��

&����� ���������� ��������� ��� ���� ����� �� ��� ������������ ������ ���)��� �������� ���� ��� ����� �� "������� ��� ��������

(� ����� ����� 3����� =�= �� �� ��� �� �� � 6�������7 �������� ������� 3�� "������� ���������� ��� ���� ����� �� ����� ������ �������� �� ������� �� ��� ������ �� ��� (( ������ ������ 3����� ���� ����� ������� � ������ �������� ������ ��� ���������� �� ������������ ���� ����� ���� ���� !��������� 3���� ������� ������� ��� ������ ���� ��� ������������� �� ���)������� ����� ��� �� ,# ��� ,�-�

���� ��� ������ � � ����� ��� ������ ��������� ����������� �� ������ �� ����

� �����������

#�� ����� I���:� ������ ���� ���� �� � ��!�� ��� ������ ������� �� �� ��� ���������� �� ��� ���� �� ����� ��!��� �� �� 3��� ����� ��������� ��� ����� ��� "�������� �������� ;����� ��� ;����:� �/0��

(� � ���� ���� � ��� � �������� �� ���� ��� ������ ������� �� ���� � ���*

;����� ��� ;����:� ��: ���� "������� �� ��� ������ �� � ������ �������� ��� ��� ����������������� 3��� ��� � ��� ������� ���� �������� ���������� (� ���� ������ � ��� ���� ��� ������ ������ ��������� ����� ��� �������� �� ��� ���� ���� �������� ��� ������������ (� ����� �� ������ �������� ������� ��� ��� �������� ��������� �� ;����� ��� ;����:����� ���� ���� �� ����� ��� ������� ��������� �� ���� ����� ��� �������� ������� ��� ����������� �� ��� ��������� ������� 6�� ;����:� ��� ���� �/J� ��� ��� ������� �������7� (� �0=�&���� ����� �� ������ �� � �������� ���� ���� ��� ��� � �� ������ �� �������������� �� �������� ��� �������� ��� ��� ��������� �������� (� ���� ����� � ����� &����'������� �� ��� ���� ����������� ��� ��� !���� ���� �������� �� ��� �������� �� � �������� ���� �������� ��� �������� 3��� �������� ������ � ��� � �� �/0��

#�� ����� ���� ���� � ���� ��� � ����� �������� � ��!�� � ����������� 6�7� ������ �������� �� �� �� ��� �������� �� � ��� ����� ���� �� � ����� !��������� ���� � ��

�@ �� �� 3� ��� ��� ���������� �� I���� ��� %����� �/K� ��� ;�����<�� ��<<� ���C���� �/��� � ��� ���� � �������� �� ����� ��� �������������� 6�7�� �� � ��������� � ������ �������� �� �������� ��� 6������7 ������������� ��� � �� �� � �� ��� �� ���������<����� �� �� �� ������ ��� ����� ���� ���� �� ��� �������<����� �� �� %������ ������ �������� ������� ��� � ����� �� ����� ������ ����� �������� �������� ������� ���� ���� ��

2

49

Page 59: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

&������ ������� ���� �������� ���� �� � ���� ���� � ����� !��������� ���� �� ��������� !�������� ��� �������� �� � �� �� �������� !������ �������<����� ���� �� �� ��������6� �� ��������� �=�7� %���� ��� ��������� ������� ������ �� � ����� ������ �� ��� ��� ��!��������� ��� ������� �� ����� ���� �� ��� ��� !�� �������� ��� ���� ���� ����������� ����������� � �� �� � ��� ���� ��������� 6������7 ������������� ��� !���� �������<�������� %� ������� ��� ���� ���� ������ ��� ���� �� ������ �������� �& ����� !��������� ������������ ��� �������� ��� �& �� ��� ������ ������� �&�

3�� �����"�� ��������� �� ���� ���� �� ��� ����� ���� � �� �� ������ ����� ������ ���������� ������� ���� ��� ������ ������ �� ����� ����� ��� �������� �� �� �/� ��� ����������� ��� !���� �������<������� �� ).#� �0/�� #���������� � ����� ���� �� � ��� �,#� ��� ,�-� ���� ������ ������������� ���� ����������� �������� �� ��� �� � :�� ������� ������������� ��� ��� ���� ������� �� ��� ������ �� ������ ���� ������� #������� ���������� ��!� ��� ��������� ���� ������ ������� �� � ���� ���� �� ��� ���� ���� ���������� ��� ���������

)��������

�� �� �� ���� � � ��� ����

.���� ��� ��� ( ���� ��� ���� ������ �������� �� ��� ������ �� ��������� ��� ���� �������� ��� ������� ��� ����������� ���������� (� ��� ������� ����� � ��� �� ������� ��� ����� ���������� ������ ��������� 3�� ����� ������ �������� �� �� ��������� �� ��� �������� �������� ��� �������� #������� ����� � �������� ��� �� ����������� ������� �������� � �������� ��� �� ������� ���� ��� � !���� ��� �� ��������� ���� ��� ����� �������������� �� ��!��� ��

� ��? � � � � �� � � � � � ����

���� � � ����� � � ��� ��� � � ���� %���� ��� ����� ���� � ��: ��� ��� ��� ����� �� �������� ���� ����� � ��� ��� ����� �������� �� ��� ������ ��� ������ ����� ���� �� � ���"�� ����� �� ��� ����� (� ����� ����� � ���� �� � ���� 6�� � 7� ���� � ��� ���� ��� � �� � ��������� ��� � ��� �� 6�7� ? / ��� ��� � � ���� ����� ��� ���� �������"�������� �� ������ ��� ��� �� ��� �������� (� ���������� ��� ����� ��!������ ���������� �� ��:� ���� �� �������� ����������� ��������

6�� � 7� � � �@ � � 6�7

3�� ��������� ��"������� �� ��� ��������� �� ������� ����� ���� �� �� ����������� #��������� 6� ��� � ��7� � �� ����� �� ��� ����� 6�� � ����� �� ��$����� ��� ��� ��� ��������������� �� ���� ��� ����� ������ �����!�� �7� ����� ����� ������� ��� ���� ���������� �����!����� ���� ��� ��� ��!����� �� ��� ���� ���� ��������� #�� �������� ����$��������� ��!��� �� � � ��� 6:��� �� ��� ���� ���� �������� ��� ������ ������� �� ��!��� ����������� "�����!����� ���� ����� ����������� ����� ���������� ���� ������ �� ��� ������ ���������� ���7�

;���� � ���� ���� �� � ��� ��� ��� �� ������ ��� ��� �� ������� �� ��� ����� �������������� ���� ��� ����� �� � 6�� ����� ����� ��� ����� ���� � �7� �6�7 �� ��� ��� �� ����������� ����� �� ��

+� � ������� � ��� ����� � I���:�)������ ���� ���� �� ��� ��� ��� �� ��� ���� ������� �� ��� �� �� ����� 6���� ���� � �� ������ ��� ���7� %� ��� ��� �� �� � ������������ ������ ����� ��� ������ ���� �� ��� ���� ���� ��!��� �� �� %� ��� �� ��� ������������� �� ��� ���� ���� ��

+���� ���� ���� �� ��� ��� ���� �������� �� �� ��!�� ������ �������� ��� ���� ������(� ���������� �� � ���� ���� � �� ������ ��� ������� ���� ������� ��� ������ ����� ������� ������ ���� ��� �� ��� ���� ��� ��� ���� ������� ��� ������ ���� �� ����� ���������� ���� ��� ��� ������ �� �� +������������ ��� ���� ���� � ���� ������� ����

J

50

Page 60: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3�� ��� "������� � ������� �� ���� ����� �� ��� ����� ����

%��� ���������� �� ����� ��� ��������� ����� ������� ��� � �� ��*

����� �� ��� �/� ��� ���� ��� �����!������� ������ ��� ��� ������ ����� ���� ��� �������)������� 3��� ������ ���� �������� ���� ��� �������� �� �������� ���� ��������!������� ������ ��� �� �� ���� ��� �)������� 3�� ���� ���� �� ������ ������& �������� � ���)���� ������� 3�� �����!������� ������ ��� �& �� ��� �)�������K�� �� �����

������� �� ��� ��������� ��� ��� �� � �&� �� ���������� ����

)���"* #�� ��� ���� ������ �� ��� �� ? ���������������� ���� � �� ��� ������ ������ �� �������� ��� � �� ������������ ��� ��������� ��� �� 6����� �������� ��� ���������� ����� ��� �� �� �6����7 ��� �������� ��� ���������� �� ��� ��� �� �� �6��� �77� 1���� ������ ��� ���� �� �� ����� �� � ���� �� � ������ ����� ���� � ����� �������� ����� ������� �� � ��������� �� � ���� ��� ���� ������� (� ����� � ����� �� ������������ �� �� �����!���� �@ � �� �������� �����!����� 3�� ������ �����!������� ������ ��� ����������� �� ��� ���� �� ������ ����� �� �����)������ �K�� ����� ��� �����!������������� ���&� �������)����� 3��� ��� ������ �� ����������� ����� � ��� ��� ������� ������� ).# ��� ������� �� �� ����� �/F�� 3� ��� ����� ��� ����� ������ ������� � �������� � ������ � �� ��� �������� �� ������� ).# �� �������� ����� ��������� � �� � �� �6� � �7�� ��� �������� ����� ������� �� � �� � �� �6� � �7��� ��� ���!��� ���� ������ �� 3��� � �� �����!���� �� ������ ����� �@ � �� �����!����� ������� ������� ).# ��� ������� �� �� ������ � ������ ���� �&� �� ���� �� ������

+���� � ��� ��� ���� ����������� ��� ��� !���� ���� �������� �� ��� �������� �������,������� ��� ��)���� �������� ��� ��������� �� ��� ��� ��� ��� � �� ��� ����� �������� �������<�� �� ��� ����� ��� &���"���� ������

������ �

������6� � ��7 6�� ��� 0 ��)��������7�

������� ��

�������6� � ��7 6�� ��� = � �)���� ��)��������7�� ����� 6�� �� ������7

)��!� ��� � � ��� ��� ����� ���� ������ ��� �� ������� ��

)���"* #����� ������� ��� ���)���� ���� �������<�� �� ��� !��� � � ������ 3��� �� �������� ���� 6����� ��� ���� �� ����� �� ����� ����� ��������7� (� ����� � �� �J� 3�����//�0F� ���� ���� ���� ��� ��� !���� ���� �������� ��� �� ��������� +� ������� ��� ���)���� ���� �& ����� �� ��� ���� ����� 3��� ���� �� ������ ��� ��� ���� �� ������������ ��� ��� !���� ���� �������� �J� ��� ��� � ��� �)������ �����!������� �������K�� &��� ����������� ��� ��� !���� ���� �������� ��� ��������� ����� ������� �//�� ��������� ����� �� �

)��!� ��� �� �� �� ��������� � ��� ��'� ��� ����� ���� �������

)���"* #�� ��� ���)���� ������ � ��� ������� ��� ��� �� ? � � ��� � ������� � ����

1�� �� ������ ��� ���� �� �� ����� �� � ���� �� � ���� � ���� � ����� �������� �� ���������� �� � ��������� �� ����� ��� �������� ��� (� ����� � ���� �

� �� �����!���� �@ � ���������� �����!����� ;����� �����!������� �� ���)���� ������� �� ��� ���� �� ����� �� ���

K

51

Page 61: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��� ����� ��� �� ��� � � �������� ��� �� ��� ����� � �)���� �������� �� �����������0=�� (� ����� � ���� �� �� �����������

8� ,�������� =�� �� ��� ( ��� ������ ������� �� ��� �������� �6>7 6��� ��������� �� ����������� ������� ���� ��� �������7 ����� �� ���� ���� ��� ���������� ���������� (� �� ��� :�� � ���� ����� ���� ��� ��� ��� !���� ���� �������� ��� �� ���������� ����������� �������� 6��� ����� ��� 3����� 2�/��7� �� �� ���������� ��������� ��� ����������� ������� ��� ��: ��� !���� ���� ��������� �

L���� ��� 3������� ���������� ���� ������ �� �� ������ ���� � ���)���� ������ �����)�������� �� ����������� ��� ��� !���� ���� ���������

(� ��� �������� �� ���� ������ � ������� �������� ������� ��� � ���� �� ����� ���� �������������� ���):�� � ���)������� ������ �������� ).#� ,# ���,�-� %� ��� ���� ��� �������� �� ������ ��������� !���� �������<�������� ������������� ��� ������ ����������������������

� /�������

(� ��� ���� �� ��� ������ � ��� ����� �� ������� ��� ����� ���� ���� !��������� +� � ��� ����$� ����� ��� ���� �� !���������

A�� � �� � ���� ����� �� � ���� � ? 6���7 ��� ��� 4 �� � ��� �� ������� ���������� ����������� H�!�� �� �"�������� �������� �� �� � ��� ���� ��� ����� � � ���� � � 4�

�� �� �? � � �? �

H����� �� �� ��� ��)�"�������� ���� ��������� ��� ��� ����� ��� ��� �� ��� ��)

�"�������� ������ �� � � H�!�� � ��������� �� �� ������ ���� ��6�7 ? ��� � �? ���

3�� ���� ���? 6���

� ��� ��7 �� ����� � !�������� �� � ������� 4 �� �� �� � �������������� �� ���

��� ���� ��� ��� � � 4 ��� �� � � �����

�� �? � �@ ���� �� �? �

(� �� ���� �� ��� ���� ����� � 0���� ���� ���

�� !���� ������� 4 �� !����� 3�� �������

? 6���� ��7 �� ����� � !�������� �� � ������� 4�

.�0���� �% (� ��� ���� � ���� ���� � ������ � ������ � � ���� ��� � � �����!���� � ����� ��� � ��� �� 4� ���������� � ������ �� � � ���� ���� � ���� ������ ? 6���7) ���� ����� � � ��� ���� ���� � ? 6�� � 7 ���� ���� �� �? �) ����� ������ � � ��� 4� �� �� ������

(� ��� ���� � ���� � ������ �� ������ � ������ � �� ������ � ������ ��� ��� ���� �4� �� �� ������ �� ��� ����� � �� (� ��� ���� � ���� ���� ������ ���� � � ������ � �������� � ������ ��� � ���� ��� � � �� ���� 4� ? ���6�7�

+��� ���� ���� ��� ��<� �� ���6�7 �� ������ �� ��� ������ �� �� ����� ���� ���� ���������� !�������� ����� ��������� !���������

������� �� * � ������ +�� ������ , ���� �- � ������) ���� �� ������ +�� ������ ,���� �- � ��������

)���"* %� ����� ����� ��� ����� !��������� �������� ������� �� ����������� �������� ������� ����� �� �� ��:�� �� ���� ��� !������� ���� �� � ������ ����� �� ����� ����� ��������� ������ ������� �� � ����� ������ �� ���� �� ������ ��� ������ &��� ��� �������� ���� �� ��� ����� �� ��� �������� ����� �� ��� ���� �� ���� �� ��� ����� �� ��� !������������ +� � ������� ���� � ������ � �� ���� �� � � ������ �� ��� !������� ����� ��� ����� ��� 3��� �? � ��� �? � �� ��� �������� ����� ��� �� ? � ��� ������ ������ ? ��� �

�� ������� /20 ��� ���(�� ��� ������� /34� �5�20 ����� � ���� �� ���� �� �� �!���� �(��� ���� ������������ ����� ��� ���� ��� �������� ���� ��!�

9

52

Page 62: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ��� ���� �����

(� ���� ������� � ������� ��� ��� ������������ �������� %� ��!�� � � ������������ ������ ����� ������ �������� �� ��� ���� ���� �������� ��� ����� ���� ���� �������� ���)��!�������� 3�� !��� ����������� ������� �� ����� ��� � ����� �������� ��� ��� ����� ���������� �� � ����� �� ����� ������ � ����� ��������

��� #��� 1�� � �� ��� �������

.�0���� ��� � ���� ���� � ��� � ����� ���� ��� ./) �� 0123 � ���� �� � ���� ��� ��6�7 ���������� �� � ��� ���������� ���� � ���� ���� � � ���� � � ����� �� �� �������� � �� �� ) �� �? �6�7 �� � �� ��� �������� �� ��� ������� � � �������� �� +�#���� ��� �) � � �� ��� �� � ����� ������ � �� �� � ����� ����� � �����-� * � ���� ��� � ����� ���� ���) �� �� �� �� �� �� �� 6��� ������ �) �� �� ��� ��� �� ��������� � �6�77�

/��� �� 2� ��� ���� � ������� ����� ��� � ����� ���� ����

4� ��� �!������� � �% ��� � ����� ���� ����

0� ��� �!������� � �� ��� � ����� ���� ����

/� ).# ��� � ����� ���� ����

5� ��� �!������� � �%��% ��� � ����� ���� ����

6� ��� �!������� � ��� ����� ���� �%� ���� ��������� ��� � ����� ���� ���+���� ��������� �� ��� �!��� �� � � � � 6� � � 7 � � 6 � ��7 � � 6� � 7-�

)���"* /� 3�:� � ?�

���� ���� � �� ��� ����� �� ��� ������

0� 3�:� � ? � ����

=� 3�:� � ? � ��� ����

�� 3�:� � ? �6����7����

�� 3�:� � ? � ���� ���� �������

2� 3�:� � ? � � �� � ����

+��� ���� � ���� ��� ���� � ����� ������� 6���� �� ������ �� ��� � �� ��� ��� ��������� ���� ������ ��� � !���� ���� ������ ��� ���� �� ���� ���: ����� � ���� ������ ��� ��������� �������7� &�������� ��� ���� ����� ���� ��� ��� ����� ���� �� ��������������� �%� ��� ��� ���� �& �� ������ ����� �� ��� ���� � ����� ��������

3�� ���� �� ����� � ��� �� ��:��� ��� �� ��� ���� �� ����� ���� ���� � ����� ���������� ���� ���� !��������� A�� � �� � I���:�)������ ���� ���� ���� ��� � ����� ���������� ���� ����� !��������� +� � ��!�� � ����������� ��� ��� �������� �� �� �� ����������� �� �� 3��� ��� � ��������� ����� ������ ������ ���� � ���� ������� #�� � ������������ �6��� � � � � ��7� ��� ��������� ������ ��� ������ �������� ��� � �� �������� ��������

�$�� ������������ �� ������ ���� ��� ��(��(� �������� �� ! ���� ! ���! ������ ���(����� ��� ���� ��� ���� ����� ��� ���� ! �� (����� �'�� �� ��� ���� ������ ! ���! ������ ���(�����

/F

53

Page 63: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��� ������ � �� � ������� �� ����������� ������ ��� � #�� � ������ ������ �6��� � � � � ��7����

�� ? ��������� ��

�� ���������

��

�6��� � �7� ��6��� � �7

���� 4����������� ��� !�������� ��� �� ����������

������� ��� ��� � �� � ���� ���� ��� � ����� ���� ��� ��� ���� ������ � ������� ��� ��� ��� ����� ��� �� ���� � �� ����������� � �� �

� �� ���������� ��

)���"*

��� &������ 6�� � 7� �? � ��� � � ���� A�� � � �� ��� ��������� ���� ������ ��� � ����� ����������� ������� ������� �� �� ��� ��� ���� � �6���7 ? � 6�7 ��� ��� ������

�� ,������� 6�� ��7� �? ���������� 3�� ����� �� ��� ����� ��D��� �� �

� �� ����� � �

����� � ������� ��� ��� ��� ���� � �6���7 �� � ��������� ��� ��� ��� � ? /� � � � � ��

��� &������ 6�� � 7� �? �� ��� � ? 6���7 � ���� %������ ���� �� ����������� � ������� ���� � �� ��������� �� 6���� ���� �� �� � ������ ���� ������7� 1�� ���: ���� ������� � ������ ���� ���������� ��

#����� � ��� !������ 6�� � 7� A�� 4 ? 4���������� &��� � ����� !��������� ����� ������ �

���� � ? 6���� ��� ��7 ��� ���� 6���

� ��7 � ��� ��� ��� ���� ��� ��� � �

��� � � 4� �� �� �? � �@ 6�� � 7� �? �� (� ���������� �� �� �? ����������

$��� � ��6���7 �������� �� ���� ��� ����� +� � ? /� � � � � �-�

)���" �" ����* &������ ��� ��� � ��6���7� 3��� � � � � 6���7� �� ��� ��!������ ��

��� &��� 6�� � 7� �? ��6��� � �7 � ��6��� � �7 ��� ��� � � 4� �� ����� � ���� � �

����� �� ������� �� 4� (������ �� �? � ���� �? ��6��� � �7� �� �? ��6��� � �7

��� ��������� � �? �� 3���� �� � ��� �� �� ? ���� �

(� ����� ��� �� ���� �� �!��� � ��� ������ ���� ��� ����� �� !������� ���� � �� �������6���

� ��7 �� �� � ������ ���� ��� �� (� �������� �� ����� ���� ���� ��� �� ��� ������� ���� �� ���� �� ����� &�� � ���� �� ������ ���� ��� ����� ��� ����� �� ������ � ����� ���� ��� �� ����� A�� � �� ��� ���D���� ����� �� � � �������� ����� �� 6���

� ��7�#�� ���������� � ��� ��� ��� ��� ��� �� ����� �� ��� � � ������� ����� �� � ������ ����

� &��� ��� �� ������ ��!������ �� �� ����� ����� ���D���� ������ ��� ������ � ���� H�!�� ��� ��������� � � ��� 6���

� ��7 �� ���� ��6�7 ? �� � � ��6�7� ���

��� ����������� ������ � ������� �� �� ��� ��� ��� ������ � ? /� � � � � ��

� �6���7 ?

������ �� ��6���7 ? ����

����� �� ��6���7 ? �

(���������� ����:���� ��� ���� ���� �� ��� ����� ���D���� ��� �� 6���� ��7 �� �� �������

������ ������ �� ���� � �� �:� ��� ��� ���� ��� ����� ������ �@����� ��� �����

�� � �� ��� (������ � ����� ����������� ������� ��� � ���� 6�� � �7� �� �? ����������

8� ����������� � � ������� �� ��� ��� � ��������� ���� .������� ��� ��� �� ��������������� �� � ��������� ������ � ������ ���� ������ ��� ���������� �����!��� �� ���� %� ������ ���� � �� �����!���� �� ����

//

54

Page 64: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

$�������� ��% ��� � �� � ���� ��� ���� ���� ���� ��� � ����� ���� ��� ��� ���� ������� ������� ��� � �� ��� ����� ��� � ���� ������ � ��� � � � � ��� ���� � �� �� ��� �� �� ������ �7 � �

�� ����

��6��� � �7� �

�6��� � �7�� ���������

�� ���� ��

)���"* &������ �� �� ��� ��� ������ � ��� ������� ��� � � � � ��� 3��� �� �� ��� �� )�����!����� ����� �� 3����� =�=�� �

�� ����

��6��� � �7� �

�6��� � �7�� ����������

�� ��� �)�����!����� ��� ���� ����

� � �

�����

����

��6��� � �7� �

�6��� � �7�� ���������

2����3 ��� %� ����: ���� ���� ��� ���):�� � ����� ��� :�� � �� ���� � ������������ ��� �� ���� ��������� !��������� %� ���� ��� �� ��� ��� ��������� ��� ���������� �%� �%�� �.%�� �%� ��� �%�� �%��� �%�� �%��� ��� �%��� �%�� �J� ���=�M ,#��� ).# ��� ���K�M ��� �� �//�M ,�- �2�� E�������� ��� �� ����� ����� ����� ��� �%����%��� ).# ��� ,�- ���� ����� !���������

�� #��� 1�� �����1 �(��

+� � ��� ���� ���� ������ ��� ���� ����)���� ���� �� ���� ��� ���� � ���������������� ����� � �����!�������)���������� ����������� ������� %� ��� � ���� ��������� �� �� ����� ������� �� � ����������� ������ �� �� ��� ���� �� �� ��� ��� ���� ���������%� ��� ��� ���� ��� ������������ ������ ����� ��� ���������� �� �� ��� ������ ������+��� ���� ����� ���)��������� ���� �/9� �� ������ � �� ��� �� ����� ����� ������ 6����� ������� �� ��� �� ����������� ������� ���7�

%� ��� ��� �� �� � ��������� ������������ �

� �� � ��������� �� � � � ��� �� ������

� ��� ���

�� � ��������� ����

� ���

#�� � ������ ������ �6��� � � � � �7 ���

�� ? ��������� ��

��

���������������

�������6��� � �7� ������6��� � �7

���� !6�7 �� ��� ���� ����� �� � ��� H�!������ 0�0K�� +��� ���� ��� ������ �� �� ���� ������� ����������� �� ��� ������ �� �� ��� ���� �� ��� �� ���)���� ���������� �� ������������

������� ��+ � ����� ��� � � �� �������� � �� ��� ���� ���� ����� �� �� �������� ��

/0

55

Page 65: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

)���"* 3�� ���� �� ����� ��������� �� ���� �� ������ +� ������� ���� �� �� �����!�����A�� �� �? ��� ��� � ? 6�� � 7 ��� � ? 6�� 6��7�����7� %������ ���� �� ����������� ��� ����� ���� � �� ��������� �� � A�� �� ?

�������� #�� ����� ����� � � � ���

!�67 �� ��� ����� ����� �� ��)����� �� ��� �� �������� ��� ��� ���� � ,���������� �"�������� �������� ���������������

� 3 � ����� ����� �� ���� �"�������� �������� �@ ����

������� ��� ��� ���������� �� ���������� #�� ��� ���������������)�"�������� ���� ��� ����� �

�������������� " �� � �� ��� ���� ��� ��� � � �� � ���� !�6" ��7 � !�6�7� +��� ���� ����

" �� ? � ����� ��������������� ��� �� ������� ��� ���"��� ���� ���� ���� ��� ����� � �

��� � � ���6���������7� �� �? � �@ �� " �� �? ��A�� � � ? �" �� � � ��� #�� ��� � � ���� ��!�� ��� �������� ��� �� � � �� ����

" ������" �� �@ ����� �� � � � �� ��� " �����

�� H�!�� � ��������� � � �� � � �� ������� " �� �

� �6�7 �@ � � 6�7 ��� ��� � � ���6���������7� A�� �� ? 6� �� 6���7�����7 ��� �

� ? 6��� � �7�

$��� � 8� ��� � � ���6���������7 ��� ��� ����� �� ) �� " �� �? � �� ��� " �� �? �

)���" �" ����* 8� �������� �� ��� �������� �� �� (� � �� � ������������� ������� ������� ��� ����� �� ��� ��!������ �� � �� 3�� 8������ ���� ��� �������� #������� ��� � ? �#���� ��� � � ����

��� &������ ���� �� " �� �? �#� 3��� ����� �� � ����� � � � ��� ���� " ����� ���

�� � �? #� &��� # � ���6���������7 ��� � ���������������" ���� � ���� ������ " ��� �? #� 8�

��� �������� ����������� �� ����� � ������� " ��� �? #� #������� � ���� ���� " �����" ������ ��� ��!������ �� ��

�� ����� ��� " �� �? �#�

��� &������ ���� ��� " �� �? �#� 3��� ����� �� �� " ��� � � � ��� ���� " �����" ��� ������ " ��� �? #� 8� ��� �������� ����������� �� " ��� �? #� ����� �� ��� ��!������

�� ���� ����� ��� �� � �� � ��� ��� ���� " �����

�� &��� # � ���6���������7 ����� ���������������

" ���� �� ����� � ���� �� �� �? #� %� ������ ���� �� " �� �? �#�

A�� �� ��!�� !�� ������ �� !�� +��� ���� �� ���� ��� �� �����)��������� ������� #�� �����

" �� �� � ���� ��� ��� �������� ��� ��� ���� " �� ? � � ��� !��6" ��7 ?��

$��� !�6" ��7 � !��6" ��7) � � ��

)���" �" ����* (� !��6" ��7 ?�� ��� ��� ��������� ������ 1���� ���� ��� ����� ��������� �������� �� !��6" ��7� 3�� ���� ���� ��� !��6" ��7 ? F� ���� ������� �� " �� ? � �� ��� ��� ��� ��� ������ ������ +���� ������� !��6" ��7 ? �N/� 8� ��!������� ����� ����� � ���� �� ��� ���

" �� ? ������ � � �

������ " ����������� " ��

��� ��� ��� � � � ����� � ���� (� ����� � ���� !��6" ���7 � �� ��� ���� �� ��� ������������������� !�6" ���7 � !��6" ���7 � �� &��� " �����

�" ��� �� ��� ��!������ �� ��� � ����

���� ����� �� � � � �� ��� ���� " ������� 3��� ������ ���� !�6

�7 � �N/� 8� ��� ��!�������� " � � :�� ���� !�6" ��7 � !�6

�7� ������ � � ��� 3��������� !�6" ��7 � �N /� �

$��� � 8� � � ? / � � � �) ���� �� �� ���� ��� �� � " �� �� � ���� ���� !��6" ��7 � !6�7��� ��� " �� �? ��� �

/=

56

Page 66: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

)���" �" ����* &������ ��� " �� �? ��� ��� ��� " ��� �? ��� � ��� !��6" ��7� !��6" ���7 � !6�7� 8� ,��� 0� !�6" ��7� !�6" ���7 � !6�7� #���������� �� " �� �? ��� ��� �� " ��� �?��� � 8� ��� ������� ���������� �� �? ��� ���� " �� ������� " ���� ��� ������ ����" �� ? " ���� �

#�� ,��� /� � ��������� ����� ������� �? ���������� 3�� ��������� �� ��� � � � � � ��� ��

�� ��������� �� ��� ����� ��� ����� � !6�7 ������ �@����� ��� ����� �� ��������� �� �(� ���� ��� � ������ ���� ����� ��� �� ���� �� �� ��� ��� ����� #������� �������� ��� ���������� �� �� ��� ����� �� 3����� =�=� � ������ ���� ��� �������� ������ ������ � �������!����� �

A�� � �� � ���� ���� ��!��� �� !������ ��� ������ ������ ��� ��� � ������ �������6��� � � � � ��7� ���

�� ? ��������� ��

�� ��

�������6��� � �7� ������6��� � �7

���� 4 ������� �� ��� ���������� �� � ���� ��� ����� ���������� �� ��� 6!������ ���7������ ����� �� � 6����� ���� � ���� ������ �� ����� �� �� ������� �� ����������� �������7�

������� ��4 ��� � �� ��� ���� ��� ���� ���� �!��������� �� ����� � ���� ��� �� �!������ � ����� ��� � � �� ����������� � �� �

� �� ���������� ��

)���"* %� ��� ��� ��� ���������� �� �� ��� ����� �� 3����� =�2� ��� �� � ��� � �����!�������� ���� ���� ���� ������� ��� ����� ���������� �� ��� ������ ����� �� �� (� ���� ����� ���� ��� ��������� ���� �� �� �� �)����� A�� � � �� � ��������� �� ��� ��� ��� � �� �

�� ��� ���� 6��� � �7� � �? �� H�!�� � �� � ��� ���� � � 6�7 �@ " �� � � �6�7� %� ������� ��� ��� ������ ����� # �� � ��� ��� ��� �� � 6�� � 7� " �� �? # �@ 6��� � �7� " �� �? #�

3���� � ����� �� �������� �� #� +��� ���� # �� ������ � ��� ���� � �� ����� ����# �� ��������� �� ��� ����� ��� �������� ��!�������

# ��? � � � � �# � #��#� � ��� ���� � �� ��� 8������ ��������� �� ������������������ ��� ����� ��������

3�� ���� ���)������� ��� �� ��� �������� �� ��� # �� �� ��� ��� �� ���� � �� � 8��������������� �� ����������� ������� ��� ����� �������� (� ���� ���� � ������ �� ����� ��

��� &������ 6�� � 7� " �� �? ��� 3��� ����� �� � � � � ��� ���� " ����� ��� 6�� � 7� � �?�� 8� ��� ��!������ �� � ��� ��� ��� ���� ��� ����� ���������� �� � ��� �� ���!�������� ���� �� ����� � ���� 6��� � �7� " ��� �? �� 8� ��!������ �� ���� " ���

��" ���� �����

6��� � �7� " �� �? ���

��� &������ 6��� � �7� " �� �? ��� 3��� ����� �� �� " ��� � � � ��� ���� 6��� � �7� " ��� �? ���� " �����" ���� 8� ��!������ �� �

��� ����� �� � �

� � ��� ��� ���� " ������� 8� ���

��!������ �� � ��� ��� ��� ���� ��� ����� ���������� �� � ��� �� ��� !�������� ���� ������� � ���� 6�� � 7� �� �? �� ����� 6�� � 7� " �� �? ���

3��� ����� ����� �������<�� ����� ��$������� 6�� � �7� �������� 6��7 ��� ������� ���� 6� � �

����� �

�����6� � ��77�

+��� ����� ���� ��� ������ �� �� �� �� ������� ����������� �� ��� ������ �� �� ��� ���� ����� �� ���)���� ���������� �� �� ����������

/�

57

Page 67: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

$�������� ��5 ��� � �� � ���� ��� �������� ���� �!��������� � �� ����� � ���� ��� ���!����� ��� � �� ��� ����� ��� � ���� ������ � ��� � � � ��� ���� � �� ����� �� �� ���� ����� ��� � �� ���� ��7� �

�� ��

������6��� � �7� ������6��� � �7�� ���������

���� 4 �������� � � ������ �� � � � �� � � ���� ������ �� � ��� +����� � ����-��� �� �!���� � �

)���"* �� ��� ,�������� =��� �

% 6!!������ �" ��� ���� �����

#�� ��� ��:� �� �������� �� ���� ������ ��� ������� ��� ������ ������ ������ 3�� ������ ���������� �� �0� ��� ��� �������

#�� 3����� =�= ��� =�J� �������� ��� ��� ����������� ���� ��� ����� �������� ����)����� !��������� ��� ������ �� �� �� ��������� �� ��� ������ �� �� � ������ ��� ����� ����������� ������ ��� ���������

$�������� %�� ��� � �� � ���� ��� ���� ���� ��������� ��� � ��� � ����� ����������7

+�- � ��� � ����� ���� ��� ��� ������ �� ������ � �������

+�- � �� � �������� ���� ������ �� ����� � ���� ��� �� �!�����

���� ������������ ��� �� �� ������ � ������ � �� ����������� ����

����� �� � ���� ���� � �����!�� �������� 6�7 �� 6�7� ��� �� � �� ��� �������� ��������� ����� ��������� ���������� ��� �� ��� ��� �� ������ ������� �������� ���� ���� �)�����!������� �� �� 6������ ���7 � �@ �� �� �� 6������ ���7 ��

+��� ���� ,�������� ��/ ����� �� ������ ��������<��� �����&� ��� ���)���� ���� �� ��)���� ������ ����� !��������� ��� �� 3����� 0�/� �&� �� �����)������� �������& �� ���� ��� �)�������

+���� � ��� ������ ��� ����� �� �������� �� ������������� ��� ������ �������������� #����� ������ ������ �� ��� �6�7 ��� �6�7 ������ ��� ��� �� ����������� ������� ��� �������������������� ������� �� ��

.�0���� %� ������!������ "�� �� �� ���� � � ����� ���� � ��� ������ ����� ������������� ����� � � � ��� �� � ��� � ���� ���� � �� �) ���� �� � ��� � � �������� � �� �) � �� � ��� �6�7 � �6�7 �6�7�

+��� ���� ������� �� ���� ��!������� � ���� ������ ������� ������� �� � ��� ��� �� ��� ��� ������ (� ���� ��� ������� �� ��"���� ���� ��� ������� ������� �� ��� ������������ ������ ��� ���� �� � ��� �� �� �� ����� �� ����� ��� ���� ����� ��� ������ ����� ��:���� ������ ��� �� ������������� �0���

.���� ���� � ���� ���� ������ ���� � � ������ �� �� ����� !�������� ��� ��� ����������� � � ���� 4� ? ���6�7� #�� ����� �������� ����� !��������� ������������� ����������

������� %�� ��� � �� � ���� ��� ���� ���� ��������� ��� � ��� � ����� ����������7

+�- � ��� � ����� ���� ��� ��� ������ ���� � � �������

+�- � �� ������ �� ����� � ���� ��� �� �!�����

/�

58

Page 68: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

* � ��� ������ �����) ���� �� ��� ������ ����� ��� ���������� ������

.�0���� %�% �7�"��� ����!������ "�� �� �� ���� � � ����� ���� � ��� ����������� ����� ��� ���������� ����� � � ���� ��� � � ��� ���� ����� ��� � ���������� ����� � � �6�7) ���� �� � ��� � �� ���� ����

! �6�� 7 � � ) ���

! 8� � ��� �� �) � �6�7 �6�7 � � ��� �6�7 � �6�7) ���� � �� � �� � �� ���

(� ������� �� ��� ��� ���� ������ ��� ������ ����������� �� ���� ��� ����� ��������� � ���� ������ ������� ��� ������� �� ��

������� %�� ��� � �� � ���� ��� ���� ���� ��������� ��� � ��� � ����� ����������7

+�- � ��� � ����� ���� ��� ��� ������ ���� � � �������+�- � �� ������ �� ����� � ���� ��� �� ��� ���

* � ��� ����� ������ �����) ���� �� ��� ����� ������ ����� ��� ���������� ������

(� �� :�� � ���� �� ,#� �� ��� ,�- ���� ������ ������������� 6��� �0J� ��� �/��7� #��3����� ��� ��� ��� ��� ���� ,# ��� �� ���� ����� !��������� �� ����� � ������������� �� � ��� ��� ,#� ���� ������ ������������� ���� ����������� �������� ,�- ���� ������� ����� !��������� +������������ ��� ��� �����"�� �� �� ������� �� ,�-� �� ����

������� %�+ ,�-� ��� ����� ������ ����� ��� ���������� ������

.�0���� %�4 8� ��� ���� ���� �) �� �� �� ��� � ���� �

� � ���6� � �7� ��6�� �7 � � �� � ������ ) � �� � ��� � ��� �� � ��

���� 9���� ������) ������������� ��� :���� "����������� � ���������� ������

#�� ����� ���� ���� � ����� �������� ��� ��� �� ����� ��� H�!������ ��J �������� �� ��������� ���� ���� ��6� � �7� �

�6�� �7�

������� %�5 ��� � �� � ���� ��� ���� ���� ��������� ��� � ��� � ����� ����������7

+�- � ��� � ����� ���� ��� ��� ������ � �������

+�- � �� �!��������� �� ����� � ���� ��� �� ���� ��� ���

���� �� ? �� �

2����3 %�8 &������ � �� � ���� ���� �����!�� ��� ��������� 6��� � ����� ������� �������� !���������7� #���������� ������� � �� ������ ��� � ���� ���� � � (� ������� ������ ������ ��� ��� ������� ���� �� �� ������ ��� � � ��� � :�� �� ���� �� �������� ��� ��6�7� ,������� ��� ��� �� ,#� �� �� ��� :�� �� ,# �� ��� ���� ��������� ��� ���� �� ���������� ��������� ���)������� ����� 6 ��� �� ��!���7� ��� ���� ��� ������� �� !���� ���������� ����$����� ����� 6����� ���� ��� �����7� 8� 3����� ��K � :�� ����,#� �� ������ ��� ��� ���� �� ���������� ��������� ���)������� ������ �� �� ����� ����������� ,#� �� ��� ������ ��� ��� ���� �� !���� ������ ��� ������

�6� ���7 � �6 ���7 � �6� �� 7 � �6 ���7 � �6� � 7

�� ����� �� !���� ����� ��� �� ��� ����� �� ��� ���� �� ���������� ��������� ���)������� ����������� �� �� ��� ��������� �� ,#� � %� ��D����� ���� �� ���� ������ �� ����� �� �� ������ ,#� � ��� ��������� ���� �� ������ ��� !���� ������

��! ������� ���������� ��� ���������� ������ �� ���� ���� ����������� ������ *��� ������ �+ ��� ����� ���� ������ ! �! ��������! ���� �� /60�

/2

59

Page 69: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� $����� ��

(� ���� ���� �� ��� ������ � ��������� ��� "������� �� ��� ���������� �������� ����� ���������� � ���� ���� �� ��� ������ �������� L���� ��� �����"�� �� &���� �0=�� � ��� ������ ��������� ����������� ��� ��� !���� ���� �������� �� ��� �������� �� �������� ������������� ������ � ��� � �� �/0�� +���� � �������� � ����������� ��� ������ ���� �� ��������� %� ��� �� ���� ���� ����������� ��������� �������� ��� ����� ���� ���� � ����� ���������� ���� ���� !��������� L���� ���� ������������ � ������� �������� ������� ��� ���������������������� ��� ������� ����������� %� ���� ���� � ����������� ���� ������� �� ������������<�� �� ������ �������� ����� ������� �� ��� ����� ������� �� � ����������������� �� ����� ��� ���� �� �� ��� ��� ���� ���������

�� ��� �� � ��� � ���� ,�������� ��/ �� ��� !��� ������� �������� ������ ��� ������������ L����� ������������� ��� ���� ��� ���� ������� ������ �� ��� ������ �� ������ �����#������� ���� ������� ���)��������� ������ ����� ���� ���� ������������ ������ 6�� ���������).#� �0/� /=� /F�7� ��� ������� ���������� ������� � ��� � ��� �� �/0� �� 02� ���� ����� ������������ �� �� ����� �������� ������

3���� ��� ����� ��� "�������� ��������� 3�� ����� �� ���� ���� ���� �� ��������� ������� ���������� 6/7 !�� ����� ������ 6�� ������ ��� ���� �� ����� � ��� ��:��� ���7��� ��� ��� ����������� ��:�� 607 ��� ��� ����� ���������� �������� ��� � �� �� � 6=7��������<� ����� ������� �� ����� ������ ���������� .�������� ��� !��� ������ ��� �������������� ��� ���� �� ������ ����� �& �������� ���� ��� ���� �� ����� �� ��� �������������� ������� ����� �� ������ ��������<��� ,�������� ��� ����� ������ �� �� �������� ;����� ��� ;����:� �/0� ��� �� ��� ( �� ���� ����� ���� �������� ���������� (����������"�������� ��� ������ �������������� �������� ��� ��� 8��� �������� �������� �� ��������%��� ������ �� ��� ����� ������ ��� ������� ��� �������� ��:��� �� ��������<������ �� ���������� ������� �� ������ ��������� ��� >)����������

#������� ���� ���� � ���� ���� ��!��� ��� ������ ������� �� � ���� ���� � �� ���� ������ ���� �� � (� ���� ���� � ������ ��� �������<������ ��� ����� ��� ���� ��!������ ���� ����� ���� �� �������� ���� ������ 1� ��� ����� ����� �������� �� ���������� ���� ���� ��� ��� ����������� ��� ������� ��� �� ������ �� ����������� "��������

2�"������

%&' (���� )����� *����� �������� �� +���� +��� ,����� ������- (�������$���� �������!����� �� ����������� ������� �� ����� ���� .."/#-0112&3&3 433&�

%4' 5��� ��$����� ��� �� ����� ��� (��� 6������ ������� ��� ������ ����� 2 ��� �- ��� ��� ����������������� ��������� 6������� 7����� **!4338!3. �99( 4338�

%/' *����� ��������� 5����� ����� ������ ����� ��� ������� �� ����� ���� /8-:.2;/ &00/�%8' *����� �������� +���� �� 7�<�� �� =�� >����� ����� ���� � (������� ?�� ������ *����

(������� ?� 433&�%:' *����� �������� �� ����� ��� (��� *��� ���������� ����� ����� �� ������ ��������� 6���

����� 4338�%.' @����� ������� ��� ���� �� ����������� (������� ?�� ������ *���� &00/�%1' )������� (���� �� +����� A�������� � ����� ���� � B����� ?�� ������ *���� &001�%;' (����!(�� (��� �� �!*��� 9��� 6�� ���������� �� ������������ ���� �������� �� ��� ���!

������� �� ����������� �� ������������ ���� ��������� �� )��� 5����� �� =��� +����� ���������� ��� ������� �� ��� ����� ������������� ������ �� ���� �� ����������� �� �������� ��� � ����� ;&/ �� �� ���� ����� �� ������� � ��� �� �������� &008�

%0' ������ (������� C��������� �� ������� ��������� ��- 6�� ���� �������� �� � ����� ������������ ���� ������ +����D� ������ �99( ?�� ������ �� )������� 4334�

%&3' @������� �� @������ �� �������� �� ������ ���� !��"����� #������������� ��������� *�C������ ?�� �����E �� 7�� F9 �����$G &00:�

%&&' C� @��� H)���� �����$ ��� ������ �� +����� A�������� � ������������������� ���� �$ ����� ��� %���� ������ ����� &8; �� ������� �� ���� � 5����!,����� 433/�

/J

60

Page 70: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

%&4' @����� @��� �� >������ @������ +��� ����� ���� ����� ������� �� ���������� �� ���� 44-.312./. &00/�

%&/' @����� @��� � 6�� ����������� �� ��������� ���������� ������������ ������ ����� "���!���#� ��� ������� �� ����� �� ��������� &#������' &13-84280 &0;:� )��� 5�� ���7�������� B����� >������� 6����� 5� ���������

%&8' ��� �� @������ �� +��� A�������� �������( )��� ��� ����� ����������� 6����� �� ������������ )������ 4334�

%&:' @����� @����$� (�� *�$$ �� =�� >����� ��������� ������� ���� ������ �� ����������� ������� �� ����� ���� .;"4#-8&028.4 433/�

%&.' >� @����� �� C� >����� � ���� ��� ������� �� ������ ������� ���� ������� ������� ������ ��� ���������� &&":#-1/121:8 433&� ������ ����� �� ,����� 9������ )����� (� ���������� *� "����#�

%&1' >������ @����� �� ������� *���� ?���� ��� ��� ���� �������- ���� �� ���������� ��������� ���� ��� ���������� 4"&#-:2/3 &004�

%&;' +���� ����� �� ��� ������� ��������� �� ������� ������� �� ���� ����� 2 ��� ��������� ���� � :0-&802&11 &001�

%&0' C �� 9����� ���������� ������ ������� ������ � ������ ������� �� ���������� �� ���� /-8:128.. &018�

%43' +���� +�� �� =�� >����� ���������������� ����� ���� � )������ 9���� ������� ������)������ *��������� &001�

%4&' ������� *��� �� 6���� 6����� � )� ���� �� ���������� ������ ������ ���������� ��� ��������� 0/-4./2//4 &00&�

%44' ������� 7��������� +��� ����� ������ �� ������������� ������� �� ���������� �� ���� &4-83/284 &0;/�

%4/' I���� ���� �����*�� �� ����� ���� �� *�C ������ �99( ?�� ������ �� )������� &00/�%48' ����� ��� (��� ������������ ��� �������� ���� �������� ������� �� ����� ���� 6�

����� ) ���������� ������ �� ����� ���� ������������������ �����������

%4:' =�� >����� C��� ���� ����� � ���!����� �� ���� ������ ������� �� ����� ���� :;"/#-&33/2&3/8 &00/�

%4.' *������� >�� +���� +�� �� ����� ��� (��� ���� ��� ������ ��� ������ ������ *�� � ����� ������� ���� 4338�

%41' )����� >������ ?������ ������������ �� ������ ������������ �� )+���� ,--. ����� . �� �� ��������� ���� ���� &/02&.8 ������ &00.� ���������

/K

61

Page 71: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Public Announcements and Belief Expansion

Hans van Ditmarsch∗ Wiebe van der Hoek† Barteld Kooi‡

Abstract

In this paper we study the relation between two approaches to infor-mation change: Dynamic Epistemic Logic and Belief Revision. Oneof the main differences between these approaches is that higher-orderinformation plays an important role in the field of Dynamic EpistemicLogic, whereas it does not feature in Belief Revision. In this paper westudy to which extent public announcements (a particular kind of in-formation change studied in Dynamic Epistemic Logic) can be viewedas a belief expansion (a particular kind of information change studiedin Belief Revision).

Keywords: dynamic logic, epistemic logic, belief revision.

1 Introduction

Since Hintikka’s [9] epistemic logic, the logic of knowledge, has flourished asa research area in philosophy [10], computer science [6], artificial intelligence[12] and game theory [2]. The three mentioned application areas made itapparent that in multi-agent systems higher-order information, knowledgeabout (other) agents’ knowledge, is crucial.

A natural question to ask, once the formal framework to reason aboutthe information of agents — we will use the terms ‘belief’ and ‘knowledge’interchangeably in this paper — is in place, is how the agent’s beliefs changesover time, when he is confronted with new information; be it in a static, or anevolving world. The famous paper [1] by Alchourron et al. put this changeof information, coined as Belief Revision (BR), as a topic on the philosoph-ical and logical agenda: it was followed by a large stream of publications,fine-tuning the notion of epistemic entrenchment [13], revising (finite) beliefbases [4], differences between belief revision and belief updates [11], and theproblem of iterated belief change [5].

∗Department of Computer Science, University of Otago, PO Box 56, Dunedin 9015,New Zealand, [email protected].

†Department of Computer Science, University of Liverpool, Liverpool L69 7ZF, UnitedKingdom, [email protected].

‡Department of Philosophy, University of Groningen, A-weg 30, 9718 CW Groningen,The Netherlands, [email protected].

1

62

Page 72: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

This stream of research typically represents the beliefs of the (only)agent as a theory in propositional logic, and then describes, in terms ofrationality postulates, how new information should be incorporated in it,the most interesting case being when this new information is incompatiblewith the agent’s beliefs. Hence, in this approach the dynamics is studied ona level above the informational level, not allowing reasoning about change ofagents’ knowledge and ignorance within the framework, let alone about thechange of other agents’ information from the perspective of a given agent.

Alternatively, the area of Dynamic Epistemic Logic (DEL) takes as itsstarting point modal epistemic logic in the tradition of [9], and adds a dy-namic component to it. The aim here is not only to dynamize the epistemics,but also to ‘epistemize the dynamics’: the actions that (groups of) agentsperform are epistemic actions. Different agents may have different informa-tion about which action is taking place, including higher-order information.This rather recent approach treats all of knowledge, higher-order knowledge,and its dynamics on the same foot. Following an original contribution byPlaza in 1989 [14], a stream of publications appeared around the year 2000[8, 3, 16, 17].

The relation between belief revision and dynamic modal logic is inves-tigated by Segerberg in [15]. However, this study is limited to cases wherehigher-order information plays no role. In this paper we study a case withhigher-order information, namely how a public announcement, a specificepistemic action in DEL, can be viewed as belief expansion, one of the threedistinguished operations in BR. As such, our paper can be conceived as anext step in making the relation between the approaches explicit.

In Section 2 and 3 we briefly introduce public announcement logic andbelief expansion respectively. Section 4 demonstrates that belief expansioncan be seen as a special case of public updates. In section 5 conclusions aredrawn and directions for further research are indicated.

2 Public announcement logic

Public announcement logic (PAL) was first developed by Plaza [14]. It ex-tends epistemic logic to allow reasoning about information change due topublic announcements: an epistemic action [ϕ] where the whole group ofagents are aware that they learn ϕ.

Definition 1 (Language of PAL) Let a finite set of propositional vari-ables P and a finite set of agents A be given. The language LPAL is givenby the following BNF:

ϕ ::= p | ¬ϕ | ϕ1 ∧ ϕ2 | �aϕ | [ϕ1]ϕ2

where p ∈ P, a ∈ A. Besides the usual abbreviations we use Eϕ as anabbreviation for

∧a∈A �aϕ. �

2

63

Page 73: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A formula of the form [ϕ]ψ has to be read as “ψ holds after the announce-ment of ϕ”. This language is interpreted in models for epistemic logic.These are Kripke models with an accessibility relation for each agent. Allthe logical languages presented in this paper are interpreted in these models.

Definition 2 (Epistemic models) Let a finite set of propositional vari-ables P and a finite set of agents A be given. An epistemic model is a tripleM = (W,R, V ) such that:

• W �= ∅; a set of possible worlds;

• R : A → 2W×W ; assigns an accessibility relation to each agent;

• V : P → 2W ; assigns a set of worlds to each propositional variable. �

In epistemic logic R is usually restricted to equivalence relations. In thispaper we treat the weakest modal case where there are no restrictions onR, consequently most results also hold for stronger logics. The semanticsare defined with respect to models with a distinguished world (the actualworld): (M,w). We also call this a model.

The methodology used in dynamic epistemic logics is to regard epistemicactions as state transformers, i.e. they convey us from one model to another.The semantics are justified by showing how these state transformers work inexamples, and by showing that the results agree with one’s intuitions aboutthese kinds of information change. The idea of public announcements is thatall the agents simultaneously learn the announced formula, moreover it iscommon knowledge among them that they learn it. We assume that onlytrue announcements can be made. This leads to the following semantics.

Definition 3 (Semantics of PAL) Let a model (M,w) withM = (W,R, V )be given. Let p ∈ P, a ∈ A, and ϕ,ψ ∈ LPAL.

(M,w) |= p iff w ∈ V (p)(M,w) |= ¬ϕ iff (M,w) �|= ϕ(M,w) |= ϕ ∧ ψ iff (M,w) |= ϕ and (M,w) |= ψ(M,w) |= �aϕ iff (M,v) |= ϕ for all v such that (w, v) ∈ R(a)(M,w) |= [ϕ]ψ iff (M,w) |= ϕ implies (M |ϕ,w) |= ψ

The model M |ϕ = (W ′, R′, V ′) is defined by restricting M to those worldswhere ϕ holds. Let [[ϕ]] = {v ∈ W |(M,v) |= ϕ}. Then W ′ = [[ϕ]], and theaccessibility relations and valuation are restricted to [[ϕ]] as well: R′(a) =R(a)∩ [[ϕ]]2 and V ′(p) = V (p)∩ [[ϕ]]. If Γ is a set of formulas, then Cn(Γ) ={ϕ| for all (M,w), if (M,w) |= Γ, then (M,w) |= ϕ}. �

3

64

Page 74: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Definition 4 (Proof system for PAL) The proof system for PAL consistsof all the axioms and rules of K plus the following axioms:

Atoms [ϕ]p ↔ (ϕ→ p) (atoms)PF [ϕ]¬ψ ↔ (ϕ→ ¬[ϕ]ψ) (partial functionality)Distr [ϕ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ) (distribution)Ramsey [ϕ]�aψ ↔ (ϕ→ �a[ϕ]ψ) (Ramsey axiom)

Completeness for this proof system is easy: One shows that every formulaof the language of public announcements can be translated to a provablyequivalent formula without announcement operators.

Definition 5 (Translation) The translation function t takes a formulafrom the language of PAL and yields a formula in the language of epis-temic logic, in such a way that t(p) = p, and t distributes over ¬,∧ and �a.The announcement operator is treated as follows:

t([ϕ]p) = t(ϕ) → pt([ϕ]¬ψ) = t(ϕ) → ¬t([ϕ]ψ)t([ϕ](ψ ∧ χ)) = t([ϕ]ψ) ∧ t([ϕ]χ)t([ϕ]�aψ) = t(ϕ) → �at([ϕ]ψ)t([ϕ][ψ]χ) = t([ϕ]t([ψ]χ))

Lemma 1 (Plaza) � ϕ↔ t(ϕ) for every formula ϕ. �

Some have commented on the fact that every formula containing announce-ments is equivalent to a formula without any announcements, by remarkingthat announcements are merely syntactic sugar. However, applying the sameline of argument to the study of propositional logic, one would be forced tosay that one can make do with a language containing Sheffer’s stroke as theonly logical operator. Although this is technically feasible, it makes no sensephilosophically, and practically it is quite bothersome.

The next definition and lemma will be used in section 4.

Definition 6 (Epistemic depth) The epistemic depth of a formula is afunction d : LPAL → N such that:

d(p) = 0d(¬ϕ) = d(ϕ)d(ϕ ∧ ψ) = max(d(ϕ), d(ψ))d(�aϕ) = d(ϕ) + 1d([ϕ]ψ)) = d(ϕ) + d(ψ)

4

65

Page 75: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Lemma 2 For all formula ϕ ∈ LPAL, ϕ and t(ϕ) have the same epistemicdepth. �

Proof By strong induction on n. The base case is easy. For the inductionstep one can simply check every clause of the translation. �

Example 1 For an example, consider the formula [�ap]�bq. Its transla-tion into epistemic logic is t([�ap]�bq) = t(�ap) → �bt([�ap]q) = �ap →�b(t(�ap) → q) = �ap → �b(�ap → q). According to the definition ofmodal depth, d([�ap]�bq) = 2 and this indeed reflects that its translation�ap→ �b(�ap→ q) has a stack of two epistemic operators. �

The following lemma will be used in the the proof of Theorem 3 in Section 4.

Lemma 3 Let d(ψ) = n. Then: � Enϕ→ ([ϕ]ψ ↔ ψ) �

Proof If d(ψ) = 0, it follows straightforwardly from atoms, partial func-tionality and distribution.

If d(ψ) = n + 1, the interesting case is when ψ is of the form �aχ.Then by the Ramsey axiom � [ϕ]�aχ ↔ (ϕ → �a[ϕ]χ). By propositionalreasoning we get � (En+1ϕ → [ϕ]�aχ) ↔ (En+1ϕ → (ϕ → �a[ϕ]ψ)).Observe that � En+1ϕ → ϕ and � En+1ϕ → �aE

nϕ. With some modalreasoning we get � (En+1ϕ → [ϕ]�aχ) ↔ (En+1ϕ → �a(Enϕ ∧ [ϕ]ψ).The induction hypothesis tells us that � Enϕ → ([ϕ]ψ ↔ ψ). Therefore� (En+1ϕ→ [ϕ]�aχ) ↔ (En+1ϕ→ �a(Enϕ ∧ ψ)). With some more modaland propositional reasoning we finally get Enϕ→ ([ϕ]�aχ↔ �aχ). �

3 Belief expansion

In belief revision the beliefs of an agent are not represented by means of aKripke model, but with a set of propositional formulas, called a belief set.Therefore all the logical notions refer to propositional logic.

Definition 7 (Belief sets) A belief set K is a set of formulas closed underlogical consequence, i.e. K = Cn(K). �

One of the types of information change studied in belief revision is beliefexpansion. When an agent accepts the information that ϕ, the agent is saidto expand his or her beliefs with ϕ. The methodology used in this branch oflogic, is to state properties that such a change of information satisfies, as-suming that the agent performing the information change is rational. Theseare called rationality postulates, which can be justified by arguments thatappeal to one’s intuition about the information change. Then one can setout to prove that these postulates uniquely characterize a certain opera-tion on belief sets. In [1] the expansion operation + is characterized by thefollowing rationality postulates.

5

66

Page 76: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Definition 8 (Rationality postulates for belief expansion)

K1 K + ϕ is a belief set.

K2 ϕ ∈ K + ϕ.

K3 K ⊆ K + ϕ.

K4 If ϕ ∈ K, then K + ϕ = K.

K5 If K ⊆ K ′, then K + ϕ ⊆ K ′ + ϕ.

K6 K + ϕ is the smallest belief set that satisfies K1-K5. �

This operation turns out to be fully characterized by these six postulates.This leads to the following theorem.

Theorem 1 (Gardenfors [7]) The expansion function + satisfies K1-K6iff K + ϕ = Cn(K ∪ {ϕ}) �

So the operation of expansion is simply adding the new information to theagent’s belief set and closing under logical consequence. In the next sectionwe show that under certain conditions, public announcements can also beseen as expansions.

4 Positive knowledge

So far, we have seen two logical approaches that provide an analysis of thesame kind of information change: incorporating new information. Thereforeone would expect that, although technically they are worked out differently,there are great similarities. However, the similarities turn out to be limitedto a very special case. The postulates of expansion, discussed in Section 3,do not straightforwardly translate to properties in the context of dynamicepistemic logic. First of all, many logics for knowledge and belief assumethe properties of positive (�ϕ → ��ϕ) and negative introspection (¬�ϕ →�¬�ϕ) and this makes it very hard to compare belief sets, because thesecannot even be expressed in terms of belief sets of propositional formulas.But even it that were the case, if we take a belief set to be some set K(M,w) ={ϕ | (M,w) |= �ϕ}, for a given model (M,w). The assumption that K ⊆ K ′

for two belief sets K and K ′ then immediately implies that K = K ′, sinceits negation would allow for a formula ψ ∈ K ′ and ψ �∈ K. The firstof these yields, with positive introspection, that �ψ ∈ K ′, while negativeintrospection and the fact that K ⊆ K ′ also gives ¬�ψ ∈ K ′. In otherwords, fully introspective agents can never add or remove beliefs, withoutmaking their belief set inconsistent!

Secondly, postulate K2 of expansions (ϕ ∈ K + ϕ, also called success),is hard to enforce in a setting where we allow for epistemic operators in the

6

67

Page 77: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

object language. To see this, for any epistemic logic satisfying the weakvariant �¬�ϕ→ ¬�ϕ of veridicality, it not easy to see how an expansion ofK with, say, p∧ ¬�p should look like. By the postulate of success, we have(p ∧ ¬�p) ∈ K which is impossible to fulfill in combination with our weaknotion of veridicality and the requirements K �� ⊥ and K = Cn(K).

In this section we show that public announcements do behave like be-lief expansions, when we restrict ourselves to positive knowledge, and theformulas that are learned are successful. The problem (of beliefs sets beingincomparable) and the analysis below is not unlike the one presented in [18],where the driving question is what it means that an agent claims to ‘onlyknow’ some fact ϕ.

For this purpose we look explicitly at the levels of knowledge that anagent possesses. This leads to the following inductive definition of the posi-tive fragment of the language of public announcement logic.

Definition 9 (Lpos) The languages L npos (for each n ∈ N) are defined

inductively as follows, where k is any value smaller than or equal to n+ 1:

ϕ0 ::= p | ¬p | ϕ0 ∨ ϕ0 | ϕ0 ∧ ϕ0

ϕn+1 ::= ϕn | �aϕn | [¬ϕk]ϕ(n+1)−k | ϕn+1 ∨ ϕn+1 | ϕn+1 ∧ ϕn+1

Lpos =⋃

n∈NL n

pos. �

Note that if ϕ ∈ L npos, then d(ϕ) ≤ n. Also note that if ϕ ∈ L n

pos, thenthere is a formula ψ without announcement operators, which is equivalentto ϕ, such that ψ ∈ L n

pos, by lemma’s 1 and 2. Therefore in the remainderproofs by induction on L n

pos, we omit the case for [ϕ]ψ.

Definition 10 (Finite simulation) Let two models M = (W,R, V ) andM ′ = (W ′, R′, V ′), and two worlds w ∈ W and w′ ∈ W ′ be given. S ⊆W ×W ′ is a simulation up to 0 for w and w′ iff wSw and for all p ∈ P it isthe case that w ∈ V (p) iff w′ ∈ V ′(p). S ⊆ W ×W ′ is a simulation up ton+ 1 for w and w′ iff

atoms S is a simulation up to 0 for w and w′, and

forth for every a ∈ A and v ∈ W , if wR(a)v, then there is a v′ ∈ W ′ suchthat w′R′(a)v′ and S is a simulation up to n for v and v′.

If there is a simulation up to n for w and w′ we write (M,w)→n(M ′, w′). Ifthere is a simulation up to every n ∈ N for we write (M,w)→ω(M ′, w′) �

The following lemma holds due to the fact that we only have a finite numberof propositional variables and only a finite number of agents, and becausewe only look at formulas up to a certain depth.

Lemma 4 (Propositional finiteness) The number of different proposi-tions that can be expressed by formulas of L n

pos is finite for every n. Moreformally: logical equivalence yields a finite partition on L n

pos. �

7

68

Page 78: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

So althought there are infinitely many different formulas in the language upto n, only finitely many things can be expressed with it.

Definition 11 Let Thnpos(M,w) = {ϕ ∈ L n

pos|(M,w) |= ϕ} �

Theorem 2 (M,w)→n(M ′, w′) iff Thnpos(M

′, w′) ⊆ Thnpos(M,w) �

Proof By induction on n, the base case being trivial. So suppose thetheorem holds up to n.

(⇒) Suppose (M,w)→n+1(M ′, w′). Let ϕ ∈ L n+1pos . We proceed by

induction on ϕ. If ϕ ∈ L npos we can simply apply the induction hypothesis.

Suppose ϕ is of the form �aψ. By contraposition. Suppose (M,w) �|= �aψ.Then there is a world v such that wR(a)v and (M,v) �|= ψ. From forth itfollows that there is a v′ such that w′R(a)v′ and (M,v)→n(M ′, v′). Sinceψ ∈ L n

pos, we can apply the contrapositive of the induction hypothesis.Therefore (M ′, v′) �|= ψ, and therefore (M ′, w′) �|= �aψ.

(⇐) Suppose that Thn+1pos (M ′, w′) ⊆ Thn+1

pos (M,w). Now let S be, for allworlds: S = {(v, v′)|Thn+1

pos (M ′, v′) ⊆ Thn+1pos (M,v)}. We prove that S is a

simulation up to n+1. The atoms clause follows directly. Suppose toward acontradiction that there is a wR(a)v and for all v′ ∈W ′ such that w′R′(a)v′

the relation S is not a simulation up to n for v and v′. By the inductionhypothesis there is a formula ψvv′ ∈ L n

pos such that (M ′, v′) |= ψvv′ and(M,v) �|= ψvv′ . Let Γ = {ψvv′ |w′R′(a)v′}. We can assume this set is finitewithout loss of generality by Lemma 4. Therefore (M ′, w′) |= �a

∨Γ and

(M,w) �|= �a∨

Γ. But this formulas is in L n+1pos , therefore (M,w) |= �a

∨Γ.

And so we arrive at a contradiction. �

Example 2 The supermodel relation yields a simulation up to an arbitrarydepth. This also shows that the submodel relationship is not a simulationbeyond a certain level.

For example, consider the following four-state epistemic state (M,w):the set of possible worlds of the model M is {w,w′, v, v′}; there is one factp, which is true in w,w′ only, access for agent b is universal, and access for ainduces the partition {w,w′}, {v}, {v′}. The model represents an epistemicstate where agent b suspects agent a of knowing the truth about p. Wefurther consider the submodels (M ′, w) and (M ′′, w): M ′ is the submodel ofM for restricted domain {v,w}, and M ′′ is the restriction of M to {w} only.These models are visualized in Figure 1, including their relevant simulationrelations. We now observe the following, which is a direct application ofTheorem 2.

The formula �ap is true in (M,w). Therefore, by Theorem 2, andbecause of (M ′′, w)→ω(M ′, w)→ω(M,w), it is also true in (M ′, w) and(M ′′, w). The formula �b¬p is false in (M ′′, w). Therefore, because of(M ′′, w)→ω(M ′, w)→ω(M,w), it is also false in (M ′, w) and (M,w). Toshow that there is no simulation for (M,w) and (M ′, w′) up to 2, it suffices

8

69

Page 79: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

0 1

0 1

(M,w)

b

bb

a, b

�→2

←ω0 1

(M ′, w)

b

�→1

←ω1

(M ′′, w)

Figure 1: Simulation relations between three example models. All are defined forone atom p only, and two agents a and b. The point of the model isunderlined. In states named 0 fact p is false, and in states named 1 true.All access is symmetric and transitive, and reflexive access is assumedas well. For example, in the leftmost model access for b is the universalrelation; and in the rightmost model access for both a and b is reflexive.The last depicts the situation where they both know that p (and wherethey know from each other that they know that, etc.).

to observe that �b(�ap ∨ �a¬p) ∈ Th2pos(M ′, w) but �b(�ap ∨ �a¬p) �∈

Th2pos(M,w). There is no simulation for (M ′, w) and (M ′′, w) up to 1, be-

cause �bp ∈ Th1pos(M

′′, w) but �bp �∈ Th1pos(M

′, w). �

Let Cnnpos(Γ) = {ϕ ∈ L n

pos| for all (M,w), if (M,w) |= Γ, then (M,w) |=ϕ}.Theorem 3 Let ϕ ∈ L n

pos, and let ¬ϕ be successful, i.e. |= [¬ϕ]¬ϕ, then

Thnpos(M |¬ϕ,w) = Cnn

pos(Thnpos(M,w) ∪ {En¬ϕ})

Proof (⇒) Let ψ ∈ L npos. Suppose ψ ∈ Thn

pos(M |¬ϕ,w), i.e. (M |¬ϕ,w) |=ψ. Let (M ′, w′) be an arbitrary model such that (M ′, w′) |= Thn

pos(M,w) ∪{En¬ϕ}. Therefore Thn

pos(M,w) ⊆ Thnpos(M

′, w′). By Theorem 2, thisis equivalent to (M ′, w′)→n(M,w). Therefore (M ′|¬ϕ,w′)→n(M |¬ϕ,w).Since ψ ∈ L n

pos, by Theorem 2, (M ′|¬ϕ,w′) |= ψ. Therefore (M ′, w′) |=[¬ϕ]ψ. Since (M ′, w′) |= En¬ϕ, by Lemma 3, (M ′, w′) |= ψ.

(⇐) It is sufficient to show that (M |¬ϕ,w) |= Thnpos(M,w)∪{En¬ϕ}. It

is easy to see that (M |¬ϕ,w)→n(M,w). Therefore by Lemma 3 (M |¬ϕ,w) |=Thn

pos(M,w). Suppose that (M |¬ϕ,w) �|= En¬ϕ. Then there is a world vaccessible from w with a path of length at most n such that (M |¬ϕ, v) |= ϕ.This contradicts |= [¬ϕ]¬ϕ. Therefore (M |¬ϕ,w) |= En¬ϕ. �

This theorem shows that those positive formulas which are true after asuccessful announcement of a negation of a positive formula ϕ are exactlythose positive formulas which follow from the positive knowledge before theannouncement together with knowledge of ¬ϕ up to a certain depth. In orderto make the link with belief expansion more clear, we state the following twocorollaries, which look at the positive knowledge of one agent.

9

70

Page 80: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Corollary 1 Let Kn(M,w)(a) = {ϕ ∈ L n

pos|(M,w) |= �aϕ}. Let ϕ ∈ L npos,

and let ¬ϕ be successful, then Kn(M |¬ϕ,w)(a) = Cnn

pos(Kn(M,w)(a) ∪ {En¬ϕ})

Propositional formulas are always successful. Therefore we do not need itas an explicit assumption for L 0

pos. Also, we can use the ordinary notionof consequence. This leads to the following corollary, saying that publicannouncements are the same as expansions on a propositional level.

Corollary 2 For L 0pos, K

0(M |¬ϕ,w)(a) = Cn(K0

(M,w)(a) ∪ {¬ϕ}) �

5 Conclusion and further research

In this paper we investigated to what extent public announcement and be-lief expansion are the same. We showed that what positive knowledge isconcerned the two notions coincide, but it seems difficult to extend this tobroader sublanguages of public announcement logic. This indicates thateven in the simple case of learning something new, one’s intuitions mayfail. It also poses the question which methodology is to be preferred whenstudying information change.

There are three obvious main directions for further research the shortterm. First of all, now we have determined a counterpart of expansions inDEL, a natural next enterprise is to extend DEL to a framework in whichone can characterize a notion of contraction and, once that is in place, anotion of revision. The other two directions are partly suggested by similarproblems when studying only knowing [18]. First, it is not entirely clearat the moment how our results generalize to or are dependent on specificaxiomatic systems. Finally, we should expand our analysis to a real multi-agent framework, allowing for learning operators for a specific agent. Wherein our current approach the agent’s knowledge can not occur in the scopeof a negation, it is not clear how this generalises to the case of more agents:maybe a formula of type �a¬�b�ap would not disturb the results provenhere.

References

[1] C. E. Alchourron, P. Gardenfors, and D. Makinson. On the logic of the-ory change: partial meet functions for contraction and revision. Journalof Symbolic Logic, 50:510–530, 1985.

[2] R. J. Aumann and A. Brandenburger. Epistemic conditions for Nashequilibrium. Econometrica, 63(5):1161–1180, 1995.

10

71

Page 81: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[3] A. Baltag, L.S. Moss, and S. Solecki. The logic of public announce-ments, common knowledge and private suspicions. Manuscript, origi-nally presented at TARK 98, 2003.

[4] S. Benferhat, D. Dubois, H. Prade, and M.A. Williams. A practicalapproach to revising prioritized knowledge bases. Studia Logica, 70(1),2002.

[5] A. Darwiche and J. Pearl. On the logic of iterated belief revision.Artificial Intelligence, 89(1-2):1–29, 1997.

[6] R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi. Reasoning AboutKnowledge. MIT Press, Cambridge, Massachusetts, 1995.

[7] P. Gardenfors. Knowledge in Flux, modeling the dynamics of epistemicstates. MIT Press, 1988.

[8] J. Gerbrandy and W. Groeneveld. Reasoning about information change.Journal of Logic, Language, and Information, 6:147 –196, 1997.

[9] J. Hintikka. Knowledge and Belief, An Introduction to the Logic of theTwo Notions. Cornell University Press, Ithaca & London, 1962.

[10] J. Hintikka. Reasoning about knowledge in philosophy. In J. Y. Halpern,editor, Proceedings of Theoretical Aspects of Reasoning About Knowl-edge, pages 63–80. Morgan Kaufmann Publishers, 1986.

[11] H. Katsuno and A. Mendelzon. On the difference between updatinga knowledge base and revising it. In Proceedings of the Second In-ternational Conference on Principles of Knowledge Representation andReasoning, pages 387–394, 1991.

[12] J.-J.Ch. Meyer and W. van der Hoek. Epistemic Logic for AI andComputer Science. Cambridge University Press, Cambridge, 1995.

[13] T.A. Meyer, W.A. Labuschagne, and J. Heidema. Refined epistemicentrenchment. Journal of Logic, Language and Information, 9:237–259,2000.

[14] J. A. Plaza. Logics of public communications. In M. L. Emrich, M. S.Pfeifer, M. Hadzikadic, and Z. W. Ras, editors, Proceedings of the4th International Symposium on Methodologies for Intelligent Systems,pages 201–216, 1989.

[15] K. Segerberg. Two traditions in the logic of belief: bringing themtogether. In H. J. Ohlbach and U. Reyle, editors, Logic, Language andReasoning: essays in honour of Dov Gabbay, volume 5 of Trends inLogic, pages 135–147. Kluwer Academic Publishers, Dordrecht, 1999.

11

72

Page 82: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[16] H. P. van Ditmarsch. Descriptions of game actions. Journal of Logic,Language and Information, 11(3):349–365, 2002.

[17] H.P. van Ditmarsch, W. van der Hoek, and B.P. Kooi. Concurrentdynamic epistemic logic. In V.F. Hendricks, K.F. Jørgensen, and S.A.Pedersen, editors, Knowledge Contributors, pages 45–82. Kluwer Aca-demic Publishers, Dordrecht, 2003. Synthese Library Volume 322.

[18] J. Jaspars W. van der Hoek and E. Thijsse. Persistence and minimalityin epistemic logic. Annals of Mathematics and Artificial Intelligence,27(1–4):25–47, 1999 (printed 2000).

12

73

Page 83: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Model Checking, Preprocessing, and BDD Size

Andrea Ferrara Paolo Liberatore Marco Schaerf

Dipartimento di Informatica e Sistemistica, Universita di Roma “La Sapienza”,

Via Salaria 113, Roma, Italia. Email: [email protected]

Abstract

Temporal Logic Model Checking is a verification method in whichwe describe a system, the model, and then we verify whether someproperties, expressed in a temporal logic formula, hold in the system.It has many industrial applications. In order to improve the perfor-mances, some tools allow the preprocessing of the model, verifyingon-line a set of properties reusing the same compiled model; we provethat the complexity of the Model Checking problem, without any pre-processing or preprocessing the model or the formula in a polynomialdata structure, is the same. Thus preprocessing cannot dramaticallyimprove the performances.

Symbolic Model Checking algorithms work by manipulating setsof states, and these sets are often represented by BDDs. It has beenobserved that the size of BDDs may grow exponentially as the modeland formula increase in size. As a side result, we formally prove thata superpolynomial increase of the size of these BDDs is unavoidablein the worst case. While this exponential grow has been empiricallyobserved, to the best of our knowledge, it has never been proved sofar. This result not only holds for all types of BDDs regardless of thevariable ordering, but also for more powerful data structures, such asBEDs, RBCs, MTBDDs, and ADDs.

Keywords: Model Checking, Complexity, Compilability, Succinct-ness, BDDs.

1 INTRODUCTION

Temporal Logic Model Checking [14] is a verification method for discretesystems. In a nutshell, the system, often called the model, is described bythe possible transitions of its components, while the properties to verifyare encoded in a temporal modal logic. It is used, for example, for theverification of protocols and hardware circuits [3]. Many tools, called modelcheckers, have been developed to this aim. The most famous ones are SPIN[21] and SMV [27] (with its many incarnations: NuSMV [12], RuleBase [4]),VIS [16], and FormalCheck [20].

1

74

Page 84: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

There are many languages to express the model; the most widespreadones are Promela and SMV. Two temporal logics are mainly used to definethe specification: CTL [14] and LTL [28]. In this paper we focus on thelatter.

In many cases, the two inputs of the model checking problem (the modeland the formula) can be processed in a different way. If we want to verifyseveral properties of the same system, it makes sense to spend more timeon the model alone, if the verification of the properties becomes faster.Many tools allow to build the model separately from checking the formula[11, 32, 22]; in this way, one can reuse the same model, compiled into a datastructure, in order to check several formulae.

In the same way, we may wish to verify the same property on differ-ent systems: the property is this time the part we can spend more timeon. Many tools allow populating a property database [11, 32, 22], i.e., acollection of temporal formulae which will be checked on the models. Weimagine a situation in which we early establish the requirements that oursystem must satisfy, even before the system is actually designed. As a re-sult, and we can fill a database of temporal formulae, but we do not yetdescribe the system. While the design/modeling of the system goes on, wecan preprocess the formulae (without knowledge of the model, which is notyet known). Whenever the system is specified, we can then use the result ofthis preprocessing step to check the model against the formulae.

In this paper, we analyze whether preprocessing a part of the modelchecking problem instances improve the performances. The technical toolwe use is [10, 24] the compilability theory. This theory characterizes thecomplexity of problems when the problem instances can be divided into twoparts (the fixed and the varying part), and we can spend more time onthe first part alone, provided that the result of this preprocessing step haspolynomial size respect the fixed part. We show that the Model Checkingproblem remains PSPACE-hard even if we can preprocess either the modelor the formula, if this preprocessing step is constrained to have a polynomialsize. These theorems hold for all model checkers.

Finally, we answer to a long-time standing question in Symbolic ModelChecking. It has been observed that the BDDs that are used by SMV andother Symbolic Model Checking systems become exponentially large in somecases. However, it has not yet been established whether this size increase isdue to the choice of variable ordering, or to the kind of BDDs employed, orit is intrinsic of the problem. We show that, if PSPACE 6⊆ Πp

2∩ Σp

2, such a

grown is, in the worst case, unavoidable. This result is independent from theparticular class of BDDs and from the variable order of the BDDs. It alsoholds for all decision diagrams representing integer-value functions whoseevaluation problem is in the polynomial hierarchy, such as BEDs [33], RBCs[1], MTBDDs [13], and ADDs [2].

2

75

Page 85: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 Preliminaries

2.1 Model Checking

In this section, we briefly recall the basic definitions about model checkingthat are needed in the rest of the paper. We follow the notation of [30, 29].LTL (Linear Temporal Logic) is a modal logic aimed at encoding how statesevolve over time. It has three unary modal operators (X, G, and F ) andone binary modal operator (U). Their meaning is: Xφ is true in particularstate if and only if the formula φ is true in the next state; Gφ is true if andonly φ is true from now on; Fφ is true if φ will become true at some timein the future; φUψ is true if ψ will eventually become true and φ stays trueuntil then. We indicate with L(O1, . . . , On) the LTL fragment in which theonly temporal operators allowed are O1, . . . , On; for instance, L(F,X) is thefragment of LTL in which only F and X are allowed.

The semantics of LTL is based on Kripke models. In the following, foran ’atomic proposition’ we mean a Boolean variable. Given a set of atomicproposition, a Kripke structure for LTL is a tuple 〈Q,R, `, I〉, where Q is aset of states, R is a binary relation over states (the transition relation), ` isa function from states to atomic propositions (it labels every state with theatomic propositions that are true in that state), I is a set of initial states.A run of a Kripke structure is a Kripke model. A Kripke model for LTL isan infinite sequence of states, where the transition relation links each statewith the one immediately following it in the sequence. The semantics of themodal operators is defined in the intuitive way: for example, Fφ is true ina state of a Kripke model if φ is true in some following state.

The main problem of interest in practice is to verify whether all runsof a Kripke structure (all of its Kripke models) satisfy the formula; this isthe Universal Model Checking problem. The Existential Model Checkingone is to verify whether there is a run of the Kripke structure that satisfiesthe formula. In formal verification, we encode the behavior of a system asa Kripke structure, and the property we want to check as an LTL formula.Checking the structure against the formula tells whether the system satisfiesthe property. Since the Kripke structure is usually called a “model” (whichis in fact very different from a Kripke model, which is only a possible run),this problem is called Model Checking.

In practice, all model checkers describe a system by the Kripke structureof its components. A Kripke structure can be seen as a transition system.Thus the global system is obtained by parallel composition of the transitionsystems representing its components and sharing some variables [26]; usingthis approach, we can give results valid for all model checkers.

3

76

Page 86: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2.2 Composition of Transition Systems

Each component of the global system is modeled using a transition system,which is a formal way to describe a possible transition a system can gothrough. Intuitively, all is needed is to specify the state variables, the pos-sible initial states, and which transitions are possible, i.e., we have to saywhether the transition from state s to state s′ is possible for any pair ofstates s and s′. The formal definition is as follows [26].

Definition 1 A finite-state transition system is a triple (V, I, %), where V ={x1, . . . , xn} is a set of Boolean variables, I is a formula over V , and %(V, V ′)is a formula over V ∪ V ′, where V ′ = {x′1, . . . , x

′n} is a set of new variables

in one to one relation with elememts of V .

Intuitively, V is the set of state variables, I is a formula that is true on atruth assignment if and only if it represents a possible initial state, and % istrue on a pair of truth assignments if they represent a possible transition ofthe system. The set of variables V ′ is needed because % must refer to boththe value of a variable in the current state (xi) and in the next state (x′i).In other words, in this formula xi means the value of xi in the current state,while x′i is the value of the same variable in the next state. For example,the fact that xi remains true is encoded by % = xi → x′i: if xi is true now,then x′i is true, i.e., xi is true in the next state.

Formally, a state s is an assignment to the variables; a state s′ is successorof a state s iff 〈s, s′〉 |= %(V, V ′). A computation is an infinite sequence ofstates s0, s1, s2, . . . , satisfying the following requirements:

Initiality: s0 is initial, i.e. s0 |= I

Consecution: For each j ≥ 0, the state sj+1 is a successor of the state sj

For the sake of simplicity, without loss of any generality, we only considerBoolean variables and Boolean assertions.

In order to model a complex system, we assume that each of its partscan be modeled by a transition system. Clearly, there is usually some inter-action between the parts; as a result, some variables may be shared betweenthe transition systems. In the following, we consider k transition systemsM1, . . . ,Mk. Every Mi is described by ((V L

i ∪ V Si ), Ii(Vi), %i(Vi, V

′i )) for i

1 ≤ i ≤ k where V Li is the set variables local to Mi, V

Si is the set of shared

variables of Mi, and Vi = V Li ∪ V S

i . A group of transition systems canbe composed in different ways: synchronous, interleaved asynchronous, andasynchronous. The third way is not frequently used in Model Checking. Inthe following, a process is any of the transition systems Mi.

Due the lack of space we define only the interleaved case, but all ourresults, with very similar proofs, holds also for the synchronous case.

4

77

Page 87: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The basic idea of the interleaved asynchronous parallel composition isthat only one process is active at the same time. As a result, a global tran-sition can only result from the transition of a single process. The variablesthat are not changed by this process must maintain the same value.

Definition 2 The interleaved asynchronous parallel composition of M1, . . . ,Mk

is the transition system M = (V, I, %):

V =⋃k

i=1Vi I(V ) =

∧ki=1

Ii(Vi)

%(V, V ′) =∨k

i=1

[

%i(Vi, V′i ) ∧

∧kj=1

j 6=i

V Li = V L

i′]

The interleaved asynchronous parallel composition of M1, . . . ,Mk, is de-noted by M1| . . . |Mk.

A model can be described as the composition of transition systems. Asa result, we can define the model checking problem for concurrent transitionsystems as the problem of verifying whether the model described by thecomposition of the transition systems satisfies the given formula.

2.3 Complexity and Compilability

We assume that the reader knows the basic concepts of complexity theory[31, 18]. What we mainly use in this paper are the concepts of polynomialreduction and the class PSPACE.

The Model Checking problem is PSPACE-complete, and is thus in-tractable. On the other hand, as said in the Introduction, it makes sense topreprocess only one part of the problem (either the model or the formula), ifthis reduces the remaining running time. The analysis of how much can begained by such preprocessing, however, cannot be done using the standardtools of the polynomial classes and reductions. The compilability classes[10] have to be used instead.

The way in which the complexity of the problem is identified in thetheory of NP-completeness is that of giving a set of increasing classes ofproblems. If a problem is in a class C but is not in an inner class C′,then we can say that this problem is more complex to solve that a problemin C′. A similar characterization, with similar classes, can be given whenpreprocessing is allowed. For example the class ‖;P is the class of problemsthat can be solved in polynomial time after a preprocessing step. Crucial tothis definition are two points:

1. which part of the problem instance can be preprocessed?

2. how expensive is the preprocessing part allowed to be?

5

78

Page 88: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The first point depends on the specific problem and on the specific set-tings: depending on the scenario, for example, we can preprocess either themodel or the formula for the model checking problem. The second questioninstead allows for a somehow more general answer. First, we cannot limitthis phase to take polynomial time, as otherwise there would be no gainin doing preprocessing from the point of view of computational complexity.Second, we cannot allow the final result of this part to be exponentiallylarge, for practical reasons; we bound the result of the preprocessing phaseonly to take a polynomial amount of space.

In order to denote problems in which only one part can be preprocessed,we assume that their instances are composed of two parts, and that the partthat can be preprocessed is the first one. As a result, the model checkingproblem written as 〈M,φ〉 indicates that M can be preprocessed; written as〈φ,M〉 indicates that φ can be preprocessed.

The “complexity when preprocessing is allowed” is established by char-acterizing how hard a problem is after the preprocessing step. This is doneby building over the usual complexity classes: if C is a “regular” complexityclass such as NP, then a problem is in the (non-uniform) compilability class‖;C if the problem is in C after a preprocessing step whose result takespolynomial space. In other words, ‖;C is “almost” C, but preprocessing isallowed and will not be counted in the cost of solving the problem. Moredetails can be found in [10].

In order to identify how hard a problem is, we also need a concept ofhardness. Since the regular polynomial reductions are not appropriate whenpreprocessing is allowed, ad-hoc reductions (called nu-comp reductions in[10]) have been defined.

In this paper, we do not show the hardness of problems directly, butrather use a sufficient condition called representative equivalence. For ex-ample, in order to prove that model checking is ‖;PSPACE-hard, we firstshow a (regular) polynomial reduction from a PSPACE-hard problem tomodel checking and then show that this reduction satisfies the condition ofrepresentative equivalence.

Let us assume that we know that a given problem A is ‖;C-hard and wehave a polynomial reduction from the problem A to the problem B. Can weuse this reduction to prove the ‖;C-hardness of B ? Liberatore [24] showssufficient conditions that should hold on A as well as on the reduction. Ifall these conditions are verified, then there is a nucomp reduction from ∗Ato B, where ∗A = {〈x, y〉 |y ∈ A}, thus proving the ‖;C-hardness of B.

Definition 3 (Classification Function) A classification function for aproblem A is a polynomial function Class from instances of A to nonnegativeintegers, such that Class(y) ≤ ||y||.

Definition 4 (Representative Function) A representative function for

6

79

Page 89: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

a problem A is a polynomial function Repr from nonnegative integers to in-stances of A, such that Class(Repr(n)) = n, and that ||Repr(n)|| is boundedby some polynomial in n.

Definition 5 (Extension Function) An extension function for a problemA is a polynomial function from instances of A and nonnegative integersto instances of A such that, for any y and n ≥ Class(y), the instancey′ = Exte(y, n) satisfies the following conditions:

1. y ∈ A if and only if y′ ∈ A;

2. Class(y′) = n.

Let us give some intuitions about these functions. Usually, an instanceof a problem is composed of a set of objects combined in some way. Forproblems on boolean formulas, we have a set of variables combined to form aformula. For graph problems, we have a set of nodes, and the graph is indeeda set of edges, which are pairs of nodes. The classification function gives thenumber of objects in an instance. The representative function thus gives aninstance with the given number of objects. This instance should be in someway “symmetric”, in the sense that its elements should be interchangeable(this is because the representative function must be determined only fromthe number of objects). Possible results of the representative function canbe the set of all clauses of three literals over a given alphabet, the completegraph over a set of nodes, the graph with no edges, etc. Let for example Abe the problem of propositional satisfiability. We can take Class(F ) as thenumber of variables in the formula F , while Repr(n) can be the set of allclauses of three literals over an alphabet of n variables. Finally, a possibleextension function is obtained by adding tautological clauses to an instance.Note that these functions are related to the problem A only, and do notinvolve the specific problem B we want to prove hard, neither the specificreduction used. We now define a condition over the polytime reduction fromA to B. Since B is a problem of pairs, we can define a reduction from Ato B as a pair of polynomial functions 〈r, h〉 such that x ∈ A if and only if〈r(x), h(x)〉 ∈ B.

Definition 6 (Representative Equivalence) Given a problem A (hav-ing the above three functions), a problem of pairs B, and a polynomial re-duction 〈r, h〉 from A to B, the condition of representative equivalence holdsif, for any instance y of A, it holds:

〈r(y), h(y)〉 ∈ B iff 〈r(Repr(Class(y)), h(y)〉 ∈ B

The condition of representative equivalence can be proved to imply thatthe problem B is ‖;C-hard, if A is C-hard [24]. As an example, we show

7

80

Page 90: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

these three functions for the PLANSAT ∗1 problem. PLANSAT ∗

1 is the fol-lowing problem of planning: giving a STRIPS [17] instance y = 〈P,O, I,G〉in which the operators have an arbitrary number of preconditions and onlyone postcondition, is there a plan for y? PLANSAT ∗

1 is PSPACE-Complete[7]. Without loss of generality we consider y = (P,O ∪ o0, I, G), where o0 isa operator which is always usable (it has no preconditions) and does nothing(it has no postconditions). We use the following notation: P = {x1, . . . , xn},I is the set of conditions true in the initial state, G = 〈M,N〉. A state inSTRIPS is a set of conditions. In the following we indicate with φh

i the hthpositive precondition of the operator oi, with φi all its the positive precon-ditions, with ηh

i its hth negative precondition, and with ηi all its negativepreconditions; αi is the positive postcondition of the operator oi, βi is thenegative postcondition of the operator oi. Since any operator has only onepostcondition, for every operator i it holds that ‖αi ∪ βi‖ = 1.

Since we shall use them in the following, we define a classification func-tion, a representative function and a extension function for PLANSAT ∗

1 :

Classification Function: Class(y) = ‖P‖. Clearly it satisfies the conditionClass(y) ≤ ‖y‖.

Representative Function: Repr(n) = 〈Pn, ∅, ∅, ∅〉, where Pn = {x1, . . . , xn}.Clearly it is polynomial and satisfies the following conditions: (i)Class(Repr(n))=n, (ii) ‖Repr(n)‖ ≤ p(n) where p(n) is a polynomial.

Extension Function: Let y = 〈P,O, I,G〉 and y′ = Exte(y, n) = 〈Pn, O, I,G〉.Clearly for any y and n s.t. n ≥ Class(y) y′ satisfies the following con-ditions: (i)y ∈ A iff y′ ∈ A, (ii) Class(y′) = n.

Given the limitation of space we cannot give the full definitions for com-pilability, for which the reader should refer to [10] for an introduction, to[9, 8] for an application to the succinctness of some formalisms, to [24] forfurther applications and technical advances.

3 Results

The Model Checking problem for concurrent transition systems is PSPACE-complete [23]. In Section 3.1, we prove that the following problems remainPSPACE-hard even if preprocessing is allowed (in other words, they are‖;PSPACE-hard):

1. model checking on the synchronous and interleaved asynchronous com-position of transition systems, where the transitions systems are thefixed part of the problem and the LTL formula is the varying part;

2. the same problem, where the LTL formula is the fixed part and thetransition system is the varying part;

8

81

Page 91: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3. given a set of transition systems and a formula as the fixed part, astate as the varying part, checking whether the state is a legal initialstate.

We can conclude that preprocessing the model or the formula does notlead to a polynomial algorithm for model checking. We recall that thefixed part is preprocessed off-line in a polynomial data structure during thepreprocessing phase, and the varying part is given on-line.

The relevance of the first two problems is clear: in formal verification, itis often the case that many properties (formulae) have to be verified over thesame system (the model, in this case modeled by the transition systems);on the other hand, it may also be that the same property has to be verifiedon different systems.

The result about the third problem is less interesting by itself. On theother hand, we use it to prove that the superpolynomial growth of the size ofthe data structures (e.g. OBDDs) currently used in model checkers based onthe Symbolic Model Checking algorithms [27] (such as SMV and NuSMV)cannot be avoided in general. The result is independent from its variableordering, and it holds for others data structures that can be employed. Weshow these results in Section 3.2.

We point out that most of Temporal Logic Model Checking algorithms[14] fall in one of three classes: Symbolic Model Checking algorithms, whichwork on symbolic representation of M ; algorithms based on Bounded ModelChecking [5] (i.e. based on reduction from Model Checking into SAT); algo-rithms that work on an explicit representation of M (e.g. [19]). Our resultsconcerning the size of the BDD (or some other decision diagrams) are validfor all algorithms of the first class.

3.1 Preprocessing Model Checking

We now identify the complexity of the Model Checking problem when thepreprocessing of the model (represented as the composition of transitionsystems) is allowed, in the interleaved case. With a similar proof, the resultholds also for the synchronous case.

Theorem 1 The model checking problem for k interleaved concurrent pro-cess MCasyn = 〈(M1| . . . |Mk), ϕ〉 where ϕ ∈ LTL is ‖;PSPACE-Complete,and remains ‖;PSPACE-hard for ϕ ∈ L(F,X).

Proof. We show a reduction, that translates an instance y ∈ PLANSAT ∗1

into an instance 〈r(y), h(y)〉 ∈ Masyn, satisfying the condition of represen-tative equivalence. Given y = 〈P,O, I,G〉 ∈ PLANSAT ∗

1

- r(y) defines a concurrent transition systems M1, . . . ,Mn, where eachMi is obtained from a variable xi ∈ P and it is described by:

9

82

Page 92: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Vi = {xi}

Ii(Vi) = (xi) ∨ (¬xi)

%i(Vi, V′i ) = (xi = 0 ∧ x′i = 0) ∨ (xi = 0 ∧ x′i = 1) ∨

(xi = 1 ∧ x′i = 0) ∨ (xi = 1 ∧ x′i = 1)

The process M = M1‖ . . . ‖Mn represents all possible computations,starting from all possible initial assignments, over the variables x1, . . . , xn.

- h(y) = h(I,G,O) = ¬(φI ∧ φG ∧ φO)where:

ϕI =∧

i∈I

xi ∧∧

i/∈I

¬xi

ϕG = F (∧

i∈Mxi ∧

i∈N¬xi)

ϕO = Gm∨

i=0

[‖φi‖∧

h=1

φhi ∧

‖ηi‖∧

h=1

¬ηhi ∧Xγi ∧

n∧

j 6=ij=1

(xj ↔ Xxj)]

where

γi =

{

αi if αi 6= ∅¬βi if βi 6= ∅

ϕI adds constraints about the initial states of y represented by I.

ϕG adds constraints about the goal states of y represented by G: it tellsthat a goal state will be reached.

ϕO describes the operators in O: globally (i.e. in every state) one of theoperators must be used to go in the next state; ϕO also describes thenop operator o0.

Now, we prove that y ∈ PLANSAT ∗1 iff 〈r(y), h(y)〉 ∈ Masyn. Given

y = 〈P,O, I,G〉, a solution for y is a plan which generates the followingsequence of states: (s1, . . . , sp) where s1 is an initial state and sp is a goalstate. This sequence of states is obtained applying a sequence of operators(oh1

, . . . , ohp) chosen in O = {o1, . . . , om} in the following way: for all i s.t.

1 ≤ i ≤ p, preconditions for ohiare included in the state si, and the state

si+1 is obtained from the state si modifying the postcondition associatedwith ohi

. We remark that a state in STRIPS is the set of conditions.The model M = r(y) = r(P ) represents all possible traces starting from

all possible initial configurations, over the variables x1, . . . , xn. Thus, in thiscase the Existential Model Checking problem 〈M,ϕ〉 reduces to the satisfia-bility problem for ϕ: we check whether ther exists a trace among all tracesover the variables x1, . . . , xn that satisfies the LTL formula ϕ. Therefore,we have to prove that y ∈ A iff ϕ = h(y) is satisfiable:

10

83

Page 93: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

⇒. Given a solution for y ∈ A, we show a model for ϕ = h(y); byconstruction such a model has:

- initial state sM1 s.t. `(s1) = I ∪ {¬xi|xi /∈ I}

- a state sMp s.t. `(sp) ⊆ M∪ {¬xi|xi /∈ N}

- given a state sMi , sM

i+1 is successor of sMi iff

- `(sMi ) ⊆ Precond(ohi

), where Precond(ohi) = {xj |xj ∈ φhi

} ∪{¬xj |xj ∈ ηhi

}- `(sM

i+1) = `(sMi ) ∪ αi − βi

where αi is the positive postcondition of ohiand βi is the negative

postcondition of ohi.

- an infinite number of states: when the state sp is reached this stateis repeated for at least once or for ever (applying the nop operatoro0), or it is possible, it depends from y, to apply any operators whosepreconditions are satisfied by `(sM

p ).

⇐. Let (sM1 , . . . , s

Mp , . . .) a model for ϕ, and let sp the goal state, that

the first state satisfying ϕG. We obtain the sequence of states visited by aplan which is a solution for y, by cutting the states after the goal state sp

and assigning si = `(sMi ); thus this sequence of states (s1, . . . , sp), associated

with the plan, has by construction:

- initial state s1 s.t. s1 = I ∪ {¬xi|xi /∈ I}

- a state sp s.t. sp ⊆ M∪ {¬xi|xi /∈ N}

- given a state si, si+1 is successor of si iff

- si ⊆ Precond(ohi)

- si+1 = si ∪ αi − βi

where αi is the positive postcondition of ohiand βi is the negative

postcondition of ohi.

Now we show the complexity results, in the interleaved case, when theformula can be preprocessed. With a similar proof the result holds also forthe synchronous case.

Theorem 2 The model checking problem for k interleaved concurrent pro-cess MC ′

asyn = 〈ϕ, (M1| . . . |Mk)〉 where ϕ ∈ LTL is ‖;PSPACE-Complete,and remains ‖;PSPACE-hard for ϕ ∈ L(F ).

Proof. The proof is similar to the Theorem 1: we carry out a reduction fromthe PLANSAT ∗

1 problem, that satisfies the conditions of representativeequivalence.

11

84

Page 94: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Now we introduce the decision problem MCs0= 〈[M,ϕ], s0〉, where M

is specified by the interleaved parallel composition of k transition systemsM1, . . . ,Mk, ϕ ∈ L(F ), and s0 is a specific state. MCs0

is true if the modelchecking problem for concurrent transition system 〈M,ϕ〉 has solution ands0 is a legal initial state i.e., is an initial state belonging to M that satisfiesϕ.

Theorem 3 MCs0is ‖;PSPACE-complete.

Proof. The hardness follows from a polynomial time reduction from theproblem 〈(P,O,G), I〉, that can be easily shown ‖;PSPACE-complete onthe basis of the results in [25]. We sketch the reduction. We encode eachoperator in O into each process Mi, and the goal G into the formula ϕ. Weencode the set of initial states I using s0.

3.2 The Size of BDDs

In this section we prove that the size of BDDs and others data structuresincreases superpolynomially with the size of the input data, in the worstcase, when are used in a Symbolic Model Checking algorithm.

Let M a model specified by k concurrent transition systems M1, . . . ,Mk,and let ϕ an LTL (or a CTL or CTL*) formula.

Theorem 4 If PSPACE 6⊆ Πp2∩Σp

2, then there is not always a BDD of any

kind and with any variable order that is polynomially large and representsthe set of initial states consistent with M and ϕ.

Proof. The evaluation problem for any kind of BDD, i.e. giving a BDDand an assignment of its variables evaluate the BDD, is in PIf there exists apoly-size BDD representing the set of initial states consistent with M andϕ, then we can compile M and ϕ in the BDD and evaluate the assignment(representing a initial state) in polynomial time. This implies that MCs0

is in ‖;P. We know from Theorem 3 that MCs0is ‖;PSPACE-complete.

Therefore if such a BDD exists, then ‖;PSPACE=‖;P. Now, by applyingTheorem 2.12 in [10], we conclude that there is no poly-size reduction fromMCs0

to the evaluation problem for a BDD, if PSPACE 6⊆ Πp2∩ Σp

2.

Symbolic Model Checking algorithms work by building a representationof the set of the initial states of M that satisfy ϕ. In particular, this setis represented by BDDs. Therefore, the last theorem proves that these al-gorithms, in the worst case, end up with a BDD of superpolynomial size.This result does not depend on the kind of BDD used (free, ordered, etc.)and on the variable ordering. On the contrary, it holds also when the statesare labeled with enumerative variable; in other words it holds not only for

12

85

Page 95: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

BDD but also for any decision diagram, provided that the evaluation prob-lem over this representation of the states is in a class of the polynomialhierarchy. More formally, we consider an arbitrary representation of a setof states. The evaluation problem is that of determining whether a statebelongs to a set.

Theorem 5 Given a method for representing a set of states whose evalua-tion problem is in a class Σp

i of the polynomial hierarchy, it is not alwayspossible to represent in polynomial space the set of legal initial states of amodel M and a formula ϕ, provided that Σp

i+16= Πp

i+1.

Instances of such data structures, currently used in Symbolic ModelChecking tools, are BDDs, Boolean Expression Diagrams (BEDs) [33] andReduced Boolean Circuits (RBCs) [1]. Our results hold also for data struc-tures used to represent integer-value functions, like Multi terminal binarydecision diagrams (MTBDDs) [13], Algebraic Decision Diagrams (ADDs)[2]; see for details the survey [15].

Now, we discuss related works. There are several results on the expo-nential growth of the BDD size respect to a particular problem (e.g. integermultiplication [6]); moreover there are results dealing with the size growthof other decision diagrams [15] respect to particular problems. While theseresults are not conditional to the collapse of the polynomial hierarchy asthe ones reported in this paper, they are also more specific, as they concernonly specific kinds of data structures (e.g. OBDDs) respect to particularproblems (e.g. integer multiplication). On the other hand, it is also possi-ble to prove that the above two theorems cannot be stated unconditionally:indeed if P = PSPACE, then there is a data structure of polynomial sizeallowing the representation of the set of initial states in such a way decidingwhether a state is in this set can be decided in polynomial time. As a result,the non-conditioned version of the above two theorems implies a separationin the polynomial hierarchy.

References

[1] P.A. Abdullah, P. Bjesse, and N. Een. Symbolic reachability analisysbased on SAT-solvers. In TACAS’00, 2000.

[2] R.I. Bahar, E.A. Frohm, C.M. Gaona, C.M. Hachtel, G.D. Macii, andF. Somenzi. Algebraic decision diagrams and their applications. InCAD, pages 188–191, 1993.

[3] I. Beer, S. Ben David, D. Geist, R. Gewirtzman, and M. Yoeli. Method-ology and system for pratical formal verification of reactive hardware.In CAV’94, volume 818 of LNCS, pages 182–193. Springer, 1994.

13

86

Page 96: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[4] I. Beer, S. Ben David, and A. Landver. On the fly model checkingfor rctl formulas. In CAV’98, volume 1427 of LNCS, pages 184–194.Springer, 1998.

[5] A. Biere, A. Cimatti, E.M. Clarke, and Yunshan Zhu. Symbolic modelchecking without BDDs. In TACAS’1999, volume 2031 of LNCS, pages193–207. Springer, 1999.

[6] R.E. Bryant. On the complexity of vlsi implementations and graphrepresentations of boolean functions with application to integer multi-plication. IEEE Trans. on Computers, 40:205–213, 1991.

[7] T. Bylander. Complexity results for planning. In IJCAI, LNCS, pages274–279. Morgan Kaufmann, 1991.

[8] M. Cadoli, F.M. Donini, P. Liberatore, and M. Schaerf. Space efficencyof propositional knowledge representation formalisms. J. Artif.Int. Re-search, 13:25–64, 1999.

[9] M. Cadoli, F.M. Donini, P. Liberatore, and M. Schaerf. The size of arevised knowledge base. Artif. Intell., 115:1–31, 2000.

[10] M. Cadoli, F.M. Donini, P. Liberatore, and M. Schaerf. Preprocessingof intractable problems. Infor.Comp., 176:89–120, 2002.

[11] R. Cavada, A. Cimatti, E. Olivetti, M. Pistore, and M. Roveri. NuSMV2.1 User’s Manual. IRST, http://nusmv.irst.itc.it, 2002.

[12] A. Cimatti, E.M. Clarke, E. Giunchiglia, F. Giunchiglia, M. Pistore,M. Roveri, R. Sebastiani, and A. Tacchella. NuSMV 2: An opensourcetool for symbolic model checking. In CAV’02, 2002.

[13] E. Clarke, M. Fujita, P. McGeer, K.L. McMillan, J. Yang, and X. Zhao.Multi terminal binary decision diagrams: An efficient data structure formatrix representation. In Int. Work. Logic Synth., pages 1–15, 1993.

[14] E.M. Clarke, O. Grumberg, and D.A. Peled. Model Checking. MITPress, 2000.

[15] R. Drechsler and D. Sieling. Binary decision diagrams in theory andpractice. STTT, 3(2):112–136, May 2001.

[16] R.K. Brayton et al. VIS: a system for verification and syntesis. InCAV’96, volume 1102 of LNCS, pages 428–432. Springer, 1996.

[17] R. Fikes and N. Nilson. Strips: a new approach to the application oftheorem proving to problem solving. Artif. Intell., 2:189–209, 1971.

14

87

Page 97: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[18] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guideto the Theory of NP-Completeness. W.H. Freeman and Company, 1979.

[19] R. Gerth, D. Peled, M.Y. Vardi, and P. Wolper. Simple on-the-fly au-tomatic verification of linear temporal logic. In Symposium on ProtocolSpecification, Testing and Verification, pages 3–18, 1995.

[20] R.H. Herdin, Z. Har’el, and R.P. Kurshan. Cospan. In CAV’96, volume1102 of LNCS, pages 423–427. Springer, 1996.

[21] G.J. Holzmann. The model checker spin. IEEE Trans. Soft. Eng.,23(5):279–295, 1997.

[22] IBM Haifa, http://vlsi.colorado.edu/∼vis/usrDoc.html. RuleBaseUser’s Manual, 2003.

[23] O. Kupferman, M.Y. Vardi, and P. Wolper. An automata theoreticapproach to branching-time model checking. JACM, 47(2):312–360,2000.

[24] P. Liberatore. Monotonic reductions, representative equivalence, andcompilation of intractable problems. JACM, 48(6):1091–1125, 2001.

[25] Paolo Liberatore. On the complexity of case-based planning. TechnicalReport cs.AI/0407034, 2004.

[26] Z. Manna and A. Pnueli. Temporal Verification of Reactive Systems -Safety. Springer Verlag, 1995.

[27] K.L. McMillan. Symbolic Model Checking. Kluwer Academic, 1993.

[28] A. Pnueli. The temporal logic of programs. In FOCS’77, pages 46–57,1977.

[29] Ph. Schnoebelen. The complexity of temporal logic model checking. InAiML’02, volume 4, pages 1–44. World Sc. Publ., 2002.

[30] A.P. Sistla and E.M. Clarke. The complexity of propositional lineartemporal logics. JACM, 32(3):733–749, 1985.

[31] L. J. Stockmeyer. The polynomial-time hierarchy. Theor. Comp. Sci.,3:1–22, 1976.

[32] T. Villa, G. Swarny, and T. Shiple. VIS User’s Manual. VIS Group,http://vlsi.colorado.edu/∼vis/usrDoc.html, 2003.

[33] P.F. Williams, A. Biere, E.M. Clarke, and A. Gupta. Combining deci-sion diagrams and SAT procedures for efficient symbolic model check-ing. In CAV 2000, volume 1855 of LNCS, pages 124–138. Springer,2000.

15

88

Page 98: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Complete Axiomatizations for Logics of Knowledge and

Past Time

Tim French1 Ron van der Meyden2 Mark Reynolds13

Abstract

Sound and complete axiomatizations are provided for two different logicsinvolving modalities for knowledge and both past and future time modalities.The logics considered allow for multiple agents with unique initial state and syn-chrony. Such semantic restrictions are of particular interest in the context ofpast time modalities since both synchrony and unique initial state restrictionsare not expressible without past time modalities. The synchrony restrictiongives every agent access to a system clock.

Keywords: Combinations of Modal Logics, Proof Theory.

1 Introduction

There has been significant interest in multi-modal logics combining operators forknowledge and time in recent years [1, 3, 4, 6]. With only a few exceptions [3], thisliterature deals with future time temporal operators. In this paper we consider theeffect of adding past time operators to such logics.

There are some compelling reasons to consider this extension. One of the topicsof interest in the literature has been the interaction between knowledge and timewhen a variety of semantic properties are assumed, such as uniqueness of initialstates, synchrony, perfect recall and no learning (a dual of perfect recall) [7]. Theseproperties lead to interaction axioms, which involve both epistemic and temporal op-erators. Halpern, van der Meyden and Vardi [6] provide complete axiomatizationsfor all the axiomatizable cases arising out of combinations of these assumptions.However, their results indicate that in some cases, some of the properties have no

1School of Computer Science & Software Engineering, University of Western Australia, Perth6009, {tim,mark}@csse.uwa.edu.au

2School of Computer Science and Engineering, University of New South Wales, Sydney 2052,[email protected]

3This work was partially completed whilst the author was employed at Murdoch University,Western Australia

1

89

Page 99: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

impact on the axiomatization. For example, [6] obtains identical complete axiomati-zations for the cases of no assumptions, synchrony alone, unique initial states alone,and for both synchrony and unique initial states. This indicates that the logic withfuture time axioms is too weak to fully express the properties of unique initial statesand synchrony. It has also been noted that past time operators allow for a muchcleaner axiom for perfect recall in an asynchronous setting [11].

Another reason to consider knowledge in combination with past time operators isthat knowledge-based programs [2] are better behaved with past-time operators thanwith future time operators. A knowledge-based program is like a standard programwith formulas expressing the knowledge of the agent allowed to occur as conditionsin conditional statements. A concrete implementation of such a program replaces theknowledge conditions by concrete conditions of the agent’s local state. Knowledge-based programs behave somewhat like specifications, and in general, may have zero,one, or many different implementations. However, it is possible to provide conditionsunder which there is guaranteed to be a unique implementation [2]. One of theseconditions is when the system is synchronous, and all knowledge tests involve onlypast time operators.

We would like to have an interaction axiom for each of the properties mentionedabove, such that combinations of properties can be handled by combining theircorresponding axiom. In this paper, we take a step in this direction by providingaxioms which individually characterize the properties of unique initial states andsynchrony. (We will deal with combinations in future work.) As already remarked,a past time axiom for perfect recall is already given in [11]. The property of nolearning is best captured by the future time axiom in [6].

The synchrony restriction is particularly interesting since the axiomatization ap-pears to require a complex automaton-based rule. We sketch the rather interestingcompleteness proof here, based on completeness proofs given in [6]. We show wecan construct a canonical model for any consistent formula by induction over thenestings of knowledge operators. To enforce the synchrony constraint we introducetransducers to represent sufficiently detailed information about the time. This trans-ducer can be encoded in a characteristic formula, and we use the new rule, SYNC,to show that the synchrony constraint is maintained.

2 Semantics

The language is given by the abstract syntax:

α = x | ¬α | α ∧ α′ | ©α | αU α′ | ©wα | αS α′ | Kiα

2

90

Page 100: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

where x ∈ V is some propositional atom, and 1 ≤ i ≤ k is the index of an agent.The operators are respectively not, and, tomorrow, until, weak yesterday, sinceand i-knows, and have their usual meaning. Along with the usual propositionalabbreviations (true, false, ∨, →) we will also use the temporal abbreviations: ©−α =¬©w¬α; 3α = trueU α; 3− α = trueS α; α = ¬3¬α; and α = ¬3− ¬α, andthe epistemic operator, Liα = ¬Ki¬α.

For the semantics, we suppose a model is given by a set of runs, and each formulais evaluated with respect to some time in some run. Formally, a model is given by:

M ⊆ {r | r : ω −→ ℘(V)× L1 × . . .× Lk} = R, (1)

where L1, . . . ,Lk are the local states of each agent. The semantics are given withrespect to one run r ∈ M and one moment of time, n ∈ ω. We inductively defineM, r, n |= α as follows:

M, r, n |= x ⇐⇒ x ∈ r(n)0M, r, n |= ¬α ⇐⇒ M, r, n 6|= α

M, r, n |= α ∧ α′ ⇐⇒ M, r, n |= α and M, r, n |= α′

M, r, n |= ©α ⇐⇒ M, r, n+ 1 |= α

M, r, n |= αU α′ ⇐⇒ ∃m ≥ n, M, r,m |= α′ and n ≤ j < m⇒M, r, j |= α

M, r, n |= ©wα ⇐⇒ n = 0 or M, r, n− 1 |= α

M, r, n |= αS α′ ⇐⇒ ∃m ≤ n, M, r,m |= α′ and m < j ≤ n⇒M, r, j |= α

M, r, n |= Kiα ⇐⇒ M, r′,m |= α ∀r′ ∈M,∀n ∈ ω where r(n)i = r′(m)i

for each agent i.This gives the most general description of a language that describes knowledge

and past time. However there are several useful restrictions we will consider:

• We say a model has a unique initial state if for all runs r, r′ ∈ M , for alli ∈ {1, . . . , k}, we have r(0)i = r′(0)i;

• We say a model is synchronized if for all runs r, r′ ∈ M , for all n,m ∈ ω, forall i ∈ {1, . . . , k}, we have r(n)i = r′(m)i =⇒ n = m;

There are several other semantic restrictions that can be applied to combinationsof temporal and modal logic, including perfect recall and no learning. We have chosento focus on synchronization and unique initial state restrictions in this paper as theyare especially relevant to temporal logics with past. The synchronization and uniqueinitial state restrictions have little effect in logics without past operators, as theserestrictions do not alter the set of valid formulas.

3

91

Page 101: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Once past operators are added to the language, the synchronization restrictionhas a dramatic affect on the set of valid formulas. Since every agent knows the time,an axiomatization must allow reasoning about which formulas can be true at whichtimes. For example, if there is some formula, α, that is true at only even times,then if an agents even suspects that α might be true at some time, then that agentshould know that every formula that is true at only odd times must be false. Thissituation is captured in the following formula, which is a validity in the synchronizedsemantics.

Li(x ∧ (x↔©w¬x)) → Ki( (y ↔©w¬y) → y) (2)

3 Axioms

In this section, we describe the axioms and inference rules that we need for reasoningabout knowledge and time for various classes of systems, and state the completenessresults.

For reasoning about knowledge alone, the following system, with axioms K1–K5and rules of inference R1–R2, is well known to be sound and complete [1, 8]:

K1. All propositional tautologies K2. Kiϕ ∧Ki(ϕ→ ψ) → Kiψ, i = 1, . . . , kK3. Kiϕ→ ϕ, i = 1, . . . , k K4. Kiϕ→ KiKiϕ, i = 1, . . . , kK5. ¬Kiϕ→ Ki¬Kiϕ, i = 1, . . . , kR1. From ϕ and ϕ→ ψ infer ψ R2. From ϕ infer Kiϕ, i = 1, . . . , k

This axiom system is known as S5m.For reasoning about the temporal operators individually, the following system

(together with K1 and R1), can be shown to be sound and complete [9]:

F1. ©(ϕ→ ψ) → ©ϕ→ ©ψ P1. ©w (ϕ→ ψ) →©wϕ→©wψF2. ©(¬ϕ) ↔ ¬©ϕ P2. ©−¬ϕ→ ¬©−ϕF3. ϕU ψ ↔ ψ ∨ (ϕ ∧©(ϕU ψ)) P3. ϕS ψ ↔ ψ ∨ (ϕ ∧©− (ϕS ψ))FP. ϕ→ ©©−ϕ P4. trueS ©w falsePF. ϕ→©w©ϕRT1. From ϕ infer ©ϕ RT2. From ϕ′ → ¬ψ ∧©ϕ′ infer ϕ′ → ¬(ϕU ψ)RP1. From ϕ infer ©wϕ. RP2. From ϕ′ → ¬ψ ∧©wϕ′ infer ϕ′ → ¬(ϕS ψ)

This set of axioms gives a sufficient axiomatization of knowledge with past time.To allow for the unique initial states restriction, we add the following axiom:

UIS. (©w false → Kiα) → Kj (©w false → α), i, j = 1, . . . k.

4

92

Page 102: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For the rules required for synchronized time, we must first define a character-istic formula for a transducer (or deterministic linear automaton over a one letteralphabet).

Definition 3.1: A characteristic formula is a formula of the form

α = 3−

©w false ∧ a0 ∧∧a⊆X

(a→ ©a′)

,

where: X is an arbitrary finite set of propositional atoms; x0 ⊆ X; for each a ⊆ X,a is the formula

∧x∈a x ∧ ¬

∨x∈X\a x; and ′ : ℘(X) −→ ℘(X) is some function.

It should be clear to see that a characteristic formula is always satisfiable. It simplydeclares which atoms should be true at which times in a deterministic manner. Welet var(α) be the set of propositional atoms appearing in a formula, α.

In the case of synchronized time we require two new rules.

AUT. From χ→ β infer β, where var(χ)∩ var(β) = ∅ and χ is a characteristicformula.

SYNC. From α→ β infer α→ Kiβ, where var(α) ∩ var(β) = ∅, i = 1, . . . , k.

The rule, AUT, is interesting in that it does not use any knowledge operators.Such a rule is valid in temporal logics with past but has rarely been used in proofsystems (see for example, the AA rule of [10]). We require the rule to add extrapropositions into a proof when the propositions already in the proof do not yieldsufficient information about the system clock.

Just as the AUT rule does not use knowledge operators, the SYNC rule doesnot explicitly use any temporal operators. In fact the complete axiomatizationfor synchronized time has no axiom or rule that uses both temporal and epistemicoperators. This may be surprising since the language clearly has a strong interactionbetween time and knowledge. However the SYNC rule allows an implicit interactionbetween time and knowledge. Suppose that ` α → β and var(α) ∩ var(β) = ∅.Since α and β do not share any propositional atom, we can only infer β from αif α describes some structural property (ie some property that is independent ofpropositions, like “this is an initial state”). By examining the language we can seethat the only structural information that can be expressed are basic conditions onthe time, such as x ∧ (x↔©w¬x) (“the time is even”). Thus we can only infer βfrom α if for every time that α could be true, β must be true. Since agents know thetime, and are logically omniscient, if α is true then any agent will be able to deduceβ. Thus the SYNC rule captures the interaction between knowledge and time in asynchronized system.

5

93

Page 103: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For an example of the effectiveness of the Sync rule, consider the formula (2).Since it is clear that

` (x ∧ (x↔©w¬x)) → ( (y ↔©w¬y) → y), (3)

by the completeness of the temporal rules and axioms, the provability of (2) followsdirectly from the SYNC rule and some propositional tautologies.

4 Soundness for unique initial states

Suppose the axiom was not sound. Then there would be some model M , such thatfor some r ∈ M and some j, M, r, j |= (©w false → Kiα) ∧ ¬Ki (©w false → α).Therefore there must be some r′ ∈ M such that r(j)i = r′(j)i such that M, r′, j |=¬ (©w false → α). Thus M, r′, 0 |= ¬α, and M, r, 0 |= Kiα contradicting the uniqueinitial states requirement of the model.

5 Completeness for unique initial states

To prove the axiom system augmented with UIS is complete we use a standardHenkin-style construction with finite sets of formulas. Given a consistent formula,ψ, we show that ψ has a model generated from the maximal consistent subsetsof some closure set (see, for example [5]). We define the closure set in two stages.Given ψ, let Γψ = {α,¬α,©w false| α ⊆ ψ}. As usual we let Σ be the set of maximallyconsistent sets of formulas, and Sψ = Σ ∩ Γψ. We let S0

ψ = {s ∈ Sψ | ©w false ∈ s}.For the next stage, we let Γ = Γψ ∪ {3− (©w false ∧ s) | s ∈ S0

ψ} where s is theconjunction of the formulas in s. We define S = Σ ∩ Γ and define the relations;,∼i⊆ S × S as:

• s ; t if and only if there exists ∆,∆′ ∈ Σ such that s = ∆ ∩ Γψ, t = ∆′ ∩ Γψand for all α ∈ ∆′, ©α ∈ ∆;

• s ∼i t if and only if there exists ∆,∆′ ∈ Σ such that s = ∆ ∩ Γψ, t = ∆′ ∩ Γψand for all Kiα ∈ ∆, Kiα ∈ ∆′.

Note that ∼i is an equivalence relation, and for all s ∈ S, we let [s]i be thecorresponding equivalence class. We let

R = {r : ω → S | ∀i, r(i) ; r(i+ 1) and αU β ∈ r(i) ⇒ ∃d ≥ i, β ∈ r(d)}.

From R we can derive a model M = {πr : ω → ℘(V) × L1 × . . . × Lk | r ∈ R}where πr(j)0 = r(j) ∩ V, and πr(j)i = [r(j)]i. Finally, for every r ∈ R we let

6

94

Page 104: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Mr ⊂M be defined to be the smallest set such that πr ∈Mr, and for every πt ∈Mr,{πu ∈M | ∃i, j s.t. t(j) ∼i u(j)} ⊆Mr.

The standard approach here is to extend ψ to a maximal consistent set and usethis to find a run r with a state containing ψ. We then prove a truth lemma on Mr,i.e. for every j we show α ∈ r(j) if and only if M,πr, j |= α. Therefore to completethe proof all we have to do is show that the resulting model satisfies the uniqueinitial states constraint. We use the following tautology:

Lemma 5.1: For all s ∈ S,

` ©w false → (s→ (Ki (©w false → ¬Kj¬s)))

Proof: Letγ = ©w false ∧ (s ∧3 (Li3− (©w false ∧Kj¬s)))

By taking the contrapositive of UIS we have ` ¬Ki¬3− (©w false∧¬α) →3− (©w false∧¬Kjα). Let α = ¬Kj¬s. Applied to γ we have

UIS ` γ → (©w false ∧ s ∧33− (©w false ∧ ¬Kj¬Kj¬s)) (4)K5 ` ¬Kj¬s→ Kj¬Kj¬s (5)K1 ` ¬Kj¬Kj¬s→ Kj¬s (6)

LTL ` γ → (©w false ∧ s ∧33− (©w false ∧Kj¬s)) (7)K1 ` γ → s ∧ ¬s (8)K1 ` ¬γ (9)

Since ¬γ is equivalent to ©w false → (s → (Ki (©w false → ¬Kj¬s))), the proofis complete.

Corollary 5.2: The model Mr satisfies the unique initial states constraint.

Proof: If this were not true there would be some runs with non-unique initialstates. Thus there would be some s(0), t(0), s(u), t(v) ∈ S (where s, t ∈ R) such thats(u) ∼i t(v), but s(0) 6∼j t(0). However, by the above lemma,

` s(0) → Ki (©w false → Lj s(0)).

Given the definition of the closure, Γ, it follows that ` s(u) → 3− (©w false ∧ s(0))).Therefore ` s(u) → Ki (©w false → Lj s(0)), and since s(u) ∼i t(v) we must have` t(v) → (©w false → Lj s(0). Therefore it follows that t(0) ∧ Lj s(0) must beconsistent, contradicting the assumption s(0) 6∼j t(0).

7

95

Page 105: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 Soundness for synchronized time

The soundness of the rule AUT is straightforward, and is left to the reader. To showSYNC is sound, suppose that α and β do not share propositional atoms, α → β isa validity, but α∧Li¬β has some model, M . Therefore there are runs r, s ∈M andsome j such that M, r, j |= α, and M, s, j |= ¬β, and r(j)i = s(j)i. Note that theinterpretation of α, and the interpretation of β can only depend on the propositionalatoms that appear in α or β (this can be seen by the recursive definition of the |=relation).

Now let M+ be a new model defined by M+ = {rij |ri, rj ∈ M}, where the runrij is defined by rji (u) = (a, l1, . . . , lk) where

• a = (ri(u)0 ∩ var(α)) ∪ (rj(u)0 ∩ var(β))

• lm = (ri(u)m, rj(u)m)

Note that this requires that the runs are synchronized.We can show that M, ri, u |= α if and only if M+, rji , u |= α for all j, and

M, ri, u |= β if and only if M+, rij , u |= β for all j. (This is done by induction overthe complexity of formulas, using the semantic descriptions given, and is left to thereader). If we let r = ra and s = rb, it follows that M+, rba, j |= α∧¬β, contradictingthe fact that α→ β is a validity.

7 Completeness for synchronized time

We use the strategy used in [6] to construct the model as a series of levels, whereeach level defines the depth of nestings of knowledge operators in a formula. Givenany consistent formula, ψ, we will create a model (a set of runs) by taking sequencesof maximally consistent subsets of a closure set of ψ. We will then show that anyformula that appears in a maximal consistent subset will be true at the correspondingstate in the model. To create such a model we need to find a sequence of maximallyconsistent sets, where one of the sets contains ψ. We then need to provide additionalruns to ensure that if Liγ appears in some set, γ appears in some other run. Howeverthese additional runs can be defined over a smaller closure set since we are onlyinterested in the formulas that appear in the scope of a knowledge operator. We canapply this process recursively until we only have to add runs defined over a closurecontaining no knowledge operators.

When we add additional runs, we have to ensure the knowledge relations conformto the synchrony constraint as well as the normal rules for epistemic logic. To do this,at each level of the construction we include the characteristic formula of a transducer.The state of the transducer at a given level provides sufficient information about the

8

96

Page 106: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

time for us to be able to deduce which of the sets in a lower level will be inconsistentwith the current time. The SYNC rule allows us to use this information to ensurethe model satisfies the synchrony constraint. The following definitions contribute tothis construction.

Each level in the model is represented by a string of agent indexes. We call suchstrings knowledge sequences. We let λ refer to the empty string, τi be the stringτ , concatenated with the index i, and τ\i be the largest string µ such that µi is aprefix of τ , or λ if such a string does not exist. We also use <, ≤ as relations whereτ ≤ σ (τ < σ) indicates that τ is a (proper) prefix of σ.

We define the following hierarchy of languages: We let L be the language definedabove (for k agents), and define the hierarchy over sequences of agents (the nestingsof knowledge operators).

1. Lλ = {α ∈ L | ∀β ∈ L,∀i Kiβ 6⊆ α}.

2. Lτi = {α ∈ L | Kjβ ⊆ α⇒ either j = i and β ∈ Lτ or Kjβ ∈ Lτ}.

We can see that Lλ is the set of all pure temporal formulas, and let σ be the smalleststring such that ψ ∈ Lσ.

We will now define the closure of a formula, ψ.

Definition 7.1: Given a formula, ψ, we let Γψ be the closure of ψ where Γψ isrecursively defined such that:

• ψ ∈ Γψ.

• α ⊆ ϕ implies α,¬α ∈ Γψ

• α ∈ Γψ implies ¬Kiα ∈ Γψ and Kiα ∈ Γψ for i = 1, . . . ,m.

Given a knowledge sequence, τ , we define the τ -closure of ψ to be Γτψ = Γψ ∩ Lτ .

To be able to create a model we require that maximal consistent subsets of theclosure contain sufficient information about the time. We do this as follows. Let Σbe the set of maximally consistent sets of formulas taken from the language (withrespect to the axioms given and the two rules for synchronization), and given aset X of formulas, we let SX be the set of maximally consistent subsets of X, (ieSX = {∆ ∩X | ∆ ∈ Σ}).

We then define the temporal relation ;⊆ SX × SX (X will be assumed fromcontext) by s ; t if and only if

∧α∈s α ∧©

∧α∈t α is consistent.

The knowledge relations are quite complex, and will be constructed using thefollowing definitions and lemmas. These constructions are given so that if we areconsidering formulas in Lτi, then the closure includes an additional formula, χτ ,

9

97

Page 107: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

that describes a transducer, Aτ . A run of this transducer associates a state witheach moment of time and this state describes the set of maximal consistent subsetsof Lτ which are consistent with the given time. We do this by induction, where thebase case is

Xλ = Γλψ ∪ Γλ3− ©w false.

Given Xτ , for any τ , we can then define Sτ (the maximally consistent subsets ofXτ ), Aτ (a transducer showing which subsets are consistent with which times), χτ(the characteristic formula of the transducer), and Xτi (the inductive step). This isdone as follows:

• Sτ = SXτ .

• For all τ , given Sτ and ; (defined above) we let Aτ be a transducer given bythe tuple (Qτ , pτ , δτ ) where:

– Qτ = ℘(Sτ ) is the set of states;

– pτ = {s ∈ Sτ | ©w false ∈ s}– δτ : Qτ → Qτ is the transition function defined byδτ (q) = {t | ∃s ∈ q, s ; t}.

This transducer is defined to identify states which are reachable in the con-structed model at a given time. The run of Aτ is the sequence from Qτ ,(pτ , δτ (pτ ), δ2τ (pτ ), . . .).

• χτ is the characteristic formula of Aτ . To define χτ , for each s ∈ Sτ , let xsbe a propositional atom not appearing in Γτ , and for all q ∈ Qτ , let q =∧s∈q xs ∧ ¬

∨s/∈q xs. Then

χτ = 3−

©w false ∧ pτ ∧∧q∈Qτ

(q → ©δτ (q))

.

• Xτi = Γτiψ ∪ Γλχτ.

We will now restrict our attention to sets s ∈ Sτi, such that χτ ∈ s. By theconstruction of the set Sτi and the rule, AUT, every consistent formula in Γτiψ mustbe an element of some set in Sτi. Let Tλ = Sλ and Tτi = {s ∈ Sτi | χτ ∈ s}.Given any set, t of formulas, we let ti = {α |Kiα ∈ t}. We require some additionaldefinitions to allow us to compare maximal consistent subsets at different levels.Given a knowledge sequence, τ = µj, we let τ− = µ.

10

98

Page 108: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Definition 7.2: For all τ 6= λ, we define the relation ≺i⊂ Tτ\i× Tτ and say t ≺i s(t i-supports s) if

• For some q ∈ Qτ− , q ∈ s and there is some a ∈ q such that t ⊆ a.

• ti ⊆ si ⊆ t.

This definition is constructed such that for all s ∈ Tτ , and for all t ∈ Tτ\i, t ≺i sif and only if s ∧ Lit is consistent. To use this property we require the followinglemma.

Lemma 7.3: For all τ , for all j, for all s ∈ δjτ (pτ ) ∩ Tτ , if Liγ ∈ s, then there issome t ∈ δjτ\i(pτ\i) ∩ Tτ\i such that γ ∈ t and t ≺i s.

Proof: We sketch the main steps of the proof here. The transducer, Aτ identifiesall sets of formulas in Sτ that are consistent with a given time. To make use of thiswe first derive the tautology:

` χτ ∧ q → Ki

∨s∈q

s (10)

This can be shown using the construction of Aτ and the SYNC rule.In order to transfer the tautology, (10) across different levels of knowledge of

knowledge depth we must prove for all knowledge sequences, τ , for all j ∈ ω and forall i ≤ m:

` χτ ∧ δjτ (pτ ) →(χτ\i → δjτ\i(pτ\i)

)(11)

This can be shown using the completeness of the temporal axioms and rules, andthe rule AUT.

We use the tautologies (10) and (11) to prove the lemma as follows. Suppose forcontradiction that there exists τ , j ∈ ω, s ∈ δjτ (pτ )∩ Tτ and some Liγ ∈ s such thatfor all t ∈ δjτ\i(pτ\i) ∩ Tτ\i, if t ≺i s, then γ /∈ t. We can convert this statement intoa formula and use the proof theory to derive a contradiction.

We note that for all t ∈ δjτ\i(pτ\i) ∩ Tτ\i, t 6≺i s implies ` t → Ki¬s. Therefore

it follows that for all t ∈ δjτ\i(pτ\i), either t 6≺i s, or γ /∈ t, or t /∈ Tτ\i. Thus thefollowing would be a tautology:

`∧

t∈δjτ\i

(pτ\i)

(t→ (Ki¬s ∨ ¬γ ∨ ¬χτ\i)

). (12)

11

99

Page 109: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Since s ∈ δjτ (pτ ) ∩ Tτ it follows that χτ− , δjτ−(pτ−) ∈ s. By the tautologies (10),

(11) and the AUT rule we can deduce

` s→ Ki

∨t∈δj

τ\i(pτ\i)

t (13)

Putting this together with (12) we can show

` s→ Ki(Ki¬s ∨ ¬γ ∨ ¬χτ\i). (14)

We can then apply basic epistemic reasoning and the AUT rule to show ` ¬s,contradicting the fact that s is consistent.

Lemma 7.3 gives us the sufficient machinery to complete the proof. If ψ isconsistent, then for some knowledge sequence, σ, ψ must belong to Γσψ and ψ mustbe consistent with χτ , for all τ . Therefore we can find some s ∈ Tσ such thatψ ∈ s. It is clear that the relation ; can be restricted to Tσ for all σ, so we can usethis to create a σ-history (an infinite ;-sequence in Tσ) where all eventualities aresatisfied. For every set in this history we can then satisfy any knowledge formulasusing Lemma 7.3.

The construction we will use here is given as follows: A ranked set of height σ isa disjoint union R =

⋃τ≤σ Rτ , where for each τ ≤ σ, Rτ is a set of labels. For each

r in Rτ we associate a τ -history via a labeling, described in the following definition:

Definition 7.4: A labeling, `, of a ranked set R of height σ is a collection of partialfunctions `τ : Rτ × ω −→ Tτ for τ ≤ σ where:

1. for all r ∈ Rτ , `τ (r, 0) ∈ pτ ;

2. for all r ∈ Rτ , for all j ∈ ω, `τ (r, j) ; `τ (r, j + 1);

3. for all r ∈ Rτ , for all j ∈ ω for all αU β ∈ `τ (r, j) there is some i ≥ j suchthat β ∈ `τ (r, i).

Hence, for any labeling `, for any r ∈ Rτ , `τ (r, 0)`τ (r, 1)`τ (r, 2) . . . will be a τ -history.The construction must also satisfy all the knowledge formulas. To do this we usethe observation that if Liγ appears at some level (say, Liγ ∈ `τ (r, j) where r ∈ Rτ ),then a history labeled by an element of Rτ\i is all that is required to satisfy thisformula. To facilitate this we use the following definition:

Definition 7.5: A system of support, ρ, for a ranked set R of height σ equippedwith a labeling ` consists of, for all τ < σ, for all agents i, a partial functionρiτ : Rτ\i ↪→ Rτ × ω, such that

12

100

Page 110: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. for all r ∈ Rτ , for all j ∈ ω, if ρiτ (t) = (r, j), then `τ\i(t, j) ≺i `τ (r, j).

2. for all r ∈ Rτ for all j ∈ N , if Liγ ∈ `τ (r, j) then exactly one of the followingholds:

• there is some t ∈ Rτ\i such that ρiτ (t) = (r, j) and γ ∈ `τ (t, j).• there is some t ∈ Rµ such that µ\i = τ , and ρiµ(r) = (t, j).

This gives us enough to define the basic structure.

Definition 7.6: Let ψ ∈ Γn be a formula, and σ an index such that ψ ∈ Lσ. Aψ-Frame is a triple (R, `, ρ) where R is a ranked set of height σ, ` is a labeling ofR and ρ is a system of support for R and `, such that for some r ∈ Rσ, and somej ∈ ω, we have ψ ∈ `σ(r, j).

Lemma 7.7: Given any consistent formula, ψ, there exists a ψ-frame.

This is left to the reader. The existence of i-τ -supports follows from lemma 7.3, andthe existence of the τ -labellings follows from the usual reachability arguments.

Given a ψ-frame F , we can now construct a model, MF ⊆ R (see (1)) as follows.

• We let the local states for each agent be taken from R× ω.

• For all τ ≤ σ, for each r ∈ Rτ we define a function πr : ω → ℘(V)×L1×. . .×Ldby πr(j) = (a, l1, . . . , ld) where:

– a = `τ (r, j) ∩ V;

– for each i = 1, . . . d if ρiτ (r) = (t, j), then li = πt(j)i, and otherwiseli = (r, j).

It is clear that the model MF is synchronized.

Lemma 7.8: For all τ ≤ σ, r ∈ Rτ , for all j ∈ ω, and for all ϕ ∈ Γτ

MF , πr, j |= ϕ ⇐⇒ ϕ ∈ `τ (r, j).

Proof: This is shown in the usual way, by induction over the complexity of for-mulas. The only interesting case is the inductive step for the knowledge operator,where ϕ = Kiα. We assume by the inductive hypothesis that for all τ ≤ σ, r ∈ Rτ ,for all j ∈ ω, and for all ϕ ∈ Γτ , MF , πr, j |= α ⇐⇒ α ∈ `τ (r, j).Suppose that MF , πr, j |= ϕ. Therefore, MF , πr, j |= α and for all t such thatπr(j)i = πt(j)i we have MF , πt, j |= α. By the construction of MF , we have twopossibilities:

13

101

Page 111: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. For all t, such that πr(j)i = πt(j)i, t ∈ Rτ\i and ρiτ (t) = (r, j). By theinduction hypothesis, for all such t where ρiτ (t) = (r, j), α ∈ `τ\i(t). Supposefor contradiction ϕ /∈ `τ (r, j). Then Li¬α ∈ `τ (r, j) and by the definition ofρiτ there would be some t such that ρiτ (t) = (r, j) and α /∈ `τ\i, giving us therequired contradiction. Thus ϕ ∈ `τ (r, j).

2. There is some µ and some s ∈ Tµ such that µ\i = τ and for all t, such thatπr(j)i = πt(j)i, we have ρiµ(t) = (s, j) and hence α ∈ `τ (t, j). Since ϕ ∈ Γµ\i

we must have Kiϕ ∈ Γµ. If Li¬ϕ ∈ `µ(s, j), by K4 we have Li¬α ∈ `µ(s, j)and by the definition of ρiµ there must be some t ∈ Rτ such that ρiµ(t) = (s, j)and α /∈ `τ (t, j), contradicting the induction hypothesis. Therefore we musthave Kiϕ ∈ `µ(s, j) and hence ϕ ∈ `µ(s, j)i. By the definition of ≺i it followsthat ϕ ∈ `τ (r, j).

For the converse, suppose ϕ ∈ `τ (r, j). Again we consider two possibilities:

1. For all t 6= r such that πr(j)i = πt(j)i, t ∈ Rτ\i and ρiτ (t) = (r, j). In this caseby the definition of ρiτ and ≺i, we have `τ (r, j)i ⊆ `τ\i(t, j)i. Consequentlyα ∈ `τ\i(t, j)i, and by K3, α ∈ `τ (r, j). By the inductive hypothesis, for all tsuch that πr(j)i = πt(j)i, we have MF , πt, j |= α, so MF , πr, j |= Kiα.

2. There is some µ and some s ∈ Tµ such that µ\i = τ and for all t, such thatπr(j)i = πt(j)i, we have ρiµ(t) = (s, j). For all such t, `τ (t, j) ≺i `µ(s, j), so bythe definition of ≺i, Kiα ∈ `µ(s, j) implies α ∈ `τ (t, j). Since Kiα ∈ `τ (r, j)implies Kiα ∈ `µ(s, j), it must be that for all t, such that πr(j)i = πt(j)i, α ∈`τ (t, j) and by the inductive hypothesis MF , πt, j |= α. Thus MF , πr, j |= Kiα.

8 Conclusion

In this paper we have presented sound and complete axiomatizations for logics ofknowledge and past time with the synchronization and unique initial states con-straints. We note here that the axiomatization for Synchronization and UIS is astraightforward combination of the two and that the proof of completeness can beeasily modified to accommodate this.

For future work we will be investigating combinations of knowledge and pasttime given the semantic restrictions of perfect recall (where an agent retains theknowledge of previous times), and no learning (where an agent’s knowledge can notincrease over time) [1]. We will also look at incorporating common knowledge intothe language, and extending the axiomatizations to combinations of these semanticrestrictions.

14

102

Page 112: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

References

[1] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning about knowledge.MIT Press, 1995.

[2] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Knowledge-based programs.Distributed Computing, 10(4):199–225, 1997.

[3] M. Fisher, M. Wooldridge, and C. Dixon. A resolution-based proof method fortemporal logics of knowledge and belief. In Proceedings of the InternationalConference on Formal and Applied Practical Reasoning (FAPR), 1996.

[4] D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharayashev. Many DimensionalModal Logics: Theory and Applications. Elsevier, 2003.

[5] D. M. Gabbay, A. Pnueli, S. Shelah, and J. Stavi. On the temporal analysisof fairness. In 7th ACM Symposium on Principles of Programming Languages,Las Vegas, pages 163–173, 1980.

[6] J. Halpern, R. van der Meyden, and M. Vardi. Complete axiomatizations forreasoning about knowledge and time. SIAM Journal on Computing. to appear.

[7] J. Halpern and M. Vardi. The complexity of reasoning about knowledge andtime, i:lower bounds. Journal of Computer and System Science, 38(1):195–237,1989.

[8] J. Hintikka. Knowledge and Belief. Cornell University Press, 1962.

[9] L. Zuck O. Lichtenstein, A. Pnueli. The glory of the past. Lecture Notes inComputer Science, 193:196–218, 1985.

[10] M. Reynolds. An axiomatization of full computation tree logic. Journal ofSymbolic Logic, 63(3):1011–1057, 2001.

[11] R. van der Meyden. Axioms for knowledge and time in distributed systemswith perfect recall. In Logic in Computer Science, pages 448–457, 1994.

15

103

Page 113: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Products of ‘transitive’ modal logics

without the (abstract) finite model property

D. Gabelaia, A. Kurucz, and M. Zakharyaschev

Department of Computer ScienceKing’s College London

Strand, London WC2R 2LS, U.K.{gabelaia, kuag, mz}@dcs.kcl.ac.uk

Abstract

It is well known that many two-dimensional products of modallogics with at least one ‘transitive’ (but not ‘symmetric’) componentlack the product finite model property. Here we show that products oftwo ‘transitive’ logics (such as, e.g., K4 ×K4, S4 × S4, Grz ×Grzand GL×GL) do not have the (abstract) finite model property either.These are the first known examples of 2D modal product logics withoutthe finite model property where both components are natural unimodallogics having the finite model property.

Keywords: multi-modal logic, finite model property, product logics.

1 Introduction

Products of modal (in particular, temporal, spatial, epistemic, etc.) logics isa very natural and clear construction arising in both pure logic and variousapplications; see, e.g., [14, 5, 2, 15, 9, 4, 19]. Introduced in the 1970s [17, 18],products have been intensively studied in the last decade (for a comprehens-ive exposition see [8]). The obtained results that are relevant to the decisionproblem for two-dimensional products can be briefly summarised as follows:

• We know that the product of two first-order definable and recursivelyenumerable logics is recursively enumerable [9].

• We know that 2D products with logics like K and S5 are usually de-cidable (but of high—sometimes extremely high—computational com-plexity) [15, 9, 12, 8].

• We know that products of two ‘linear transitive’ logics like K4.3 orGL.3 are mostly undecidable [13, 16].

1

104

Page 114: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

• Yet, despite all efforts, we still have no clue to the computationalbehaviour of products of two ‘transitive’ (but not ‘symmetric’) logicswhere at least one component logic has branching frames (say, K4.3×K4 or S4 × S4). The only known results involve linear componentsthat are either Noetherian or discrete. For example, it is known thatLog(N, <)×K4 and Log(N, <)×S4 are not recursively enumerable [8](and the available proof heavily uses the discreteness of (N, <)).

Not only have we no idea about solutions to these decision problems, but—unlike the unimodal case [3]—very little is known about the frames formultimodal transitive logics with interacting modal operators in general.Without exaggeration one can say that the study of multimodal transitivelogics in general, and 2D product logics and commutators in particular, isone of the most challenging and intriguing topics of modern modal logic.

The aim of this note is twofold. First, we show that products (in fact,already the commutators) of two standard transitive modal logics are ratherexpressive: in particular, they can say that their models must be infinite.Although it is very tempting to try to show that finitely axiomatisable logicslike K4×K4 are decidable by proving first that they enjoy the finite modelproperty (the authors have made several attempts in this direction), oneshould not yield to the temptation—these logics do not have the finite modelproperty.

Second, we hope that the formulas below that require infinite transitive,commutative and Church-Rosser frames will either help in encoding someundecidable problem and showing that K4×K4-type logics are undecidable,or give a hint on how their infinite models can be represented by some finitemeans, say, using mosaics or quasimodels, in order to prove decidability.1

2 Definitions

Given unimodal Kripke frames F1 = (W1, R1) and F2 = (W2, R2), theirproduct is defined to be the bimodal frame

F1 × F2 = (W1 ×W2, Rh, Rv),

where W1×W2 is the Cartesian product of W1 and W2 and, for all u, u′ ∈ W1,v, v′ ∈ W2,

(u, v)Rh(u′, v′) iff uR1u′ and v = v′,

(u, v)Rv(u′, v′) iff vR2v′ and u = u′.

1Added on 14 July 2004: The ‘no fmp’ formulas presented in this paper can indeedbe used to prove that products of basically all ‘transitive’ modal logics with frames ofarbitrarily finite or infinite depth—in fact, all logics located between the commutators ofsuch logics and their 2D products—are undecidable. A draft paper with this and other‘negative’ results is available at http://www.dcs.kcl.ac.uk/staff/mz/prod.pdf.

2

105

Page 115: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Let L1 be a normal (uni)modal logic in the language with the box B and thediamond b. Let L2 be a normal (uni)modal logic in the language with thebox A and the diamond a. Assume also that both L1 and L2 are Kripkecomplete. Then the product of the logics L1 and L2 is the normal (bi)modallogic L1 ×L2 in the language ML2 with the boxes B, A and the diamondsb, a which is characterised by the class of product frames F1 × F2, whereFi is a frame for Li, i = 1, 2. (Here we assume that B and b are interpretedby Rh, while A and a are interpreted by Rv.)

Although product logics L1 × L2 are Kripke complete by definition, ofcourse there can be (and in general there will be) other, non-product, framesfor L1×L2. This gives rise to two different types of the finite model property.As usual, a bimodal logic L (in particular, a product logic L1 × L2) is saidto have the (abstract) finite model property (fmp, for short) if, for everyML2-formula ϕ /∈ L, there is a finite frame F for L such that F 6|= ϕ. (By astandard argument, this means that M 6|= ϕ for some finite model M for L;see, e.g., [3].) And we say that L1×L2 has the product finite model property(product fmp, for short) if, for every ML2-formula ϕ /∈ L1 × L2, there is afinite product frame F for L1 × L2 such that F 6|= ϕ.

Clearly, the product fmp implies the fmp. Examples of 2D product logicshaving the product fmp (and so the fmp) are K×K, K×S5, and S5×S5(see [8] and references therein). On the other hand, there are product logics,such as K4×S5 and S4×S5, that do enjoy the (abstract) fmp [9], but lackthe product fmp [8]. In general, it is well known that many product logicswith at least one ‘transitive’ (but not ‘symmetric’) component do not havethe product fmp (see, e.g., Theorems 5.32, 5.33 and 7.10 in [8]). Here werecall an example of an ML2-formula that can be used to show the manysuch logics do not have the product fmp:

B+ap ∧B+A(p → bB+¬p)

(here B+ψ abbreviates ψ ∧ Bψ). Note that this formula (as well as theothers known so far) is satisfiable in an appropriate finite (in fact, verysmall) non-product frame.

Product logics are defined in a semantical way: they are logics determ-ined by classes of product frames. So a good starting point in understandingtheir behaviour is to find basic principles that hold for every product frame(W1 ×W2, Rh, Rv):

• left commutativity : ∀x∀y∀z (xRvy ∧ yRhz → ∃u (xRhu ∧ uRvz)

),

• right commutativity : ∀x∀y∀z (xRhy ∧ yRvz → ∃u (xRvu ∧ uRhz)

),

• Church–Rosser property : ∀x∀y∀z (xRvy∧xRhz → ∃u (yRhu∧zRvu)

).

3

106

Page 116: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

These properties can also be expressed by the ML2-formulas

abp → bap, bap → abp, bAp → Abp. (1)

Given Kripke complete unimodal logics L1 and L2, their commutator [L1, L2]is the smallest normal modal logic in the language ML2 which contains L1,L2 and the axioms (1).

As product frames satisfy the commutativity and Church-Rosser prop-erties, we always have [L1, L2] ⊆ L1×L2. For some logics, in particular K4or S4, the converse also holds: for example, K4×K4 = [K4,K4]; see [9, 8].On the other hand, e.g., [K4.3,K4] $ K4.3×K4; see Theorem 5.15 in [8].In general, [L1, L2] can even be Kripke incomplete.

3 Results

From now on we only consider products of ‘transitive’ (uni)modal logics,that is, extensions of K4. Our aim is to show that products of two logicssuch as K4, K4.3, S4, Grz or GL do not have the (abstract) fmp. Infact, these are the first known examples of 2D modal product logics withoutthe fmp where both components are standard (uni)modal logics having thefmp. A preceding example of such a product, where one of the componentsis bimodal (Lin× S5), can be found in [15] (this result is generalised a bitin Theorem 5.30 of [8]).

We remind the reader that a frame (W,R) is called Noetherian if thereis no infinite strictly ascending chain x0Rx1Rx2R . . . of points from W (i.e.,no R-chain such that xi 6= xi+1, for all i < ω).

Theorem 1. Let L1 and L2 be Kripke complete normal (uni)modal logicscontaining K4 and such that both L1 and L2 have among their frames arooted Noetherian linear order with an infinite descending chain of distinctpoints. Then all bimodal logics L in the interval

[L1, L2] ⊆ L ⊆ L1 × L2

lack the (abstract) fmp.

Corollary 1.1. Let L1 and L2 be any logics from the list

K4, K4.1, K4.2, K4.3, S4, S4.1, S4.2, S4.3, GL, GL.3, Grz, Grz.3.

Then neither [L1, L2] nor L1 × L2 have the (abstract) fmp.

Proof of Theorem 1. Let ϕ be the conjunction of the following bimodal

4

107

Page 117: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

formulas:

BA((h ∨ah → Ah) ∧ (¬h ∨a¬h → A¬h)

), (2)

BA((v ∨bv → Bv) ∧ (¬v ∨b¬v → B¬v)

), (3)

¬h ∧ ¬v ∧ab(h ∧ v ∧BA(h ∧ v)

), (4)

BA(D⊥ ∧E⊥ → p), (5)

De(¬p ∧Ep), (6)

Ed(p ∧D¬p), (7)

BA(p → E(p ∧dp)

), (8)

BA(¬p → D(¬p ∧e¬p)

), (9)

where

eψ =[h → b

(¬h ∧ (ψ ∨bψ))] ∧ [¬h → b

(h ∧ (ψ ∨bψ)

)],

dψ =[v → a

(¬v ∧ (ψ ∨aψ))] ∧ [¬v → a

(v ∧ (ψ ∨aψ)

)],

Eψ = ¬e¬ψ, and Dψ = ¬d¬ψ.

On the one hand, it is easy to see that ϕ is satisfiable in a productof two rooted Noetherian linear orders each of which contains an infinitedescending chain of distinct points (such a product frame is a frame for Lbecause L ⊆ L1 × L2). Indeed, let Fi = (Wi, <i), i = 1, 2, be such frameswith infinite descending chains

x0 1 x1 1 x2 1 . . . and y0 2 y1 2 y2 2 . . .

of points in W1 and W2, respectively. Define a valuation V in F1 × F2 bytaking:

V(h) = {(x, y) | x0 <1 x} ∪ {(x, y) | xn+1 �1 x ≤1 xn, n < ω, n is even},V(v) = {(x, y) | y0 <2 y} ∪ {(x, y) | yn+1 �2 y ≤2 yn, n < ω, n is even},V(p) = {(x, y) | x1 �1 x} ∪ {(x, y) | xn+1 �1 x, y ≤2 yn, n > 0}

(see Fig. 1). Since F1 is rooted and Noetherian, there is a <1-greatest pointz1 in F1 such that z1 <1 xn for all n < ω. Similarly, there is a <2-greatestpoint z2 in F2 such that z2 <2 yn for all n < ω. The reader can easily checkthat, under the valuation V, we have (z1, z2) |= ϕ.

On the other hand, we will now show that ϕ is not satisfiable in anyfinite frame for L. To this end, suppose that

(M, r) |= ϕ

for a root r of a model M based on a frame F = (W,R1, R2) for L. Then,in view of K4 ⊆ Li and [L1, L2] ⊆ L, we know that

5

108

Page 118: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

rz1

z2 r r r r rx4 x3 x2 x1 x0

r

r

r

r

r

y4

y3

y2

y1

y0

ppppppppppppppp pppp

ppppppppppp pppp

ppppppppppp pppp

ppppppppppppppppppppppppppppppp

qqq

qqq

qqq

qqq

qqq

qqq

q q q

q q q

q q q

q q q

q q q

q q q

º

¹

·

¸F1

º

¹

·

¸

F2

h

h

h

h

h

h

h

h

v

v

v

v

v

v

v

v

p p p p

p p p

p p

p

ϕ

Figure 1: Satisfying ϕ in an infinite product frame.

• both R1 and R2 are transitive,

• R1 and R2 commute, and

• R1 and R2 are Church–Rosser.

Define new relations Ri, for i = 1, 2, by taking for all x, y ∈ W :

xR1y iff ∃z ∈ W[xR1z and

((M, x) |= h ⇐⇒ (M, z) |= ¬h

)and (either z = y or zR1y)

],

xR2y iff ∃z ∈ W[xR2z and

((M, x) |= v ⇐⇒ (M, z) |= ¬v

)and (either z = y or zR2y)

].

Then, by the transitivity of Ri, we have Ri ⊆ Ri (i = 1, 2). It is readilychecked that, for all x ∈ W ,

(M, x) |= eψ iff ∃y ∈ W (xR1y and (M, y) |= ψ),

(M, x) |= dψ iff ∃y ∈ W (xR2y and (M, y) |= ψ).

It is also straightforward to see that, in view of the properties of the Ri

6

109

Page 119: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

mentioned above and by (2)–(3), we have

both R1 and R2 are transitive, (tran)R1 and R2 commute, and (com)R1 and R2 are Church–Rosser. (chro)

We will be using the following notation. For every formula ψ, ` ∈ {e,d}and @ ∈ {E,D}, let

`0ψ = @0ψ = ψ

and, for n < ω, let

`n+1ψ = ``nψ and @n+1ψ = @@nψ.

Claim 1.1. The following formulas are true in M, for all n < ω:

BA(¬p → e>), (10)

BA(p → Endnp), (11)

BA(¬p → Dnen¬p), (12)

BA(D¬p → Dn+1en¬p), (13)

BA(Ep → En+1dnp). (14)

Proof. Formula (10) is a straightforward consequence of (5), (9) and (com).We prove (11) and (12) by induction on n. The case n = 0 is obvious.

Suppose (11) holds for some n. Take some w with (M, w) |= p andz1, . . . , zn, zn+1 such that

wR1z1R1 . . . R1znR1zn+1.

Then (M, zn) |= p by (tran) and (8), and by IH there are w1, . . . , wn suchthat

znR2w1R2 . . . R2wn and (M, wn) |= p.

By (chro), there are s1, . . . , sn such that wiR1si, for i = 1, . . . , n, andzn+1R2s1R2 . . . R2sn. Since wnR1sn, by (tran) and (8), (M, sn) |= p andthere exists sn+1 such that

snR2sn+1 and (M, sn+1) |= p,

from which (M, zn+1) |= dn+1p.Now suppose that (12) holds for some n. Take some w with (M, w) |= ¬p

and z1, . . . , zn, zn+1 such that

wR2z1R2 . . . R2znR2zn+1.

7

110

Page 120: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Then (M, zn) |= ¬p by (tran) and (9), and, by IH, there are w1, . . . , wn suchthat

znR1w1R1 . . . R1wn and (M, wn) |= ¬p.

By (chro), there exist s1, . . . sn such that wiR2si, for i = 1, . . . , n, andzn+1R1s1R1 . . . R1sn. Since wnR2sn, by (tran) and (9), (M, sn) |= ¬p andthere exists sn+1 such that

snR1sn+1 and (M, sn+1) |= ¬p,

which shows that (M, zn+1) |= en+1¬p.Now, by R2 ⊆ R2 and the transitivity of R2, (12) actually implies

BAD(¬p → Dnen¬p).

So (13) follows by the modal axiom K for D. (14) follows from (11) in asimilar way. @

We define inductively two infinite sequences

x0, x1, x2, . . . and y0, y1, y2, . . .

of points in W such that, for every i < ω,

(i) (M, xi) |= p ∧D¬p,

(ii) (M, yi) |= ¬p ∧Ep,

(iii) there exists a point ui such that rR2ui, uiR1xi and uiR1yi, and

(iv) if i > 0 then there exists a point vi such that rR1vi, viR2xi andviR2yi−1.

(We do not claim at this point that, say, all the xi are distinct.)To begin with, by (2)–(4), there are u0, x0 such that rR2u0R1x0 and

(M, x0) |= E⊥ ∧D⊥. (15)

By (5), (M, x0) |= p. By (6), there is y0 such that u0R1y0 and

(M, y0) |= ¬p ∧Ep.

So (i)–(iii) hold for i = 0.Now suppose that, for some n < ω, xi and yi with (i)–(iv) have already

been defined for all i ≤ n. By (iii) for i = n and by (com), there is vn+1

such that rR1vn+1R2yn. So by (7), there is xn+1 such that vn+1R2xn+1 and

(M, xn+1) |= p ∧D¬p.

8

111

Page 121: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

rr

rvi+1

rvi+2

rui+2

rxi+2

rui+1 rxi+1

ryi+1 r xi

rui ryi

-»»»»»»»:

»»»»»»»:

»»»»»»»:-

-XXXXXXXXXXXXXXz

pppppppppppppppppppppppppppppppppppppp6

pppppppppppppppppppppppO

pppppppppppppppppppppppppppppppppppppp6

pppppppppppppppppppppppO

pppppppppppppppppppppppppppppppppppppp6

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppº

ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppº

- R1

p p p p p p- R2

Figure 2: Generating the points xi and yi.

Now again by (com), there is un+1 such that rR2un+1R1xn+1. So by (6),there is yn+1 such that un+1R1yn+1 and

(M, yn+1) |= ¬p ∧Ep,

as required (see Fig. 2).Next, we show that (i), (ii), and (10)–(14) imply the following claim:

Claim 1.2. For all i, n < ω,

(M, xi) |= en> ↔ dn>, (16)

(M, yi) |= en+1> ↔ dn>. (17)

Proof. If n = 0 then (16) is obvious, and (17) follows from (ii) and (10).So we may assume that n > 0.

To prove (16), assume first that we have xi |= en>. Then there is apoint z such that xiR

n1z. By (i), xi |= p. So, by (11), xi |= Endnp, and

therefore, z |= dnp. Thus we have a point u such that zRn2u. Now, using

(com), we find a point v such that xiRn2v and vRn

1u, from which xi |= dn>.Conversely, suppose xi |= dn>, that is, there is a point z such that xiR

n2z.

By (i), xi |= D¬p, and so, by (13), xi |= Dnen−1¬p. Therefore, z |= en−1¬p

and we have a point u such that zRn−11 u and u |= ¬p. So by (10), u |= e>

and we have a point v such that uR1v, from which zRn1v. Using (com), we

find a point w such that xiRn1w and wRn

2v. It follows that xi |= en>.

9

112

Page 122: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

To show (17), assume first that we have yi |= en+1>. Then there is apoint z such that yiR

n+11 z. By (ii), yi |= Ep. So, by (14), yi |= En+1dnp,

and therefore, z |= dnp. Thus we have a point u such that zRn2u. Now,

using (com), we find a point v such that yiRn2v and vRn+1

1 u, from whichyi |= dn>. Conversely, suppose yi |= dn>, that is, there is a point z suchthat yiR

n2z. By (ii), yi |= ¬p and, by (12), yi |= Dnen¬p. Therefore,

z |= en¬p and we have a point u such that zRn1u and u |= ¬p. So by (10),

u |= e> and we have a point v such that uR1v, from which zRn+11 v. Using

(com), we find a point w such that yiRn+11 w and wRn

2v. It follows thatyi |= en+1>. @

Finally, the following claim shows that the xn are all distinct, and so theframe F must be infinite:

Claim 1.3. For every n < ω,

(M, xn) |= en> ∧En+1⊥.

Proof. We proceed by induction on n. For n = 0 the claim holds by thedefinition of x0 (see (15)).

Now suppose that the claim holds for some n < ω. Then,

(M, xn) |= en> (by IH)

(M, xn) |= dn> (by (16))

(M, yn) |= dn> (by (iii), (com) and (chro))

(M, yn) |= en+1> (by (17))

(M, xn+1) |= en+1> (by (iv), (com) and (chro)).

On the other hand, we also have

(M, xn) |= En+1> (by IH)

(M, xn) |= Dn+1⊥ (by (16))

(M, yn) |= Dn+1⊥ (by (iii), (com) and (chro))

(M, yn) |= En+2⊥ (by (17))

(M, xn+1) |= En+2⊥ (by (iv), (com) and (chro)).

as required. @This completes the proof of Theorem 1. @

It is worth noting that the proof above does not go through for ‘productswith expanding domains’ where only one, say, the left commutativity prin-ciple holds. ‘Expanding products’ with S4 are closely related to intuitionisticmodal logics, e.g., to the transitive analogue of the Fischer Servi logic [6, 7]

10

113

Page 123: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

the decidability of which has remained open for a decade. They are alsovery close to dynamic topological logics interpreted in topological spaceswith continuous functions; see, e.g., [1, 10]. On the one hand, it is not hardto see that ‘expanding product logics’ can always be reduced to productlogics; see [11]. Thus, in principle ‘expanding products’ can be computa-tionally simpler than the standard ones. However, no example is known sofar where the product of two logics is undecidable, while their ‘expandingproduct’ is decidable.

Acknowledgments

The work on this paper was partially supported by U.K. EPSRC grants no.GR/R45369/01 and GR/R42474/01. The work of the second author wasalso partially supported by Hungarian Foundation for Scientific Researchgrant T35192.

References

[1] S. Artemov, J. Davoren, and A. Nerode. Modal logics and topologicalsemantics for hybrid systems. Technical Report MSI 97-05, CornellUniversity, 1997.

[2] F. Baader and H.-J. Ohlbach. A multi-dimensional terminologicalknowledge representation language. Journal of Applied Non-ClassicalLogics, 5:153–197, 1995.

[3] A. Chagrov and M. Zakharyaschev. Modal Logic, volume 35 of OxfordLogic Guides. Clarendon Press, Oxford, 1997.

[4] J. Davoren and R. Gore. Bimodal logics for reasoning about continu-ous dynamics. In F. Wolter, H. Wansing, M. de Rijke, and M. Za-kharyaschev, editors, Advances in Modal Logic, Volume 3, pages 91–112. World Scientific, 2002.

[5] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning about Know-ledge. MIT Press, 1995.

[6] G. Fischer Servi. Semantics for a class of intuitionistic modal calculi. InM. L. Dalla Chiara, editor, Italian Studies in the Philosophy of Science,pages 59–72. Reidel, Dordrecht, 1980.

[7] G. Fischer Servi. Axiomatizations for some intuitionistic modal logics.Rendiconti di Matematica di Torino, 42:179–194, 1984.

[8] D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-Dimensional Modal Logics: Theory and Applications, volume 148 ofStudies in Logic. Elsevier, 2003.

11

114

Page 124: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[9] D. Gabbay and V. Shehtman. Products of modal logics. Part I. LogicJournal of the IGPL, 6:73–146, 1998.

[10] P. Kremer and G. Mints. Dynamic topological logic. Manuscript, 2004.

[11] A. Kurucz and M. Zakharyaschev. A note on relativised products ofmodal logics. In P. Balbiani, N.-Y. Suzuki, F. Wolter, and M. Za-kharyaschev, editors, Advances in Modal Logic, Volume 4, pages 221–242. King’s College Publications, 2003.

[12] M. Marx. Complexity of products of modal logics. Journal of Logicand Computation, 9:197–214, 1999.

[13] M. Marx and M. Reynolds. Undecidability of compass logic. Journalof Logic and Computation, 9:897–914, 1999.

[14] J. Reif and A. Sistla. A multiprocess network logic with temporal andspatial modalities. Journal of Computer and System Sciences, 30:41–53,1985.

[15] M. Reynolds. A decidable temporal logic of parallelism. Notre DameJournal of Formal Logic, 38:419–436, 1997.

[16] M. Reynolds and M. Zakharyaschev. On the products of linear modallogics. Journal of Logic and Computation, 11:909–931, 2001.

[17] K. Segerberg. Two-dimensional modal logic. Journal of PhilosophicalLogic, 2:77–96, 1973.

[18] V. Shehtman. Two-dimensional modal logics. Mathematical Notices ofthe USSR Academy of Sciences, 23:417–424, 1978. (Translated fromRussian).

[19] F. Wolter and M. Zakharyaschev. Qualitative spatio-temporal repres-entation and reasoning: a computational perspective. In G. Lakemeyerand B. Nebel, editors, Exploring Artifitial Intelligence in the New Mil-lenium, pages 175–216. Morgan Kaufmann, 2002.

12

115

Page 125: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

������������ �� ��� ������������ ���� �� ��

���� ������

�� �� ����

��������� ���� � �� ��� �� ����������� � �� ��� �� ���������� ���

��������

�� ���� ���� � ��� �� �� ���� ��� ��� �� ���� �� ��������������� ���� �� ����� �� ����������� ���������� �� ��� ������������ ���� �� ��� ������ �� ���� ����� ��������� ������� �� � ������ ����� ������ �� ����� ���� � �� ����� ���� �� ��� �� ��� ��� ����� ������������� �� ������������� �� ��� ������ ��� � �������� ������ �� ������� ��� � ��� ���� ����� � ���������������

��� ���� �� ������ �� �� !�� ��������������� �� ��� ��� �� ������� ����� "�� ���� !��� �������� �� �����#� $����� �� �� ����������� "��� ����� �������� ��� ���� �# � ����� ���� �� ��� �� ������ �� ��� ��� �� �������������� ���� ��� ���� �� �� ���� ��� ������� �� � ��� ��������� �� ��� ���� ��� �������� �� ����� ���� ������������

$� ��� ���� �� ���� �� �������������� �� ��� ���� �� �� �� ����� ������������� ��� ������ %� ����� ��� ���� ���� �� ���� �� ������� �� � ��� ������������ ������� &�� ���� ��� ������� �� ������ � ������� � �� �� ������ ��� ������� � ��� �� ������������������ �� � ����� �� � �� � ���� ��� ��� ������������ ������� $���� �������� ���� ���������� �� � �� ��������������� ����� ���"�#��� � �� ������� ������� ��� �������� ���� ������� ��� �� ��������������������� $� ��� ���� ���"�# ���� ��� �� � ��� ������������������� &� ������� �� �� �� ����� �� �������� �����������

�� �� � ������ �� � �� �� ��������� � �� ���� ��� ����� �� ���������� �� �� ��� �� �� ����� ����� ��� ����������� � ���� ��� �! ��� ���"�� "������� � #$� %� � ���� ������ �� ��� �������� "���� !��"&��� �� ����� �� ��� ������ � �� �� ����� �� ��� ����� �� ��"�� ��� ��������� ��� ���� �� � �������� !�� ��"� ' �� �� ��"�� ��� ������ � � #$ �&��� �&� ���� � ��"����� � �"�� (�� �� !�� �� � ��"��� �� �)��*+��,���-�� �� � ������ �� ���&����� � ������ �� �����&� �� ��� "���� ��������� ��� � � �� ���" �� �� ��"�� ��� "��� ��� .�� �&�� � ���� �� � ���������� � ����� � ����� ����&��� ����� ��� ��� ��� ��� �� � � ���� "���� �������� �

����������� ������ ������� ����� �������� ��� ���� ������� �� �������

/

116

Page 126: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� ������ ��*���00�1� 2�� �����&�� ��� �� ��"�� ��� "��� �� �! �� � ��������� �����3� %����&�� #$ ��4��� ��� ��������� �� �������� �,���-�� �� �!���&��� �����&�� � ���� ���&�� ��"�� "�� ��0%5�1�� � ��� ���� �� !�� ����������� � �� ��� �� 2 ������ ������� � 6������ � ���� #$ ���"� �� �� ������ ���������� �! � ������� ���� � � 6����� ���������� � �� ��� �� !�� � 6������ � 3� %� ��������� �� ��� �� ���� �� �� ��� ���� ������ � ����! �����" �� ��� ��� "���� ��&�� �! ���������� � �� ��� �� � ������� ������ ��"��� ��0�����#0�7�� � " ��� �� �! ������� �� �����" �� !&����� � ���� ��� ������ ��� ��������� ���� 8� � �� � �" �� �� � 9&��� �� � �� � ����� !�� ������� � : ���

�� )��� �� / �� �����&�� ���������� � �� ��� � ��� "��� �� ��� "� ����&���� �� )��� �� ; �� � � ��"� "��� ���� ��� "�� �� �� !�� �&� ��&����� ������� �� � �"��� !�� !� �&�� �! ��������� �� � �$�� 8� ����� �� ��"������ ��� � )��� �� -� �� )��� �� 7 �� !��"&���� ��� �&������� � ���� �� �! ���������� � �� ��� �� � � � "���� ��� �� ��"�� ��� ��"����� � �"�� (�� ���� �� �� ���� �$�2�3�� ����! ���� �� � ��� � ���� ��� ��� ��� ��������� ���������� ��� �� ���� ��� ��� � ��� � ����� ����� �! �$�2�3 ���� ���"� ���� ��� ������ !�� �� ��

� ������������ ����� ���������� �����

��������� ��� �� ��������� �$��������� ��� �& �� &� &� �� ��"� ' ����� �! ������ � ���� �� ������ ��� ������ � ���� ������� ��� � &���� "������������ � ��� � � ���� "���� �������� ��

8 �� ������ �� �� �� � �� � ������ � " ����� �� �� �����&�� � � ����������� ���� �� )� ��� �� "���� 2���3�� ��� �� ��� "������ 2� � �3� %� &�&�� �� &�� � �� �� ����� �� �� !�� ���� 2%� �� � ����� ����� �� ��� &�� �� �� �� ���� �� �� !�� �����3

$�� &� ������ ��� �� ��� � ���� ��� �� ��"�� ��� "��� �� �! ��������"���� !��"&��� �� �$�!��"&��� 2� ���� �� ��'�� �� � �� �� ��������� �������3� �! � ��� � ��� "���� !��"&��� ��� �� ��� �� �� ��"�� ��� �������� <�� ��"�� ��� "��� ��= �! � ��� � ������� ���� ���� ��� �� ��"�� ���"��� �� �! ��� � � !��"�� (�� �� �!�

��>�� ��������� ��>���

�� �������� � ������ � ��������� � ������ � ! ����� � ��� � �������� �� �!!��"&��� �! � ��� !��"&��� �! � �&�� ���� � ����� ��� ��� 2�������� ��� �!���3 ������"� �! �� ?�� � ���� �� !��"&��� �� ��� ��00�1��� ��*�� 8� � �� ���

��� �� ��������� � ��� ���� ���� ���������� �� �� �� ����� ����� ��� ���������� ���������������� � �!� ���� ��� ���� ���������� ������� � "�#$%�� "&$'��

��� ������ �� �� ��� �(��� � �����!� ���� ����� �)�*�+ ��� ���� �� ,��)����� ��� ��� �� -�!��� ��

;

117

Page 127: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

������ � �� �� � ���� � ��� �� � �� �� ��� �� �������� �&� ��� �� �9& ������� ��"�� ��� ���� �� �! � �������

��������� ��� ��� 8 �� �$ �� � �� ��!�� �� ��� !����� �� ��� �! � �"����"����

�� �2�� �3� 2��� ��3

�� ��� ���

�� �2��� �3� ��

�� �2�� �3� ���

�� 2���3 � 2� � �3� �� �

�� 2�� �3 � 2� � �3� � �� � �

� 2���3� 2��� ��3

� ����

8� � �� ���� � � ��� � �$ �� ��� �� ��� �������� �! ��� ���� ����"��������� ��� ������ � ���� ��� � � ��� ��� ���� ����&���� ��� ����� �6 &���������� ��� �� ��� "��&� ������� 8� �� �� �$ � � !�� � � <��� ��� � �$=�8 ���&� ������ �! ���!&� �� �� ����� �! �$ ���� �� "��� ��� ��� � �$�

,�!��� �� "�� �� ��� &� ����� � ���� ����� 2!�� � ����! ��� ��00�1�3�&� "������� ��""��

����� ���� �� �� � � ��� �$ ��

�� � ��� ����

�� � �� ����

�� � ��� ����

@��� ���� �� ���" - ����� ���� ���� �� � �� � ���� ��� �! �$� �� �������� � �� � ��'��� ��"����

��� ����� �! �� � ���������� � �� �� �� ���� !�� ����� � ����� �� ��� ��� �"�� (�� �� ��4� � �� �� �$ ������� ��� � �" ����"���� �� � � ������!�� �� �� �&�� � ����"� ��� ���� ���� ����

��������� ��� ��� ��� 8 �� � �� ������ ��� ����"��

��� � � ��� �� ����

�$� � ��� ��� �! ����"��� ��> �$ ��� �� ���� � � ��� � �$� �� ��� ����� &� ����� ����&�� �! 2��� ����"���3 �$�� ����� ��� ������ � ���� ��� � �������� �6 &���� ������ ��� �� ��� "��&� ������� %�� � �� �� �� �$� ������ "��� ��� ��� � �$��

�������������� �� !�� � � �� �!������ ���!� �(�����.� !�����!�� ����������� �!�

-

118

Page 128: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

8� ��� ���&��� �$�!��"&��� �� ����"�� !��"�� ��� � ����"�� "������

��������� ��� ������� ������� % ������ ������ �� 4&�� ������ � ��� ��� � : ��� �� �����

/� ��� � � #$�!��"� 2���� � � � ��� ��� � � ����� � �� ��������������!�&���� � ���� ����� �� �� � 3�

;� � � � ������� ����� �� ��� � 8 �� �� �� ��� ����� ��� � ���� ����� ���2� �3 � 2�� � �3 � �� %�� � ������� �� ��9& �� !�� ��� � �� � �� � ������� !����� �� ������

2�3 ��� � A ��

2�3 �� ���

2�3 � ��

������� @��"���� ��� ��� ��� ��� � ���� ��9& ��"��� ���� ���� �� � ������ � �� ?�� ����� ��� ������ �� ���������� �� � �������� ��� ��� !��� ������� �� ��'��� ������

��������� ��� ������� ������� % ������ ������ �� � "��� ������ �� 9&���&���� : ��� ���� ����� ��� �� � � ����"�� !��"� ��� � � ������ �� ������� ���"���� �! � ��� �$�!��"&��� ���� ��� �'�� ��� !����� ����9& ��"�����

/� � � ��� 6 !�� ���� �� �&�� ���� � � � ����� � ��� �2��3�� �&��

���� � � �

;� � ��""&��� � �� ������� ������� ��� ���� � � � �� 6 � � � ���� � �

8� ��� ���� �� �$�!��"&�� � � ��� � �� � ����� � : ��� ��� ����� �� � �: �� ������� !�� ��� ����"�� "���� � : ��� ���� ��� ���� �� 2�� ��� ���� �&�� � "���� � ����� �� � 3 �� ��� � � ��

��������� ��� ��� ����� ���������� % !��"� ���� ��� � ������ ���� �'�� ��� !����� �� �������� � ������ �� �$��!��"��

���� 22��3��� �3

%� �$��"���� � � ����"�� "���� ���� � ����� �� � �$��!��"��

� ����� ��! "���� #��$�������� � � ��� �� %�&�'()%�&&'!)����� ��� ����� ������� � �� ��� � ������ ���

�� � � ��� � �� ��� ������ � �$ � �

�� � � ��� � �� ��� �$�������� � �$� � �

7

119

Page 129: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ����������

%����&�� �$� � "������ ��"����� ��� � �"�� (�� ��� ��� �� ������� �� ��� ����� ����� �� �� � � ������"� �� ���� ��� ��� ��� ��������� �� ����������� � ������" � ��� �� ��������� ���� �� �� ��� �� ������� �� ��"�� ������� �� �! ��

�� ���� !������ �� � �� �� �� B� ���� �� "��� B�2��3� ���� �� �� ������� � � � B��!��"&�� 6 !�� ��"� B�C�!��"&�� � 2���� � � �! ��� !��" ���������� � �� ��� 9&��� '��� ��� ��&����3 �� ��� �� � � � ��

��������� ��� B��#�����*���*��+�� $�� � ��� � �� '� �� � ���� ��� �!��� 8� ��� ���� � � B���������� �� ��� � ! !�� ��� B����������� � ����� � � � � � ��

,�� ��� �� �� ��� ��� � �! ���������� � ��� �$� ������� �� �� ��� ��� � �!B����������� �� �! �� �� ���� 2 � !���� �� � ���&�� !��" .��� ��� 5�4�� � �� ����� ��� ��� �� �� ��� ��� ���� ��00�1�3� ���� � ��� �� � ����� ������ ?�.� !��"&���

�� � B�2�2� D��3� �2� D��33�

���� &���� �� � ���� �� �! 2��� "����3 �� �$� � �"�� (�� ��� ��� ������������� "���� !��"&����

�� ����� �� ������� !�� ��� ������ � �! ��� !� �&�� �! ��������� �� � �$���� &� ��� ��� 2���� ������ �� �� �!3 ��� � ����"� � ��&�� 8� ���� ����!�� ��� �� �� ��

�� � � ��� � � ���� 2/3

)&����� �� �� %��� �&����� ���� � � ��"� B� �������� ������� � ��>� � ��� ���� �� > � ����� ��� B� �������� �� � � 2�� � �������� �B� � ��� �� � � E�3� ��� ��&� �� > � ����� �� � � �� ����� �����&� �����> � ��� ����� ��

@�� ��� &� ���� ��� � ���� ����� ��&����� �"��� �� ��*��� �&� �� ������ �� !�� ��������� �� � �$��

�2�� ��3� 2� � �� � � �� � � �3 2;3

)&����� �� ��&�� � ����� E������ �� � �$��!��"&�� � �� ��� ������ � ���� ����� ��� E�2�3� ��� ���� ����! ���� ��� � ����"� � ��&� ���&��������� ���� ��� !����� �� ����"� � ��&��

E�2�3� 2��� � � �� �� � �3�

�/� ����� 0� � ��� ��������� ��!������ ��!��� � ��!� �������� �� � �� 1�*���� ���+� � ��� �� ������ � �� ��

��� �&��$2� �� �� ���� ���� �*�� ��+� *�� �� ��� �� �� �+ �� � !������(������ ������������ ��� �� ��3� ��� ��� � �*� � ��+ � *� � � � � � �� � � �+���� �� ���� ���������� !������(����� �� � ������� �� �����

F

120

Page 130: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

G������ ���� ���&"��� ��� �� ���� �� �&� � �� ��� ���� "����� �� ��'�� � ��� ���� ����� �� �� ���� ��� �� ��"�� ����� �� �� ) ��� �2� � ��3 �E�2�3 � �� ��"�� ��� �� � �� ����� ��� �$� � ����� �� ���� ��� �� ��"�� ������ �� � !��"&���� �� ��&�� ��� �� ���������� !�� 2;3 2��"��� E�2�33�

��� "� � �&�4��� �! �� � ����� � �� ���� ���� ��� ���� � � �� �! ��4� �� �� �� �������� �� �$�� !�� �� �� �� �� �&���� ���� �� �� ��"�� ��� "��� �� � E�������

� ��������

�� �� � ���� �� �� ����� �� ��"� ����� ��� ��� ������ ����H���� ���� ���������� ������ � � � ����� ��� "���� !��"&���� ��� �����

���� ���������� � �� � � � � � �� �� �� � � � ������ ������ � ���� �� ����� ��� ����� � !��"�� ��� "����� 2�� ���!&� �� � �� �� ��3�

?�� "����� � �� � �� &�� ��� ����� �� � !�� ���� ��� "���� ��� ����"� �� � " ����� !�� !��"��� �! � : ��� �� ���� �� �� �� �� !�� � �� !�� � �� !�� �� 8� ��'�� ��� ��� ��� ������� � ������ ��� ��� ���������� � ����� �� !��"�� �� ���!��" �� ���" �� ��� � ��"�������� ?�� ���"����� �� ��� �� � " ��� ������ ��� ���� �&� �! ��&��� �� � ���� "��������� ���� � ������ "�� �� ���� !�� ��� ������ � �� �� ���� � �� ���� � � ��2�3� � � ���2�3� �! � � � "���� !��"&�� ���� �� :�� �� ���8��� �� ���� �� ����� �� ��������� ��� �� �� ��� !��"&��� � ��"� '� ����� I� ���� �� �� �� �I�

? ����� ��� ���&�� ���� ���� ��"� ��4���� ���&�� �� � �� � ����� ������ ����� � 6����� ��"�� ���� ���� � � � ��� � ������ �������� ��� �#���-������ �� �$�2�3 ��� E��$�� $� ��� E�$� �$��!��"&��� ��� E��$��!��"&������ �2�3 ��� �2E�3�

� ��� ����� ������

8� ������ � "���� ��� � �� �� ��� ���� ���&� B���������� �� ��� E������� 8� � � � "������ ��"����� ��� �� ��"�� ����� ��"����� � �"�� (��� �� ��� � � � ��&���� � �"��� !�� ��������� ���

��������� ��� ��� �������� ��� ��������� �$���������� ��� �& �� &�&� �� ��"� ' �� ��� �! ������ � �� �� ������ ��� ������ � ���� ������� ���&���� "���� ��������� � ��� E� ��� ��� � ���� �������� �� �$��!��"&������� �� ��� ����� � � � �� �� ��!���� �� �� $��!��"&����

��������� ��� ���� 8 �� $� �� ������ ��� !����� �� ��� �! ����"����

�/ E�� �E�� � E�2� ��3�

�; E�� �E�� � E�2� ��3�

+

121

Page 131: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�- E�� ��2�� �3� E���

�7 E��� �E���

�F E���

�+ E����

�* E�E���

�1 E��� �2�� ��3�

,� �������� �� "���� �$��!��"&��� ���� ���� ����� � � ��� E� ��������� � ��� �� �&� ��� �� �� ��"�� ��� !��"&��� 2���� �� �� �������� �� ��� ���"���� �������� E� �� � !��"&�� E�2�3� �� �� ������� ���� � � ��� ���� �!� E� !��"&��3 ��� ��� ���� ���� #$ �&�"����� � �� $�� � �� ��"�� �������"����� �#���-�� 8� � �� �� � �� � ���� ��� ""�� ����� "�� �� � �� ���!&�� ����� �! �$��!��"&����

��������� ��� ��2�3�� �$�2�3 � ��� ��� �! ����"��� �$ > $� ��������� �� ��� ����"� �2�3�

E�� � 2���3� � � � �� � ��

�� ��� � �$�2�3 � ��� �"������ ��� �! �$�2�3�!��"&��� �� �� ����� �� ���&� ����� ����&�� �! ��� ��� �$�2�3 ����"��� ��� � ������ &���� "��&������� ��� ������ ��� ��� 8� �� �� �$�2�3 � �� �� � �� !�� � � � ��� �

�$�2�3�

8� � �� ���&��� �$��!��"&��� �� ����"�� !��"�� ��� ����"�� "��������� ������� ��� ����"�� !��"�� !�� �&� ��� � �&��� �&� �� �� ��� ����� �!�$��!��"���

��������� ��� ��2�3� ����� ��2�3�������� % �$�2�3������ � ���$��!��"�� %� �$�2�3�"���� � � 9&���&��� ��� ���� �&�� ���� ��� �� � � �$�2�3�!��"� ��� � � � ����� �� ������� ���"���� �! � ��� �$��!��"&��� ���� ��� �'�� ��� !����� �� ��9& ��"�����

/� ?�� ��� ������� ������� �� ��� � ��� ��"� ���&��� �� !�� ����"��"����� ������

;� � � E�� 6 !�� ��� ����� �&�� ���� �2 ����� ��3

��� �� ���� � � � � ��

.! ��&��� �� ��� ���� � �� � ���� �� �! !��"� �� � �� �� ����� %�� ����� ��� !����� �� ������"�

� ����� ��� "���� #��$���������� �$�2�3 � � � � �: � ��� ���

�$�2�3������ � �

*

122

Page 132: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��� �� !����" 2?�� � !&�� ����! ��� �#���-��3 8� '��� ����� ��� ��&������� ���� ��� ?��"� �� � �� � �������� &���� "��&� ������ ��� ������ ���� ��� J������ ��� ������ � ���� ��&����� �� ��� �� � �� ��� !��"�� �� � ��&� ��� �� ���� ���� ��� ��� �$�2�3 ����"��� ��� �� � �� ���� !��"���� �� ��� �� � ������ ��������

��� ��"��������� � ���� �� ��� �� ����� �� ������&�� �� � "���� �&� �� � ������������ "������ �

��� ����������

� ����� ��� ������� � �����$��������� ���� �# �� �������� � ���

� ��� � �$�2�3 � � � �� �� ��� �� ������� � � � ���� �� ����

!��!�� �� ��� ����� ���� ��� � �� � ��� � �� ��� �$�2�3 � � � �

��� �$�2�3 � � � ���

J��� ��� ��� !����� �� ��� !��"&����

�� : �2�� �� �3 :�� �E�� �E�� ��2�� �3 ��2� � � � �3�

�� : �2�� �� �3 :�� �E�� �E�� ��2� � �3 ��2�� � � � �3�

,�!��� �� ���� ���� ����� !��"&��� ����� �&�� � ��&���� � �"��� !�� ��� ��������� �� �������� ��� &� ��� ���� ��&�� �� ������ !�� �� ���������� ��� ���� % �&� ��� ��� � �� !�� �� ���������� � E�� ��������� � �� �����-��

��2�� �3 : ��2E�2�3 ��2�� �3 ��2� � �33�

?�� ��� "��� �� � !��" ������ � �� �� ����� �� 2#K���� �&"���� �!3 �� ���"�� ��� ��������� ��������

�� � �2��� ��� ��3� ��2�� � ��� ��3 2-3

����� � �2��� ��� ��3� ��

2��� � ��� ��3�

G�������� � E��

� � ��2��� ��3�

������ ��� ��� ��$ �� ��� �� �� � ��2��� �3 � ��

2��� �3 � ��2�� �

��� �3�

%����� L����� � ��� %��&"� ���� !�� ��"� ��� E�2��3� �� � �� � �� ����� � �� � �� %�� ���&"� ���� !�� ��"� ��� E�2��3� �� � �� � �� ����� � �� � �� ���� �� � �� � �� � � ��� �� � �� � �� � �� � ��� ) ���E�2�� � ��3� �� � �����&��� ��� ����!� �

1

123

Page 133: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

)�� �� ��� ���� ��""��

�� � ��2��� � ��� ��3 � ��

2�� � ��� ��3� ��2��� ��3�

)��� � �E��

� � ��2��� � ��� ��3� ���

2�� � ��� ��3�

%�� ��&��� � �2��� ��� ��3� ���

2�� � ��� ��3� 273

)�� ��"� � �� 2-3 ��� 273� ! �� ��&�� � ����� E�� ��������� � ��� �� �2"����3 !��"&�� ��

2�� �3 ���� ���� ��� !��"&�� ��2� � �� �3 � �� �����

������ !�� �� � ����$�� &� ��� �&�� �� ��� ����! �! ������" 7�+� % ��""�� �������� ��

��� �� �&�� � ������" � �� ��'�� �� "����� ��� ��� � ���� ����� ��!��"�� !�� �� �� �$�2�3 � ��&��� ���� ��� ����� ���� � � ��� �

��

�� �&� � ��� �� !���� � ����� ��� ��"� "���� !��"&��� ���� ���� &�������� � �� �� ����� �"��� ��� � 2!�� ������� �� ���� �� ���� ����� � ���� ��� ��� � "&��� �� ���� ��������� � ����3� 8� � �� !����� �� � ���� ������������ "��� ���� � ����� ��� � ��� !����� �� ��'� � ��� ��� ��""���

��������� ��� ���,������������� $�� � ��� � � �� $��"����� ���

� ��� �! ������ � �� �� ������ % � ���� ����� �� ! � � � � � � � $��� � ���� �� ������ ! ��� !����� �� ���� � ��� ��� "���

/� �! �!�� ���� !�� ���� � � � � � � 2�3� �� � � �2�3�

;� �! �!�� ���� !�� ��� �� � � � � �! ��� � ��� ����� ���� ����� � ����� �� �� � �&�� ���� �!��� �!�� ���� ���� ��� ��������

-� )�"� �� ; � �� � ��� � � ������������

��������� ��! ��,������������� $�� � ��� � � �� ��� "����� ��� � ��� �! ������ � �� �� ������ % ����� �� ! �� �� � � �� �"�� � ���� ������� ! ��� !����� �� ���� � ��� ��� "���

/� �! �!�� ���� !�� ���� � � � � � � 2�3� �� �� 2�3�

;� �! �!�� ��� �� ���� ����� � ��� �� �&�� ���� �!��� ���� ��� !������ �� � �� ������� ����� � ��� ��"� �� �&�� ���� �!�� ��� �����

-� )�"� �� ; � �� � ��� � � ������������

�! � : ��� �� � � � � �$�2�3�"���� ���� � �� �� �� ������ ���"���� ��� ��� � �� ����� �� � ��� &� 9&� ������� ����� �� �� � �&������ !�� ���� � �� �� ��� 2��3� : 2��3

�����!� �! ��� !����� �� ��� ��""�� ��� �� !�&�� � �#���-��

124

Page 134: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� �

�� �

�� �

�� �

�� �

�� �

�� �� �

�� �

�� �� �

�� �

�� �

�� �� �

��

��

��

��

? �&�� /� ��� $��� � "&��� ��� ������ ��� ��

�� �

��

��

��

��

�� �

�� �

�� �

? �&�� ;� 2� �3� : 2# �3��

����� ��'� &� ���$ � ����$ #�$ ��� # �� $� �� �$�2�3������� ���

� 2� �3� : 2# �3�� '� ��� �# �� �� �$�� � ���� �� ����� (���!� $��� � ���� ��) � ������� � ��� �� ��� � � ��� ��� $ ��� $�� ��� ���

��� �$�������� (���!� $���������) � � ���� ���� �� !��!�� �� ��� �����

���� $ �� ��� � � � � ��� ��

����� ���(� '� % : %2��� � � � � ��� ��� � � � � ��3 � � �$��������� ��� �

��� �� ��� �� ���� ����� ����� � ���!� �� � � �� E� ��"

�$�2�3 � ��� �

�%2�� ���� � � � � �� ����E���� � � � �E���3� %2�� � � � ��3�

%���� �� ������ *�+� J��� ��� ? �&�� /� .�� ��� �� ������ ���� ��� !�&�"����� ��� �$�2�3�"����� 2!�� ���� �� �� ��� �" ���� ��� ��M� � � ������ ���3� ��� ���� ��� � �� ��� #�$ � ��� 8� �� �2�� �3 !�� � !��"&�� ������ "��� ����� �� ��� �� ����� � ��� �� 8� ��� �� ���� ��� !����� ���

!�� ��� �2�� �3� ��� � �2�� �3� #�$ � �2�� �3 2F3

/�

125

Page 135: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� � � ���&�� � ��� ! � : �2�� �3 ��� �$�2�3 � �� � � � ��� ������� � �� �� #�$ � � ��� #�$ � ���

@�� ��� &� ���� 2F3� ,� $�""� 7�/� �� ��� ���� �� ����� "�������� �$��!��"&�� � �9& ����� �� � !��"&�� � �� �� ������ "���� � ��� ���� � �9& ����� �� � ������� ��"� ��� �� �! �$�!��"&��� ��� $��!��"&������ �����!��� �&��� �� ���� ���� ��� ��� #�$ !���� � ����� ��� ��"�$��!��"&��� �2�� �3 ��� ��� ��� #�$ !���� � ����� ��� ��"� �$�!��"&����2�� �3�

? ��� �� ���� ����2� �3� : 2# �3��

8� �� � ����� !��" ? �&�� ;� )� �� $�""� 7�� � � �&���� �� ���� ��������� � ��� �� �$�� � "&��� �� ������ ��� � �� ���� �� � $��� � "&��� �������� ��� � � ���� ����� ������� �� � � �� �� ��� $ � �� $��

�! �� � ���� ��� ����� ��� �� ����� �� ��� ������ � ��� ������� � ���� � �� ������� �� � � �� �� �� ���� �� ��� !�&� � ���"��� �� ��� � ���"���� � �� ��� � ���������� �� ��&��������� � ��� ����� "����� ���� ������&�� �� ����� �� � � $��� � "&��� �� ������ ��� �� ��� ��"� ����� !�� #��� # ��

%� �$�� � "&��� �� ������ ��� � � "��� ��� �� �������� ,������ � ���� � ������� ���� ������ �� ���� ��� !�� ��� ����� �� ��� � �� ��� �� �� �

������� � � �� �� 6 � ��� �� !���� ��� ��"� ������ � �� �� ����� � � ��� ��$ ��� �� !�� # ��� # ��

�� ����������� ����������

8� '� �� �&� �����"��� �! �$�2�3 �� "�� �� ��� �� ��"�� ��� "��� �� ��!��"&��� ���� �� ��� !��"&���� ��� �� ��"�� ��� ��"��������� ������"�

@�� ��� ��� ������� � �� �� � ��� � ��� ������ �$��!��"&��� �� ���� ��'��� ����� !��"&���� � �� ���� ��� � ��� ����&��� �! ��� >��� �� /��

8� ���&"� � ��� �� �! ��� ����� �! �� � �� �,���-�� �! � � � !��"&���� ������ �� ��� �� ����� 8 �� D� �� ������ ��� �� " � � ���&�� � !&���� �� �&�� ���� !�� ���� !��"&�� �� D���� : ����� ) " ��� ������ ��� ����!�� ��� ����� ������� ������� ��� G������ �� ���&"� � !��"�� (�� �� �!����� � �� � ��� � ���� ���� �2�3 �&�� ���� !�� ��� �������� ��

�� � �� �� � �2���3� 2+3

,��� !��� ���������� � �� � ��� - ������ ' �� 2��������3 ���&��� �&"������ 8� �� ��� "��� � � �� ��� �� ������� ���&��� �&"���� ��� �&"������@��"�� ���������� � �� $ ��� � ��� 24&��3 �� ������

H� �� � �� " � � ���&�� � !&��� �� &��'��&�2�� �� '3 �&�� ���� !�� ����!��"&�� �2�3 ��� � � (� �&�2��2�3�� �����3 : ��2�3� �,���-� �� ��'��

//

126

Page 136: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

!�� !��"&��� �2�3 � �� �� "��� � !����

����2�3�� :�� �2�&�2��2�3�� ���� �33�

�! � ��� "��� ���� ��� !��� �� ���� ���� �� ��� ������ ��� &�� �! �&� ������� � � " ��� ��'� � ��� 2��� ������� �� ����3�

�� ���� !������ �� �� �� �2�3 ��� ���� !�� �2���3 ����� ������� ) ���!�� ��������� � �� � ����� �2�3 �� � �� ���� ��� ����� ��� �2�3 ��� �������������� ��� 4&�� &�� � 2�������� &� �� �������� !�� ���&� ��3 �� "���������

,�� ��� 2+3 �� ���&"� ��� $K�� ��� �� � �� ���� � ��� �,���-�� !�� ���!��"&��� ��� �� ���

/� �� � ��� ����

;� �� � �2�� �3� 2��� ��3�

-� �� � �2��� �3� ���

H� �� ��� ����� � �� ���� ���� ��� � ���� ���� E�C2�3 !�� ������� � E������ �� ��'��

E�2�3 :�� ��2E�C2�3 ��2� D��33�

%�� �� ��'�� � � ���� ���� ���� �� � ����� �� B���������� ��� �� !�������

�� � :�� ��2B�2�3� 2�2� D��3� �2� D��333�

B�2�3 � � ��������� !�� E�2 D��3�@�� �� ��� �������� � �$��!��"&�� � ���

��������� ���� .��� ����#�� �����/������ ��� %� �� �� ��� ���� ,��

�� � � � !&��� �� <�$��!��"&���= �� <��������� �! ��= �� �� ��� �'�� ���!����� ���

/� � ��""&��� � �� ������� ������� ��� ���� 2� ��3� : 2��3 � 2��3

;� 2���3� : ����� ���

-� 2E��3� : E�2��

��3

8� ��� ���� � �$��!��"&�� � � �� �� ����� ��� � ! �� ����� �� !������ �� ��"�� ��� ���� (�� �� ��

� ����� ���� .��� ����#�� #��$���������� �$�2�3 � � � ��� ���

�� �� ��� ���� ,� ��� � �� ��� �� � ���

������ �� �� �� !����� ���� ���!� � �*+ ��� ��� ���� �*��*+�+ ������� � ��*+ �� ���*+� �����4 � ��� � �� � �*�+� /� ��3� ��� ���3�

/;

127

Page 137: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��� �� !����" 2?�� � !&�� ����! ��� �#���-��3 8� '��� ����� ��� ��&������� ���� ��� J������ ��� ����� �! �� ��"�� ����� �� � !��"&��� � ������ &����"��&� ������ ���� �� 2+3 &���� ������ ��� �� �� � � �&� ��� �� ���� ������� � �" ����"��� �! �$�2�3 ��� �� ��"�� ����� �� �� ��� ���� ��� �� ������"� � ��� �&� ��� ��&������ �! �� � � �" � ���� ������

��� ��"��������� � ���� �� � � ��"� ��� �� �! ��� �� ��"�� ��� ��"���������� ����! !�� �$� �)��11��,�������00�1� ��� ��� ��� !�� 5#$ �0���7��

� ��������� ��� ������ �������

8� ��� ���� ����� ��� ���� � � �� �! � ���� �� ��� ����&��� �! ����������� � �� ��� � � �� � "���� �������� �� �� � ������� E�N����� ��� �� �"��� ������ !�� �� � ���� ��� �� � ��� !� �&�� �! ��������� �� !�� �$���� "������� ���������� � �� ��� �� 8� ��� !��"&����� � "������ ����� ��"�� ����� ��"����� ��� � �$�2�3� )���� �� � ��� � ���� ��� ��� ���������� �� � ����� ��� ������ !�� �� � ���"� �� �� � ��� � � ����� � ������� �� " ��� �� '���� �� ��� ��� �� �! E�N ��������� � ��� ��� "��� ������� �� � ���� �� !�� !&����� �������� � �����!�� �� ��� �� � ���� ��� ����&���� �� �� �������� �� �� � ������� �� � ��� ���

! "�#��$��������

� ����� O �� �� 0����� %����� � ����� 0���� 0� 0������� G������ �� L 4����� ��� ��� �����"�&� ��!������

%� ������

�,����� %� ,�����&�� � ��� ���������� � �� ��� � �! ���� �� ��"�� ��-������ �� ����� � &�� �� FF�/�F�N/�1�� /����

�,���-� #� ,������ �� &�� � �� %����� � �� J�"�� ��� H� ��� �� �����J�"�� ���� /��-�

��0%5�1� O�5�0� �� 0����� J� %������ ��� P� 5�������� ��� ��������� ��������" !�� �$ ��� �$�� �� %������ ��� �� . /&01� .������� �

/���� &�� �� H������� )������ .������ /��1� H������ H� ��� ���

��00�1� O�5�0� �� 0���� ��� #��� 0���� �(�� ��� $�� � �! ���� � ����� )�L� ,&��� �� ���� 2������� �� %���� ������ )�&� �� � $�� ���� ��� ?�&���� ��� �! G����"�� ��� ����/-*�� ����� 7*FNF7+�P��� ��� %"������"� /��1�

/-

128

Page 138: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��0���� O�5�0� �� 0���� ��� ?� ����"��� ���� � �� ��� �� !�� ����� � ���������� � ��� �� � ����� �� ����/���� ��� ��� �$ %�������

��� �� � 2�� �� �011 ����� ����� ����� -/N7;� ���&"����� /����

��0��/� O�5�0� �� 0���� ��� %� � ����� P �� � � ' �� �� ��� � ����������� � �� ��� �� �� � &�� ��� F��-�NF�� /��/�

��0���� O�5�0� �� 0���� ��� ?� ����"��� G���� ��"��������� �! �$���� 0� #��������� G� G�� � G� L 4��� ��� Q� ����"�� �� �����3����� ��� ���� � -��� ��� 4���� �� � ����� �� �� �

56 � ����� %"������" H� ��� �� ����� %"������"� /����

�#0�7� P� #�� � ��� 0�0� 0������� G���� "������ � ���������� � �� ���� ��� $�� � #��&� ���� �� )�� �� ;;+� H� ��� �� �! H������� G����;��7�

�#���-� P� #�� �� P ���� �� �$� � �� �� �������� !�� E�N����� ��������&�� ��� �� ��� ��� H� ��� �� �! %"������"� ;��-�

�����-� ��@� ����� �� ��� ����� � �� ��� � !�� E�� ��������� � ��� .������� %��� ��� .!!� �� &�� �� +7�/N;F� /��-�

�0���7� #��� 0���� �(�� ��� ��� � �! ��� �� ��"�� ��� � ������� .�����

�� %��� ��� .!!� �� &�� �� ++�1�N//;� /��7�

�)��11� �� )���&��� ��� ��� � �! ����� � ���������� � �� ��� ������ ��"�� � 2 � L&�� ��3� ����� ��� L����� F� )������ G����"��� ��� ���� �&��� G������ /�11�

�)��*+� L�G� )������ ���� � �� ���������� ��� �! "���� ��� �� '�����

-������ �� /���� ��� ;F�;1*N-�7� /�*+�

�� ���� %� � ����� ����������� � �� ��� �� �� � � ����� �� ���� /���

�� ��� &�� �$ %������ ��� �� � �011 2�� �� 7���������� �����/*FN;/�� @�� Q���� /���� ���&" �����

�� ��*� %� � ����� %� ��� �� �! ���������� � �� ��� �� �� G� �������G� �� L 4��� ��� 5� 8��� ��� �� ����� .������� � ����� ��� �

80+� ����� -�*N-F�� J)$� &�� ��� ���� )���!���� J%� /��*�

/7

129

Page 139: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Brief Notes on the Axiomatization of Elgesem’s Logic ofAgency and Ability

Guido GovernatoriSchool of ITEE, The University of Queensland,

Brisbane, QLD 4072, [email protected]

Antonino RotoloCIRSFID, University of Bologna

Via Galliera 3, 40121, Bologna, [email protected]

Abstract

In this paper we show that the Hilbert system of agency and ability presented by DagElgesem is incomplete with respect to the intended semantics. We argue that completenessresult may be easily regained. Finally, we shortly discuss some issues related to the philo-sophical intuition behind his approach. This is done by examining Elgesem’s modal logicof agency and ability using semantics with different flavours.

1 Introduction

Modal logic of agency is a traditional research field in philosophical logic. Roughly speaking,the approach adopts the general policy to abstract from making explicit state changes and fromconsidering the temporal dimension in describing actions. In fact, actions are simply taken tobe relationships between agents and states of affairs. Thus, the conceptual qualification of theserelations is made by using suitable modal operators to represent, for example, that an agent“brings it about” or “sees to it” thatA, or that such agent is “able” to realiseA, or again that she“attempts” to achieve it.

It is well known that modal logic of agency has a number of drawbacks. As recently sum-marised in [24], the main limit of this approach, as found in the literature, is that it is “tooabstract”. For example, it does not usually capture the difference between the modal qualifica-tions “sees to it” and “brings it about”. Both expressions are in general represented by a singlemodal operator, despite the fact that the former exhibits a clear intentional character, whereas thelatter may refer as well to unintentional actions [11]. Secondly, for the purpose of analysing thestructure of multi-agent contexts it is crucial to distinguish between direct actions and indirectactions. This is necessary, for example, to account for the notions of influence and control of anagent over other agents [14, 15, 19, 20]. While these problems may be, or have been, solved byproviding suitable integrations and new operators within the same paradigm of modal logic of

1

130

Page 140: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

agency, a last drawback is inherent in the paradigm as such. In fact, “sometimes it is essential tobe able to refer to themeansby which an agent brings about a state of affairs”, as for example byreferring to specific actions performed to achieve a goal [24]. As is well-known, this shows thatmodal logic of agency is less expressive than other formal theories of action, such as dynamiclogics. On the other hand, this last limit is also an advantage. Although the abstractness ofmodal logic of agency does not make the language very expressive in itself, it allows flexibilityfor the easy combination of agency with a number of other concepts, such as powers, obliga-tions, beliefs, etc, in a multi-modal setting. This perhaps explains why the approach has beenrecently used to analyse some crucial aspects of normative and institutional domains (see, e.g.,[3, 9, 14, 15, 19, 20]).

The formal properties of modal logic of agency have been extensively investigated, and anumber of variants and axiomatizations can be found in literature (see, e.g., [1, 2, 4, 6, 7, 13, 17–22]). Despite this great variety, it is possible to identify a minimal core of axioms that seem tocharacterize indisputably some aspects of agency. The recent contributions by Dag Elgesem aremeant to work in this direction [6, 7]. We will focus here on two praxeological notions amongthose considered by him1.

The first is the idea of personal and direct action to realise a state of affairs. In the men-tioned general view, this idea is formalised by the well-known modal operatorE, such that aformula likeEiA means that the agenti brings it about thatA. Elgesem’s logic ofE is a classicalnon-normal system [5], namely is closed under logical equivalence, and is characterized by thefollowing schemas.

EiA→ A (1)

(1) is recognised as valid by almost all theories of agency. It is nothing but the usual axiom T ofmodal logic, and it expresses the successfulness of actions which is behind the common readingof “bring about” concept.

¬Ei> (2)

The axiom (2), also named No, is used to capture the very concept of agency at hand, accordingto which the occurrence of any state of affairs, in the scope ofEi , is the result of an action ofi.In other words, ifi had not behaved in the way she did, the world might have been different. Inthis perspective, at least no agent can bring about what is logically unavoidable.

(EiA∧EiB)→ Ei(A∧B) (3)

This third schema, C or Agglomeration, follows from the co-temporality of actions implicitlyassumed within the paradigm of modal logic of agency. In fact, if the agenti realisesA andB, presumably by performing two distinct actions, it can be also said thati brings it about thatA∧B only if the two actions have been performed at the same time. As it is argued by Elgesem,however, the converse of 3 must be rejected because, in presence of it, substitution of equivalents(i) plus 2 make the logic inconsistent whenever at least one action is performed, (ii) gives theusual rule RM ( A→ B/ `2A→2B), which is not acceptable in the logic forE.

1As we will see in a few moments, the two concepts are those of “bring about” and “practical ability”. Elgesemformalises them as Does and Ability respectively, such that both operators are, as expected, indexed by agents. Forthe sake of simplicity, we will adopt a different notation, which is quite common in the literature (see, e.g., [15]).Thus the first is represented by the operatorE, while the second byC. Of course, both are labelled by agents as well.

2

131

Page 141: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The second praxeological concept, analysed by Elgesem and considered here, is agents’“practical ability” to realise states of affairs. This praxeological qualification is represented bythe modal operatorC. Accordingly,CiA expresses thati is capable of realisingA. The logicfor C is quite weak. It is closed as well under logical equivalence and is characterised by thefollowing principles.

EiA→CiA (4)

This schema states a strong connection between ability and agency. Of course, the latter impliesthe former, in presence of axiom (1): Ifi realises successfullyA, it is obvious thati is able to dothis.

¬Ci> (5)

This last axiom is the natural counterpart of schema (2) forE. As we have alluded to, bothexpress jointly the idea of avoidability, namely that the occurrence of a state of affairs cannot becaused by one agent if the goal obtains in every state of the world2.

In the next sections we will analyse some aspects of Elgesem’s semantics for the aboveoperators. The focus will be then on a decisive, but quite solvable, problem arising from his ownsemantic characterisation of the logic of agency and ability.

2 An Axiomatization for Agency and Ability

Elgesem’s analysis starts from semantical considerations [6, 7]. His aim is to give a fresh accountof Sommerhoff’s theory of goal-directness. The semantics is given in terms of selection functionmodels, where a selection function modelE is a structure〈W, f ,v〉 whereW is a (non empty) setof possible worlds,f is a selection function fromP(W)×W to P(W), andv assigns to eachpropositional letter a subset ofW.3

Each formula corresponds to a set of worlds, the set of worlds where it is true, and a worlddescribes the formulas true at it; thus a formula corresponds to a state of affairs, and it determinesall worlds where the state of affairs is true. The selection function identifies then the worldsrelative to the actual world where a goal (state of affairs) has been realized.

2According to Elgesem, the full idea of avoidability requires to focus on two different, but interconnected, aspects.The first corresponds to the negative conditions stated by (2) and (5). Both schemas are aimed to state that no agentbrings about logical truths. The second claim is that “an agent’s behaviour, when he brings about something, isinstrumental in the production of that which he brings about”. This general idea corresponds to saying, positivelyand with respect to any state of affairsA, that “if the agent had not behaved in the way he did when he brought itabout thatA, then he might not have brought it about thatA”. The last requirement is rendered by defining suitabledyadic operators and principles which reflect Elgesem’s own philosophical interpretation of agency [7]. This secondaspect will not be considered here, since it does not seem relevant with regard to the aims of this paper.

3Elgesem’s semantics for the modal logic of agency and ability is a structure〈W, f1, . . . , fn,V〉 (cf. [7, p. 20]and [6, p. 54]), where eachfi , 1≤ i ≤ n is a function as in the structure described above andi is an agent. Sincethere are no interactions among the agents and all functionsfi are independent from each other and obey the sameconditions, we can restrict ourselves to the case of a single agent. Elgesem also considers some foundational aspectsof the notions he deals with and introduces some additional functions in order to capture the idea of avoidability andaccidence. However those functions do not play any relevant role in the characterisation of the modal operatorsE andC. The valuation function and the constraints on the model are given in terms of properties off . The other functionsare used to specify constraints on concrete instances off . FinallyV is a valuation function whilev is an assignment.

3

132

Page 142: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For convenience, before providing the valuation clauses for the formulas, we define thenotion of truth set, i.e., the set of worlds where a formula is true.

DEFINITION 1. The truth set of a formulaA wrt to a modelM , ‖A‖M is thus defined:

‖A‖M = {w∈W : w �M A} .

An Elgesem model is a selection function modelE satisfying the following valuation clauses:

S1. w �E p iff w∈ v(p);

S2. w �E ¬A iff w 6�E A;

S3. w �E A→ B iff w 6�E A or w �E B;

S4. w �E EA iff w∈ f (‖A‖E ,w);

S5. w �E CA iff f (‖A‖E ,w) 6= /0.4

The notion of truth in a model and validity are defined as usual.It is immediate to see that (S4) and (S5) together imply the validity of (4), namelyEA→CA.

Notice that Elgesem uses only one selection function to represent the two modal operatorsEandC. This is crucial in his philosophical approach to agency becausef (‖A‖,w) correspondsexactly to the set of worlds where an agent realizes her ability, relative to the actual worldw, tobring about the goalA. In this perspective, ability and agency are two facets of the same generalconcept.

Then Elgesem goes on and discusses the conditions required to characterise the modal oper-ators of agency (E) and ability (C); though the two operators are defined by the same selectionfunction, he treats them as independent operators (even ifC corresponds to the possibility oper-ator ofE, they are not duals, and cannot be defined in terms of each other in the present setting).

To characterise the other principles Elgesem imposes the following conditions on the selec-tion function f :

E1 f (W,w) = /0;

E2 f (X,w)∩ f (Y,w)⊆ f (X∩Y,w).

E3 f (X,w)⊆ X;

Condition E1 says that a goal that is realized in every world is not a state the agent is able tobring about. As an immediate consequence of this constraints we have the validity of (5) and(2).

Condition E2, corresponding to the agglomeration principle forE (3), is motivated by theidea that the ability needed for the intersection ofA andB is not more general than the ability todoA and the ability to doB.

4From now on, whenever clear from the context we drop subscripts and superscripts.

4

133

Page 143: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Finally E3 makes explicit the idea that in all worlds where an agent realizes her ability tobring about a goal, the goal is indeed realized. It is easy to see that it validates the successprinciple (1).

To sum up, let us recall synoptically Elgesem’s axiomatization for the logic of agency andability (let us call the resulting logicL1).

A0 propositional logic

A1 ¬C>,

A2 EA∧EB→ E(A∧B),

A3 EA→ A,

A4 EA→CA;

plus Modus Ponens andA≡ B

EA≡ EBREE

A≡ BCA≡CB

REC (6)

As we have seen Elgesem also considers the principle¬E>; however this principle is redundantsince it can be easily derived from A1 and the contrapositive of A4.

Another interesting principle, which can be derived from the success axiom for the operatorE (A3) is ¬E⊥.This principle states that nobody can realize an inconsistent (impossible) state.But, what about the corresponding principle that nobody is capable to produce an inconsistentstate?

¬C⊥ (7)

This principle is valid in the proposed selection function semantics, but, as we shall see in thenext section, is not provable inL1.

Let E be an Elgesem model. For every worldw in E we have

w �E ¬C⊥ ⇐⇒ w 6�E C⊥ ⇐⇒ f (‖⊥‖E ,w) = /0.

According to condition E3∀w∈W, f (X,w)⊆ X and,‖⊥‖E = /0; hencef (‖⊥‖E ,w)⊆ ‖⊥‖E =/0. According to the intended interpretation¬C⊥ means that an agent is not able to realize theimpossible (here with impossible we understand an inconsistent state of affairs). This readingseems appropriate in a physical (practical) conception of the notion of ability. However thereare other interpretations where such condition might be relaxed. For example Hintikka [12] pro-poses a reading where impossible worlds are worlds where we have a partial knowledge of thestructure of the world and some contradictions do not appear to be as such, unless we performa deeper analysis of them. A second interpretation whereC⊥ can be accepted is when we havea “normative” reading ofC. As we have alluded to, in the last few years logics of agency andability have been used, in conjunction with deontic logics, to model the relationships among(autonomous) agents in agent societies conceived as normative systems [3, 9, 14, 15, 20]. In thisinterpretation we can useC to describe, among other types of “normative” actions, the abilityof an agent to create a new normative position. In this perspective, it is indeed possible for alaw-maker to draft an inconsistent norm. However, its inconsistency could prevent it from being

5

134

Page 144: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

a true “legal” norm, at least in the event it is accepted the view that all norms must be logi-cally compliable with (and also violable). Of course, a lot depends on the exact axiomatizationadopted for praxeological as well as for deontic notions. IfO stands as usual for the deonticoperator of obligation, it may be argued, for example, thatO⊥ (and/orO>) are meaninglessand so self-contradictory (this view is adopted, for example, in [15]; see also [23]). But nothingin theory is against accepting that an agent may be logically able to issue bizarre norms likeO⊥, for the simple reason that any logic for the operatorO does not include the axiom T. TheexpressionCiO⊥ may be thus accepted. Perhaps, the problem at stake here is that the right wayto approach these questions requires to focus on thenormative powerto issue norms, rather thanon the practical ability to do this. But, of course, this is outside the scope of the paper.

3 Neighbourhood Models

As we have seen in the previous section¬C⊥ is valid, but, as we will see, it is not provable fromL1, henceL1 is incomplete wrt the intended semantics. To show thatL1 is incomplete wrtEwe have to provide a class of models such thatL1 is complete for it and¬C⊥ is false. Whileit is possible to devise a class of selection function models forL1 (see Section 4) we prefer tointroduce models with a different structure.5

A neighbourhood modelN is a structure〈W,NC,NE,v〉whereW is a set of possible worlds,NC andNE are functions fromW to P(P(W)), andv assign subset ofW to atomic letters.

The valuation clauses for atomic and boolean formulas are as usual while those for modaloperators are given below.

DEFINITION 2. Letw be a world inN = 〈W,NC,NE,v〉:

N1 w �N CA iff ‖A‖N ∈ NCw;

N2 w �N EA iff ‖A‖N ∈ NEw .

It is natural to add some conditions on the functionsN in neighbourhood models to validate theaxioms A1–A4.

C1 W /∈ NCw;

C2 if X ∈ NEw andY ∈ NE

w thenX∩Y ∈ NEw ;

C3 if X ∈ NEw thenw∈ X;

C4 NEw ⊆ NC

w.

THEOREM 3. `L1 A iff �N A.

Proof. The proof is a straightforward extension of that given in [5] using minimal canonicalmodels. 2

5As we shall see the difference between the two types of semantics is just in the intuition behind them; in fact,mathematically, they are equivalent and both neighbourhood semantics and selection function semantics are alsoknown as Scott-Montague semantics (cf. [10]).

6

135

Page 145: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

It is easy to provide a neighbourhood model that falsifies¬C⊥. Let W = {w}, NEw = /0 and

NCw = { /0}. Here,‖⊥‖ = /0∈NC

w, thereforew � C⊥ andw 6� ¬C⊥. Hence we have the followingresult:

PROPOSITION4. 6`L1 ¬C⊥.

An immediate consequence of Proposition 4 is thatL1 is incomplete with respect to the intendedselection function semanticsE . It is possible, however, to regain completeness by adding¬C⊥as axiom toL1 (let us call the resulting logicL2).

PROPOSITION5. Let N ′ = 〈W,NE,NC,v〉 a neighbourhood model andE = 〈W, f ,v〉 be anElgesem model satisfying the following conditions:

1. w∈ f (‖A‖E ,w) iff ‖A‖N ′ ∈ NEw ; and

2. f(‖A‖E ,w) 6= /0 and‖A‖E 6= W iff ‖A‖N ′ ∈ NCw.6

Then�E A iff �N ′ A.MoreoverE satisfies conditions E1, E2 and E3 iffN ′ satisfies conditions C1–C4, and/0 /∈ NC

w,for every w∈W.

The above proposition shows that any selection function models can be transformed in an equiv-alent neighbourhood models. However such models must satisfy the condition

C5 ∀w, /0 /∈ NCw,

which is known to correspond to the axiom¬C⊥. Hence we have the following theorem.

THEOREM 6.

1. `L2 A iff �N ′ A;

2. `L2 A iff �E A.

The above theorem proves thatE does not determineL1 but L2 (i.e.,L1 +¬C⊥). In the nextsection we will investigate whether there is a class of selection function models that characterisesL1.

4 Completeness Regained

In the previous section we have seen that it is possible to regain completeness by using neigh-bourhood semantics with two neighbourhood functions, one forC (NC) and one forE (NE) plusthe condition thatNE is included inNC. Obviously, by the well-known equivalence between se-lection function semantics and neighbourhood semantics [10], we can use a semantics with twoselection functions; but what about a selection function semantics with only a common selectionfunction for the two operators? The answer is positive, and in the rest of this section we showhow to modify the conditions on the selection functionf to recover completeness. All we haveto do is to replace the condition E3 with the following condition:

6The condition that‖A‖E 6= W is due to the axiom A1, which requires it.

7

136

Page 146: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

F1 If ‖A‖ 6= /0, then, for allw, f (‖A‖,w)⊆ ‖A‖; otherwisew /∈ f (‖A‖,w).

It is immediate to give a counter-model for¬C⊥: LetW = {w1,w2} and f ( /0,w1) = {w2}. Sincef ( /0,w1) 6= /0, andw1 /∈ f ( /0,w1) we have thatw1 � C⊥.

As a first result for this semantics we show that axioms are valid in it and the inference rulespreserve validity. We useS to denote an Elgesem model that satisfies condition F1.

THEOREM 7. If `L1 A then�S A.

Proof. The only not trivial case is that of axiom A2, since its characteristic condition E2 andF1 are entangle together in this semantics. Condition E2 takes care of the majority of cases,but we have to be careful since it is possible that the conjunction ofA and B is inconsis-tent. If w � EA∧EB, thenw ∈ ‖EA∧EB‖; thusw ∈ ‖EA‖ ∩ ‖EB‖, which means thatw ∈f (‖A‖,w) andw∈ f (‖B‖,w). According to condition F1 we have‖A‖ 6= /0 and‖B‖ 6= /0,whichimplies that f (‖A‖,w) ⊆ ‖A‖ and f (‖B‖,w) ⊆ ‖B‖. On the other hand it is possible that‖A∧B‖ = /0, which means thatw /∈ f (‖A∧B‖,w). If this is the case then‖A‖ ∩ ‖B‖ = /0;hencef (‖A‖,w)∩ f (‖B‖,w) = /0. On the other hand if‖A‖ = /0 (or‖B‖ = /0) then‖A∧B‖ = /0and sof (‖A‖,w) = f (‖A∧B‖,w). A≡ B iff ‖A‖ = ‖B‖. In particular if‖A‖ = ‖B‖ = /0, thenf (‖A‖,w) = f (‖B‖,w). 2

The proof for the completeness is based on canonical models.

DEFINITION 8. A selection function canonical modelis a structureSc = 〈W, f ,v〉 such that:

• W is the set of allL1-maximal consistent sets;

• v is an Elgesem valuation function such that, for all atomic propositionp, w � p iff p∈w.

• f : P(W)×W 7→P(W) is thus defined:

– if CA /∈ w, then f ([A]Sc,w) = /0; otherwise

– if [A]Sc = /0, then f ([A]Sc,w) = W−{w},– if [A]Sc 6= /0, then f ([A]Sc,w) = [EA]Sc.

where[A]Sc, the membership set of a formulaA, is defined as follows:

[A]Sc = {w∈W : A∈ w} .

An immediate consequence of the above construction and Lindebaum’s Lemma is the followingproposition.

PROPOSITION9. LetSc be a canonical selection function model〈W, f ,v〉, then:

• [A]Sc = /0 iff A≡⊥.

• |W|> 1.

• If A 6≡ > and A6≡ ⊥, then[EA]Sc 6= /0.

8

137

Page 147: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

LEMMA 10. For every world w∈W inSc, and every formula A, w�Sc A iff A∈ w.

Proof. We prove the lemma by induction on the complexity of the formula. The inductive baseis given by the basic condition on the valuation function for canonical models. Furthermore weconsider only the case of modal operators.

If w � EA, then by the evaluation function we havew∈ f (‖A‖,w); by the inductive hypoth-esisw∈ f ([A],w), thusw∈ [EA], thereforeEA∈ w.

If EA∈w, then this implies thatCA∈w andA∈w. Sincew is consistentA 6≡ ⊥ and[A] 6= /0;thus f ([A],w) = [EA] and thenw∈ f ([A],w). By the inductive hypothesisw∈ f (‖A‖,w), whichimpliesw � EA.

If w�CA then f (‖A‖,w) 6= /0, and by the inductive hypothesisf ([A],w) 6= /0; by constructionCA∈ w.

If CA∈ w, then eitherf ([A],w) = [EA] or f ([A],w) = W−{w}. ClearlyA cannot be>,thus, according to Proposition 9,f ([A],w) 6= /0, and by the inductive hypothesis so isf (‖A‖,w);thereforew � CA. 2

LEMMA 11. Sc satisfies conditions E1, E2, and F1.

Proof. ¬C> is an axiom, so¬C> ∈ w, for every worldw; henceC> /∈ w. By the constructionof canonical models we havef ([>],w) = /0. Since[>] = W, we havef (W,w) = /0.

If w∈ f ([A],w)∩ f ([B],w), thenw∈ f ([A],w) andw∈ f ([B],w). This means that[A] 6= /0and[B] 6= /0. From this we obtain thatEA∈w andEB∈w. ConsequentlyEA∧EB∈w and by theproperty of maximal consistent setsE(A∧B) ∈w. All we have to prove now is that[A∧B] 6= /0.To prove it we can use the same argument we have developed in the proof of Theorem 7 whenwe have shown thatEA∧EB→ E(A∧B) is valid.

If A≡ ⊥ then eitherf ([A],w) = W−{w} or f ([A],w) = /0. In both casesw /∈ f ([A],w). IfA 6≡ ⊥, then, ifCA∈ w, f ([A],w) = [EA]. But for every worldx if EA∈ x thenA∈ x; thereforef ([A],w)⊆ [A]. On the other hand ifCA /∈w, then f ([A],w) = /0, thus, trivially f ([A],w)⊆ [A].2

From the two Lemmata above we obtain thatL1 is complete with respect toS .

THEOREM 12. `L1 A iff �S A.

5 Non-normal Worlds and Relational Models

In the previous sections we examined Elgesem’s modal logic of agency and ability using se-mantics with different flavours. In general the selection function semantics and neighbourhoodsemantics give rise to the same structure: the selection function semantics focuses on the worldswhere some actions can be realized in relation to a given world, while the neighbourhood se-mantics identifies the actions (formulas) that can be completed successfully in a given world.

In Section 4 we proposed a characterization ofL1 based on models satisfying condition F1.According to the intended readingf ( /0,w) is the set of worlds where the agent realizes his/herability to bring about an impossible goal (whatever an impossible goal is). So in some senses,

9

138

Page 148: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

f ( /0,w) corresponds to a set of impossible or imaginary worlds7. At any rate, the technicalmachinery of impossible (non-normal, queer) worlds offers us the opportunity to present analternative class of Elgesem’s models forL1. All we have to do is to supplement the setW ofpossible worlds with the impossible worldw⊥, to establish that for every formulaA, w⊥ � A,and to define validity as validity at the normal worlds. The revised semantics makes explicitthe need for impossible worlds –after all, if we assume that agents might have the ability torealize the impossible, it seems plausible to have a semantic counterpart for this notion. Hanssonand Gardenfors [10] point out that it is possible to destroy the general dependency of modaloperators on the underlying semantic structure (in the case at hand the selection functionf , andthe accessibility relationR in relational models) by using non-normal/impossible worlds obeyingdifferent logical rules.

Technically non-normal worlds deny the general idea behind intensional semantics that thevalue of modal formulas at a worldw depends on the values of other formulas in other worlds,and validity is defined as validity at the normal worlds. Although the philosophical intuitionbehind non-normal worlds is sound, it commits us to postulate their existence; what is moreis that its treatment is rather unsatisfactory: they are taken as black-boxes without any furtheranalysis of their (internal) structure. In this way, we fail to recognise the potential multiplicityof types of non-normal worlds. A more appropriate solution is to recast the semantics with somemore general type of dependence relation between truth of modal formulas and truth in otherworlds [10].

Scott-Montague models were devised, originally, to overcome the drawback of non-normalworlds we just have alluded to; but, for Elgesem’s models, we have to reintroduce them, eitherimplicitly or explicitly. If we have to reinstate non-normal/impossible worlds in order to prevent¬C⊥ to be valid in Elgesem’s models, then we overstep the very own idea motivating this typeof semantics.

Since non-normal worlds are required, either implicitly or explicitly, in Elgesem’s modelsthe advantages of using a selection function semantics instead of relational models with non-normal worlds is lost. One could then ask if it is possible to devise a relational model forL1

(andL2). In the rest of this section we will investigate this issue.Classical modal logics are characterised by models with the following structure [8]:

〈W,N,R∗,v〉

whereW, v are as before,N ⊆W is the set of normal worlds, andR∗ is a set of binary relationsoverN×W. The valuation clause for2 is

w � 2A iff w∈ N and∃R∈ R∗ such that∀x(wRxiff x � A) (8)

The set of non-normal worlds is denoted byQ (whereQ = W−N). Alternatively we coulddefine a model as〈W,Q,R∗,v〉. Clearly if w ∈ Q, for any formulaA, w 6� 2A. Worlds in Qcorresponds to worlds in a neighbourhood model with empty neighbourhoods.

7It is beyond the scope of the paper to give a characterisation of impossible worlds. All we ask for is that non-normal/impossible worlds are worlds whose rules and laws are different from the rules and laws of the normal worlds.

10

139

Page 149: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Now to accommodateC andE we have to combine one model for theE component and onemodel for theC components. Fortunately the two operators are related by axiom A4, thus wecan adopt the structure8

〈W,QE,QC,RE,RC,v〉

whereW is a set of possible worlds,QE andQC are sets of non-normal worlds such thatQC⊆QE,RE and RC are sets of binary relations with signatureW−QX ×W, andv is an assignment.Moreover

R1 ∀R∈ RC∀w∃x¬(wRx) (all relations inRC are point-wise non-universal);

R2 ∀w /∈ QE∀R,S∈ RE∃T ∈ RE such thatRw∩Sw = Tw (RE is point-wise closed under inter-section);

R3 ∀R∈ RE∀w(wRw) (all relations inRE are reflexive);

R4 ∀w /∈ QE∀R∈ RE∃R′ ∈ RC such thatRw = R′w (the relations inRE are sub-relations of rela-tions inRC);

R5 ∀R∈ RC∀w∃x(wRx) (all relations inRC are serial).

As we shall seeL1 is determined by the class of relational models satisfying R1–R4, andL2

by R1–R5. To prove these results we are going to show that for each relational model thereis an equivalent neighbourhood model, and for every (finite) neighbourhood model there is anequivalent relational model.

Before proving this result we give an auxiliary lemma about sufficient conditions to ensurethe equivalence of relational and neighbourhood models. In what follows we will useRw, forR∈ RX to denote the set of worlds accessible fromw using the relationR, formally: if R∈ RX,thenRw = {w′ ∈W : wRw′}.

LEMMA 13. LetN = 〈W,NE,NC,v〉 be a neighbourhood model andR = 〈W,QE,QC,RE,RC,v〉be a relational model such that

1. ∀w∈W if NXw , then∀x∈ NX

w∃R∈ RX such that x= Rw, and

2. ∀w∈W if w /∈Q, then∀R∈ RX∃x∈ NXw such that x= Rw.

Then for all formulas A:�N A iff �R A.

Proof. The proof is by induction on the complexity ofA. The two models have the same set ofpossible worlds and the same assignment, thus they agree on every propositional variable. Forthe inductive step and the modal operators all we have to do is to apply the condition 1 and 2.2

For every relational model we can generate an equivalent neighbourhood model whereNXw ={

Rw : R∈ RX}

. For the other direction, on the other hand, we have to be careful. Beside theconstraints dictated by the internal structure of the model we have to ensure that the set ofrelations generated fromNE

w is closed under intersection and the relations are serial if we want

8From now on we will useX as a variable ranging overC,E.

11

140

Page 150: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

to satisfy R5. The idea is the same as in the other direction: we use the sets inNXw to create

instances of relations inRX. Here the problem is that given two worldsw andw′ it is very likelythat |NE

w | 6= |NEw′ |; hencew generates|NE

w | sub-relations andw′ generates|NEw′ | sub-relations,

thus there are sub-relations without elements in relation withw. A simple solution to obviatethis problem is to pick a fixed but arbitraryx∈ NE

w for all the additional relations.

THEOREM 14.

1. For every (finite) relational modelM there is an equivalent (finite) neighbourhood modelN such that ifR satisfies Rn thenN satisfies Cn (for1≤ n≤ 5).

2. For every finite neighbourhood modelN there is an equivalent finite relational modelRsuch that ifN satisfies Cn thenR satisfies Rn (for1≤ n≤ 5).

Proof. First of all the models will have the same set of worlds and the same assignment, thusall we have to show is that it is possible to generate appropriate sets of relations from the givenneighbourhood functions and appropriate neighbourhood functions from the given sets of rela-tions.

Part 1. Given a (finite) relational modelR we can generate an equivalent (finite) neighbourhoodmodel as follows:

• If w∈QX thenNXw = /0; otherwise

• NXw =

{Rw : R∈ RX

}.

It is immediate to verify that the conditions of Lemma 13 are satisfied by the models obtainedfrom the above construction, therefore the generated models are equivalent to the generatingmodels.

Part 2. To build a finite relation model from a finite neighbourhood model we use the followingconstruction.

For eachNEw andNC

w let ΣEw andΣC

w be sequences of all the elements inNEw andNC

w such thatif i ≤ |NE

w |, thenΣEw,i = ΣC

w,i (we useΣXw,i to indicate the i-th element ofΣX

w). Moreover

e= max{|NE

w | : w∈W}

c = max{|NC

w| : w∈W}

.

ThenRE =

⋃1≤i≤e

REi RC =

⋃1≤i≤c

RCi

whereRE

i ={(w,w′) : w /∈QE andw′ ∈ α(w, i)

}RC

i ={(w,w′) : w /∈QC andw′ ∈ γ(w, i)

}whereα andγ are partial functions with signatureα : W×N 7→ NE andγ : W×N 7→ NC suchthat:

α(w, i) =

undefined ifi > eor NE

w = /0ΣE

w,i if i ≤ |NEw |

ΣEw,1 otherwise

12

141

Page 151: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

and

γ(w, i) =

undefined ifi > e+c or NC

w = /0ΣC

w,i if i ≤ |NEw |

ΣCw,i−e+|NC

w |if e< i ≤ e+ |NC

w|− |NEw |

ΣCw,1 otherwise

It is easy to verify that the models obtained from the above construction obey to the conditionsof Lemma 16; consequently this construction produces equivalent models. 2

Due to the above procedure to generate such relational models, in the case of infiniteN or N ′

models we would get non-enumerable infinitary relational structures. To avoid these complexi-ties, it is sufficient to considerN andN ′ when they are finite. This is possible by preliminarilyshowing thatL1 andL2 have the finite model property wrt the neighbourhood models previ-ously defined. The fmp follows immediately from the results of Lewis [16] and [25] that everyclassical non-iterative modal logic has the finite model property.9 ClearlyL1 andL2 are non-iterative thus we have the following theorem.

THEOREM 15. L1 andL2 have the fmp.

We can now prove the completeness of theL1 andL2 with respect to the relational modelsdeveloped in this section.

THEOREM 16. Let R1 be a relational model satisfying R1–R4, andR2 be a relational modelsatisfying R1–R5; then

1. `L1 A iff �R1 A;

2. `L2 A iff �R2 A.

Proof. Let us consider onlyL1. From Theorem 3 we know that�N A→ `L1 A, which isequivalent to saying that6`L1 A→ 6�N A. SinceL1 has the finite model property, there is afinite modelNFIN and a worldw in it such thatw �NFIN ¬A According to Proposition 14 andthe generation of the corresponding relational modelw �R1 ¬A which implies 6�R1 A Then,6`L1 A→ 6�R1 A and so�R1 A ⇐⇒ `L1 A The proof forL2 andR2 is analogous. 2

Here we want to propose a simple interpretation of relational models: the capability of anagent to realize a particular stateA depends on his/her ability to perform some actions in thesituation described by the then actual world. Accordingly each accessibility relation correspondsto a concrete action. In this perspective non-normal worlds are just situations where an agenthas no possibility to perform any action.

9A modal logic is non-iterative iff it can be axiomatized by using only non-iterative axioms. A formula (axiom)A is non-iterative iff for every subformula2iB/3iB of A, B does not contain a modal operator.

13

142

Page 152: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 Discussion

When we consider the semantics developed by Elgesem we have to notice that he uses onlyone selection function to represent the two modal operators instead of the two neighbourhoodfunctions of Section 3. This amounts to say that Elgesem considers agency and ability as twofacets of the same phenomenon –the phenomenon described by the selection function. Thusto discern the two concepts he has to adopt two different valuation clauses. In particular thecondition forE is the condition for a2 operator, while that forC is the condition used for a3 operator. However these conditions, in the context of non-normal modal logic, do not implythat3 is the dual of2. On the contrary the neighbourhood semantics assumes two separate butrelated modal operators.

It has been argued that an agent can carry out an action successfully if she has the ability aswell as the opportunity do to it. Indeed Elgesem studies the relationships between ability andagency, and he correctly realizes that agency implies opportunity, i.e.,EA→OpA, whereOp isthe modal operator for opportunity. But the notion of opportunity is given in terms of agency,i.e.,OpA≡ (E¬A∨A). Therefore we believe that the semantics proposed by Elgesem does notfully capture the idea that agency consists of ability plus opportunity since those three notionsare represented by the same selection function. The other semantics do recognise that abilityalone is not enough to represent agency and that it has to be supplemented by something else.

Finally Elgesem semantics requires the introduction (either implicitly or explicitly) of non-normal worlds, but their interpretation is not satisfactory; on the contrary the interpretation wehave proposed for non-normal worlds seems to fit nicely with the intended reading of the acces-sibility relations for this type of logics.

Acknowledgments

This research was partially supported by the Australia Research Council under Discovery GrantDP0452628 on “Combining modal logics for dynamic and multi-agents systems”.

References

[1] Nuel Belnap and Michael Perloff. Seeing to it that: A canonical form for agentives.Theoria,54:175–99, 1988.

[2] Nuel Belnap and Michael Perloff. The way of the agent.Studia Logica, 51:463–484, 1992.[3] Jose Carmo and Olga Pacheco. Deontic and action logics for organized collective agency modeled

through institutionalized agents and roles.Fundamenta Informaticae, 48:129–163, 2001.[4] Brian Chellas.The Logical Form of Imperatives. Perry Lane Press, Palo Alto, 1969.[5] Brian Chellas.Modal Logic: An Introduction. Cambridge University Press, Cambridge, 1980.[6] Dag Elgesem.Action Theory and Modal Logic. Phd, Institut for filosofi, Det hisotisk-filosofiske

fakultetet, Universitetet i Oslo, 1993.[7] Dag Elgesem. The modal logic of agency.Nordic Journal of Philosphical Logic, 2(2):1–46, 1997.[8] Olivier Gasquet and Andreas Herzig. From classical to normal modal logic. In Heinrich Wansing,

editor,Proof Theory of Modal Logic, pages 293–311. Kluwer, Dordrecht, 1996.

14

143

Page 153: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[9] Jonathan Gelati, Guido Governatori, Antonino Rotolo, and Giovanni Sartor. Declarative power,representation, and mandate: A formal anaysis. In Trevor Bench-Capon, Aspassia Daskalopulu, andRadboudb Winkels, editors,Legal Knowledge and Information Systems, pages 41–52. IOS Press,Amsterdam, 2002.

[10] Bengt Hansson and Peter Gardenfors. A guide to intensional semantics. InModality, Moralityand Other Problems of Sense and Nonsense. Essays Dedicated to Soren Hallden, pages 151–167.Gleerup, Lund, 1973.

[11] Risto Hilpinen. On action and agency. In E. Ejerhed and S. Lindstrom, editors,Logic, Action andCognition: Essays in Philosophical Logic, pages 3–27. Kluwer Academic Publishers, Dordrecht,1997.

[12] Jaakko Hintikka. Impossible possible worlds vindicated.Journal of Philosophical Logic, 4:475–484, 1975.

[13] John F. Horty and Nuel Belnap. The deliberative stit. a study of action, omission, ability, andobligation.Journal of Philosophical Logic, 24:583–644, 1995.

[14] Andrew J. I. Jones and Marek Sergot. A formal characterisation of institutionalised power.Journalof IGPL, 3:427–443, 1996.

[15] Andrew J.I. Jones. A logical framework. In Jeremy Pitt, editor,Open Agent Societies: NormativeSpecifications in Multi-Agent Systems, chapter 3. John Wiley and Sons, Chichester, 2003.

[16] David Lewis. Intensional logic without iterative axioms.Journal of Philosophical Logic, 3(4):457–466, 1974.

[17] Ingmar Porn. The Logic of Power. Blackwell, Oxford, 1970.[18] Ingmar Porn. Action Theory and Social Science: Some Formal Models. Reidel, Dordrecht, 1977.[19] Filipe Santos and Jose Carmo. Indirect action. influence and responsibility. In Mark Brown and

Jose Carmo, editors,Deontic Logic, Agency and Normative Systems. Springer, Berlin, 1996.[20] Filipe Santos, Andrew J.I. Jones, and Jose Carmo. Action concepts for describing organised inter-

action. InThirtieth Annual Hawaii International Conference on System Sciences. IEEE ComputerSociety Press, Los Alamitos, 1997.

[21] Krister Segerberg. Bringing it about.Journal of Philosophical Logic, 18:327–347, 1989.[22] Krister Segerberg. Getting started: Beginnings in the logic of action.Studia Logica, 51:347–358,

1992.[23] Marek Sergot. A computational theory of normative positions.ACM Transactions on Computational

Logic, 2(581–622), 2001.[24] Marek Sergot and Fiona Richards. On the representation of action and agency in the theory of

normative positions.Fundamenta Informaticae, 48:273–293, 2001.[25] Timothy J. Surendonk. Canonicity for intensional logics without iterative axioms.Journal of Philo-

sophical Logic, 26(4):391–409, 1997.

15

144

Page 154: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A Two-Sorted Hybrid Logic with Guarded Jumps

Bernhard HeinemannFachbereich Informatik, FernUniversitat in Hagen

58084 Hagen, GermanyPhone: + 49 2331 987 2714, Fax: + 49 2331 987 319E-mail: [email protected]

Abstract

In this paper we present a two-sorted hybrid logic where formulasare interpreted in spaces of sets. I.e., the logical language we considercontains names for both points and sets. Usual nominals are thenrealized as pairs consisting of one name from each category. Accom-panying satisfaction operators are mimicked with the aid of the globalmodality. And we have a family of additional hybrid operators asso-ciated with names of sets in a natural way. These connectives makejumps possible that are ‘guarded’ in a sense, which yields in fact moreexpressiveness than admitting only unrestricted jumps (by means ofthe global modality) and the strictly controlled ones from basic hybridlogic. Below we take the first steps towards a hybrid logic comprisingthe features just indicated, by proving both a corresponding complete-ness and decidability result. Moreover, we discuss an application ofthe new system to reasoning about knowledge.KEYWORDS: sorted hybrid logic, completeness, decidability, modallogic of set spaces, reasoning about knowledge

1 Introduction

Recently, hybrid logic was recognized a useful tool for extending ‘faithfully’the expressive means of usual modal logic; cf [1] for a summary regardingthis. Thus the reason for hybridizing a modal language is obvious from thedemands for more expressive power in concrete applications. But doing so,it is less clear how to proceed. Already on the elementary stage of definingappropriate satisfaction operators there are several possibilities, dependingon the source language. In the present paper, we want to illustrate this bymeans of a sample language we are particularly interested in because it isrelated to reasoning about knowledge. (As to the latter notion, cf [6] or[12].)

In [13] (and more detailedly in [5]), a bi-modal system called topologicis developed. The operators K and � of this system represent knowledge

1

145

Page 155: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

and effort (in the course of time), respectively. Formulas are interpretedin set spaces (X,O, V ) consisting of a non-empty set X of states, a setO of distinguished subsets of X , which can be taken as knowledge statesof an agent, and a valuation V determining the states where the atomicpropositions are true. The operator K quantifies then across any knowledgestate, whereas � quantifies ‘downward’ across O. That is, descending insideO with respect to the set inclusion relation means gaining knowledge, andjust this is modelled by �.

It is less noticed that evolving knowledge has also a spatial componentbesides the obvious temporal one. In fact, knowledge can be viewed ascloseness and knowledge acquisition as getting closer proximity in set spaces.Thus topology enters the context of knowledge (justifying the name ‘topo-logic’).

Unfortunately, it turned out that several topologically interesting classesof set spaces cannot be dealt with in the framework of topologic. However,hybridizing the underlying language can help a lot here. Already the naıve,‘unsorted’ hybrid logic of set spaces allows us to characterize neatly theclasses arising from linear flows of time; cf [8] and [10].1 And following thesorting strategy from [1] makes the hybrid approach much more flexible, evenwithout having any satisfaction operators at one’s disposal; cf [9].

The hybrid logic considered in this paper is a development of the onefrom [9]. Satisfaction operators are now integrated. Actually, we have twokinds of such operators: usual ones which can be realized with the aid ofthe global modality, and new ‘guarded jump’–operators coming along withnames of sets. For technical reasons we treat here both types of hybridconnectives simultaneously. But in general, the intended application willtell us which language to prefer. The ‘guarded jump’–operators seem tobe particularly suited to a certain class of intersection closed set spaces werevisit repeatedly in this paper; cf [14].

In the next section we introduce the new hybrid language of set spaces.Afterwards we present an axiomatization and touch on the question of com-pleteness. Finally, we argue that the new logic is decidable.

Throughout this paper, most proofs are only sketched. The details willappear in the final version.

2 The language

In this section we first define precisely the syntax and semantics of the two-sorted hybrid language for set spaces indicated above. Then, we give anexample proving the expressive power of this language.

We extend the basic bi-modal language of set spaces by two sets of nom-inals on the one hand and a family of unary modalities on the other hand.

1We let nominals denote neighbourhood situations there; see Definition 2.1 below.

2

146

Page 156: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The denotation of every nominal is either a unique state or a distinguishedset of states, whereas the new modalities represent the global one2 and the‘guarded jump’–operators belonging to set names, respectively.

Let PROP = {p, q, . . .} be a denumerable set of symbols called proposi-tion letters. Moreover, let Nstat = {i, j, . . .} and Nsets = {A,B, . . .} be twofurther sets of symbols called names of states and names of sets, respectively.We assume that the sets PROP, Nstat and Nsets are mutually disjoint. Thenwe define the set WFF of well-formed formulas over PROP ∪ Nstat ∪ Nsets

by the rule

α ::= p | i | A | ¬α | α ∧ β | Kα | �α | Aα | [εA]α.

The modality [εA] is to be read ‘in (the set denoted by) A (at the actualstate)’. The missing boolean connectives �,⊥,∨,→,↔ are treated as ab-breviations, as needed. The duals of K, �, A and [εA] are denoted L, �, Eand 〈εA〉, respectively.

We give next meaning to formulas. To begin with, we define the domainswhere formulas are to be interpreted in. We let P(X) designate the powersetof a given set X.

Definition 2.1 (Set spaces with names) 1. A pair (X,O) consistingof a non-empty set X and a set O ⊆ P(X) of subsets of X such thatX ∈ O is called a set frame.

2. Let S := (X,O) be a set frame. The set of neighbourhood situationsof S is the set NS := {x, U | x ∈ U and U ∈ O}.

3. Let S be a set frame as above. An S–valuation is a mapping

V : PROP ∪ Nstat ∪ Nsets −→ P(X)

such that

(a) V (i) is either ∅ or a singleton subset of X for every i ∈ Nstat,and

(b) V (A) ∈ O for every A ∈ Nsets .

4. A set space with names (or, in short, an SSN) is a triple (X,O, V ),where S = (X,O) is a set frame and V an S–valuation. One saysthen that M := (X,O, V ) is based on S.

Due to an obvious Coincidence Lemma, the requirement ‘X ∈ O’ fromitem 1 of the above definition does not result in any loss of generality. How-ever, this condition facilitates definitely the proof of Theorem 3.2 below.

2The main reason for integrating the global modality into our system is to make asmooth proof of Theorem 3.2 possible.

3

147

Page 157: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For a given SSN M we define now the relation of satisfaction, |=M ,between neighbourhood situations of the underlying frame and formulas inWFF. (The obvious clauses for negation and conjunction are omitted.)

Definition 2.2 (Satisfaction and validity) Let M = (X,O, V ) be anSSN based on the set frame S = (X,O), and let x, U a neighbourhood situ-ation of S. Then

x, U |=M p : ⇐⇒ x ∈ V (p)x, U |=M i : ⇐⇒ x ∈ V (i)x, U |=M A : ⇐⇒ V (A) = U

x, U |=M Kα : ⇐⇒ y, U |=M α for all y ∈ U

x, U |=M �α : ⇐⇒ ∀U ′ ∈ O : (x ∈ U ′ ⊆ U ⇒ x, U ′ |=M α)x, U |=M Aα : ⇐⇒ y, U ′ |=M α for all y, U ′ ∈ NSx, U |=M [εA]α : ⇐⇒ if x ∈ V (A), then x, V (A) |=M α,

for all p ∈ PROP, i ∈ Nstat , A ∈ Nsets , and α ∈ WFF. In case x, U |=M αis true we say that α holds in M at the neighbourhood situation x, U.

A formula α is called valid in M (‘M |= α’) iff it holds in M at everyneighbourhood situation of S.

Note that the meaning of both proposition letters and names of statesis independent of neighbourhoods, thus stable with respect to �. This factis reflected in two special axioms later on.

The formulas of the form i ∧ A, where i ∈ Nstat and A ∈ Nsets , can betaken as names for elements of NS . The satisfaction operator associatedwith such a name reads then E(i ∧ A ∧ . . .). I.e., pairs (i, A) can act like‘proper’ nominals in set spaces with names; cf [8].

Note that the last clause of Definition 2.2 explains what a guarded jumpmeans for set spaces: we are allowed to jump from the neighbourhood U tothe neighbourhood V (A) for evaluating α there, provided that the ‘guard’x, which is not affected by the jump, is also contained in the latter set.

The reader might wonder why a ‘guarded jump’–operator belonging toa name i ∈ Nstat does not appear here. Such an operator is missing for thesimple reason that it can be defined, by K(i→ . . .).

Now, the question comes up naturally whether [εA] can be defined aswell. But this is not the case for the following reason. One can ‘leave’ theactual neighbourhood U with the aid of the connective [εA] in a special,controlled way. However, leaving U is not at all possible by means of K or�, and not in that special way by means of E. Thus we have really got moreexpressiveness.

The example covered by the next definition and proposition points to animportant application of the new language.

4

148

Page 158: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Definition 2.3 (Directed frames) A set frame S = (X,O) is called di-rected, iff

∀U1, U2 ∈ O : (U1 ∩ U2 �= ∅ ⇒ ∃U ∈ O : ∅ �= U ⊆ U1 ∩ U2) .

An SSN M is called directed, iff it is based on a directed frame.

Proposition 2.4 (Expressing directedness) Let S = (X,O) be a setframe. Then S is directed, iff for all SSNs M based on S we have that

M |= �A∧�B → L�K (〈εA〉� ∧ 〈εB〉�) for all A,B ∈ Nsets .

Proof. We prove only that the condition is sufficient for directedness. So,let S be not directed. Then there are U1, U2 ∈ O such that U1 ∩ U2 �=∅, but U = ∅ or U �⊆ U1 ∩ U2 for all U ∈ O. Take any S–valuation Vsatisfying V (A) = U1 and V (B) = U2. Let M := (X,O, V ) and x ∈U1 ∩ U2. Then we obtain that x,X |=M �A ∧ �B. However, we havethat x,X |=M K�L ([εA]⊥∨ [εB ]⊥) because we can find a point witnessingU �⊆ V (A) ∩ V (B) in every non-empty set U ∈ O. In this way we havefound an SSN M based on S and an instance of the above formula schemawhich is not valid in M, as desired. �

Directed frames correspond to the topological notion of filter base; cf [4],Ch. I, § 6.3. Filters are crucial to the general idea of convergence; cf [4],Ch. I, § 7. As set spaces represent a model for the development of knowl-edge, directed spaces are, therefore, associated with ‘converging’ knowledgeacquisition procedures. The modal logic of knowledge arising from that wasstudied in the paper [14]. We will come back to this system later on, at theend of the next section (and in Section 4 as well).3

Concluding this section, we comment on the relevance of names andaccompanying hybrid operators to the context of knowledge. We confineourselves to names of sets here since the general usefulness of names ofstates has adequately been demonstrated elsewhere; cf, in particular, [1].As mentioned in the introduction already, the elements of O can be viewedas knowledge states of an agent, for any correspondingly given set frame(X,O). Thus our language supplies us with names of knowledge states.And the hybrid operators [εA] allow to switch over to named knowledgestates as long as the point of evaluation is fixed (A ∈ Nsets ). This additionalmeans of expression is, therefore, perfectly in accordance with the common‘external’ view on knowledge in multi-agent systems, where knowledge is‘ascribed’ to the agents; cf [6], Ch. 4.

3Originally, a good portion of work was devoted to the axiomatizability problem forthe logic of intersection closed spaces; cf [5]. Later on, in [14], it was announced that thislogic and the logic of directed spaces coincide.

5

149

Page 159: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3 Completeness

In the first part of this section we propose a system of axioms which areeasily seen to be sound for set spaces with names. Then we add a couple ofGabbay–style proof rules to the usual ones of modal and hybrid logic towardscompleteness. It turns out that this yields even extended completeness (inthe sense of hybrid logic, cf [2], Sec. 7.3). Note that we can utilize much ofthe completeness proof contained in the paper [9] as far as the satisfactionoperator–free fragment of our logic is concerned.

The axiom schemata are arranged in four groups. First, the usual axiomsfor arbitrary set spaces are listed.

1. All instances of tautologies. 6. (p→ �p) ∧ (�p→ p)2. K(α→ β) → (Kα→ Kβ) 7. � (α→ β) → (�α→ �β)3. Kα→ α 8. �α→ α

4. Kα→ KKα 9. �α→ ��α5. Lα→ KLα 10. K�α→ �Kα,

where p ∈ PROP and α, β ∈ WFF. Note that the schemata 2 – 5 representthe usual S5 axioms of knowledge, 6 captures the aforementioned stabilityof the proposition letters with respect to �, 7 – 9 say that � satisfies allthe S4 laws, and 10 was called the Cross Axiom in [5];4 cf the comments onDefinition 4.1 below. — The second group of axioms concerns names.

11. (i→ �i) ∧ (�i→ i) 14. A (A ∧ Lα→ Lβ) ∨ A (A ∧ Lβ → Lα)12. i ∧ α→ K(i→ α) 15. K(�B → �A) ∧ L�B → � (A→ L�B)13. A→ KA 16. K�A→ A,

where i ∈ Nstat , A, B ∈ Nsets and α, β ∈ WFF. Some comments on theaxioms of this group can be found in the paper [9];5 see also the sketchof the completeness proof below. — Each axiom of the third group dealswith the global modality. In particular, the interplay between A and K� isdescribed therein.

17. A(α→ β) → (Aα→ Aβ) 19. Aα→ AAα 21. Aα→ K�α18. Aα→ α 20. α→ AEα 22. E (Eα→ L�α) ,

where α, β ∈ WFF. Note that Axiom 22 expresses the fact that a set frame(X,O) is generated in a sense. To be more precise, any neighbourhoodsituation of the form x,X (where x ∈ X) can serve as a generating element,due to both the clause ‘X ∈ O’ in item 1 of Definition 2.1 and the semantics

4The latter schema plays a subtle part in multi-agent systems where agents have perfectrecall, i.e., do not forget anything of their own history in the course of time; see [6], notesto Ch. 8, for a more detailed discussion.

5Note the difference between Axiom 14 here and there.

6

150

Page 160: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

of K and �; cf also item 5 of Definition 4.1 below. — By the final group,the ‘guarded jump’–operators are handled.

23. [εA](α→ β) → ([εA]α→ [εA]β) 24. �(A∧ α) → �〈εA〉α 25. [εA]A26. �〈εA〉A ∧ L�A→ �A 27. Aα→ [εA]α,

where A ∈ Nsets and α, β ∈ WFF. As to the properties corresponding toAxioms 24 – 26, cf item 8 of Definition 4.1.

Now we define the desired logical system GJ (indicating guarded jumps).

Definition 3.1 (The logic) Let GJ be the smallest set of formulas con-taining all of the above axiom schemata and closed under application of thefollowing rules:

(modus ponens)α→ β, α

β(∆–necessitation)

α

∆α

(namestat)j → β

β(namesets)

B → β

β

(K–enrichment)E (i ∧ A ∧ L(j ∧ α)) → β

E(i ∧ A ∧ Lα) → β

(�–enrichment)E (i ∧A ∧�(B ∧ α)) → β

E(i ∧ A ∧�α) → β

(A–enrichment)E (i ∧A ∧ E(j ∧ B ∧ α)) → β

E(i ∧ A ∧ Eα) → β

([εA]–enrichment)E (i ∧ A ∧ 〈εA〉(j ∧ α)) → β

E(i ∧A ∧ 〈εA〉α) → β,

where α, β ∈ WFF, i, j ∈ Nstat, A,B, C ∈ Nsets , ∆ ∈ {K,�,A} ∪ {[εA] |A ∈ Nsets}, and j, B are new each time (i.e., do not occur in all the othersyntactic building blocks of the respective rule).

The reader can easily check that GJ is sound with respect to the classof all SSNs.

In order to obtain completeness one proceeds via the canonical model ofGJ. For a start, take a maximal consistent set Γ containing the negationof a given non-derivable formula. Then, extend Γ to a named and enrichedmaximal consistent set Γ′ in the language increased by enough new con-stants. As in usual hybrid logic, this can be done in such a way that everyexistential demand with respect to any of the modalities is realized in asubmodelM′, ‘yielded’ by Γ′, of the ‘big’ canonical model. Note that this isthe place where the new rules (and Axioms 11 – 14 and 25) come into play.

Finally, the remaining axioms of the second group along with Axiom 22ensure that a set space structure is definable on M′. Moreover, we obtainan appropriate Truth Lemma (in which Axioms 24 and 26 are needed for

7

151

Page 161: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

the [εA]–case). All in all, this gives us in fact ‘canonical’ completeness ofour system.6

Theorem 3.2 (Completeness) Every formula α ∈ WFF that is valid inall SSNs is contained in the logic GJ.

And we obtain even more. The fact that we are dealing with namedobjects enables us to extend the completeness proof to other classes of setspaces, by adding simply a suitable defining correspondent. In this way weget, eg:

Theorem 3.3 (Extended completeness) Let D be the formula schemafrom Proposition 2.4. Then the system GJ + D is sound and complete withrespect to the class of all directed SSNs.

A corresponding result holds for quite a lot of different classes of setspaces.

Regarding the issues of the paper [14], Theorem 3.3 is somewhat sur-prising at first glance. It is proved there that the modal theory of directedset spaces is not finitely axiomatizable. As we have just shown it is in thehybrid setting, actually;7 see also the end of the following section.

4 Decidability

Subsequently it is shown that GJ is a decidable set of formulas. In a prepara-tory step we consider a certain class of auxiliary Kripke structures which area bit involved.

Definition 4.1 (GJ–models) A quintupel

M :=(

W,L−→ ,

�−→ ,E−→ , {〈εA〉−→}A∈Nsets , V

)

is called a GJ–model, iff the following conditions are satisfied:

1. W is a non-empty set,

2. the relation L−→ ⊆W ×W (belonging to K) is an equivalence,

3. the relation �−→ ⊆W ×W (belonging to �) is reflexive and transitive,

6It is worth mentioning that Axiom 26 can be strengthened in such a way that we candispense with the rule [εA]–enrichment. However, we get then into difficulties provingthe decidability of the resulting logic.

7One of the referees pointed recent unpublished work by Blackburn and ten Cate out tous, showing that schemata like our above enrichment rules can be viewed as a substitutefor infinite axiomatizations in a sense.

8

152

Page 162: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4. for all u, v, w ∈ W such that u �−→ vL−→ w there exists t ∈ W such

that u L−→ t�−→w,

5. M is generated in the following sense: there is some w0 ∈ W suchthat W = {v | (w0, v) ∈ ( L−→∪ �−→ )∗},

6. the relation E−→ ⊆W ×W (belonging to A) is universal,

7. V : PROP ∪ Nstat ∪ Nsets −→ P(W ) is a mapping such that

(a) for all c ∈ PROP∪Nstat and u, v ∈W satisfying u �−→v, it holdsthat u ∈ V (c) iff v ∈ V (c),

(b) for all i ∈ Nstat , the intersection of V (i) with any L−→–equivalenceclass is either empty or a singleton set,

(c) for all A ∈ Nsets , the set V (A) equals either ∅ or a unique L−→–equivalence class, and

8. for every A ∈ Nsets , the relation〈εA〉−→ (belonging to [εA]) satisfies

(a) ∀u, v ∈W : (u〈εA〉−→v ⇒ v ∈ V (A)),

(b) ∀u ∈ W : if ∃ v ∈ V (A) : u �−→ v, then ∀w ∈ W : (u �−→w ⇒w

〈εA〉−→v),

(c) ∀u ∈ W : if ∃v, w : u �−→ v〈εA〉−→w and ∃x, y : u L−→x

�−→ y andy ∈ V (A), then ∃z ∈ V (A) : u �−→z.

The fourth item above is the ubiquitous cross property, correspondingto Axiom 10 above, from [5]; (a variant of) this property is used decisivelyin completeness proofs for all set space logics. — Item 7 suits the notion ofhybrid valuation from Definition 2.1 to GJ–models. — Finally, certain prop-erties, for which one can easily draw corresponding diagrams, are imposed

on the relation〈εA〉−→ by (b) and (c) of the last item.

Though every SSN induces a GJ–model in a rather natural way (vizby taking neighbourhood situation as points and defining the accessibilityrelations and the valuation accordingly), we cannot directly use Theorem3.2 for our purposes because, unfortunately, not every GJ–model validatesnecessarily all of the above axioms. But, for a start, we get at least:

Proposition 4.2 Apart from, possibly, Axioms 15 and 16 each of the aboveschemata is valid in every GJ–model.

9

153

Page 163: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

We call a GJ–model M faithful, iff all the axioms are valid in M. Wewant to show now that GJ satisfies the strong finite model property (cf [2],Def. 6.6) with respect to the class of all faithful GJ–models. With that thedesired decidability of GJ follows easily, by using [2], Th. 6.7.

To establish this finite model property we use the method of filtration,followed by an appropriate model surgery. So, let α ∈ WFF be a consistentformula for which we want to find a model of size at most f(|α|), where fis some computable function and |α| denotes the length of α. Moreover, letsf(α) be the set of all subformulas of α. We construct a suitable filter set Σvia the following sets of formulas. We first let

Σ0 :=

sf(α) ∪ {�¬A | A ∈ Nsets occurs in α}∪ {[εA]A, [εA]¬A | A ∈ Nsets occurs in α}∪ {�¬(A ∧ ¬β) | [εA]β ∈ sf(α)},

and secondly Σ¬ := {¬β | β ∈ Σ0}. Then we take the set Σ′ of all finiteconjunctions of pairwise distinct elements of Σ0 ∪ Σ¬. Afterwards we closeΣ′ under single applications of the operator L and take, finally, the set of allsubformulas of the resulting set. Let Σ be the union of all these intermediatesets of formulas. Then Σ is subformula closed, and 2c·|α| is an upper boundof the cardinality of Σ (for some constant c).

Let C be the submodel of the canonical model generated by a maximalconsistent set realizing α. (Here is the point where we use a part of thecompleteness proof.) Moreover, let the Kripke model

M :=(

W,L−→ ,

�−→ ,E−→ , {〈εA〉−→}A∈Nsets , V

)

be obtained from C as follows:

• W is the filtration of the carrier set of C with respect to Σ,

• L−→ , �−→ and E−→ are the smallest filtrations, and〈εA〉−→ is the largest

filtration of the accessibility relations of C belonging to the respectivemodalities (where A ∈ Nsets), and

• V is induced by the canonical valuation.

Then, exploiting the structure of the filter set Σ and the fixings for therelation filtrations we get the following crucial proposition.

Proposition 4.3 By modifying some of the relations〈εA〉−→ and the valuation

V suitably, the just defined model M can be turned into a faithful GJ–modelM

′ which is semantically equivalent to M with respect to α.

10

154

Page 164: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Proof. The model M′ differs from M in that the denotation of every ele-ment of PROP ∪ Nstat ∪ Nsets not occurring in α is the empty set and the

corresponding relations〈εA〉−→ are empty as well. — Presently, we focus only

on the relations〈εA〉−→ . We must prove that the conditions from item 8 of

Definition 4.1 are valid forM′. To this end, we may assume that A actuallyoccurs in α, for otherwise the desired properties can easily be verified.

(a) Let u, v ∈ W satisfy u〈εA〉−→ v, and let u, v be any representatives of

u, v, respectively, contained in C. Then we have that C, u |= [εA]β ⇒C, v |= β for all formulas [εA]β ∈ Σ, due to the definition of filtration.Now, [εA]A is in Σ, and C, u |= [εA]A is in fact true because of Axiom25. It follows that C, v |= A. This impliesM′, v |= A, i.e., v ∈ V (A).

(b) Let u, v, u, v be as above, and let v ∈ V (A) and u�−→ v be the case.

Furthermore, take any w ∈ W such that u �−→w. We have to show

that w〈εA〉−→v. For that it suffices to prove that C, w |= [εA]β ⇒ C, v |= β

for all formulas [εA]β ∈ Σ, where w is any representative of w; this isdue to the fact that we are dealing with the largest filtration. So, letC, w |= [εA]β be valid. Then, also M′, w |= [εA]β holds. Since �−→ isa smallest filtration and [εA]β ∈ Σ, there are representatives u, w ofu, w, respectively, such that w is accessible from u with respect to themodality � and �[εA]β ∈ u. With the aid of Axiom 24 we conclude�¬(A ∧ ¬β) ∈ u from that. As this formula is likewise contained inΣ we get that M′, u |= �¬(A ∧ ¬β). Now M′, v |= β follows, whichimplies C, v |= β, as desired.

(c) Let u, v, w, x, y ∈W satisfy u �−→v〈εA〉−→w, u L−→x

�−→y and y ∈ V (A).In order to prove that ∃z ∈ V (A) : u �−→ z, it suffices to show that�A ∈ u for some representative u of u since �A ∈ Σ. We will clearlyuse Axiom 26 for that. First, we get from y ∈ V (A) and �A ∈ Σthat �A ∈ x for all representatives x of x. Second, it is a peculiarityof our filtration that for every representative u of u there is somerepresentative of x which is accessible from u with respect to K.8

Thus we obtain L�A ∈ u for all u. Third, we have thatM′, w |= A, asM

′, v |= [εA]A. Thus even M′, v |= 〈εA〉A is valid. Since this formulais contained in Σ, too, we infer (similarly as above) that �〈εA〉A ∈ u

for some representative u of u. Now Axiom 26 can be applied, yielding�A ∈ u, as desired.

Therefore, α is satisfiable in a finite model of the axioms, of which thesize is in O

(

2c·|α|) (where c ∈ N is some constant). This gives us the first

8This property is proved, eg, in [5], Sec. 2.3.

11

155

Page 165: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

main result of this section.

Theorem 4.4 (Decidability) The logic GJ is decidable.

Now we turn to the system GJ+D. By changing the filter set Σ0 slightlyand performing a similar filtration-and-model-surgery procedure, we obtainthe following theorem.

Theorem 4.5 (Extended decidability) The logic GJ + D is decidable.

For proving this result we have to guarantee that the schema D remainsvalid by passing to the accordingly defined filtrated structure. Again, onlythe nominals occurring in the formula for which we want to construct a finitemodel have to be taken into account in the process.

As an immediate consequence of Theorem 4.5 we get that the modallogic of directed spaces is decidable, too.

Corollary 4.6 The modal logic of directed spaces is decidable.

This result constitutes a striking application of hybrid logic to reasoningabout knowledge. In fact, it solves the decidability problem raised in [14],Sec. 6.

Concluding this section we comment on the complexity of the GJ–satisfiability problem. Unfortunately, we can say only little about this. Theabove decidability proof yields obviously NEXPTIME as an upper bound.On the other hand [3], Theorem 4.5, gives us good reason for suspecting thatEXPTIME is a lower bound. (Note that [εA] is in fact a partially functionalmodality, for every A ∈ Nsets .)

5 Summary, comparison, and outlook for furtheroptions

In the present paper we developed the fundamental matters of a two-sortedhybrid logic, GJ, for set spaces with names. A family of new ‘guardedjump’–operators assigned to names of sets was considered, in particular. Weobtained a completeness as well as a decidability theorem for GJ, and couldextend these results to the logic of directed spaces (which are significant toreasoning ‘topologically’ about knowledge).

It should be remarked that hybrid logics for spaces of sets received littleattention in the literature up to now. Nevertheless, a (usual) hybrid exten-sion of the classical topological semantics of modal logic (going back to [11])was briefly considered in [7] (with regard to expressiveness).

Which other classes of set spaces should be studied from the point ofview of hybrid logic? — Concluding the paper, we touch on the class of

12

156

Page 166: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

frames (X,O) where O is closed under complementation. This class is ofsome interest to various applications.

Addressing the complement of the actual neighbourhood at a situationx, U means, in particluar, jumping completely outside U . Thus a ‘comple-mentation operator’ appears, which goes beyond all the previous abilities.Now, it is not too hard to write down some formulas describing certainproperties of such an operator and its interaction with K and �. In orderto obtain nice meta-results, however, it seems more promising to connectcomplementation operators with names. In a future research project we willdevelop a system like that.

References

[1] Patrick Blackburn. Representation, reasoning, and relational structures: a hybridlogic manifesto. Logic Journal of the IGPL, 8:339–365, 2000.

[2] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal Logic, volume 53of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press,Cambridge, 2001.

[3] Patrick Blackburn and Edith Spaan. A modal perspective on the computationalcomplexity of attribute value grammar. Journal of Logic, Language and Information,2(2):129–169, 1993.

[4] Nicolas Bourbaki. General Topology, Part 1. Hermann, Paris, 1966.

[5] Andrew Dabrowski, Lawrence S. Moss, and Rohit Parikh. Topological reasoning andthe logic of knowledge. Annals of Pure and Applied Logic, 78:73–110, 1996.

[6] Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi. Reasoningabout Knowledge. MIT Press, Cambridge, MA, 1995.

[7] David Gabelaia. Modal definability in topology. Master’s thesis, ILLC, Universiteitvan Amsterdam, 2001.

[8] Bernhard Heinemann. Axiomatizing modal theories of subset spaces (an example ofthe power of hybrid logic). In HyLo@LICS, Proceedings, pages 69–83, Copenhagen,Denmark, July 2002.

[9] Bernhard Heinemann. Extended canonicity of certain topological properties of setspaces. In M. Vardi and A. Voronkov, editors, Logic for Programming, ArtificialIntelligence, and Reasoning, volume 2850 of Lecture Notes in Artificial Intelligence,pages 135–149, Berlin, 2003. Springer.

[10] Bernhard Heinemann. The hybrid logic of linear set spaces. Logic Journal of theIGPL, 2004. To appear.

[11] J. C. C. McKinsey. A solution to the decision problem for the Lewis systems S2 andS4, with an application to topology. Journal of Symbolic Logic, 6(3):117–141, 1941.

[12] J.-J. Ch. Meyer and W. van der Hoek. Epistemic Logic for AI and Computer Sci-ence, volume 41 of Cambridge Tracts in Theoretical Computer Science. CambridgeUniversity Press, Cambridge, 1995.

[13] Lawrence S. Moss and Rohit Parikh. Topological reasoning and the logic of knowl-edge. In Y. Moses, editor, Theoretical Aspects of Reasoning about Knowledge (TARK1992), pages 95–105, San Francisco, CA, 1992. Morgan Kaufmann.

[14] M. Angela Weiss and Rohit Parikh. Completeness of certain bimodal logics for subsetspaces. Studia Logica, 71:1–30, 2002.

13

157

Page 167: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

On modularity of theories

Andreas Herzig Ivan Varzinczak∗

Institut de Recherche en Informatique de Toulouse (IRIT)118 route de Narbonne

F-31062 Toulouse Cedex 4 (France)e-mail: {herzig,ivan}@irit.fr

http://www.irit.fr/recherches/LILAC

Keywords: modularity, interpolation

1 Introduction

In many cases knowledge is represented by logical theories containing multi-ple modalities α1, α2, . . . Then it is often the case that we have modularity,in the sense that our theory T can be partitioned in a union of theories

T = T ∅ ∪ T α1 ∪ T α2 ∪ . . .such that

• T ∅ contains no modal operators, and

• the only modality of T αi is αi.

We call these subtheories modules. Examples of such theories can be foundin reasoning about actions, where each T αi contains descriptions of theatomic action αi in terms of preconditions and effects, and T ∅ is the setof static laws (alias domain constraints, alias integrity constraints), i.e. thoseformulas that hold in every possible state of a dynamic system. For ex-ample, T marry = {¬married → 〈marry〉>, [marry]married}, and T ∅ =

∗Supported by a fellowship from the government of the Federative Republic of Brazil.Grant: CAPES BEX 1389/01-7.

1

158

Page 168: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

{¬(married ∧ bachelor)}. Another example is when mental attitudes suchas knowledge, beliefs or goals of several independent agents are represented:then each T αi contains the respective mental attitude of agent αi.

1

Let the underlying multimodal logic be independently axiomatized (i.e.the logic is a fusion and there is no interaction between the modal opera-tors), and suppose we want to know whether T |= ϕ, i.e. whether a formulaϕ follows from the theory T . Then it is natural to expect that we only haveto consider those elements of T which concern the modal operators occur-ring in ϕ. For instance the proof of some consequences of action α1 shouldnot involve laws for other actions α2; querying the belief base of agent α1

should not require bothering with that of agent α2. Moreover, intensionalinformation in any T αi should not influence information about the laws ofthe world encoded in T ∅.

Similar modular design principles can be found in structural and object-oriented programming. We have advocated and investigated the case of rea-soning about actions in [3].

2 Preliminaries

Let MOD = {α1, α2, . . .} be the set of modalities. Formulas are constructedin the standard way from these and the set of atomic formulas ATM. Theyare denoted by ϕ, ψ, . . .. Formulas without modal operators (propositionalformulas) are denoted by PFOR = {A,B,C, . . .}.

Let mod(ϕ) return the set of modalities occurring in formula ϕ, and letmod(T ) =

⋃ψ∈T mod(ψ). For instance mod([α1](p → [α2]q)) = {α1, α2}. If

M ⊆ MOD is a set of modalities then we define

T M = {ϕ ∈ T : mod(ϕ) ∩M 6= ∅}

Hence T ∅ is a set of formulas without modal operators. Another example is

T {marry,divorce} =

{ ¬married→ 〈marry〉>, [marry]married,married→ 〈divorce〉>, [divorce]¬married

}

We write T α instead of T {α}.1Here we should assume more generally that [αi] is the only outermost modal operator

of T αi ; we think that this case could be analyzed in a way that is similar to ours. Thingsget just more complicated.

2

159

Page 169: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

We suppose from now on that T is partitioned, in the sense that {T ∅} ∪{T αi : αi ∈ MOD} is a partition of T . We thus exclude T αi containing morethan one modal operator.

Models of the logic under concern are of the form M = 〈W,R, V 〉, whereW is a set of possible worlds, R : MOD −→W×W associates an accessibilityrelation to every modality, and V : W −→ 2ATM associates a valuation toevery possible world.

Satisfaction of a formula ϕ in world w of model M (M,w |= ϕ) and truthof a formula ϕ in M (noted M |= ϕ) are defined as usual. Truth of a setof formulas T in M (noted M |= T ) is defined by: M |= T iff M |= ψ forevery ψ ∈ T . T has global consequence ϕ (noted T |= ϕ) iff M |= T impliesM |= ϕ.

We suppose that the logic under concern is compact.

3 Modularity

Under the hypothesis that {T ∅} ∪ {T αi : αi ∈ MOD} partitions T , we areinterested in the following principle of modularity:

Definition 3.1 A theory T is modular if for every formula ϕ,

T |= ϕ implies T mod(ϕ) |= ϕ

Modularity means that when investigating whether ϕ is a consequence of T ,the only formulas of T that are relevant are those whose modal operatorsoccur in ϕ.

This is reminiscent of interpolation, which more or less2 says:

Definition 3.2 A theory T has the interpolation property if for every for-mula ϕ, if T |= ϕ then there is a theory Tϕ such that

• mod(Tϕ) ⊆ mod(T ) ∩mod(ϕ)

• T |= ψ for every ψ ∈ Tϕ• Tϕ |= ϕ

2We here present a version in terms of global consequence, as opposed to local conse-quence or material implication versions that can be found in the literature [4, 5]. We wereunable to find such global versions in the literature.

3

160

Page 170: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Our definition of modularity is a strengthening of interpolation because itrequires Tϕ to be a subset of T .

Contrarily to interpolation, modularity does not generally hold. For ex-ample, let

T = {p ∨ [α]⊥, p ∨ ¬[α]⊥}Then T ∅ = ∅, and T α = T . Now T |= p, but clearly T ∅ 6|= p.

Being modular is a useful feature of theories: beyond being a reasonableprinciple of design that helps avoiding mistakes, it clearly restricts the searchspace, and thus makes reasoning easier. To witness, if T is modular then con-sistency of T amounts to consistency (in classical logic) of the propositionalpart T ∅.

4 Propositional modularity

How can we know whether a given theory T is modular? The followingcriterion is simpler:

Definition 4.1 A theory T is propositionally modular if for every proposi-tional formula A,

T |= A implies T ∅ |= A

And it will suffice to guarantee modularity:

Theorem 4.1 Let T be a partitioned theory. If T is propositionally mod-ular then T is modular.

Proof: Let T be propositionally modular. Suppose T mod(ϕ) 6|= ϕ. Hencethere is a model M = 〈W,R, V 〉 such that M |= T mod(ϕ), and there is somew in M such that M,w 6|= ϕ. We prove that T 6|= ϕ by constructing from Ma model M ′ such that M ′ |= T and M ′, w 6|= ϕ.

First, as we have supposed that our logic is compact, propositional mod-ularity implies that for every propositional valuation val ⊆ 2ATM which is amodel of T ∅ there is a possible worlds model Mval = 〈Wval, Rval, Vval〉 suchthat Mval |= T , and there is some w in Mval such that Vval(w) = val. In otherwords, for every propositional model of T ∅ there is a model of T containingthat propositional model.

4

161

Page 171: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Second, taking the disjoint union of all these models we obtain a ‘bigmodel’ Mbig such that Mbig |= T , and for every propositional model val ⊆2ATM of T ∅ there is a possible world w in Mbig such that V (w) = val.

Now we can use the big model to adjust those accessibility relations R(α)of M whose α does not appear in ϕ, in a way such that the resulting modelsatisfies the rest of the theory T \ T mod(ϕ): let M ′ = 〈W ′, R′, V ′〉 such that

• W ′ = {uv : u ∈ W, v ∈ Wbig, and V (u) = Vbig(v)}• if α ∈ mod(ϕ) then uvR

′(α)u′v′ iff uRu′

• if α 6∈ mod(ϕ) then uvR′(α)u′v′ iff vRv′

• V ′(uv) = V (u) = Vbig(v)

W ′ is nonempty becauseM ′ is ‘big enough’ and contains every possible propo-sitional model of T ∅. Then for the sublanguage constructed from mod(ϕ) itcan be proved by structural induction that for every formula ψ of the sub-language and every u ∈ W and v ∈ Wbig, M,u |= ψ iff M ′, uv |= ψ. Thesame can be proved for the sublanguage constructed from MOD \ mod(ϕ).As T ∅ and each of our modules T α are in at least one of these sublanguages(in both sublanguages in the case of T ∅), we have thus proved that M ′ |= T ,and M ′, wv 6|= ϕ for every v.

5 Action theories

In the rest of the paper we investigate how it can be automatically checkedwhether a given theory T is modular or not. We do this for a particular kindof theories that are commonly used in reasoning about actions. For suchtheories we also show how the parts of the theory that are responsible for theviolation of modularity can be identified. First of all we say what an actiontheory is.

Every action theory contains a representation of action effects. We calleffect laws formulas relating an action to its effects. Executability laws in turnstipulate the context where an action is guaranteed to be executable. Finally,static laws are formulas that do not mention actions and express constraintsthat must hold in every possible state. These are our four ingredients thatwe introduce more formally in the sequel.

5

162

Page 172: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Static laws Frameworks which allow for indirect effects make use of logicalformulas that link invariant propositions about the world. Such formulascharacterize the set of possible states. They do not refer to actions, and wesuppose they are formulas of classical propositional logic A,B, . . . ∈ PFOR.

A static law 3 is a formula A ∈ PFOR that is consistent. An example isWalking→ Alive, saying that if a turkey is walking, then it must be alive [10].

Effect laws Here MOD is the set of all actions. To speak about actioneffects we use the syntax of propositional dynamic logic (PDL) [2]. Theformula [α]A expresses that A is true after every possible execution of α.

An effect law4 for α is of the form A→ [α]C, where A,C ∈ PFOR. Theconsequent C is the effect which obtains when α is executed in a state wherethe antecedent A holds. An example is Loaded→ [shoot]¬Alive, saying thatwhenever the gun is loaded, after shooting the turkey is dead. Another one is[tease]Walking: in every circumstance, the result of teasing is that the turkeystarts walking.

A particular case of effect laws are inexecutability laws of the form A →[α]⊥. For example ¬HasGun → [shoot]⊥ expresses that shoot cannot beexecuted if the agent has no gun.

Executability laws With only static and effect laws one cannot guaranteethat shoot is executable if the agent has a gun. 5 In dynamic logic the dual〈α〉A, defined as ¬[α]¬A, can be used to express executability. 〈α〉> thusreads “the execution of action α is possible”.

An executability law6 for α is of the form A → 〈α〉>, where A ∈ PFOR.

3Static laws are often called domain constraints, but the different laws for actions thatwe shall introduce in the sequel could in principle also be called like that.

4Effect laws are often called action laws, but we prefer not to use that term here becauseit would also apply to executability laws that are to be introduced in the sequel.

5Some authors [9, 1, 7, 10] more or less tacitly consider that executability laws shouldnot be made explicit, but rather inferred by the reasoning mechanism. Others [6, 11] haveexecutability laws as first class objects one can reason about. It seems a matter of debatewhether one can always do without, but we think that in several domains one wants toexplicitly state under which conditions a given action is guaranteed to be executable, e.g.that a robot should never get stuck and should always be able to execute a move action.In any case, allowing for executability laws gives us more flexibility and expressive power.

6Some approaches (most prominently Reiter’s) use biconditionals A ↔ 〈α〉>, calledprecondition axioms. This is equivalent to ¬A ↔ [α]⊥, which illustrates that they thusmerge information about inexecutability with information about executability.

6

163

Page 173: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For instance HasGun→ 〈shoot〉> says that shooting can be executed when-ever the agent has a gun, and 〈tease〉> says that the turkey can always beteased.

Action theories S ⊆ PFOR denotes the set of all static laws of a domain.For a given action α ∈ MOD, Eα is the set of its effect laws, and Xα is the setof its executability laws. We define E =

⋃α∈MOD Eα, and X =

⋃α∈MODXα.

An action theory is a tuple of the form 〈S, E ,X〉. We suppose that S, E andX are finite.

6 Checking modularity of action theories

How can we check whether a given action theory T = 〈S, E ,X〉 is modular?Assuming T is finite, the algorithm below does the job:

Algorithm 6.1 (Modularity check)

input: S, E ,Xoutput: a set of implicit static laws SISI:= ∅for all α do

for all J ⊆ Eα doAJ:=

∧{Ai : Ai → [α]Ci ∈ J}CJ:=

∧{Ci : Ai → [α]Ci ∈ J}if S ∪ {AJ} 6` ⊥ and S ∪ {CJ} ` ⊥ then

for all B → 〈α〉> ∈ X doif AJ ∧B 6` ⊥ thenSI:= SI ∪ {¬(AJ ∧B)}

Theorem 6.1 An action theory 〈S, E ,X〉 is modular iff SI = ∅.

The proof of this theorem relies on a sort of interpolation theorem formultimodal logic, which basically says that if Φ |= Ψ and Φ and Ψ haveno action symbol in common, then there is a classical formula A such thatΦ |= A and A |= Ψ.7

7The detailed proof can be found in http://www.irit.fr/ACTIVITES/LILaC/Pers/Herzig/P/AiML04.html

7

164

Page 174: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Remark 6.1 In [3] a monotonic solution to the frame problem has been in-tegrated in such an algorithm. This makes the algorithm a bit more complexas it involves computing of prime implicates. For the sake of simplicity thishas not been done here.

7 Discussion and conclusion

In the perspective of independently axiomatized multimodal logics we haveinvestigated several criteria of modularity for simple theories. We havedemonstrated the usefulness of modularity in reasoning about actions, wherewe have given an algorithmic checking for modularity of a given action theory.

We can have our criterion of modularity refined by taking into account po-larity. Let mod±(ϕ) be the set of modalities of MOD occurring in ϕ togetherwith their polarity. For instance mod±([α1]([α2]p → q)) = {+α1,−α2}.mod±(T ) is defined accordingly. If M is a set of modalities with polaritythen we define: T M = {ϕ ∈ T : mod±(ϕ) ∩M 6= ∅}.

Definition 7.1 A theory T is ±-modular if for every formula ϕ,

T |= ϕ implies T mod±(ϕ) |= ϕ

There are other theories that are modular but not ±-modular, e.g.

T = {¬[α]p, [α]p ∨ [α]¬p}

Indeed, T |= [α]¬p, but T +α 6|= [α]¬p.For the restricted case of action theories this has been proved in [3].

References

[1] P. Doherty, W. Lukaszewicz, and A. Sza las. Explaining explanation clo-sure. In Proc. Int. Symposium on Methodologies for Intelligent Systems,Zakopane, Poland, 1996.

[2] D. Harel. Dynamic logic. In D. M. Gabbay and F. Gunthner, editors,Handbook of Philosophical Logic, volume II, pages 497–604. D. Reidel,Dordrecht, 1984.

8

165

Page 175: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[3] A. Herzig and I. Varzinczak. Domain descriptions should be modu-lar. In L. Saitta, editor, Proc. Eur. Conf. on Artificial Intelligence(ECAI’2004), Valencia, Spain, aug. 2004.

[4] M. Kracht and F. Wolter. Properties of independently axiomatizablebimodal logics. J. of Symbolic Logic, 56(4):1469–1485, 1991.

[5] M. Kracht and F. Wolter. Simulation and transfer results in modal logic:A survey. Studia Logica, 59:149–177, 1997.

[6] F. Lin. Embracing causality in specifying the indirect effects of actions.In Mellish [8], pages 1985–1991.

[7] N. McCain and H. Turner. A causal theory of ramifications and quali-fications. In Mellish [8], pages 1978–1984.

[8] C. Mellish, editor. Proc. 14th Int. Joint Conf. on Artificial Intelligence(IJCAI’95), Montreal, 1995. Morgan Kaufmann Publishers.

[9] L. K. Schubert. Monotonic solution of the frame problem in the situationcalculus: an efficient method for worlds with fully specified actions. InH. E. Kyberg, R. P. Loui, and G. N. Carlson, editors, Knowledge Rep-resentation and Defeasible Reasoning, pages 23–67. Kluwer AcademicPublishers, 1990.

[10] M. Thielscher. Computing ramifications by postprocessing. In Mellish[8], pages 1994–2000.

[11] D. Zhang and N. Y. Foo. EPDL: A logic for causal reasoning. InB. Nebel, editor, Proc. 17th Int. Joint Conf. on Artificial Intelligence(IJCAI’01), pages 131–138, Seattle, 2001. Morgan Kaufmann Publish-ers.

9

166

Page 176: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECTRECALL

TAPANI HYTTINEN AND TERO TULENHEIMO

Department of Mathematics and Statistics Department of PhilosophyP.O. Box 68 P.O. Box 9University of Helsinki University of HelsinkiFIN-00014, Finland FIN-00014, [email protected] [email protected]

Abstract. IF Modal Logic of Perfect Recall is obtained from basicmodal logic by allowing modal operators to be logically independent ofsyntactically superordinate dual operators. It is proven that the satisfi-ability and validity problems of this logic are decidable.

Keywords: Decidability, informational independence, modal logic, IF logic

1. Plan of the Paper

The aim of this paper is to establish decidability of the satisfiability andvalidity problems of the logic (to be called ‘IF Modal Logic of Perfect Recall’)obtained from basic modal logic by allowing modal operators to be logicallyindependent of syntactically superordinate dual operators.

In Section 2, IF Modal Logic of Perfect Recall (or, IFML+) is defined indetail. It is an extension of basic modal logic, allowing diamonds (boxes) tobe logically independent of syntactically superordinate boxes (diamonds).Truth and falsity of formulas of this logic are defined in terms of evalua-tion games. In game-theoretical terminology, these evaluation games aresimultaneously games of imperfect information, and games of perfect recall.The former feature of the games is used for implementing the possibility ofmodal operators to be logically independent of syntactically superordinatemodal operators. The latter feature follows from the syntactic restrictionthat modal operators may only be marked as independent of their duals.IFML+ is a variant of the IF modal logic (IFML) introduced [13]. It isobserved that all results proven in [13] and [14] about the expressive powerof IFML are true also of IFML+.

In Section 3 winning strategies of a player are shown to have history-freenormal form: if a player has a w.s., the player has a w.s. whose value on ahistory only depends on the last known position in the history.

In Section 4 it is shown that IFML+ has the strong finite model property,whereby its satisfiablity problem is decidable. It is observed in Section 5 thatfor IFML+, the validity problem is not the dual of the satisfiablity problem.However, the validity problem of IFML+ is proven to be decidable as well.

167

Page 177: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 TAPANI HYTTINEN AND TERO TULENHEIMO

Section 6 concludes the paper by pointing out related works, discussingthe motivation behind IF modal logics, and mentioning open problems.

2. IF Modal Logic of Perfect Recall

2.1. Syntax. We begin by defining a class L of strings; IFML+ is thenintroduced as a class of strings from L meeting certain additional conditions.The class L of strings is defined recursively as follows. Every string ϕ in Lis simultaneously associated with its set of indices, |ϕ|.

• If p ∈ prop, then p,¬p ∈ L; and |p| = |¬p| = {0}.

• If ϕ,ψ ∈ L and n = max(|ϕ| ∪ |ψ|) + 1, then ϕ ∨n ψ ∈ L andϕ ∧n ψ ∈ L; and |ϕ ∨n ψ| = |ϕ ∧n ψ| = {n} ∪ |ϕ| ∪ |ψ|.

• If ϕ ∈ L, n = max|ϕ| + 1, i < k, and Wn ⊆ ]n,∞[, then(♦i,n/Wn)ϕ ∈ L and (�i,n/Wn)ϕ ∈ L; and |(♦i,n/Wn)ϕ| =|(�i,n/Wn)ϕ| = {n} ∪ |ϕ|.

For every string ϕ in L there corresponds in an obvious way a syntactictree, whose nodes consist of modal operators, propositional connectives and(negated) propositional atoms. For instance, the syntactic tree of the string�4(�1p ∧3 ((¬q ∧1 r) ∨2 (♦1/{4})s)) is the one depicted in Figure 1:

�4

∧3

��

∨2

��???

????

�1

������

��

p��

∧1

������

���

♦1/{4}��?

????

?

r��?

????

??

¬q����

����

s��

∨2

∧1

������

���

♦1/{4}��?

????

?

r��?

????

??

¬q����

����

s��

Figure 1 Figure 2

We call the ordering relation (≺ϕ) of the syntactic tree of the string ϕ (syn-tactic) superordination. In Figure 1, �4 for instance is superordinate to♦1/{4}; conversely ♦1/{4} is said to be subordinate to �4. Whenever Ois superordinate to O′, there is a path from O to O′, namely the sequence〈t1, . . . , tn〉, where t1 = O, tn = O′, and ti ≺ϕ tj iff i < j. For exam-ple, (∧3,∨2,∧1, r) is a path from ∧3 to r in the tree depicted in Figure 1.Operators appearing in a string and ordered according to the relation ofsuperordinateness are said to be nested. If (T,≺ϕ) is the syntactic tree of ϕand t ∈ T , the set {x : t = x ∨ t ≺ϕ x} ordered by the relation ≺ϕ is a fullsubtree of (T,≺ϕ). Hence for instance the tree depicted in Figure 2 is a fullsubtree of the tree represented in Figure 1.

The first index (i) of a modal operator (Oi,n/Wn) appearing in stringϕ ∈ L identifies a modality type, i < k; while the second (n) identifies the

168

Page 178: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 3

occurrences of nested modal operators. For each ϕ, we single out a subset||ϕ|| of the set |ϕ|, the set of indices of modal operators of ϕ:

||ϕ|| := {x : an operator of the form (Oi,x/Wx) appears in ϕ}.

IF Modal Logic of Perfect Recall with k modality types, or IFML+[k], isdefined as the class of strings ϕ ∈ L meeting the following conditions:

(1) Closedness: If O ∈ {♦,�} and (Oi,n/Wn) appears in ϕ, thenWn ⊆ |ϕ|.

(2) Diamonds independent only of boxes: If (♦i,n/Wn) appears in ϕ,j ∈ Wn and Or,j (with some r < k) is superordinate to (♦i,n/Wn),then Or,j = �r,j .

(3) Boxes independent only of diamonds: If (�i,n/Wn) appears in ϕ,j ∈ Wn, and Or,j (with some r < k) is superordinate to (�i,n/Wn),then Or,j = ♦r,j .

Condition (1) rules out formulas with operators (Oi,n/Wn), where someindices in Wn refer to no operator appearing in ϕ at all. Notice that themore subordinate operators an operator has, the greater its index; and thatdue to conjunctions/disjunctions, there may well be several operators withthe same index. The semantic correlate of the syntactic restrictions (2) and(3) will be that the evaluation games for IFML[k] are games of perfect recall.The motivation for this restriction is that while it certainly makes sense tospeak of an operator (say, a diamond) being logically independent of its dual(a box), it is not evident that there is such a properly logical phenomenonas the independence of a diamond (box) from another diamond (box).

2.2. Semantics. The semantics is defined relative to k-ary modelsM = (D,R0, . . . , Rk−1, h), where D 6= ∅, Ri ⊆ D2 for all i < k, andh : prop −→ Pow(D). By stipulation we will call pointed k-ary models〈M, w〉 simply ‘models’. We associate with every formula ϕ, k-ary modelM, and element w ∈ D, a game G(ϕ,M, w) between two players, ∀ and ∃.First we specify game rules for something a bit more complicated, namelythe games G(ϕ,M, w, πϕ, σϕ) proceeding from the extra information codi-fied in the partial functions

πϕ : ω + 1 ⇀ dom(M) and σϕ : ω ⇀ {0, 1}.

• If ϕ ∈ {p,¬p}, no move is made. If ϕ = p and w ∈ h(p), or ϕ = ¬pand w /∈ h(p), ∃ wins the play and ∀ loses it. Otherwise ∀ wins and∃ loses.

• If ϕ = θ ∧n ψ, ∀ chooses χ ∈ {θ, ψ}, and the play continues asG(χ,M, w, πχ, σχ), where πχ := πϕ and σχ := σϕ ∪ {(n, ]χ)}, where]χ = 0 if χ = θ, and ]χ = 1 otherwise.

• The case ϕ = θ ∨n ψ is dual to the case for ϕ = θ ∧n ψ.• If ϕ = (�i,n/Wn)ψ, ∀ chooses, if possible, v ∈ dom(M) satisfyingRi(w, v). The play continues as G(ψ,M, v, πψ, σψ), where πψ :=πϕ ∪{(n, v)} and σψ := σϕ. If such a choice is not possible, the playends in ∀’s choice FAIL, ∃ wins and ∀ loses. (No choice of v withRi(w, v) is possible precisely in the case that w has no Ri-successor.

169

Page 179: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 TAPANI HYTTINEN AND TERO TULENHEIMO

If no other choice is possible, the player chooses by stipulation theobject FAIL /∈ dom(M).)

• The case for ϕ = (♦i,n/Wn)ψ is dual to the case for ϕ = (�i,n/Wn)ψ.

The evaluation games G(ϕ,M, w) are now defined via the following stipu-lation:

Stipulation 2.1. The game rules for G(ϕ,M, w) are the game rules ofG(ϕ,M, w, {(ω,w)}, ∅).

Accordingly, the evaluation of ϕ in 〈M, w〉 starts off in a situation where thepoint w is indexed by the ordinal ω and no conjunctive/disjunctive moveshave been made: πϕ = {(ω,w)} and σϕ = ∅. As a consequence, ω will be inthe domain of all functions πχ corresponding to subformulas χ of ϕ.

Plays of a game are also called its histories. An m-move play of gameG(ϕ,M, w) is a sequence 〈(πϕ0 , σϕ0), . . . , (πϕm , σϕm)〉, satisfying πϕ0 ⊆ . . . ⊆πϕm and σϕ0 ⊆ . . . ⊆ σϕm , with ϕ0 = ϕ. The play can be identified withthe last member of the sequence, i.e. the pair (πϕm , σϕm). Terminal playsare plays that end with a (negated) propositional atom ϕm, or else with theappropriate player’s inability to move.

A token of a subformula χ of ϕ is uniquely determined by fixing theleft/right choices for the conjunctions/disjunctions in ϕ to which χ is sub-ordinate. Hence the tokens of χ may be identified with pairs (χ, σ) suchthat for some π, (π, σ) is a play. When no confusion may arise, we refer tosubformula tokens equally as subformulas.

A strategy of a player in G(ϕ,M, w) is a function specifying a move forevery play at which it is this player’s turn to move. Hence if f is ∃’s strategyand (πχ, σχ) is a play, f(πχ, σχ) ∈ {0, 1}, if χ = ψ ∨n θ, and f(πχ, σχ) ∈dom(M) ∪ {FAIL}, if χ = (♦i,n/Wn)ψ.

Given a subformula (χ, σχ), we write Vσχ for the common domain of allfunctions πχ such that (πχ, σχ) is a play ofG(ϕ,M, w). For every subformula(χ, σχ) with χ of the form (♦i,n/Wn)ψ, let us partition the set of plays

{(πχ, σχ) : πχ is a function Vσχ −→ dom(M)}

into equivalence classes under the following equivalence relation ∼σχ

∃ :

(π′χ, σχ) ∼σχ

∃ (π′′χ, σχ) iff π′χ(i) 6= π′′χ(i) =⇒ i ∈Wn.

This partition simply gathers together plays at which a given formula χ :=(♦i,n/Wn)ψ is evaluated and which differ at most for the moves made tointerpret an operator (�j,m/Wn) with m ∈ Wn. Finally, we define ∼∃ asthe union of all equivalence relations ∼σχ

∃ , where σχ identifies a subformula(χ, σχ) with χ of the form (♦i,n/Wn)ψ. An equivalence relation ∼∀ is definedin the analogous fashion. A strategy f of ∃ is a winning strategy (w.s.) forher, if there exists a set S of plays (called a plan of action) such that:

(i) S contains the play ({(ω,w)}, ∅).(ii) S is closed under applying f .(iii) S is closed under all moves of ∀ made in accordance with the game

rules.(iv) f is uniform, i.e. if h ∼∃ h

′, then f(h) = f(h′).(v) Every terminal play in S is a win for ∃.

170

Page 180: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 5

Semantics of a formula ϕ ∈ IFML+[k] is given as follows:

• Truth: M |=+ ϕ[w] iff there exists a w.s. for ∃ in G(ϕ,M, w).• Falsity: M |=− ϕ[w] iff there exists a w.s. for ∀ in G(ϕ,M, w).• Indeterminacy: M |=0 ϕ[w] iff M 2+ ϕ[w] and M 2− ϕ[w].

When no confusion may arise, we will simply write |= for the relation |=+.

Example 2.2. Think of the models 〈M, a〉 and 〈M′, a′〉 depicted as follows:

���s

a@

@@scsb

@@@

���

sje

M

ds

@@@ f

s�

��

���s

a′@

@@s c′sb′

@@@

���

se′

M′

d′ sj@

@@f ′sj

���

Figure 3

In Figure 3, the circles indicate the points at which the propositional atomp is true in the relevant model. Consider evaluating the formula ϕ :=

�0,2♦0,1/{2} prelative to the two models. We observe:

(a) ϕ is true in 〈M, a〉. In fact, the function f defined byf(a, b) = f(a, c) = e

is a w.s. for ∃ in G(ϕ,M, a).

(b) ϕ is indeterminate in 〈M′, a′〉. (i) Suppose for contradiction thatsome strategy f is a w.s. for ∃ in G(ϕ,M′, a′). Hence f is uniform.Now, if the value x = f(a′, b′) = f(a′, c′) is distinct from e′, x con-stitutes an illegal move for one of the histories (a′, b′) and (a′, c′),and so f is not winning. If, again, x = e′, f does not lead to a winfor ∃ in any play of the game G(ϕ,M′, a′). (ii) Observe that alsoneither of the two possible strategies g1(a′) = b′ and g1(a′) = c′ of∀ is winning: ∃ can extend both histories (a′, b′) and (a′, c′) with achoice which yields a win for her.

(c) The models 〈M, a〉 and 〈M′, a′〉 are bisimilar. Hence they are notdistinguished by any formula of basic modal logic, i.e. the logic gen-erated by closing the class of (negated) propositional atoms under ∧,∨ and applications of the unary modal operators � and ♦.

2.3. Known Results of Expressive Power. In [13] a logic termed ‘IFmodal logic of k modality types’ (or, IFML[k]) was introduced. Its eval-uation games are not games of perfect recall: a player always knows thestrategy he has followed up to any given stage in the game, but may fail todistinguish between two or more individual moves he himself has made inaccordance to that strategy. IFML+[k] is trivially embeddable in IFML[k].

Basic modal logic of k modality types (or, ML[k]) and basic tense logicof k temporal modality types (or, TL[k]) are generated by closing the classof (negated) propositional atoms under conjunction, disjunction and appli-cations of unary modal operators �i and ♦i resp. �i, ♦i, �−1

i and ♦−1i ,

with i < k; these modal operators have their usual semantics. Syntactically

171

Page 181: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 TAPANI HYTTINEN AND TERO TULENHEIMO

IF Tense Logic of Perfect Recall (or, IFTL+[k]) equals IFML+[2k], and itis interpreted relative to models with 2k accessibility relations of which thei-th and the (i+ k)-th are each other’s converses for all i < k, and which allare irreflexive and transitive. We write Ck for the class of all k-ary models,and Tk for the class of all k-ary temporal models. Directly by (the proofsof) the corresponding theorems in [13], we have:

Theorem 2.3.(a) The class of all IFML+[k]-formulas that are everywhere either true

or false (the “determined fragment” of IFML+[k]) coincides withML[k] over Ck.

(b) For all k ≥ 1, IFML+[k] has greater expressive power than ML[k]over Ck.

(c) For all k ≥ 1, IFTL+[k] has greater expressive power than TL[k]over Tk.

(d) The expressive powers of IFTL+[1] and TL[1] are the same over theclass of all unary linearly ordered temporal models.

In passing it may be noted that the above Example 2.2 in effect providesa proof of Theorem 2.3 (b). Concerning the relation between the logicsIFML+[k] and IFML[k], it is not difficult to prove the following:

Theorem 2.4. For all k ≥ 1, IFML[k] has greater expressive power thanIFML+[k] over Ck.

Hence the IF modal logic of [13] is a proper extension of the IF modal logicof the present paper. However, the extra expressive power of IFML[k] is notneeded to establish the main expressivity results concerning this logic: as iswitnessed by Theorem 2.3 (b,c), these results are obtained already withinits fragment IFML+[k].

3. History-Free Strategies

If (χ, σχ) is a subformula of ϕ, we write Sσχ for the set of all histories(plays) of the game G(ϕ,M, w) that end with (χ, σχ). That is,

Sσχ = {(πχ, σχ) : (πχ, σχ) is a play and πχ is a function Vσχ −→ dom(M)}.

Definition 3.1. (History-free strategy) We say that a strategy f of ∃ ishistory-free, if for any subformula (χ, σχ) of ϕ with χ ∈ {θ∨nψ, (♦i,n/Wn)ψ},and any histories (π′χ, σχ), (π

′′χ, σχ) ∈ Sσχ, we have:

π′χ(K) = π′′χ(K) =⇒ f(π′χ, σχ) = f(π′′χ, σχ),

where

K :=

{min{x ∈ ||ϕ|| ∪ {ω} : x > n}, if χ := θ ∨n ψmin{x ∈ ||ϕ|| ∪ {ω} : x > n & x /∈Wn}, if χ := (♦i,n/Wn)ψ

Hence the value of a history-free strategy of ∃ for a history (πχ, σχ) dependsonly on the last ‘known’ member of πχ. The following can be proven:

Theorem 3.2. Let ϕ ∈ IFML+[k], M, and w be arbitrary. If thereis a w.s. for ∃ in G(ϕ,M, w), there is a history-free w.s. for her in thisgame. �

172

Page 182: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 7

Given a formula, let us write ||ϕ||∀ for the set of indices of boxes,

||ϕ||∀ := {x : an operator of the form (�i,x/Wx) appears in ϕ}.By Theorem 3.2 any IFML+[k]-formula ϕ has the following normal formsϕ+ and ϕ◦:

Corollary 3.3. For all ϕ ∈ IFML+[k], M, and w, we have:

M |= ϕ+[w] ⇐⇒M |= ϕ[w] ⇐⇒M |= ϕ◦[w],

where ϕ+ is obtained from ϕ by replacing any operator (♦i,n/Wn) by theoperator (♦i,n/Vn), and ϕ◦ is obtained from ϕ by replacing any operator(♦i,n/Wn) by the operator (♦i,n/Un), where for K := min{x ∈ ||ϕ|| ∪ {ω} :x > n & x /∈Wn}, Vn = [n+ 1,K − 1] ∩ ||ϕ||∀ and Un = ([n+ 1,K − 1] ∪[K + 1,∞[) ∩ ||ϕ||∀. �

If for instance ϕ is the formula �7♦6�5♦4�3�2(♦1/{2, 5})p, it is equiva-lent to both of the formulas ϕ+ and ϕ◦:

ϕ+ := �7♦6�5♦4�3�2(♦1/{2})p ;ϕ◦ := �7♦6�5♦4�3�2(♦1/{2, 5, 7})p.

By Theorem 3.2, when evaluating any of the formulas ϕ, ϕ+ and ϕ◦, thechoice for ♦1 can be made as a function of the last known move in therelevant history, i.e. depending only on the choice for �3. Hence allowingthe choice for ♦1 to depend not only on the move for �3 but also on all earliermoves by the opponent (ϕ+), or requiring this choice to be independent ofall of these earlier moves (ϕ◦), simply results in the truth-condition of ϕ.

4. Satisfiability Problem

Supposing that ϕ ∈ IFML+[k] is satisfied in 〈M, w〉, (1) we construct atree structure T out of the model 〈M, w〉; (2) we generate a certain labeledsubtree T f of T ; (3) we produce a model M∗ of ϕ out of T f ; (4) wegenerate, by a stepwise construction, from 〈M∗, w〉 a finite model 〈N , v〉 ofϕ, whose size has a computable upper bound. For the rest of the section,let ϕ be an arbitrary satisfiable IFML+[k]-formula, and let 〈M, w〉 be itsmodel.

Definition 4.1. With each formula ϕ ∈ IFML+[k] we associate a naturalnumber, md(ϕ), to be called the modal depth of ϕ:

• md(p) = 0 = md(¬p).• md(ϕ ∨n ψ) = max{md(ϕ),md(ψ)} = md(ϕ ∧n ψ).• md((♦i,n/Wn)ϕ) = md(ϕ) + 1 = md((�i,n/Wn)ϕ).

4.1. Towards a Finite Model: Preparations. For each n ≤ md(ϕ), putTnw := {(w, i1, s1, . . . , im, sm) : Ri1(w, s1), . . . , Rim(sm−1, sm)

for some m ≤ n and some i1, . . . , im ∈ {0, . . . , k − 1}}.Further, define relations R′

i (i < k) on the set⋃n≤md(ϕ) T

nw as follows:

(w, i1, s1, . . . , im, sm) R′i (w, i′1, s

′1, . . . , i

′m′ , s′m′) ⇐⇒

m′ = m+ 1 and (ij , sj) = (i′j , s′j) [0 ≤ j ≤ m] and i′m′ = i.

Finally, fix an assignment h′ by putting, for all p ∈ prop:(w, i1, s1, . . . , im, sm) ∈ h′(p) ⇐⇒ sm ∈ h(p).

173

Page 183: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

8 TAPANI HYTTINEN AND TERO TULENHEIMO

We have obtained a tree structure

〈T , w〉 = 〈⋃

n≤md(ϕ)

Tnw , R′0, . . . , R

′k−1, h

′, w〉.

〈T , w〉 is not a model of ϕ, if ϕ is not equivalent to any ML[k]-formula.1

Definition 4.2. A ϕ-label is a set {σϕ1 , . . . , σϕn} of partial functions σϕi :ω ⇀ {0, 1} such that ϕi is a subformula of ϕ, and dom(σϕi) consists ofindices of disjunctions and conjunctions appearing in |ϕ|\|ϕi|. When noconfusion may arise, we refer to ϕ-labels as labels.

We will write x := (w, s0, . . . , sm), if x = (w, i0, s0, . . . , im, sm). Let f bea w.s. of ∃ in game G(ϕ,M, w), and define a labeling on T as follows:

• First let all points in T be labeled with the empty label ∅.• Label the root w of T with {∅}.• If a point x in T has the label Σ = {σϕ1 , . . . , σϕN } with ϕm =

(�i,n/Wn)ψ (m ∈ {1, . . . , N}), and if Σ′ is the label of an Ri-successor of x, replace Σ′ by the label Σ′ ∪ {σψ}, where σψ := σϕm .

• If a point x in T has the label Σ = {σϕ1 , . . . , σϕN } with ϕm = (ψ∧jθ)(m ∈ {1, . . . , N}), replace the label Σ itself by the label (Σ\{σϕm})∪{σψ, σθ}, where σψ := σϕm ∪ {(j, 0)} and σθ := σϕm ∪ {(j, 1)}.

• If a point x in T has the label Σ = {σϕ1 , . . . , σϕN } with ϕm =(♦i,n/Wn)ψ (m ∈ {1, . . . , N}), then take the Ri-successor y :=f(x, σϕm) of x, and if Σ′ is the label of y, replace Σ′ by the labelΣ′ ∪ {σψ}, where σψ := σϕm .

• If a point x in T has the label Σ = {σϕ1 , . . . , σϕN } with ϕm =(ψ ∨j θ) (m ∈ {1, . . . , N}), then take the disjunct χ ∈ {ψ, θ}given by f(x, σϕm) ∈ {0, 1}, and replace the label Σ itself by thelabel (Σ\{σϕm}) ∪ {σχ}, where σχ := σϕm ∪ {(j, f(x, σϕm))}.

We define T f as the subtree of T containing precisely those of its elementsthat have a non-empty label under the above labeling.

4.2. Model M∗ of ϕ. Next we turn the labeled tree T f into a model M∗ ofthe formula ϕ. Specify the model M∗ := 〈D∗, R∗

0, . . . , R∗k−1, h

∗〉, by setting:

• D∗ := {w} ∪ {(n, i, s) : for some (w, i1, s1, . . . , im, sm) ∈ T f ,n = m & i = im & s = sm}.

• For all j < k, (n, i, s)R∗j (n′, i′, s′) ⇐⇒

n′ = n+ 1 and j = i′ and Ri′(s, s′).2

• For all p ∈ prop, (n, i, s) ∈ h∗(p) ⇐⇒ s ∈ h(p).

Claim 4.3. M∗ |= ϕ[w]. �

4.3. Finite model N of ϕ. We move on to build up a finite model 〈N , v〉of ϕ out of the model 〈M∗, w〉. We observe first the following:

Observation 4.4. (a) For a fixed ϕ, the number mϕ of different ϕ-labels isfinite: if r is the number of subformulas of ϕ, and n is the maximal numberof nested conjunctions/disjunctions in ϕ, then mϕ ≤ 2r·2

n.

1For 〈T , w〉, cf. the proof of the finite model property of basic modal logic in [1].2Here Ri′ is the relation of type i′ of the original model M.

174

Page 184: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 9

(b) The labeling of the tree T f induces a labeling of the model M∗. Thelabel of an element (n, i, s) ∈ D∗ is obtained as the union of the labels ofelements of the set T(n,i,s) :=

{(w, i1, s1, . . . , im, sm) ∈ T f : m = n & im = i & sm = s}.Roughly, the idea in the construction of the model N is that we partitionthe domain of M∗ into md(ϕ) ‘levels’, the level of an element (N, i, s) ∈ D∗

being determined by its ‘height’, N . Then on each level we ‘identify’ as manyelements as it goes without affecting the fact that the resulting structureremains a model of ϕ. As a result of the identification, a finite structurewill be yielded. Now let us write M := md(ϕ). We will define, for allN ∈ {1, . . . ,M}, a model MN := (DN , RN0 , . . . , R

Nk−1, h

N ). We begin bydefining an equivalence relation ≡M on D∗:

(n, i, s) ≡M (n′, i′, s′) iff (n, i, s) = (n′, i′, s′) ∨n = n′ = M & i = i′ & ∀p ∈ prop[s ∈ h(p) ⇐⇒ s′ ∈ h(p)].

The model MM is then specified by putting:• DM := D∗/ ≡M .• [ξ]≡M RMj [ζ]≡M iff there is ζ ′ such that ζ ≡M ζ ′ and R∗

j (ξ, ζ′).

• For all p, [ξ]≡M ∈ hM (p) iff ξ ∈ h∗(p).It can be checked that the definitions of hM and the RMj are independent ofthe choices of representatives of the equivalence classes involved.

Observation 4.5. The classSM := {ξ ∈ DM : ξ = [(M, i, s)]≡M for some i, s}

is finite, in fact it has at most k · 2|prop[ϕ]| elements, where prop[ϕ] is theset of propositional atoms actually appearing in ϕ.

Claim 4.6. There is a history-free w.s. for ∃ in G(ϕ,MM , [w]≡M ). �

Let then N > 1 be fixed. Let us stipulate that DM+1 := D∗. Supposethat for all K ∈ {N, . . . ,M}, we already have defined the following threethings: (a) an equivalence relation ≡K ⊆ DK+1 × DK+1; (b) a modelMK := (DK , RK0 , . . . , R

Kk−1, h

K) with DK := DK+1/ ≡K ; and (c) a history-free winning strategy fK for ∃ in GK := G(ϕ,MK , [[w]≡M . . .]≡K ). We goon to define an equivalence relation ≡N−1 on DN by putting

ξ ≡N−1 ξ′

if and only if either ξ = ξ′, or else there are n, i, s, n′, i′, s′ such thatn = n′ = N − 1, andξ = [[(n, i, s)]≡M . . .]≡N and ξ′ = [[(n′, i′, s′)]≡M . . .]≡N ,

and the following conditions (i) to (v) hold:(i) i = i′.(ii) ∀p ∈ prop[s ∈ hN (p) ⇐⇒ s′ ∈ hN (p)].(iii) The points (n, i, s) and (n′, i′, s′) have the same label in M∗.(iv) For all i < k and all ζ, RNi (ξ, ζ) ⇐⇒ RNi (ξ′, ζ).(v) For all subformulas (χ, σχ) of ϕ, and all plays (π′χ, σχ) and (π′′χ, σχ)

of GN in which ∃ has applied fN , if the last choices from the domain‘known’ to ∃ are, respectively, π′χ(K) = ξ and π′′χ(K) = ξ′, and forall l < K, π′χ(l) = π′′χ(l), then fN (π′χ, σχ) = fN (π′′χ, σχ).

175

Page 185: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

10 TAPANI HYTTINEN AND TERO TULENHEIMO

For condition (v), notice that given any subformula (χ, σχ) of ϕ, and anyplays (π′χ, σχ) and (π′′χ, σχ) of GN , we necessarily have: dom(π′χ) = dom(π′′χ).Further, the index K of the last modal operator ‘known’ to ∃ is obtained as

K :=

{min{x ∈ ||ϕ|| ∪ {ω} : x > n}, if χ = ψ ∨n θmin{x ∈ ||ϕ|| ∪ {ω} : x > n & x /∈Wn}}, if χ = (♦i,n/Wn)ψ

As fN is history-free, its values on the plays (π′χ, σχ) and (π′′χ, σχ) do notdepend on the values π′χ(l) resp. π′′χ(l) for l > K. Now the model MN−1 isspecified as follows:

• DN−1 := DN/ ≡N−1.• [ξ]≡N−1 R

N−1j [ζ]≡N−1 iff there is ζ ′ s.t. ζ ≡N−1 ζ

′ and RNj (ξ, ζ ′).• For all p, [ξ]≡N−1 ∈ hN−1(p) iff ξ ∈ hN (p).

It can be checked that the definition of the model MN−1 is independentof the choices for representatives of equivalence classes involved.

Claim 4.7. For all K ∈ {1, . . . ,M}, we have:

(a) The size of the set

SK = {ξ ∈ DK : ξ = [[(K, i, s)]≡M . . .]≡K for some i, s}has a recursive upper bound.

(b) There is a history-free w.s. for ∃ in the game

GK = G(ϕ,MK , [[w]≡M . . .]≡K ).

Proof. The inductive proofs of both statements are straightforward.

To finish the construction, we put: N := M1 and v := [[w]≡M . . .]≡1 .

4.4. Satisfiability problem is decidable. We obtain:

Theorem 4.8. For all k ≥ 1,

(a) IFML+[k] has the strong finite model property.(b) The satisfiability problem of IFML+[k] is decidable.

Proof. (a) Suppose ϕ ∈ IFML+[k] is satisfied in 〈M, w〉, and let 〈N , v〉be the model constructed out of 〈M, w〉 as explained above. By Claim 4.7(a), 〈N , v〉 is a model of ϕ. And directly by Claim 4.7 (b) 〈N , v〉 has arecursive upper bound, whose size is roughly exponential in the size of theformula ϕ. The statement (b) is immediate from (a).

5. Validity Problem

5.1. Basic observations. Formulas of IFML+[k] are in negation normalform: negation symbol ¬ appears only as prefixed to propositional atoms.Closing IFML+[k] under ¬, and interpreting ¬ as initiating a ‘role switch’between the players of evaluation games (from verifier to falsifier and viceversa), results in the following:

• ∃ has a w.s. in G(¬ϕ) ⇐⇒ ∀ has a w.s. in G(ϕ)• ∀ has a w.s. in G(¬ϕ) ⇐⇒ ∃ has a w.s. in G(ϕ).

176

Page 186: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 11

The negation which is hence given a game-theoretical interpretation may betermed dual negation. As there are indeterminate IFML+[k]-formulas, i.e.formulas ϕ such that neither ∃ nor ∀ has a w.s. in the associated evaluationgame, it is immediately seen that dual negation does not have the force ofcontradictory negation (∼), whose semantics is simpy this:

• ∼ ϕ is true in 〈M, w〉 iff ϕ is not true in 〈M, w〉.In particular, then, if ϕ is indeterminate in 〈M, w〉, ∼ ϕ is true therein.

Fact 5.1. The satisfiability and validity problems of IFML+[k] are not dualsto one another.

Proof. If they were duals, we would have:ϕ is valid ⇔ ¬ϕ is not satisfiable.

Let ψ ∈ IFML+[k] be any formula which is indeterminate in a model〈M, w〉. Then the formula ¬(ψ ∨ ¬ψ) is not satisfiable, but still the for-mula (ψ ∨ ¬ψ) is not valid, since it is indeterminate in 〈M, w〉.

We will prove that the validity problem of IFML+[k] is decidable. Itshould be noticed that due to Fact 5.1, this result neither implies nor isimplied by Theorem 4.8.

5.2. Reducing the problem. Let us define a syntactic transformationϕ 7→ ϕ∗ in the class IFML+[k] as follows. Recall the definition of syn-tactic tree and a path in a syntactic tree from Subsect. 2.1. Now given anyIFML+[k]-formula ϕ, consider all paths of the form

(*) 〈(�i′,n′/Wn′), β1, . . . , βk, (♦i′′,n′′/Wn′′)〉in its syntactic tree (k ≥ 0), where the βi are from {∨,∧} and the diamondis independent of the box (that is, n′ ∈ Wn′′). For each such path, replacethe full subtree with the diamond at its root by the symbol ⊥. The resultof these replacements is denoted as ϕ∗. We have:

Claim 5.2. For any ϕ ∈ IFML[k]+, ϕ is valid ⇔ ϕ∗ is valid.

Proof. Let ϕ ∈ IFML[k]+ be arbitrary. The validity of ϕ∗ triviallyimplies the validity of ϕ. For the converse direction, suppose that ϕ is valid.We assume that the syntactic tree of ϕ contains at least one path of the form(∗), for otherwise there is nothing to prove. LetM = (D,R0, . . . , Rk−1, h) bean arbitrary model, and w ∈ D any point. We must show that M |= ϕ∗[w].

Let N := 22](ϕ), where ](ϕ) is the number of symbols appearing in the

formula ϕ. We construct a model M∗ = (D∗, R∗0, . . . , R

∗k−1, h

∗) as follows:• D∗ := {(η, d) : d ∈ D and η : n→ N for some n < ω}• (η, d) R∗

i (η′, d′) iff (d Ri d′ & η′ = η_〈k〉 for some k < N)

• (η, d) ∈ h∗(p) iff d ∈ h(p).Because ϕ is valid, we have in particular:

M∗ |= ϕ[(∅, w)].Let f be a w.s. of ∃ in the corresponding evaluation game. We prove:

Subclaim 5.3. For any d1, . . . , dn ∈ D there are numbers m1, . . . ,mn < Nsuch that the following holds: if a play of game G(ϕ,M∗, (∅, w)) is playedso that

177

Page 187: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

12 TAPANI HYTTINEN AND TERO TULENHEIMO

(i) ∃ has followed the strategy f , and(ii) ∀ has chosen for each box an element (η_〈mi〉, di), where i is the

least j ≤ n with mj /∈ dom(η), and the previous element chosen fromD∗ is (η, d) for some d ∈ D,

then assuming that the play has gone according to the rules, there is no pathof the form (∗) in the syntactic tree of ϕ such that the play has proceeded tothe diamond (♦i′′,n′′/Wn′′).

Proof of the Subclaim. The proof is by induction on the number n ofboxes reached in a play of game G(ϕ,M∗, (∅, w)). The base case of 0 boxesholds trivially. Suppose then that the claim holds for the case of n boxes.

Let h be a play of game G(ϕ,M∗, (∅, w)) involving choices interpretingn+1 boxes. Suppose that it is ∀’s turn to move at h to interpret the (n+1)-th box. Assume that ∃ has followed f when playing h. Given that the first nmoves from D∗ by ∀ are (η_〈mi〉, di) [i := 1, . . . , n] and that they are madeaccording to the game rules, by the induction hypothesis we may assumethat the numbers mi are chosen so that the play h has not proceeded alongany path of the form (∗) to a diamond (♦i′′,n′′/Wn′′).

Let dn+1 ∈ D be any element satisfying d Ri dn+1, where d ∈ D isgiven by the previous choice (η, d) from D∗ in h, and consider the differentelements (η_〈p〉, dn+1) with p < N that ∀ may choose in order to interpretthe (n+ 1)-th box (�i′,n′/Wn′) according to the game rules.

Assume that from (�i′,n′/Wn′) there begins a path of the form (∗) in thesyntactic tree of ϕ . Now the sequence β1, . . . , βk contains some finite numberm ≤ k of conjunctions, and hence there are at most 22m

< 22](ϕ)different

ways in which the values of the strategy f can be chosen for the disjunctions(if any) in the sequence β1, . . . , βk. But then there are numbers p1, p2 < Nwith p1 6= p2, and plays h1, h2, such that up to the box (�i′,n′/Wn′), h1 andh2 both coincide with h; in h1 the box is interpreted as (η_〈p1〉, dn+1) andin h2 as (η_〈p2〉, dn+1); and proceeding from both points (η_〈p1〉, dn+1) and(η_〈p2〉, dn+1), the strategy f gives the same choices for the disjunctions (ifany) in β1, . . . , βk.

Choose mn+1 := min{p1, p2}. Suppose now for contradiction that thereare choices for the conjunctions in β1, . . . , βk, and replies to them by f ,so that these choices b ∈ {0, 1}k extend both histories h1 and h2 in sucha way that the diamond (♦i′′,n′′/Wn′′), which is independent of the box(�i′,n′/Wn′), is reached. By the uniformity of f , we have then:

f(h1_b) = f(h2

_b).But then f is not a w.s. for ∃, since from distinct points in M∗ it is onlypossible to access distinct points along R∗

i .Given the truth of Subclaim 5.3, the strategy f clearly induces a w.s.

for ∃ in game G(ϕ,M, w), and so M |= ϕ∗[w]. This completes the proofClaim 5.2.

5.3. Validity problem is decidable. We obtain:

Theorem 5.4. For all k ≥ 1, the validity problem of IFML+[k] is decidable.

Proof. Let us define a syntactic transformation ϕ 7→ ϕ− from IFML[k]+

to ML[k] by letting ϕ− be the result of replacing any operator of the form

178

Page 188: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 13

(Oi,n/Wn) in the formula ϕ by the operator Oi,n. By Theorem 3.2 it isimmediate that for any ϕ ∈ IFML[k]+ and for any model 〈M, w〉,

M |= ϕ∗[w] ⇐⇒M |= (ϕ∗)−[w],

where the superscript ∗ stands for the transformation ϕ 7→ ϕ∗ fromIFML+[k] to IFML+[k] defined above. The transformations ϕ 7→ ϕ∗ 7→(ϕ∗)− are effective; hence to decide whether any given ϕ ∈ IFML+[k] isvalid, it suffices to apply an algorithm deciding the validity problem of ML[k]to the formula (ϕ∗)−.

6. Concluding Remarks

Related work. Independence-Friendly (IF) first-order logic was introducedby Hintikka and Sandu in [10]. Its basic properties were studied by Sandu in[11]. The basic idea behind this logic is to allow arbitrary relations of logicaldependence and independence between quantifiers. One way of thinking howto accomplish this is by separating the syntactic relation of subordinatenessand the semantic relation of logical dependence, so that in particular aquantifier can be logically independent of some quantifiers in whose syntacticscope it nevertheless lies. The class of IF first-order sentences has the sameexpressive power as the class of Σ1

1-sentences, and so it is a proper extensionof usual first-order logic. This logic has been discussed in connection withfoundations of mathematics, for instance in the monograph [7] of Hintikka,and in the papers [8] and [9] by Hyttinen and Sandu.

Sandu and Vaananen showed in [12] how to extend the idea of informa-tional independence from the case of quantifiers to propositional connectives.Bradfield was in [2] the first to define a version of IF modal logic; his logicwas specifically designed to be used when studying transition systems withconcurrency. With Froschle, Bradfield studied further properties of this logicin [5]. Recently, in [3] and [4], Bradfield has extended IF first-order logicinto an IF fixpoint logic, and shown how to obtain an IF modal mu-calculus.

Motivation. Given the existence of IF first-order logic and the fact that itsexpressive power exceeds that of usual first-order logic, a systematic ques-tion arises: how to formulate an IF modal logic. A natural way to define IFmodal logic is simply by modifying basic modal logic so as to allow modaloperators to be logically independent of syntactically superordinate dual op-erators. The result is the logic IFML+ defined in the present paper. Therelevant notion of logical independence is implemented by the requirementof uniformity of players’ winning strategies. In the corresponding evaluationgames, a player may be ignorant of earlier moves of the opponent, but alwaysremembers his or her own previous moves. Allowing this kind of indepen-dence in a modal logical setting typically allows expressing the existence ofa common endpoint of several transitions: for instance the requirement thatvarious computations end in the same state becomes expressible.

A more general motivation for studying IF modal logics is that they pro-vide one possible − and arguably natural − way of producing relativelyexpressive modal languages. The existence of such languages is interestingnot only for modal logic itself, but in particular for finite model theory, in

179

Page 189: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

14 TAPANI HYTTINEN AND TERO TULENHEIMO

which modal logics are of interest in connection with different definabilityissues (cf., e.g., [6]).

Changing basic modal logic by allowing more logical independencies re-sults in powerful logics. As noted in the present paper, IFML+ is moreexpressive than basic modal logic. On the other hand, it is possible to showthat this logic is translatable into first-order logic. (In [14] such a transla-tion is provided to a fragment of this logic; the proof generalizes easily tothe whole of IFML+.) By contrast, in [14] it is shown that generalizingIFML+ by allowing diamonds (boxes) to be independent of conjunctions(disjunctions) results in a modal logic that has genuine second-order expres-sive power.

Open problems. In the present paper the ‘perfect recall’ fragment IFML+

of the IF modal logic (IFML) of [13] was studied. The answer to theproperly philosophical question which of the two logics better deserves to betermed ‘IF modal logic’ depends on whether or not it makes sense to speakof a diamond (box) being logically independent of a diamond (resp. box). Itis not obvious how to separate cases where the game-theoretical frameworkmerely serves to model logical phenomena from cases where this tool addsto the picture something that does not really belong to the domain of logic.Is independence of a diamond from a diamond really a logical phenomenon,as is suggested by IFML, or are we simply misled to think so because wecan find a game-theoretical correlate to this phenomenon?

It is an open problem whether the satisfiability and validity problemsof IFML[k] are decidable. In particular, it is an open question whetheran analogue of Theorem 3.2 can be proven for IFML[k], i.e. whether thewinning strategies of ∃ in associated evaluation games allow a history-freenormal form.

References

[1] Blackburn, P., de Rijke, M. & Venema, Y., 2002. Modal Logic, Cambridge UniversityPress, Cambridge.

[2] Bradfield, J., 2000. “Independence: Logics and Concurrency,” Lec-ture Notes in Computer Science vol. 1862, available electronically ashttp://www.dcs.ed.ac.uk/home/jcb/Research/csl00.ps.gz

[3] Bradfield, J., 2004a. “Parity of Imperfection or Fixing Independence” in M. Baaz andJ.A. Makowsky (eds.), Computer Science Logic, Lecture Notes in Computer Science,2803, pp. 72-85.

[4] Bradfield, J., 2004b. “On Independence-Friendly Fixpoint Logics,” Philosophia Scien-tiae 2/2004, to appear.

[5] Bradfield, J. & Froschle, S., 2002. “Independence-Friendly Modal Logic and True Con-currency” in Nordic Journal of Computing, Vol. 9, No. 2, pp. 102-117.

[6] Gradel, E., Kolaitis, P. G., et al., 2004. Finite Model Theory and Its Applications,Springer-Verlag.

[7] Hintikka, J., 1996. The Principles of Mathematics Revisited, Cambridge UniversityPress, New York.

[8] Hyttinen, T. & Sandu, G., 2000. “Henkin Quantifiers and the Definability of Truth,”Journal of Philosophical Logic, Vol. 29, pp. 507-527.

[9] Hyttinen, T. & Sandu, G., 2001. “IF-logic and foundations of mathematics,” Synthese,Vol. 126, pp. 37-47.

180

Page 190: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

DECIDABILITY OF IF MODAL LOGIC OF PERFECT RECALL 15

[10] Hintikka, J. & Sandu, G., 1989. “Informational Independence as a Semantical Phe-nomenon,” in J. E. Fenstad et al. (eds.), Logic, Methodology and Philosophy of Science,Vol. 8, Amsterdam, North-Holland, pp. 571-589.

[11] Sandu, G., 1993. “On the Logic of Informational Independence and its Applications,”Journal of Philosophical Logic, 22, pp. 29-60.

[12] Sandu, G. & Vaananen, J., 1992. “Partially ordered connectives,” Zeitschrift furMatematische Logik und Grundlagen der Mathematik, 38, pp. 361-372.

[13] Tulenheimo, T., 2003. “On IF Modal Logic and its Expressive Power” in P. Balbianiet al. (eds.): Advances in Modal Logic, Vol. 4, King’s College Publications, London, pp.475-98.

[14] Tulenheimo, T., 2004. Independence-Friendly Modal Logic: Studies in its ExpressivePower and Theoretical Relevance, Philosophical Studies from the University of Helsinki4 (Doctoral dissertation).

Tapani HyttinenDepartment of Mathematics and StatisticsP. O. Box 68 (Gustaf Hallstromin katu 2b)00014 University of Helsinki, FinlandE-mail: [email protected]

Tero TulenheimoDepartment of PhilosophyP. O. Box 9 (Siltavuorenpenger 20 A)00014 University of Helsinki, FinlandE-mail: [email protected]

181

Page 191: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

On dynamic topological and metric logics

B. Konev1, R. Kontchakov2, F. Wolter1, and M. Zakharyaschev2

1Department of Computer Science, University of Liverpool,

Liverpool L69 7ZF, U.K.

�b.konev, frank�@csc.liv.ac.uk,

2Department of Computer Science, King’s College London,

Strand, London WC2R 2LS, U.K.

�romanvk, mz�@dcs.kcl.ac.uk

Abstract

The first result of this paper shows that some dynamic topological log-ics interpreted in various topological spaces with homeomorphisms are notrecursively enumerable (and so are not recursively axiomatisable). Thisgives a ‘negative’ solution to a conjecture of Kremer and Mints [12]. Sec-ond, we prove the non-elementary decidability of the dynamic metric logicwith distance operators of the form ‘somewhere in the ball of radius a,’ fora � �� , interpreted in arbitrary metric spaces with distance preserving auto-morphisms.

1 Introduction

Dynamic topological logics were first introduced in 1997 (see, e.g., [10, 11, 13, 2,12]) as a logical formalism for describing the behaviour of dynamical systems, e.g.,in order to specify liveness and safety properties of hybrid systems [6]. Dynamicalsystems [4, 9] are usually represented by some ‘mathematical’ space W (modellingpossible system states) and a function f on W (modelling the evolution of thesystem), with one of the main research problems being the study of iterations of f ,in particular, the orbits O�w� � �w� f �w�� f2�w�� � � �� of states w �W .

A natural logical formalism for speaking about such iterations is a variant oftemporal logic. For example, given a subset V of W , we can introduce the standardtemporal operators � (‘at the next moment’), �F (‘always in the future’), and itsdual �F (‘eventually’) by taking

�V � f�1�V �� �FV ��

0�n�ωf�n�V � and �FV �

0�n�ωf�n�V ��

Using this language we can describe in a succinct and transparent way propertieslike

1

182

Page 192: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� starting from a state in some region P, one will never leave a region Q:P��FQ;

� starting from a state in a region P, one will eventually reach a state in Q:P��FQ;

� w ‘visits’ P ever and ever again: w ��F�FP.

To speak about the structure of the underlying space W —important examplesare (subspaces of) the Euclidean spaces �n , general topological spaces, metricspaces, and measure spaces—as well as the type of the intended functions f , onemay require different non-temporal operators. So far, research has mainly been fo-cused on topological spaces with continuous mappings. The corresponding logicalconstructors are those of the modal logic S4 which can be regarded also as the topo-logical closure and interior operators—we denote them by C and I, respectively.For example, a property similar to Poincare’s recurrence theorem corresponds inthis language to the validity of the formula C�Ip ���FIp� in spaces based onthe unit disc with measure preserving continuous mappings.

Metric operators were suggested in [16] in order to formulate safety properties.For example, using the operator ��a, where a is a positive rational number, theformula P��F��

�aQ states that, having started from a point in P, one can neverreach the a-neighbourhood of some ‘unsafe’ area Q.

The resulting combinations of temporal and topological/metric logics are ofa clear ‘two-dimensional character,’ which makes it very difficult to analyse theircomputational properties (see, e.g., [7]). Perhaps this is the main reason why in thefield of dynamic topological systems no (un)decidability or axiomatisability resultshave been obtained yet for the full language containing both� and the infinitary�F .

This note provides answers to some of the open problems. First, we show thatsome dynamic topological logics introduced in [12] and interpreted in various topo-logical spaces with homeomorphisms are not recursively enumerable (and so arenot recursively axiomatisable). This result gives negative solutions to Conjectures2.7 (ii) and 2.7 (iv) from [12]. Second, we prove the non-elementary decidabil-ity of the dynamic metric logic with distance operators of the form ��a from [14]interpreted in arbitrary metric spaces with distance preserving automorphisms.

Although numerous problems remain open, the obtained results clearly indicatethat the logics for dynamic systems are very sensitive to the available operators(say, topological vs metric) as well as the constraints imposed on the spaces �W� f �(e.g., the proof of the undecidability result mentioned above does not go throughfor continuous functions, while the decidability proof only works for arbitrarymetric spaces, but not for, say, compact ones).

The missing proof can be found in the full version of the paper available athttp://www.dcs.kcl.ac.uk/staff/mz/.

2

183

Page 193: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 Definitions

Syntax. The language DT L of dynamic topological logic (or dynamic topo-logic, for short) [2, 12] is constructed from a countably infinite set of propositionalvariables using the Booleans and �, the modal operators I and C (for topologicalinterior and closure), and the temporal operators � (for ‘next’), �F and �F (for‘always’ and ‘eventually’). By DT L� we denote the fragment of DT L whichdoes not use �F and �F . We write ��

F ϕ for ϕ�Fϕ and dually ��F ϕ � ϕ�Fϕ,

for every DT L-formula ϕ.

Semantics. In this paper, by a dynamic topological structure (or DTS, for short)we understand a pair of the form � � ��� f �, where � � �T��� is a topologicalspace with an interior operator � (satisfying the standard Kuratowski axioms) andf is a homeomorphism1 (i.e., a bijective continuous and open mapping) on �. Adynamic topological model (or DTM) is a pair �� �����, where � is a DTS and�, a valuation, associates with each propositional variable p a subset ��p� of T .The truth-relation ���w� �� ϕ, for a DT L-formula ϕ, is defined as follows:

���w� �� p iff w ���p��

���w� �� Iϕ iff w � ��v� T � ���v� �� ϕ�����w� �� Cϕ iff w � � �v � T � ���v� �� ϕ�����w� �� �ϕ iff ��� f �w�� �� ϕ����w� ���Fϕ iff ��� f n�w�� �� ϕ for every n � 0�

���w� ���Fϕ iff ��� f n�w�� �� ϕ for some n � 0�

Here f n�w� �

n� �� �f � � � f �w�. If ���w� �� ϕ for some w � T , then we say that ϕ is

satisfied in �. A DT L-formula ϕ is satisfiable in a DTS � if ϕ is satisfied in aDTM based on �.

Given a class K of dynamic topological structures, we denote by ���K (re-spectively, ����K ) the logic of K in the language DT L (or DT L�), i.e., theset of all DT L-formulae (respectively, DT L�-formulae) ϕ such that ���w� �� ϕholds for every model � based on a structure in K and every point w in�.

We remind the reader that every preorder � � �W�R� (R is a reflexive andtransitive relation on W ) gives rise to a topological space �� � �W����, where, forevery X �W , ��X � �x � X � y �W �xRy � y � X��.

Such spaces are known as Aleksandrov spaces. Alternatively they can be de-fined as topological spaces where arbitrary (not only finite) intersections of opensets are open; for details see [1, 3]. Clearly, for �� ����� f � ��� we have

���w� �� Iϕ iff ���v� �� ϕ for every v �W with wRv�

���w� �� Cϕ iff there is v �W such that wRv and ���v� �� ϕ�

1In a more general setting, f can be a continuous mapping.

3

184

Page 194: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

It should be also clear that a function f : W �W is a continuous mapping on ��if, for all w�v �W , wRv implies f �w�R f �v�� The function f is a homeomorphismon �� if f is bijective and the converse implication holds as well.

Let �n denote the standard Euclidean space of dimension n and � is the realline. For n� 1, a unit ball is a DTS�n � �Bn� f �, where Bn is a ball in �n of radius1, and f is a measure preserving homeomorphism on Bn.

The results of the theorem below were explicitly proved in or easily followfrom [2, 13, 12].

Theorem 1. The four dynamic topo-logics listed below coincide, have the finitemodel property, are finitely axiomatisable, and so decidable:

1. �������� f � � ��� f � a DTS�,

2. �������n � f � � ��n � f � a DTS� n � 1�,

3. �������� f � � ��� f � a DTS� � an Aleksandrov space�,

4. ������n ��n a unit ball� n� 1�.

Later on we will use the fact that ��������x �� x�1�� coincides with all of thelogics above as well (see [12]).

We show now that the computational behaviour of dynamic topo-logics be-comes completely different if we allow the use of the operators �F and �F .

3 Undecidability and non-axiomatisability

Theorem 2. No logic from the list below is recursively enumerable:

1. ������� f � � ��� f � a DTS�,

2. ������n � f � � ��n � f � a DTS� n � 1�,

3. ������� f � � ��� f � a DTS, � an Aleksandrov space�,

4. �����n ��n a unit ball� n� 1�.

Remark 3. Before proceeding to the proof, we note that all logics mentioned inthis theorem are different. As was shown in [18], the formula I�F�pCI�p� is notsatisfiable in any DTS of the form ��n � f �, while it is clearly satisfiable. Accordingto [12], the formula C�Ip ���FIp� is valid in all unit balls, but refuted in botha DTS based on an Aleksandrov space and ��n �x �� x�1�. Finally, the formula�FCp � C�F p is valid in DTSs based on Aleksandrov spaces, but refuted bothin ��n � f � and in all unit balls.

We prove Theorem 2 by reduction of Post’s correspondence problem or PCP,for short [17]. (Cases (2) and (4) will only be proved for n � 1 and n � 2, respec-tively.) The idea of the proof is taken from [7]. Let A be a finite alphabet and P afinite set of pairs �v1�u1� � � � � ��vk�uk� of nonempty finite words

vi ��bi

1� � � � �bili

�� ui �

�ci

1� � � � �ciri

��i � 1� � � � �k�

4

185

Page 195: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

over A. For a sequence of indices i1� � � � � iN ranging over 1� � � � �k, let

mj � li1 � � � �� li j and nj � ri1 � � � �� ri j �

for 1 � j � N. The following problem is undecidable (for a proof see, e.g., [8]):given a set P of pairs of words as above, decide whether there exist an N � 1 anda sequence i1� � � � � iN of indices such that

vi1� � � � �viN � ui1� � � � �uiN � (1)

where � is the concatenation operation. If the condition above holds for a PCPinstance P then we say that P has a solution. Later, we will use a consequence ofthis result, namely, that the set of PCP instances without solutions is not recursivelyenumerable.

The reduction formula φA�P is constructed using the propositional variables r,s, left and right, lefta and righta, for every a � A, as well as pairi, for every pair�vi�ui� in P, 1 � i� k.

The variable s is used to introduce a new operator S (which can be interpreted asa ‘strict diamond’ in Kripke preordered frames). Namely, for every DT L-formulaψ, we put

Sψ ��s� C��sCψ�

���s� C�sCψ�

��

Denote by Sm a string of m operators S (so that S0ψ � ψ). The variable r is usedto ‘relativise’ �F in the following ways: ��r

F ψ � ��F ��Fr � ψ� and ��r

F ψ ���F ��

�F r � ψ�.

Now φA�P is defined as the conjunction

φA�P � ϕeq ϕpair ϕstripe ϕleft ϕright�

where ϕeq ��F�r�

a�A

I�lefta � righta���

ϕpair ���rF

� �1�i�k

pairi �

1�i� j�k

��pairipair j��

ϕstripe ���rF I�s ��s��

ϕleft is the conjunction of (2)–(8), for 1 � i � k,�

a ��ba�b�A

��rF I�

�lefta leftb

� ��r

F I�left �

a�A

lefta�� (2)

a�A

��rF I�lefta � �lefta�� (3)

I�left ��rF I��left ��Sleft�� (4)

��rF

�pairi � I��left ���Sli left�

�� (5)

��rF

�pairi �

j�li

�I��S jleft�S j�1left�� leftbi

li� j

��� (6)

pairi � �lwi� (7)

��rF

�pairi � I��left�Sleft�� S�lwi�

�� (8)

5

186

Page 196: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

where lwi � leftbi1S

�leftbi

2S�leftbi

3 �� �Sleftbi

li� � � �

�(remember that li is the length of the word vi �

�bi

1� � � � �bili

�), and the conjunct

ϕright—the ‘right counterpart’ of ϕleft—is defined by replacing in ϕleft all occur-rences of left with right, lefta with righta, li with ri, etc.

We also require DT L�-formulae φnA�P, n � 0, which are defined similarly to

φA�P by replacing� ϕeq with �n�

a�A I�lefta � righta�, and

� every occurrence of ��rF ψ (or ��r

F ψ) with ��nF ψ (respectively, ��n

F ψ),

where��nF ψ � ψ�ψ��ψ �� ��nψ

and�

�nF ψ � ψ�ψ��ψ �� ��n�1ψ�

Evidently, there is a correspondence between subformulae of φA�P and φnA�P; for

every subformula φ of φA�P, we denoted the corresponding subformula of φnA�P by

φn.

Lemma 4. If φA�P is satisfiable in ��� f � then there is n � 0 such that φnA�P is satis-

fiable in ��� f �.

Lemma 5. If P has a solution, then the following hold:

(i) φA�P is satisfiable in a DTS ��� f �, where � is an Aleksandrov space;

(ii) φA�P is satisfiable in a DTS;

(iii) φA�P is satisfiable in�n, n � 1;

(iv) φA�P is satisfiable in ��� f �, where f : x �� x�1.

Proof. Suppose that P has a solution

vi1� � � � �viN � ui1� � � � �uiN � (9)

Let vi1� � � � �viN � �b0� � � � �bmN�1�.

(i) Define a preorder �� �W�R� by taking W � �0� � � � �2mN���, where � isthe set of integers, and �x�y�R�x��y�� iff x � x� and y � y�. Define f : W �W bytaking f �x�y� � �x�y� 1�. Clearly, f is a homeomorphism on ��. Finally, define� by taking

��s� � ��2n�z� � 0 � 2n � 2mN � z � ��� ��r� � ��0�N���

��pairi� � ��0� j�1� � i � i j� for some j � N��

��lefta� � ��2k� j� � 0 � j � N� k � mj� bk � a�� ��left� ��

a�A��lefta��

��righta� � ��2k� j� � 0 � j � N� k � nj� bk � a�� ��right� ��

a�A��righta��

Let � � ������. It is an easy exercise to show ����0�0�� �� φA�P. We leavethis to the reader. (Informally, at every moment of time, there exists an S-related

6

187

Page 197: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

sequence of worlds representing parts of the left and right words; lefta is true on anj-th element of the sequence iff the j-th letter of the left word is a (same for righta);pairi is true iff in the i-th moment of time, we extend the left and right words withvi and ui resp.; r marks the end of the solution string.)

(ii) follows from (i).(iii) We only consider the two-dimensional unit ball �2 �

�B2�g

�, where g is

the rotation of B2 clockwise by some rational angle α such that 0 � α � 2π�N�1and N is given by (9) (for n � 2, the construction is similar: we rotate the ballaround a fixed axis by the same angle α, for n � 1 the proof differs: one has to con-sider a measure preserving piecewise-linear function on a unit interval). Obviously,g is a measure preserving homeomorphism.

Let E be an open set, say, a smaller open ball contained in the sector��α�2�α�2� of B2 and let Ei � gi�E�, for i � ω. Note that E � E0�E1� � � � �EN arepairwise disjoint sets. Let �� ��0� � � � �2mN����. By the main result of McKinseyand Tarski [15], there exists a continuous mapping hi from Ei onto ��. More-over, one may assume that, for every x � Ei, we have hi�x� � hi�1�g�x�� and thathi�ei� � 0, where ei is the centre of the ball Ei. Define a valuation � on �2 bytaking

��s� ��N

i�0�h�1i �2n� � 0 � 2n � 2mN�� ��r� � �eN��

��pairi� � �e j�1 � i � i j� for some j � N��

��lefta� ��N

j�1�h�1j �2k� � k � mj� bk � a�� ��left� �

�a�A��lefta��

��righta� ��N

j�1�h�1j �2k� � k � nj� bk � a�� ��right� �

�a�A��righta��

As α is rational, we have gj�e0� � eN iff j � N. It is not hard to check now that������ �e0� �� φA�P.

(iv) We know from (i) and Lemma 4 that there exists n � 0 such that φnA�P is

satisfiable in a DTS ��� f �. Then, by the remark following Theorem 1 and sinceφn

A�P is a DT L�-formula, ���w� �� φnA�P, for some model � � ���� f � ��� and

f : x �� x�1. Define a new valuation �� on � which coincides with � except foronly one case: now we set ���r� � � f n�w��. (Note that r does not occur in φn

A�P.)Let�� � ���� f � ����. Then clearly ����w� �� φA�P.

Lemma 6. Suppose that there exists n � 0 such that φnA�P is satisfiable in a DTS

based on an Aleksandrov space. Then P has a solution.

Proof. Suppose that ���w01� �� φN

A�P for some DTM � � ����� f � ���, where�� �W�R� is a preorder, f a homeomorphism on �� and w0

1 �W . For j � ω, let

Wj � �w �W � f j�w01�Rw��

As ���w01� �� �N�

a�A I�lefta � righta�, we have

��� f N�w01�� ��

a�A

I�lefta � righta�� (10)

7

188

Page 198: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Since ���w01� �� ϕN

stripe, for each w �W0 and each j � N, we have ���w� �� s iff��� f j�w�� �� s.

Denote by Sj, j�N, the transitive binary relation on Wj defined by taking wSjviff there is u �Wj such that wRuRv and ���w� �� s iff ���u� ��� s. Then we clearlyhave that, for every j � N and every w �Wj,

���w� �� Sψ iff there is v �Wj such that wSjv and ���v� �� ψ�

Note that, since f is a homeomorphism and in view of ���w01� �� ϕN

stripe, for allw�v �W0 and i � N, we have wS0v iff f i�w�Si f i�v�.

Let i1� � � � � iN be the sequence of indices such that, for 1 � j � N, we have��� f j�1�w0

1�� �� pairi j(ϕN

pair ensures that there is a unique sequence of this sort).We claim that (1) holds for this sequence.

For every j with 1 � j � N, let

W Lj � �w �Wj � ���w� �� left��

Call a sequence �w1� � � � �wl� of (not necessarily distinct) points from WLj an Sj-path

in W Lj of length l if w1Sjw2Sj � � �Sjwl , and set

leftwordj�w1� � � � �wl� � �a1� � � � �al� �

where the ai are the (uniquely determined by (2)) symbols from A such that���wi� �� leftai

.We show now that there is a sequence π1� � � � �πN such that, for every j with

1 � j � N, the following hold:

(a) π j � �wj1� � � � �w

jmj� is an Sj-path in WL

j of length mj, and there is no Sj-path inW L

j of length � mj;

(b) f �w01� � w1

1 and if j � 1 then wjm � f �wj�1

m �, for all m, 1 �m� mj�1;

(c) leftwordj�wj1� � � � �w

jmj� � vi1� � � ��vi j ;

(d) for every Sj-path �w1� � � � �wmj� in WLj of length mj, we have

leftword j�w1� � � � �wmj� � vi1� � � ��vi j .

Indeed, by ���w01� �� pairi1 , (7), (4) and (5), there exists an S1-path π1 in W L

1 suchthat (a)–(c) hold. Condition (d) follows from (6).

Now assume inductively that conditions (a)–(d) hold for some j� 1 with 1 �j�1 � N. Let π j�1 � �wj�1

1 � � � � �wj�1mj�1� be an Sj�1-path in WL

j�1 for which (a)–(d)

hold. By (3), the sequence�

f �wj�11 �� � � � � f �wj�1

mj�1��

is an Sj-path in WLj . Since

���wj�1mj�1� �� left�Sleft and ���wj�1

1 � �� pairi j, (8) means that there exists a

sequence wjmj�1�1� � � � �w

jmj�1�li j

of points in WLj such that

π j ��

f �wj�11 �� � � � � f �wj�1

mj�1��wj

mj�1�1� � � � �wjmj�1�li j

8

189

Page 199: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

is an Sj-path in WLj of length mj � mj�1 � li j such that

leftword j�wjnj�1�1� � � � �w

jmj�1�li j

� � vi j . By (5) and the induction hypothesis,

there is no Sj-path in WLj of length � mj. Thus, (a) and (b) hold for πj, (c) follows

from (3), and (d) from (6) and the induction hypothesis.Now define sets WR

j in the same way as WLj , but with left replaced by right, in-

troduce the notion of an Sj-path in WRj , and, for every sequence w1� � � � �wl of points

from W Rj , set rightwordj�w1� � � � �wl� � �a1� � � � �al� � where the ai are the uniquely

determined (by ‘right analogue’ of (2)) elements of A such that ���wi� �� rightai.

In precisely the same way as above we show that there is a sequence π�1� � � � �π�Nsatisfying properties analogous to (a)–(d) above. Now it is easy to see that (10)means that

vi1� � � ��viN � leftwordN�wN1 � � � � �w

NmN

�� rightwordN�wN1 � � � � �w

NnN��ui1� � � ��uiN �

as required.

Theorem 2 now follows immediately.

4 Dynamic metric logic

The language DM L of dynamic metric logic is defined in the same way as DT Lwith the exception that the topological operators are replaced by the metric op-erators ��a, for a � �� , where �� is the set of positive rational numbers. Theintended semantics of this logic is defined as follows.

A dynamic metric structure (DMS, for short) is a pair � � ��W�d� � f �, where�W�d� is a metric space (with a metric d) and f : W �W is a metric automorphism,i.e., a bijection on W such that d�x�y� � d� f �x�� f �y�� for all x�y�W . For example,the map x �� x� 1 on � and the rotation g on B2 considered above are metricautomorphisms on the respective spaces with the Euclidean metric.

A dynamic metric model (or DMM) is a pair � � �����, where � is a DMSand� a valuation defined in precisely the same way as in the topological case. Thetruth-relation is also defined in the same manner as for DTMs with the exceptionthat the truth-conditions for the topological operators I and C are replaced by

���x� �� ��aϕ iff there exists y �W such that d�x�y� � a and ���y� �� ϕ.

In contrast to the topological case, now we have the following:

Theorem 7. The set of DM L-formulae that are valid in all DMSs is decidable.However, the decision problem is not elementary.

Roughly, the idea of the decidability proof is similar to that of Theorem 13.6from [7]: first we represents DMMs in the form of quasimodels and then show thatquasimodels can be encoded in monadic second-order logic. The main novelty ofthis proof is the rather involved notion of a quasimodel.

9

190

Page 200: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Given a DM L-formula ϕ, denote by γϕ the maximal numerical parameter oc-curring in ϕ and by M�ϕ���� the smallest set containing all numerical parametersin ϕ and closed under the rule

��� if a�b �M�ϕ� and a�b� γϕ, then a�b �M�ϕ�.

Clearly, M�ϕ� is finite. Let M��ϕ� � M�ϕ���2 � γϕ�.Define the metric depth mtd�ϕ� of ϕ inductively by taking:

mtd�p� � 0� mtd���aϕ� � mtd�ϕ��a�

mtd��ϕ� � mtd�ϕ�� mtd��ϕ� � mtd�ϕ��mtd�ϕ1ϕ2� � max�mtd�ϕ1��mtd�ϕ2��� mtd��F ϕ� � mtd�ϕ��

Our first observation is that every satisfiable DM L-formula ϕ can be satisfiedin a DMS that is based on the metric space generated by some intransitive labelledtree.

Given an intransitive tree �T��� and a function δ labelling the edges of �T���with positive real numbers, we denote by �T�δ�� the tree metric space induced by�T��� and δ, i.e., for any x �� y in T , δ��x�y� is the sum of labels on the edgesoccurring in the shortest path from x to y in �T���, and δ��x�x� � 0. If δ��x�y� isbounded, then the number

max��r�x� � r the root and x � T�

is called it the radius of the tree metric space �T��.

Lemma 8. A DM L-formula ϕ �with mtd�ϕ� � 0� is satisfiable iff it is satisfiablein a DMS of the form �� � ��T ���d�� � f ��, where

� ��T��� �δ� is a labelled intransitive tree such that δ�x�y� � M��ϕ�, for allx�y � T with x � y, and �T�δ�� is of radius � mtd�ϕ�;

� �T ���d�� is the metric space with

d���x� i� ��y� j�� �

�x�y�� if i � j�

3 �mtd�ϕ�� otherwise;

� f ���x� i�� � �x� i�1�, for all �x� i� � T ��.

Proof. Suppose that ���u0� �� ϕ for some model � � ����� with � ���W�d� � f � and some point u0 �W . For any u�v �W , put

d0�u�v� � min��2 � γϕ���a �M�ϕ� � d�u�v� � a�

��M��ϕ��

Now define the required labelled tree ��T��� �δ� by taking

T � ��u0�u1� � � � �un� �n

∑i�1

d0�ui�1�ui�� mtd�ϕ�� u1� � � � �un �W��

x � y iff x � �u0� � � � �un� and y � �u0� � � � �un�un�1� �

�u0� � � � �un� ��u0� � � � �un�un�1�� � d0�un�un�1��

10

191

Page 201: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Clearly, the radius of �T�δ�� is � mtd�ϕ�.Let �� � ��T ���d�� � f ��, where d� and f � are defined as above. It is easy

to see that �T ���d�� is indeed a metric space and f� : T ��� T �� a metricautomorphism. Put, for every propositional variable p,

���p� � ���u0� � � � �un� � i� � T �� � ��� f i�un�� �� p�

and �� � �������.Denote by x0 the root �u0� of �T���. We show now by induction that, for every

ψ � sub ϕ, every i � � and every x � �u0� � � � �un� � T such that

δ��x0�x�� mtd�ϕ��mtd�ψ�� (11)

we have��� f i�un�� �� ψ iff �����x� i�� �� ψ� (12)

The basis of induction (propositional variables) follows immediately from the def-inition of ��. The case of the Booleans is trivial.

Case ψ � ��aψ�. Let ��� f i�un�� �� ��aψ�. Since f is a metric automor-phism, there is un�1 � W such that d�un�un�1� � d� f i�un�� f i�un�1�� � a and��� f i�un�1�� �� ψ�. Take y � �u0� � � � �un�un�1�. By (11), y � T . By the defini-tion of δ, we have δ��x�y� � a. By the triangle inequality,

δ��x0�y� � δ��x0�x��δ��x�y� � mtd�ϕ��mtd�ψ���

By IH, �����y� i�� �� ψ� and, since d���x� i� ��y� i�� � a, we finally obtain�����x� i�� �� ��aψ�.

Conversely, suppose �����x� i�� �� ��aψ�. Then there is �y� j� � T �� withd���x� i� ��y� j�� � a and �����y� j�� �� ψ�. By the definition of d�, we must havej � i, and so δ��x�y� � a. By the triangle inequality, δ��x0�y� � δ��x0�x� �δ��x�y� � mtd�ϕ��mtd�ψ��. By IH, ��� f i�vm�� �� ψ� for �u0�v1� � � � �vm� � y.As f is a metric automorphism, d� f i�un�� f i�vm�� � d�un�vm� and, by the defi-nition of δ�, we have d�un�vm� � δ��x�y�. Therefore, d� f i�un�� f i�vm�� � a and��� f i�un�� �� ��aψ�.

Case ψ ��ψ�.

��� f i�un�� �� �ψ� iff ��� f i�1�un�� �� ψ�

iff �����x� i�1�� �� ψ� �by IH�

iff �����x� i�� ���ψ� �by definition of f ���Case ψ ��Fψ� is similar.

It follows that �����x0�0�� �� ϕ.

Define the closure clϕ of ϕ to be the set

�ψ��ψ � ψ � subϕ�����aψ����aψ � ψ � subϕ and a �M�ϕ���

where subϕ is the set of all subformulae of ϕ. A type for ϕ is a subset ttt of clϕ suchthat

11

192

Page 202: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� for every �ψ � clϕ, ψ � ttt iff �ψ �� ttt;

� for every ψ1ψ2 � clϕ, ψ1ψ2 � ttt iff ψ1 � ttt and ψ2 � ttt.

We are now in a position to define the notion of a quasimodel for a givenformula ϕ as a set of quasistates connected by runs.

A quasistate qqq for ϕ is a triple qqq � ��Tqqq��qqq��δqqq�tttqqq�, where

� �Tqqq��qqq� is an intransitive tree with a labelling function δqqq : ��u�v� � Tqqq�Tqqq �u �qqq v� �M��ϕ� such that the radius of

�Tqqq�δ�qqq

�is bounded by mtd�ϕ�;

� tttqqq is a function associating with each u � Tqqq a type tttqqq�u� for ϕ, and

(qs1) for every u� Tqqq and every ��aψ� clϕ, we have ��aψ� tttqqq�u� iff thereis v � Tqqq such that δ�qqq�u�v� � a and ψ � tttqqq�v�;

(qs2) for no u � Tqqq, there exist two isomorphic substructures generated byimmediate �qqq-successors v1 and v2 of u and such that δqqq�u�v1� �δqqq�u�v2�.

We say that a point u � Tqqq is of index �a1� � � � �an� if u0 �qqq u1 �qqq � � � �qqq un � u,where u0 is the root of

�Tqqq��qqq

�, and ai � δqqq�ui�1�ui�, for all i, 1 � i � n. The

index of the root, u0, is ��.Let q be a function associating with each i � � a quasistate q�i� �

��Ti��i� �δi�ttt i� for ϕ. A run of index �a1� � � � �an� through q is a function r map-ping each i � � to a point r�i� � Ti of index �a1� � � � �an� such that, for every i � �

� and every �ψ � clϕ, �ψ � ttti�r�i�� iff ψ � ttti�1�r�i�1��;

� and every �Fψ � clϕ, �Fψ � ttt i�r�i�� iff ψ � ttt j�r� j�� for all j � i.

Given a set � of runs, we denote by ��a1�����an� its subset of all runs of index�a1� � � � �an�.

A quasimodel for ϕ is a pair �q���, where for every i � �, q�i� ���Ti��i� �δi�ttt i� is a quasistate for ϕ such that

(qm2) ϕ � ttt0�u0�, where u0 is the root of �T0��0�,

and � is a set of runs through q satisfying the following condition

(qm3) ��� �� /0 and, for all r � ��a1�����an�, i � � and u � Ti, if r�i� �i u andδi�r�i��u� � an�1 then there is a run r� � ��a1�����an�an�1� such that r��i� � uand r�i� �i r��i� for all i � �.

Lemma 9. A DM L-formula ϕ (with mtd�ϕ�� 0) is satisfiable in a DMM iff thereis a quasimodel for ϕ.

We can now deduce the decidability of the satisfiability problem for DM L-formulae by translating into monadic second-order logic the statement that thereexists a quasimodel for a given formula ϕ. We require a number of auxiliary

12

193

Page 203: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

formulae. Denote by Σ the set of all quasistates for ϕ. Given a quasistateqqq� ��Tqqq��qqq��δqqq�tttqqq� from Σ and a point u in Tqqq we denote the index of u by idxqqq�u�.

Introduce a unary predicate variable Pqqq for each quasistate qqq � Σ and a unary

predicate variable R�a1�����an�ψ for each ψ � clϕ and index �a1� � � � �an� with ∑n

i�1 ai �mtd�ϕ�. Given a type ttt for ϕ and such an index �a1� � � � �an�, let

χttt�R�a1�����an��x�� ��

ψ�ttt

R�a1�����an�ψ �x�

ψ�ttt

�R�a1�����an�ψ �x��

saying that the type ttt at ‘moment’ x of index �a1� � � � �an� is defined with the help of

R�a1�����an��x� ��

R�a1�����an�ψ �x�

��� ψ � clϕ �

For each index �a1� � � � �an� with ∑ni�0 ai � mtd�ϕ�, let ���0�P�x��R�a1�����an��x�� de-

note the conjunction of the three formulae�

qqq�Σ x�Pqqq�x��

u�Tqqq

idx�u���a1�����an�

χtttqqq�u��R�a1����an��x��

��

�Fψ�cl ϕ x�R�a1�����an��Fψ �x� � y

�x � y� R�a1�����an�

ψ �y����

�ψ�clϕ x�R�a1�����an��ψ �x� � R�a1�����an�

ψ �S�x���

—this is intended to say that R�a1�����an��x� defines a run of index �a1� � � � �an� through

a sequence of quasistates defined with the help of P�x� ��

Pqqq�x���� qqq � Σ

.

However, we have to refine this definition in order to ensure that condition(qm3) holds. To this end, we define, by ‘backwards’ induction on the length ofthe index, another formula ����P�x��R�a1�����an��x�� as follows. If �a1� � � � �an� ismaximal (that is, we are at the ‘leaf-level’) then take

����P�x��R�a1�����an��x�� � ���0�P�x��R�a1�����an��x���

Suppose, inductively, that for all proper extensions �a1� � � � �am� of �a1� � � � �an�

(that is, m � n) we have already defined ����P�x��R�a1�����am��x��. Then����P�x��R�a1�����an��x�� is the following formula:

���0�P�x��R�a1�����an��x�� �

qqq�Σ

u�Tqqqidx�u���a1�����an�

x�Pqqq�x� χtttqqq�u��R

�a1�����an��x�� �

v�Tqδqqq�u�v��a

�ψ�clϕ

R�a1�����an�a�ψ

�����P�x��R�a1�����an�a��x�� χtttqqq�v��R

�a1�����an�a��x��

sss�Σ

u��Tsssidx�u����a1�����an�

z�

Psss�z� χtttsss�u���R�a1�����an��z�� �

v��Tsssδsss�u��v���a

χtttsss�v���R�a1�����an�a��z��

��

13

194

Page 204: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Finally, we define a monadic second-order sentence ϕ� by taking

ϕ� � �qqq�Σ

Pqqq

� x�

qqq�Σ

�Pqqq�x�

qqq��Σqqq��qqq�

�Pqqq��x�

sss�Σ

u�Tsssidx�u����ϕ�tttsss�u�

�x�

Psss�x� �ψ�cl ϕ

R��ψ�����P�x��R���x�� χtttsss�u��R

���x���

Evaluated in �����, the first line of ϕ� says that the sets Pqqq � � (qqq � Σ) form apartition of �. By defining the map q : �� Σ as

q�i� � qqq iff i � Pqqq

we obtain a quasimodel �q��� for ϕ: the second line of ϕ� states condition (qm2);condition (qm3) is satisfied by the formulae ����P�x��R�a1�����an��x��. Therefore, itis easy to see that the following holds:

Lemma 10. For every DM L-formula ϕ, mtd�ϕ� � 0, ����� �� ϕ� iff there is aquasimodel for ϕ.

Clearly, Σ can be constructed from ϕ by an algorithm. So we can now applythe result of Buchi [5] stating the decidability of monadic second-order logic over�����.

The non-elementary lower bound can be proved by a trivial polynomial reduc-tion of the satisfiability problem for the product modal logic PTL�K (which isnon-elementary by Theorem 6.37 from [7]) to the satisfiability problem for DM L-formulae in DMSs. We leave this to the reader.

Open problems. Interesting and challenging open problems are (i) the decid-ability of dynamic topological logics interpreted in various topological spaces withcontinuous mappings, and (ii) the decidability of dynamic metric logics interpretedin various compact metric spaces; for a justification and more details see, e.g., [12].

Acknowledgements. The work on this paper was partially supported byU.K. EPSRC grants no. GR/S61973/01, GR/S63175/01, GR/S61966/01 andGR/S63182/01, and the U.S. Civilian Research & Development Foundation forthe Independent States of the Former Soviet Union (CRDF) award no. RM1-2409-ST-02.

References

[1] P. S. Alexandroff. Diskrete Raume. Matematicheskii Sbornik, 2 (44):501–518, 1937.

[2] S. Artemov, J. Davoren, and A. Nerode. Modal logics and topological semantics forhybrid systems. Tech. Rep. MSI 97-05, Cornell University, 1997.

14

195

Page 205: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[3] N. Bourbaki. General topology, Part 1. Hermann, Paris and Addison-Wesley, 1966.

[4] J. R. Brown. Ergodic Theory and Topological Dynamics. Academic Press, 1976.

[5] J. R. Buchi. On a decision method in restricted second order arithmetic. In Logic,Methodology and Philosophy of Science: Proceedings of the 1960 InternationalCongress, pages 1–11. Stanford University Press, 1962.

[6] J. Davoren and A. Nerode. Logics for hybrid systems. Proceedings of the IEEE,88:985–1010, 2000.

[7] D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-Dimensional ModalLogics: Theory and Applications, volume 148 of Studies in Logic. Elsevier, 2003.

[8] J. Hopcroft, R. Motwani, and J. Ullman. Introduction to Automata Theory, Lan-guages, and Computation. Addison–Wesley, 2001.

[9] A. Katok and B. Hasselblatt. Introduction to Modern Theory of Dynamical Systems,volume 54 of Encyclopedia of mathematics and its applications. Elsevier, 1995.

[10] P. Kremer. Temporal logic over S4: an axiomatizable fragment of dynamic topologi-cal logic. Bull. of Symb. Logic, 3:375–376, 1997.

[11] P. Kremer and G. Mints. Dynamic topological logic. Bull. of Symb. Logic, 3:371–372,1997.

[12] P. Kremer and G. Mints. Dynamic topological logic. Manuscript, 2003.

[13] P. Kremer, G. Mints, and V. Rybakov. Axiomatizing the next-interior fragment ofdynamic topological logic. Bull. of Symb. Logic, 3:376–377, 1997.

[14] O. Kutz, H. Sturm, N.-Y. Suzuki, F. Wolter, and M. Zakharyaschev. Logics of metricspaces. ACM Transactions on Computational Logic, 4:260–294, 2003.

[15] J.C.C. McKinsey and A. Tarski. The algebra of topology. Annals of Mathematics,45:141–191, 1944.

[16] T. Moor and J. Davoren. Robust controller synthesis for hybrid systems using modallogics. In M.D. Di Benedetto and A. Sangiovanni-Vincentelli, editors, Hybrid Sys-tems: Computation and Control �HSCC’01�, volume 2034 of LNCS, pages 433–446.Springer, 2001.

[17] E. Post. A variant of a recursively unsolvable problem. Bulletin of the AMS, 52:264–268, 1946.

[18] S. Slavnov. Two counterexamples in the logic of dynamic topological systems. Tech-nical Report TR–2003015, Cornell University, 2003.

15

196

Page 206: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Reduction axioms for epistemic actions

Barteld Kooi¤ and Johan van Benthem†

Abstract: Current dynamic epistemic logics often become cumbersome and opaque whencommon knowledge is added for groups of agents. We propose new versions that extend theunderlying static epistemic languages in such a way that completeness proofs for the fulldynamic systems can be obtained by perspicuous reduction axioms.

Keywords: dynamic logic, epistemic logic.

1 Introduction

Epistemic logic typically deals with what agents consider possible given their currentinformation. This includes knowledge about facts, but also higher-order information

about information that other agents have. A prime example is common knowledge.A formula ϕ is common knowledge if everybody knows ϕ, everybody knows thateverybody knows that ϕ, and so on.

The aim of dynamic epistemic logics is to analyze changes in basic and higher-order information. Completeness proofs for such logics are either easy, or hard.For instance, the logic of public announcements without common knowledge has aneasy completeness proof due to axioms such as [ϕ]2ià ↔ (ϕ → 2i[ϕ]Ã). We callthese reduction axioms, because the announcement operator is “pushed through”the epistemic operators. The completeness proof works by way of a translationthat follows the reduction axioms. Formulas with announcements are translated toprovably equivalent ones without announcements. Then completeness follows fromthe known completeness of the epistemic base logic. This approach also is taken in[2] and [1] for more general epistemic actions.

Completeness proofs for dynamic epistemic logics with common knowledge arehard. Reduction axioms are not available, as the logic with epistemic actions is moreexpressive than the logic without them [1]. In this paper we extend the base languagewith static operators in such a way that reduction axioms do work. Section 2 doesthis for public announcement logic, Section 3 for general epistemic actions. Section 4draws conclusions and indicates directions for further research. We see our proposalas more than a technical trick for smoothening completeness proofs. It also addressesa significant issue of independent interest: what is the best epistemic language fordescribing information models of a group of agents?

2 Public announcement logic

Section 2.1 is an introduction to public announcement logic. In Section 2.2 we givea new logic of relativized common knowledge. It ties in closely to the idea of viewingupdates as a kind of relativization, first introduced in [6]. This logic is expressiveenough to allow a reduction axiom for common knowledge. A proof system is definedin Section 2.3, and shown to be complete in Section 2.4. The system is extendedwith reduction axioms for public announcements in Section 2.5.

∗Department of Philosophy, University of Groningen, A-weg 30, 9718 CW Groningen, TheNetherlands, [email protected]

†ILLC, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Nether-lands & Philosophy Department, Stanford University, [email protected]

1

197

Page 207: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2.1 Language and semantics

Public announcement logic (PAL) was first developed by Plaza [5]. A public an-nouncement is an epistemic action where all agents commonly know that they learna certain formula. This is modeled by a modal operator [ϕ]. A formula of the form[ϕ]Ã is read as “Ã holds after the announcement of ϕ”. This language LPAL isinterpreted in models for epistemic logic.

Definition 1 (Epistemic models) Let a finite set of propositional variables Pand a finite set of agents N be given. An epistemic model is a triple M = (W,R, V )such that W 6= ∅ is a set of possible worlds, R : N → ℘(W × W ) assigns anaccessibility relation to each agent, and V : P → ℘(W ) assigns a set of worlds toeach propositional variable. ¤

In epistemic logic R is usually restricted to equivalence relations. In this paper wetreat the general modal case. The semantics are defined with respect to models witha distinguished ‘actual world’: (M,w).

Definition 2 (Semantics of PAL) Let a model (M,w) with M = (W, R, V ) begiven. Let i ∈ N , B ⊆ N , and ϕ,Ã ∈ LPAL. For atomic propositions, negations,and conjunctions we take the usual definition.

(M,w) |= 2iϕ iff (M, v) |= ϕ for all v such that (w, v) ∈ R(i)(M,w) |= CBϕ iff (M, v) |= ϕ for all v such that (w, v) ∈ R(B)¤

(M,w) |= [ϕ]Ã iff (M,w) |= ϕ implies (M |ϕ,w) |= Ã

where R(B) =⋃

i∈BR(i), and R(B)¤ is its reflexive transitive closure. The updated

model M |ϕ = (W ′, R′, V ′) is defined by restricting M to those worlds where ϕ

holds. Let [[ϕ]] = {v ∈ W |(M, v) |= ϕ}. Now W ′ = [[ϕ]], R′(i) = R(i) ∩ [[ϕ]]2, andV ′(p) = V (p) ∩ [[ϕ]]. ¤

A completeness proof with reduction axioms is impossible for this logic.

2.2 Relativized common knowledge

For public announcement logic there is no reduction axiom for formulas of the form[ϕ]CBÃ, given the results in [1]. However the semantic intuition is clear. If ϕ is truein the old model, then every B-path in the new model ends in a à world. This impliesthat in the old model every B-path that consists exclusively of ϕ-worlds ends in a[ϕ]à world. To facilitate this, we introduce a new operator CB(ϕ,Ã), which expressesthat every B-path which consists exclusively of ϕ-worlds ends in a à world. We callthis operator relativized common knowledge. The crucial clause in the semantics ofthe logic of relativized common knowledge, RCL, is: (M,w) |= CB(ϕ,Ã) iff

(M, v) |= Ã for all v such that (w, v) ∈ (R(B) ∩ [[ϕ]]2)¤

where (R(B)∩[[ϕ]]2)¤ is the reflexive transitive closure of R(B)∩[[ϕ]]2. The semanticsof the other operators is standard. Ordinary common knowledge can be defined withthe new notion: CBϕ ≡ CB(>, ϕ). The new operator is like the “until” of temporallogic. A temporal sentence “ϕ until Ô is true iff there is some point in the futurewhere à holds and ϕ is true up to that point.

2

198

Page 208: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2.3 Proof system

Relativized common knowledge still resembles common knowledge, and so we needjust a slight adaptation of the usual axioms1.

Definition 3 (Proof system for RCL) The proof system for RCL contains thefollowing axioms and rules:

Taut all instantiations of propositional tautologies

Dist 2i(ϕ→ Ã) → (2iϕ→ 2iÃ) (distribution)

Dist CB(ϕ,Ã → χ) → (CB(ϕ,Ã) → CB(ϕ, χ)) (distribution)

Mix CB(ϕ,Ã) ↔ (ϕ→ (Ã ∧ EB(ϕ→ CB(ϕ,Ã)))) (mix)

Ind ((ϕ→ Ã) ∧ CB(ϕ,Ã → EB(ϕ→ Ã))) → CB(ϕ,Ã) (induction)

MPϕ ϕ→ Ã

Ã(modus ponens)

Necϕ

2iϕ(necessitation)

Necϕ

CB(Ã,ϕ)(necessitation)

In the mix axiom and the induction axiom EBϕ is an abbreviation of∧

i∈B2iϕ

(everybody knows ϕ). A proof consists of a sequence of formulas such that each iseither an instance of an axiom, or it can be obtained from formulas that appearearlier in the sequence by applying a rule. If there is a proof of ϕ, we write ` ϕ. ¤

The soundness of the proof system can easily be shown by induction on the lengthof proofs, and we do not provide it explicitly.

2.4 Completeness for the static language

To prove completeness for our extended static language, we follow [4]. We takemaximally consistent sets with respect to finite fragments of the language that forma canonical model for that fragment. In particular, for any given formula ϕ we workwith a finite fragment called the closure of ϕ.

Definition 4 (Closure) The closure of ϕ is the minimal set Φ such that 1. ϕ ∈ Φ,2. Φ is closed under taking subformulas, 3. If à ∈ Φ and à is not a negation, then¬Ã ∈ Φ, 4. If CB(Ã, χ) ∈ Φ, then 2i(à → CB(Ã, χ)) ∈ Φ for all i ∈ B. ¤

Definition 5 (Canonical model) The canonical model Mϕ for ϕ is the triple(Wϕ, Rϕ, Vϕ) where Wϕ = {Γ ⊆ Φ | Γ is maximally consistent in Φ}; (Γ,∆) ∈ Rϕ(i)iff à ∈ ∆ for all à with 2ià ∈ Γ; and Vϕ(p) = {Γ|p ∈ Γ}. ¤

Next, we show that a formula in such a finite set is true in the canonical modelwhere that set is taken to be a world, and vice versa.

Lemma 1 (Truth Lemma) For all à ∈ Φ, à ∈ Γ iff (Mϕ,Γ) |= Ã. ¤

Proof (A sketch:) By induction on Ã. The cases for propositional variables, nega-tions, conjunction, and individual epistemic operators are straightforward. There-fore we focus on the case for relativized common knowledge.

1It is also helpful to write CB(ϕ,ψ) as a sentence in PDL: [?ϕ; (S

i∈B i; ?ϕ)∗]ψ. Our proof system

below essentially follows the usual PDL-axioms for this formula.

3

199

Page 209: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

From left to right. Suppose CB(Ã, χ) ∈ Γ. If à 6∈ Γ, then by the inductionhypothesis (Mϕ,Γ) 6|= Ã, and by the semantics (Mϕ,Γ) |= CB(Ã, χ).

Otherwise, if à ∈ Γ, take any ∆ ∈ Wϕ such that (Γ,∆) ∈ (R(B) ∩ [[Ã]]2)¤. Wehave to show that ∆ |= χ, but we can show something stronger, namely that ∆ |= χ

and CB(Ã, χ) ∈ ∆. This is done by induction on the length of the path from Γ to∆ and the Mix axiom. We omit the details.

From right to left. Let (Mϕ,Γ) |= CB(Ã, χ). Now consider the set Λ ={±∆|(Γ,∆) ∈ (R(B)∩ [[Ã]]2)¤}, where ±∆ is

{Ã|Ã ∈ ∆}.. Let ±Λ =∨

∆∈Λ±∆ We can

show that ` ±Λ → EB(Ã → ±Λ). By necessitation then ` CB(Ã, ±Λ → EB(Ã → ±Λ)).Applying the induction axiom, we get ` (Ã → ±Λ) → CB(Ã, ±Λ). Since ` ±Λ → χ,we also get ` (Ã → ±Λ) → CB(Ã, χ). Now ±Γ → ±Λ, and hence ` ±Γ → (Ã → ±Λ).Therefore CB(Ã, χ) ∈ Γ. ¤

This argument is an easy adaptation of the usual completeness proof for commonknowledge, reinforcing our idea that our language extension is a natural one: sinceexisting arguments yield more than is usually realized.

Theorem 1 (Completeness for RCL) If |= ϕ, then ` ϕ. ¤

Proof Let 6` ϕ, i.e. ¬ϕ is consistent. One easily finds a maximally consistent setΓ in the closure of ¬ϕ with ¬ϕ ∈ Γ, as only finitely many formulas matter. By theTruth Lemma, (M¬ϕ,Γ) |= ¬ϕ, i.e., (M¬ϕ,Γ) 6|= ϕ ¤

2.5 Reduction axioms

Next, let RCL+ be the epistemic dynamic logic with both relativized common knowl-

edge and public announcements. Its semantics combines those for PAL and RCL.RCL

+ is no more expressive than RCL by a direct translation.

Definition 6 (Translation) The translation function t takes a formula from thelanguage of RCL

+ and yields a formula in the language of RCL.

t(p) = p

t(¬ϕ) = ¬t(ϕ)t(ϕ ∧ Ã) = t(ϕ) ∧ t(Ã)t(2iϕ) = 2it(ϕ)t(CB(ϕ,Ã)) = CB(t(ϕ), t(Ã))

t([ϕ]p) = t(ϕ) → p

t([ϕ]¬Ã) = t(ϕ) → ¬t([ϕ]Ã)t([ϕ](à ∧ χ)) = t([ϕ]Ã) ∧ t([ϕ]χ)t([ϕ]2iÃ) = t(ϕ) → 2it([ϕ]Ã)t([ϕ]CB(Ã, χ)) = CB(t(ϕ) ∧ t([ϕ]Ã), t([ϕ]χ))t([ϕ][Ã]χ) = t([ϕ]t([Ã]χ)) ¤

Lemma 2 (Translation Correctness) For all dynamic-epistemic formulas ϕ andall models (M,w), (M,w) |= ϕ iff (M,w) |= t(ϕ). ¤

This observation underlies the soundness of the following reduction axioms, withC-Red the crucial reduction of relativized common knowledge.

Definition 7 (Proof system for RCL+) The proof system for RCL

+ is that forRCL plus the following reduction axioms:

At [ϕ]p↔ (ϕ→ p) (atoms)PF [ϕ]¬Ã ↔ (ϕ→ ¬[ϕ]Ã) (partial functionality)Dist [ϕ](à ∧ χ) ↔ ([ϕ]à ∧ [ϕ]χ) (distribution)KA [ϕ]2ià ↔ (ϕ→ 2i[ϕ]Ã) (knowledge-announcement)C-Red [ϕ]CB(Ã, χ) ↔ CB(ϕ ∧ [ϕ]Ã, [ϕ]χ) (common reduction)

as well as an inference rule of necessitation for all announcement modalities. ¤

4

200

Page 210: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The formulas on the left of these equivalences are of the form [ϕ]Ã. In At the an-nouncement operator no longer occurs on the right-hand side. In the other reductionaxioms formulas within the scope of an announcement are of higher complexity onthe left than on the right.

Theorem 2 (Completeness for RCL+) If |= ϕ, then ` ϕ. ¤

Proof The proof system for RCL is complete (Theorem 1), and every formula inL

RCL+ is provably equivalent to one in LRCL. ¤

2.6 Model comparison games

The notion of relativized common knowledge is of independent interest, just as irre-ducibly binary general quantifiers (such as Most A are B) lead to natural completionsof logics with only unary quantifiers. We provide some more information throughcharacteristic games.

Definition 8 (Relativized common knowledge game) Let two models M =(W,R, V ) and M ′ = (W ′, R′, V ′) be given. Starting from each w ∈W and w′ ∈W ′,the n-round relativized common knowledge game between Spoiler and Duplicator isgiven as follows. In each round Spoiler can initiate one of two scenarios:

2i-move Spoiler chooses a point x in one model which is an i-successorof the currentw or w′, and Duplicator responds with a matching successor y in the othermodel. Play continues with the new link x, y.

CB-move Spoiler chooses a B-path x0 . . . xn in either of the models with x0 thecurrent w or w′. Duplicator responds with a B-path y0 . . . ym in the othermodel, with y0 = w′. Then Spoiler can (a) make the end points xn, ym theoutput of this round, or (b) he can choose a world z on the M ′-path, andDuplicator must respond by choosing a matching world u on the M -path, andz, u becomes the output. ¤

The game continues with the new output states. If these differ in their atomicproperties, Spoiler wins – otherwise, a player loses whenever he cannot perform amove while it is his turn. If Spoiler has not won after all n rounds, Duplicator winsthe whole game.

Definition 9 (Modal depth) The modal depth of a formula is defined by: d(⊥) =d(p) = 1, d(¬ϕ) = d(ϕ), d(ϕ∧Ã) = max(d(ϕ), d(Ã)), d(2iϕ) = d(ϕ)+1, d(CB(ϕ,Ã)) =max(d(ϕ), d(Ã)) + 1 If two models (M,w) and (M ′, w′) have the same theory up todepth n, we write (M,w) ≡n (M ′, w′). ¤

The following result holds for most logical languages.

Lemma 3 (Propositional finiteness) For every n, up to depth n, there are onlyfinitely many different propositions up to logical equivalence. ¤

Theorem 3 (Adequacy) Duplicator has a winning strategy for the n-round gamefrom (M,w), (M ′, w′) iff (M,w) ≡n (M ′, w′). ¤

Proof The argument is by induction on n. The base case is obvious, and all in-ductive cases are also standard in modal logic, except that for relativized common

5

201

Page 211: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

knowledge. As usual, perspicuity is increased somewhat by using the dual existentialmodality CB(ϕ,Ã).

First, suppose that Duplicator has a winning strategy in games of length n+ 1,and that (M,w) |= CB(ϕ,Ã), a formula of depth n+1. By the truth definition, thereis a finite sequence of ϕ-worlds inM starting from w, and ending in a world v where Ãholds. Suppose this sequence is picked by Spoiler as his opening move. Duplicator’swinning strategy produces a matching sequence in M ′, starting at w′ and endingin some world v′. Suppose now that Spoiler outputs the end points v, v′ as thenew link. Duplicator’s winning strategy then works for the n-round game startingfrom M, v, M ′, v′, and by the inductive hypothesis, (M ′, v′) |= Ã. Suppose nextthat Spoiler chooses any s′ on the finite M ′-sequence. Then Duplicator’s winningstrategy yields a matching point s on Spoiler’s initial sequence, and again Duplicatorhas an n-round winning strategy left for (M, s), (M ′, s′). Once more by the inductivehypothesis, then, (M, s′) |= ϕ. Thus, (M ′, w′) |= CB(ϕ,Ã).

Conversely, suppose that (M,w) ≡n+1 (M ′, w′). A winning strategy for Dupli-cator in the (n + 1)-round game can be described as follows. If Spoiler makes anopening move of type [2i-move], then the usual modal argument works. Next, sup-pose that Spoiler opens with a finite sequence in one of the models, say M , withoutloss of generality. By the Finiteness Lemma, we know that there is only a finitenumber of complete descriptions of points up to logical depth n, and each point s inthe sequence satisfies one of these: say ∆(s, n). In particular, the end point v sat-isfies ∆(v, n). Let ∆(n) be the disjunction of all formulas ∆(s, n) occurring on thepath. Then, the initial world w satisfies the following formula of modal depth n+1:CB(∆(n),∆(v, n)). By our assumption, we also have (M ′, w′) |= CB(∆(n),∆(v, n)).But any sequence witnessing this by the truth definition is a response that Duplica-tor can use for her winning strategy. Whatever Spoiler does in the rest of this round,Duplicator always has a matching point that is n-equivalent in the language. ¤

Thus, games for LRCL are straightforward. But it is also of interest to look atthe extended dynamic language L

RCL+ with announcement modalities. Here, the

shift modality passing to definable submodels requires a new type of move, whereplayers can decide to change the current model. The following description of whathappens is ‘modular’: a model changing move can be added to model comparisongames for ordinary epistemic logic (perhaps with common knowledge), or for ourrelativized common knowledge game. By way of explanation: we let Spoiler proposea model shift. Players first discuss the ‘quality’ of that shift, and Duplicator can winif it is deficient; otherwise, the shift really takes place, and play continues within thenew models. This involves a somewhat unusual sequential composition of games,but perhaps one of independent interest.

Definition 10 (Public announcement move) Let the setting be the same as forthe n-round game in Definition 9.

[ϕ]-move Spoiler chooses a number r < n, and sets S ⊆ W and S ′ ⊆ W ′, with thecurrent w ∈ S and likewise w′ ∈ S′. Stage 1 : Duplicator chooses states s inS ∪ S′, s in S ∪ S′. Then Spoiler and Duplicator play the r-round game forthese worlds. If Duplicator wins this subgame, she wins the n-round game.Stage 2 : Otherwise, the game continues in the relativized models M |S,w andM ′|S′, w′ over n− r rounds. ¤

The definition of depth is easily extended to formulas [ϕ]Ã as d([ϕ]Ã) = d(ϕ) +d(Ã). For the sake of illustration, assume that the new move has been added to the

6

202

Page 212: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

relativized common knowledge game.

Theorem 4 (Adequacy) Duplicator has a winning strategy for the n-round gameon (M,w) and (M ′, w′) iff (M,w) ≡n (M ′, w′) in L

RCL+ . ¤

Proof We only discuss the inductive case demonstrating the match between an-nouncement modalities and model-changing steps.

First, let Duplicator have a winning strategy in the n-round game between M,w

and M ′, w′. Suppose that (M,w) |= 〈ϕ〉Ã, with total modal depth n. Consider thecase where Spoiler chooses r = d(ϕ) and sets S, S ′ equal to the extensions of ϕ inthe two models. Then Spoiler has a winning strategy in all r-round games in Stage1, exploiting the ϕ,¬ϕ-difference between whatever points Duplicator chooses: bythe inductive hypothesis for r = d(ϕ). Suppose that Spoiler plays such a strategy.In that case, the n − r-round subgame for the relativized submodels is reached,and Duplicator must have a winning strategy there. But that means, now by theinductive hypothesis for n− r, that (M ′|ϕ,w′) |= Ã, and hence (M ′, w′) |= 〈ϕ〉Ã.

Next, suppose that (M,w), (M ′, w′) are equivalent up to depth n. We need todescribe Duplicator’s winning strategy. Consider any opening choice of r, S, S ′ madeby Spoiler. Case 1 : Duplicator can choose two points s, s in Stage 1 giving her awinning strategy in the initial r-round game. Then we are done. Case 2 : Duplicatorhas no such winning strategy, which means that Spoiler has one – or equivalentlyby the inductive hypothesis, there is some formula of depth r distinguishing s froms. In that case, we can find a formula A defining both set S in M and S ′ in M ′. Tofind this, consider any point x in S. Using the preceding observation, we can find aformula ±x of depth n which holds at x but at no world in M −S or M ′ −S′. (Notethat there can be infinitely many worlds involved in the comparison, but finitelymany difference formulas will suffice by the Finiteness Lemma, which also holds forthis extended language.) Let ∆S be the disjunction of all these ±x. A formula ∆′

S

is found likewise, and we let A be the disjunction of ∆′

Sand ∆S . It is easy to see

that this formula of depth r defines S in M and S ′ in M ′. Now we use the givenlanguage equivalence between M,w and M ′, w′ with respect to all depth n-formulas〈A〉à where à runs over all formulas of depth n− r. We can conclude that M |A,w

and M ′|A,w′ are equivalent up to depth n− r, and hence Duplicator has a winningstrategy for the remaining game, by the inductive hypothesis. ¤

Finally, our two games must be related, since LRCL

+ has the same expressive poweras LRCL. This means that players who can win one of our games should also beable to win the other, given suitable game lengths. An explicit description of therelevant strategy conversion is beyond the scope of this paper.

2.7 Complexity

Update logics are about processes that manipulate information, and hence they raisenatural questions of complexity. In particular, all of the usual complexity questionsconcerning a logical system make sense:

Model checking : When is a formula true in a model, i.e., when do agents knowgiven propositions in an information state, or when do specific epistemic actions inthe model produce specified effects?

Satisfiability testing : When does a formula have a model, or more generally:when can we find an informational setting realizing given epistemic specifications?Or, in terms of validity : e.g., when will a given epistemic action always producesome global specified effect?

7

203

Page 213: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Model comparison: When do two given models satisfy the same formulas, orequivalently, when can Duplicator win any game over them - or: when do tworepresentations describe the same information about some group of agents?

Now technically, the translation of definition 6 combined with known algorithmsfor model checking, satisfiability, validity, or model comparison for epistemic logicyield similar algorithms for public announcement logic. But, worst case, the lengthof the translation of a formula is exponential in the length of the formula. E.g.,the translation of ϕ occurs three times in that of [ϕ]CB(Ã, χ), and hence a directcomplexity analysis is worth-while.

Lemma 4 Deciding whether a model (M,w) satisfies a formula ϕ ∈ LRCL is com-putable in polynomial time in the length of ϕ and the size of M . ¤

Proof Let ‖M‖ be the cardinality of the set of worlds of M plus that of the ac-cessibility relation of M , and | ϕ | the length of ϕ. A formula ϕ has at most | ϕ |

subformulas. We make a list ϕ0, . . . , ϕn (where ϕn = ϕ) of these such that for allformulas Ã, their subformulas occur earlier. We process the list by successively la-beling the worlds in the model where the list formulas ϕi hold. The two crucialcases are as follows. For 2iÃ, we check for a given world in the model whether Ãholds in every accessible world. Since we have already labeled the states where thesubformula à holds, this can be done in ‖M‖ steps. For CB(Ã, χ) we proceed asfollows. First label all those worlds with ¬Ã as worlds where CB(Ã, χ) holds, andlabel worlds where à and ¬χ hold as worlds where CB(Ã, χ) fails. Then iterate thefollowing step until the labeled set does not grow anymore: pick an unlabeled worldthat can reach a world labeled with ¬CB(Ã, χ) in a single i-step (for any i ∈ B) andalso label it as a world where CB(Ã, χ) fails. Each round takes at most ‖M‖ stepsfor checking accessibilities, and the total set of labelled worlds can grow at most‖M‖ steps. When the set stops growing, all still unlabelled worlds are labelled withCB(Ã, χ). By induction of formula complexity, this algorithm can be proved correct.So, the complexity of model checking for RCL is in time O(|ϕ | ×‖M‖2). ¤

This algorithm does not suffice for the case with public announcements. The truthvalues of ϕ and à in the given model do not fix that of [ϕ]Ã. We must also knowthe value of à in the model restricted to ϕ worlds.

Lemma 5 Deciding whether a model (M,w) satisfies a formula ϕ ∈ LRCL+ is com-putable in polynomial time in the length of ϕ and the size of M . ¤

Proof Again there are at most |ϕ | subformulas of ϕ. Now we make a binary treeof these formulas which splits with formulas of the form [Ã]χ. On the left subtreeall subformulas of à occur, on the right all those of χ. This tree can be constructedin time O(| ϕ |). Labeling the model is done by processing this tree from bottomto top from left to right. The only new case is when we encounter a formula [Ã]χ.In that case we first label those worlds where à does not hold as worlds where [Ã]χholds, then we process the right subtree under [Ã]χ where we restrict the model to Ãworlds. After this process we label those worlds that were labeled with χ as worldswhere [Ã]χ holds and the remaining as worlds where it does not hold. We can seeby induction on formula complexity that this algorithm is correct.

Also by induction on ϕ, this algorithm takes time O(| ϕ | ×‖M‖2). The onlydifficult step is labeling the model with [Ã]χ. By the induction hypothesis, restrictingthe model to à takes time O(| à | ×‖M‖2). We simply remove (temporarily) all

8

204

Page 214: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

worlds labelled ¬ϕ and all arrows pointing to such worlds. Again by the inductionhypothesis, checking χ in this new model takes O(|Ã | ×‖M‖2) steps. The rest ofthe process takes ‖M‖ steps. So, this step takes over-all time O(| [Ã]χ | ×‖M‖2). ¤

Moving on from model checking, the satisfiability and the validity problem of epis-temic logic with common knowledge are both EXPTIME-complete. In fact, thisis true for almost any logic that contains a transitive closure modality. Satisfiabil-ity and validity for PDL are also EXPTIME-complete. Now there is a linear timetranslation of the language of RCL to that of PDL. Therefore the satisfiability andvalidity problems for RCL are also EXPTIME-complete. For L

RCL+ and even PAL,

however, the complexity of satisfiability and validity are still unknown.Finally, the complexity of model comparison for finite models is the same as

that for ordinary epistemic logic, viz. PTIME. The reason is that even basic modalequivalence on finite models implies the existence of a bisimulation, while all ourextend languages are bisimulation-invariant.

2.8 Other logics with relativization

Languages with relativizations are very common in logic. Indeed, closure under rel-ativization is sometimes stated as a defining condition on logics in abstract modeltheory. Basic modal or first-order logic as they stand are closed under relativizations[A]ϕ, often written (ϕ)A. And the same is true for logics with fixed-point construc-tions, like PDL (cf. [6]) or the modal µ-calculus. E.g., computing a relativized leastfixed-point [A]µp.ϕ(p) yields the same result as µp.ϕ(p) ∧ A. Relativization looksmuch like restricting quantifiers, as in the earlier-mentioned shift from unary “Mostobjects are Ô to binary “Most ϕ are Ô. By ‘Conservativity’ for generalized quan-tifiers, ”Most ϕ are Ô is equivalent to ”Most ϕ are à ∧ ϕ. But note that the latterprinciple does not relativize à to evaluation wholly inside the ϕ-area! Thus, theexpressive power of the two sorts of extension is not evidently the same. A similarissue arises in our setting. We defined LRCL as an extension with binary commonknowledge in the second sense. We have shown how this allows us to define all rel-ativizations and all ordinary common knowledge operators, i.e., the language PAL.But the converse is still open.

Question: Do LRCL and LPAL have the same expressive power?

If the answer to this question is negative, we would have two competing relativization-closed versions of epistemic logic with common knowledge, even though LRCL seemsthe more elegant one of the two.

3 Logic of epistemic actions

Our proposed methodology for epistemic logic with announcements also works moregenerally. In this section, we make the same move in the general dynamic logic ofepistemic actions, which also lacks a reduction axiom for common knowledge. InSection 3.1 we introduce the logic of epistemic action LEA. In Section 3.2 we brieflyintroduce a variant of PDL, called automata PDL, which is our technical tool for cre-ating a suitably enriched base language for LEA that allows for perspicuous reductionaxioms for common knowledge. These axioms are introduced in Section 3.3.

9

205

Page 215: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3.1 Language and semantics

Dynamic actions with epistemic aspects, such as communication or other information-bearing events, are quite similar to static epistemic situations. In [1] this analogy isused as the engine for general update of epistemic models under epistemic actions. Inparticular, individual actions come with preconditions holding only at those worldswhere they can be performed.

Definition 11 (Action models) An action model for a finite set of agents N witha language L is a triple A = (W,R, pre) where W 6= ∅ is a set of actions; R : N →

℘(W × W) assigns an accessibility relation to each agent, and pre : W → L assignsa precondition in L to every action. A pair (A,w) is an action model with adistinguished actual action w ∈ W. ¤

Here L can be any language that can be interpreted in the models of definition 1.The effect of executing an action is modeled by the following product construction.

Definition 12 (Execution) Given a static epistemic model (M,w) and an actionmodel (A,w) with (M,w) |= pre(w), we say that the result of executing (A,w)in (M,w) is the static model (M · A, (w,w)) = ((W ′, R′, V ′), (w,w)) where W ′ ={(v, v) | (M, v) |= pre(v)}, R′(i) = {((v, v), (u, u)) | (v, u) ∈ R(i) and (v, u) ∈ R(i)},and V ′(u, v) = V (u). ¤

Definitions 11 and 12 provide a semantics for the logic of epistemic action LEA of[1]. The basic epistemic language LLEA is extended with dynamic modalities [A,w]ϕ,where a A is any finite action model for LLEA. These say that “every execution of(A,w) yields a model where ϕ holds”:

(M,w) |= [A,w]ϕ iff (M,w) |= pre(w) implies that (M · A, (w,w)) |= ϕ

[1] presents a proof system for this logic with a complicated completeness proof, andwithout reduction axioms for common knowledge (which were already lacking forpublic announcement actions). So, we must extend this language to get reductionaxioms after all. Again, the semantic intuition about the crucial case (M,w) |=[A,w]CBϕ is clear. It says that, if there is a B-path w0, . . . , wn (with w0 = w) inthe static model and a matching B-path w0, . . . ,wn (with w0 = w) in the actionmodel with (M,wi) |= pre(wi) for all i ≤ n, then (M,wn) |= ϕ. To express allthis in the initial static model, it turns out convenient to choose a representation ofcomplex epistemic assertions that meshes well with action models. Now, the relevantfinite paths in static models involve strings of agent accessibility steps and tests onformulas, as programs in dynamic logic are associated with regular string languages.These are the sort of object that can be recognized by a finite automaton. Butaction models resemble finite automata, too, with regular languages of accessibilitytransitions and tests for preconditions. All this leads us to automata PDL a variantof PDL where finite automata tag modalities, rather than programs.

3.2 Automata PDL

The system for APDL presented here is taken from [3, Section 10.3], where relevantbasic references can be found for what follows. Here, in our epistemic perspective,atomic programs will be viewed as agents.

Definition 13 (Finite automata) Let Σ be an alphabet. A finite automaton for

Σ is a quadruple A = (S, I, F, ±), where S is a finite set of states; I, F ∈ S are the

10

206

Page 216: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

initial state and the final state of the automaton respectively; and ± ⊆ S ×Σ× S isthe set of transitions. ¤

Intuitively, if (s0, σ, s1) ∈ ±, then s1 is reachable from s0 by symbol σ. A path fromthe initial state to the final state yields a string, which is said to be accepted by theautomaton. Ordinary finite automata theory allows more than one final state, butour set-up loses no generality (cf. [3]).

Definition 14 (Acceptance) A string σ1 . . . σn ∈ Σ¤ is accepted by A = (S, I, F, ±)iff there exists a sequence of states s0, . . . sn such that for all i < n it holds that(si, σi, si+1) ∈ ±, where s0 = I and sn = F . ¤

In APDL, automata feature as modal operators. Their alphabet consists of theatomic programs together with tests on formulas of the language itself.

Definition 15 (Language of APDL) Let a set of atomic programs Π be given.The language LAPDL is given by the following BNF:

ϕ ::= p | ¬ϕ | ϕ1 ∧ ϕ2 | [i]ϕ | [?ϕ1]ϕ2 | [A]ϕ

where i ∈ N and A is an automaton over Π ∪ {?ϕ | ϕ ∈ LAPDL}. ¤

A formula of the form [A]ϕ should be read as “ϕ holds after every execution of astring accepted by A”.

Definition 16 (Semantics of APDL) Let (M,w) be any model withM = (W,R, V ).Let i ∈ Π, and ϕ,Ã ∈ LAPDL. For atomic propositions, negations, and conjunctionswe take the usual definition. Next, we set

(M,w) |= [i]ϕ iff (M, v) |= ϕ for all v such that (w, v) ∈ R(i)(M,w) |= [?ϕ]Ã iff (M, v) |= ϕ implies that (M, v) |= Ã

(M,w) |= [A]ϕ iff (M, v) |= ϕ for all v and ~σ such that (w, v) ∈ [[~σ]]and ~σ is accepted by A

[[i;~σ]] = R(i) ◦ [[~σ]][[?ϕ;~σ]] = {(w,w)|(M,w) |= ϕ} ◦ [[~σ]][[ε]] = {(w,w) | w ∈W}

where ε is the empty string. ¤

APDL and PDL have the same expressive power [3], but the former system offersa more convenient ‘intensional’ description of actions, which we will exploit in ouraccount of reduction axioms. In particular, given the earlier-mentioned connectionbetween epistemic logic and PDL, one can also translate epistemic logic into APDL.Agents’ accessibility relations become atomic programs, and e.g., common knowledgeamong group B involves a single-state automaton ACB

= ({0}, 0, 0, ±) with ± ={0} ×B × {0}. Henceforth, we will think of APDL in this epistemic guise.

3.3 Reduction axioms

[3] has a sound and complete proof system for APDL by itself. Now we show thatadding epistemic actions to this static language does not increase expressive power,while we can also find a complete proof system with reduction axioms.

11

207

Page 217: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

p ¬p

>

p

¬p

>

Figure 1: A three-state epistemic action model and its six-node automaton. The middlenodes are only reachable by executing the appropriate preconditions, which labelthe arrows. Solid lines are transitions for i, dashed lines for j.

Definition 17 (Language of APDL+) Let a finite set of propositional variables

P and a finite set of atomic programs Π be given. The language LAPDL

+ is givenby the following BNF:

ϕ ::= p | ¬ϕ | ϕ1 ∧ ϕ2 | [i]ϕ | [?ϕ1]ϕ2 | [A]ϕ | [A,w]ϕ

where p ∈ P , i ∈ Π, A is an automaton over N ∪ {?ϕ | ϕ ∈ LAPDL

+} and (A,w) isan action model for L

APDL+ . ¤

To translate formulas of the form [A,w]ϕ to the language without epistemic ac-tions, we can use reduction axioms from the logic of epistemic actions without com-mon knowledge, but now there is an extra case, namely for sentences of the form[A,w][A]ϕ, with complex epistemic postconditions [A]ϕ such as common knowledgestatements. Our idea is to merge the two modalities using a product of two automata:one being the epistemic [A], and one for the action model [A,w].

As to the latter, we construct automata A(A,w,v) for each reachable action world, v, making paths in A(A,w,v) and (A,w) correspond. By way of explanation, actionworlds play two roles: their preconditions determine whether the action can beexecuted, but they are also serve as epistemic alternatives for agents. A path in ourautomaton takes both roles into account by having two copies of each world. Agentscan reach only one sort of copy, from which the other is accessible by executing theright precondition. From the latter, agents can again reach other states.

Definition 18 (Automata for action models) Let A = (W,R, pre) be an actionmodel, with worlds w, v. The automaton A(A,w,v) is the four-tuple (S, I, F, ±) whereS = W × {0, 1}, I = (w, 0), F = (v, 1), and ± = {((u, 0), ?pre(u), (u, 1)) | u ∈

W} ∪ {((u, 1), i, (t, 0)) | (u, t) ∈ R(i)} for all agents i. ¤

For instance, consider an epistemic action model for two agents i and j where nothinghappens or agent i is informed about the the truth of p. Agent i knows exactly whatis going on, but j does not know what is going on at all. This action model is shownon the left in Figure 1. Its automaton is shown on the right.

Now we must combine the two sorts of automata. In the simple case of epistemiccommon knowledge, the automaton for the action model itself would be virtuallywhat we want, but in general, the desired paths, and hence the combined automaton

12

208

Page 218: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

must be restricted. For this we use a kind of multiplication, which is actaully notunlike product update.

Definition 19 Let an action model A, two action worlds w, v ∈ W, and an automa-ton A be given. Then the automaton A(A,w,v) ⊗ A = (S′, I ′, F ′, ±′) is defined asfollows: S′ = W × {0, 1} × S, I ′ = ((w, 0, I), F ′ = ((v, 1, F ) and

±′ = {((u, 1, sj), i, ((t, 0, sk) | (u, t) ∈ R(i) and (sj , i, sk) ∈ ±}∪

{((u, 0, s), ?ϕ, ((u, 1), s) | pre(u) = ϕ}∪

{((u, 1, si), ?〈A, u〉ϕ, ((u, 1, sj) | (si, ?ϕ, sj) ∈ ±)

As a special case, multiplication with the epistemic automaton for common knowl-edge yields nothing new, i.e. AA,w,v ⊗ ACN

= AA,w,v. Now we can translate thelanguage of APDL

+ to the language of APDL.

Definition 20 (Translation) The translation map t takes a formula or automatonfrom the language of APDL

+ and yields a formula of APDL:

t(p) = p

t(¬ϕ) = ¬t(ϕ)t(ϕ ∧ Ã) = t(ϕ) ∧ t(Ã)t([i]ϕ) = [i]t(ϕ)t([?ϕ]Ã) = t(?ϕ) → t(Ã)t([A]ϕ) = [t(A)]t(ϕ)t([A,w]p) = t(pre(w)) → p

t([A,w]¬ϕ) = t(pre(w)) → ¬t([A,w]ϕ)t([A,w](ϕ ∧ Ã) = t([A,w]ϕ) ∧ t([A,w]Ã)t([A,w][i]ϕ) = t(pre(w)) →

(w,v)∈R(i)[i]t([A, v]ϕ)

t([A,w][?ϕ]Ã) = t([A,w](ϕ→ Ã))t([A,w][A]ϕ) =

v∈Wt([A(A,w,v) ⊗A][A, v]ϕ)

t([A,w][A′,w′]ϕ) = t([A,w]t([A′, αwone′]ϕ))

where t(A) is the automaton where every occurrence of a formula as a condition hasbeen replaced by its t-translation. ¤

Every formula is provably equivalent to its translation. This is essentially the sound-ness of the following proof system.

Definition 21 (Proof system for APDL+) The proof system for APDL

+ con-sists of all the axioms and rules of APDL plus the following axioms:

At [A,w]p↔ (pre(w) → p) (atoms)PF [A,w]¬Ã ↔ (pre(w) → ¬[A,w]à (partial functionality)Dist [A,w](à ∧ χ) ↔ ([A,w]à ∧ [A,w]χ) (distribution)KA [A,w][i]ϕ↔ (pre(w) →

(w,v)∈R(i)[i][A, v]ϕ (knowledge-action)

Red [A,w][A]ϕ↔∧

v∈W[A(A,w,v) ⊗A][A, v]ϕ (reduction axiom)

plus necessitation for action model modalities. ¤

The difficult case for the soundness of these axioms is the reduction axiom.

Lemma 6 [A,w][A]ϕ is equivalent to∧

v∈W[A(A,w,v) ⊗A][A, v]ϕ ¤

13

209

Page 219: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Proof (Sketch only:) From left to right. By contraposition. Suppose (M,w) |=∨

v∈W〈A(A,w,v) ⊗A〉〈A, v〉ϕ. Let 〈A(A,w,v) ⊗A〉〈A, v〉ϕ be a disjunct that makes the

formula true. Therefore there is a string ~σ = σ1 . . . σn accepted by A(A,w,v) ⊗A suchthat some path w0, . . . , wn runs from w to v with (w, v) ∈ [[~σ]] and (M, v) |= 〈A, v〉ϕ.Let ((w, 0, I), . . . , (v, 1, F ) be the path in A(A,w,v) ⊗ A that yields ~σ. We can nowmake two strings: a string σA(A,w,v)

that is accepted by A(A,w,v) and a string σ′A(A)

that is accepted by A. These can be “read” from the path ((w, 0), I), . . . , ((v, 1), F ).In particular, (w, v) ∈ [[σA(A,w,v)

]], because of the path w0, . . . , wn. In the productupdate model M ·A this path becomes ((w0,w0), . . . , (wn,wn)). Along this path σAcan be executed. Therefore (M,w) |= 〈A,w〉〈A〉ϕ.

From right to left. Suppose (M,w) |= 〈A,w〉〈A〉ϕ. Then (M ·A, (w,w)) |= 〈A〉ϕ.Therefore some string ~σ = σ1 . . . σn is accepted by A such that ((w,w), (v, v)) ∈

[[~σ]]M ·A and (M ·A, (v, v)) |= ϕ. Therefore there is a path w0, . . . wn from w to v anda path w0, . . . ,wn from w to v such that (M,wi) |= pre(wi). Let ~σ′ = σ′

1. . . σ′n be the

string obtained by replacing every à in ~σ with 〈A, u〉Ã. Then (w, v) ∈ [[~σ′]]M . Let~σ′′ be the string obtained from ~σ′ by prefixing it with ?pre(w), while every σ′

iof the

form i is replaced by i; ?pre(wi). It can be shown that ~σ′′ is accepted by A(A,w,v)⊗A

and that (w, v) ∈ [[~σ]]. Therefore (M,w) |=∨

v∈W〈A(A,w,v) ⊗A〉〈A, v〉ϕ. ¤

Theorem 5 (Completeness) If |= ϕ, then ` ϕ. ¤

Proof The proof system for APDL is complete and every formula in LAPDL

+ isprovably equivalent to a formula in LAPDL. ¤

4 Conclusion and further research

Dynamic-epistemic logics provide excellent means for studying exchange of factualand higher-order information. In this many-agent setting, common knowledge is anessential concept. We have presented two extended languages for dynamic-epistemiclogic that admit explicit action/common knowledge reduction axioms: one (PAL) forpublic announcement only, and one (APDL) for general action update. These systemsmake proof and complexity analysis more perspicuous than earlier attempts in theliterature. Still, PAL and epistemic APDL are just two extremes on a spectrum, andmany further natural update logics may lie in between. For instance, we found anopen question of expressiveness of LRCL versus PAL.

In addition, our methods raise new model-theoric questions about languages thatadmit of ‘update closure’ by reduction axioms and translation procedures. Exam-ples are temporal UNTIL logic, fragments of the µ-calculus, or even first-orderlogic and its fixed-point extensions. Also, our automata seem a natural setting forgeneralizing other open questions, known mainly so far for public announcement(cf. [7]). These include extensions of our current concerns, such as axiomatizingthe schematic validities of update languages, closed under arbitrary substitutionsfor proposition letters– an issue which is not solved by our reduction axioms alone.

References

[1] A. Baltag, L. S. Moss, and S. Solecki. The logic of public announcements,common knowledge, and private suspicions. Technical Report SEN-R9922, CWI,Amsterdam, 1999.

14

210

Page 220: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[2] J. Gerbrandy. Bisimulations on Planet Kripke. ILLC Dissertation Series, Ams-terdam, 1999.

[3] D. Harel, D. Kozen, and J. Tiuryn. Dynamic Logic. Foundations of Computing.MIT Press, Cambridge, Massachusetts, 2000.

[4] D. Kozen and R. Parikh. An elementary proof of the completeness of PDL.Theoretical Computer Science, 14:113–118, 1981.

[5] J. A. Plaza. Logics of public communications. In M. L. Emrich, M. S. Pfeifer,M. Hadzikadic, and Z. W. Ras, editors, Proceedings of the 4th International

Symposium on Methodologies for Intelligent Systems, pages 201–216, 1989.

[6] J. van Benthem. Update as relativization. ILLC, University of Amsterdam, 1999,manuscript.

[7] J. van Benthem. One is a lonely number: on the logic of communication. Tech-nical Report PP-2002-27, ILLC, Amsterdam, 2002. To appear in P. Kopke, ed.Colloquium Logicum, Munster, 2001, AMS Publications, Providence.

15

211

Page 221: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Strong completeness for non-compact hybrid logics

Barteld Kooi∗

Gerard Renardel de Lavalette†

Rineke Verbrugge‡

University of GroningenP.O. Box 800, 9700 AV Groningen, The Netherlands

Abstract

We provide a strongly complete infinitary proof system for hy-brid logic. This proof system can be extended with countably manysequents. Thus completeness proofs are provided for infinitary hy-brid versions of non-compact logics like ancestral logic and Segerberg’smodal logic with the bounded chain condition. This extends the com-pleteness result for hybrid logics by Blackburn and Tzakova.

Keywords: hybrid logic, strong completeness, non-compact logics,infinitary proof rules

1 Introduction

Hybrid logic is an extension of modal logic. Special propositional variablescalled nominals, which are true in exactly one possible world, are added tothe language. Therefore they could equally be taken as names of possibleworlds. Hybrid logic was initially developed by Prior in the 1960’s [6], butthere has been a flurry of activity surrounding hybrid logic in the past decade(see www.hylo.net). A textbook introduction to hybrid logic can be foundin [1].

One of the pleasant features of hybrid logic is that its correspondencetheory is very straightforward. Hybrid logic can be translated into first-order logic, where nominals are interpreted as constants. The link is sostrong that it is very easy to obtain complete proof systems for classes offrames that satisfy additional properties. This works not only for the usualproperties such as transitivity, reflexivity, and symmetry, but also for ir-reflexivity, asymmetry, and many others that cannot be characterized by

∗Department of Philosophy, [email protected]†Department of Computing Science, [email protected]‡Department of Artificial Intelligence, [email protected]

1

212

Page 222: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

modal formulas. The completeness theorem which is proved in [1] (which isa variation of a theorem from [2] by Blackburn and Tzakova) provides com-plete proof systems for many hybrid logics. It exploits the straightforwardcorrespondence theory. When the base proof system is extended with pureaxioms (axioms without propositional variables), then the new proof systemis automatically complete for the class of corresponding frames.

One would like to prove strong completeness also for logics where therelevant properties are not characterized by axioms, but by infinitary rules,such as the following rule, which characterizes the frame property that anystate is reachable from any other state by a (finite) sequence of moves alongthe accessibility relation:

{¬@i3nj | n ∈ N} ` ⊥

Although the completeness proof in [1] is very general, it is not applicableto non-compact modal logics, such as propositional dynamic logic PDL, an-cestral logic, other modal logics with (reflexive) transitive closure operators,and the ‘reachability logic’ given by the infinitary rule above.

Let us remind the reader of some relevant definitions. Strong complete-ness (also called extended completeness) with respect to a class of frames Sis the following property of a modal logical system S:

Γ |=S ϕ implies Γ `S ϕ, for all formulas ϕ and all sets of formulas Γ.

This generalizes weak completeness, where Γ is empty. Observe that weakcompleteness implies strong completeness whenever the logic in question issemantically compact, i.e. when Γ |=S ϕ implies that there is a finite Γ′ ⊆ Γwith Γ′ |=S ϕ, hence |=S

∧Γ′ → ϕ. This is, for example, the case in modal

logics such as K and S5.Propositional dynamic logic is a well-known example of a non-compact

logic: we have for the relevant class of frames S, that {[an]p | n ∈ N} |=S

[a∗]p but there is no natural number k with {[an]p | n ≤ k} |=S [a∗]p. As aconsequence, we do not have strong completeness for any finitary axiomati-zation, a fortiori not for its usual, weakly complete proof system. So strongcompleteness requires an infinitary proof system. Here, “infinitary” doesnot refer to the language (all formulas in this paper have finite length), butto the derivation relation (proof sequents may be non-standard in requiringinfinitely many premises).

Infinitary non-hybrid versions of such non-compact modal logics wereinvestigated and strong completeness proofs were given by Goldblatt [3],Segerberg [8] and the present authors [7]. In those cases a strongly com-plete infinitary proof system can be obtained by adding infinitary rules:one simply makes a rule from an example that shows non-compactness. In[5], Passy and Tinchev investigate hybrid versions of PDL and present aninfinitary proof system, which is shown to be strongly complete.

2

213

Page 223: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The main goal of this paper is to extend Blackburn and Tzakova’s strongcompleteness result for hybrid logic to hybrid systems that are not semanti-cally compact. Rather than extending the base system with pure axioms, asthey do, we allow extensions with countably many pure sequents, each withpossibly infinitely many premises. We will first prove a general result abouta basic hybrid system extended with a countable set of sequents; it turns outthat the completeness proof looks considerably different from the usual one.Especially Blackburn’s and Tzakova’s version of the Lindenbaum Lemmacannot be straightforwardly generalized. Then we show some applicationsto hybrid versions of specific modal logics.

In Section 2 we briefly introduce the language and semantics of hybridlogic. In Section 3 we provide the infinitary proof system for hybrid logic.In Section 4 we show this proof system is complete. In Section 5 we discusssome specific extensions of the basic proof system. Finally, in Section 6 wedraw conclusions and indicate directions for further research.

2 Language and semantics

There are some variants of the language of hybrid logic. We take the minimallanguage of hybrid logic, where the language of modal logic is extended withnominals and at-operators @.

Definition 1 (Language of hybrid logic)Let a countable set of propositional variables P , and a countably infinite setof nominals I be given. The language of hybrid logic L(P, I) is given by thefollowing BNF:

ϕ ::= ⊥ | p | i | ¬ϕ | ϕ1 ∨ ϕ2 | 3ϕ | @iϕ

where p ∈ P , and i ∈ I. We use the usual abbreviations. We are usuallysloppy and write L instead of L(P, I). �

Formulas of the form @iϕ, are to be read as “ϕ holds at the world namedi.” The function nom : L → ℘(I) yields the set of nominals that occur in aformula. We generalize this to sets of formulas, and later to proof sequentsand proofs.

The models used in the semantics of hybrid logic are simply models formodal logic, where the valuation of a nominal is a singleton set.

Definition 2 (Models for hybrid logic)A model for L is a triple M = (W,R, V ) such that:

• W 6= ∅; a set of possible worlds;

• R ⊆W ×W ; an accessibility relation;

3

214

Page 224: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

• V : P∪I → ℘(W ); assigns a set of possible worlds to each propositionalvariable and a singleton to each nominal.

A frame F is a tuple (W,R), where W and R are as above. �

Definition 3 (Semantics)Let a model (M,w) where M = (W,R, V ) be given. Let p ∈ P , i ∈ I, andϕ,ψ ∈ L.

(M,w) 6|=⊥(M,w) |= p iff w ∈ V (p)(M,w) |= i iff V (i) = {w}(M,w) |= ¬ϕ iff (M,w) 6|= ϕ(M,w) |= ϕ ∨ ψ iff (M,w) |= ϕ or (M,w) |= ψ(M,w) |= 3ϕ iff (M,v) |= ϕ for some v such that (w, v) ∈ R(M,w) |= @iϕ iff (M,v) |= ϕ where V (i) = {v}

Given a set of formulas Γ we write (M,w) |= Γ iff (M,w) |= ϕ for everyϕ ∈ Γ. We write Γ |= ϕ iff (M,w) |= Γ implies (M,w) |= ϕ for every modelM and world w. We write M |= ϕ iff (M,w) |= ϕ for every w. We writeM |= Γ/ϕ iff (M,w) |= Γ implies (M,w) |= ϕ for every world w. Given aframe F and a world w, we say that (F,w) |= ϕ iff ((F, V ), w) |= ϕ for everyvaluation V . Likewise for (F,w) |= Γ, F |= ϕ and F |= Γ/ϕ. �

3 The proof system Khybω

The proof system is based on sequents, i.e. expressions of the form Γ ` ϕwhere Γ is a (possibly infinite) collection of formulas. For technical reasons,only sequents Γ ` ϕ are allowed in which infinitely many nominals i ∈ I donot occur, i.e. where (I−nom(Γ, ϕ)) is infinite (recall that I is always infinite).This is a weak restriction: any sequent not satisfying it can be transformedby renaming of nominals into a sequent satisfying the restriction.

The proof system consists of axiom sequents and sequent rules, and deriv-ability is defined inductively as usual: a sequent is derivable when it is anaxiom, or when it is the conclusion of a rule with derivable sequents aspremises. Observe that, due to the infinitary cut rule, derivations may con-tain infinitely many sequents. We write 2Γ for {2ϕ | ϕ ∈ Γ}, @iΓ for{@iϕ | ϕ ∈ Γ}, and Γ ` ∆ for (Γ ` ϕ for every ϕ ∈ ∆).

Definition 4 (Proof system for hybrid logic)Γ ` ϕ is defined by the axiom sequents and sequent rules provided in Fig-ure 1.

The soundness of the proof system (Γ ` ϕ, then Γ |= ϕ) can be shown byinduction on the length of the proof. We do not provide an explicit proof.

4

215

Page 225: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Taut ` ϕ if ϕ is an instance of a propositionaltautology

K2 ` 2(ϕ→ ψ) → (2ϕ→ 2ψ) (distribution)

K@i` @i(ϕ→ ψ) → (@iϕ→ @iψ) (distribution)

SD ` @iϕ→ ¬@i¬ϕ (self-dual)

Intr ` i ∧ ϕ→ @iϕ (introduction)

T@i` @ii (reflexivity)

B@i` @ij ↔ @ji (symmetry)

Nom ` @ij ∧@jϕ→ @iϕ (nom)

Agree ` @i@jϕ↔ @jϕ (agree)

Back ` 3@iϕ→ @iϕ (back)

MP ϕ,ϕ→ ψ ` ψ (modus ponens)

SNec2 if Γ ` ϕ, then 2Γ ` 2ϕ (strong necessitation)

SNec@iif Γ ` ϕ then @iΓ ` @iϕ (strong necessitation)

InfCut if Γ ` ∆ and Γ′,∆ ` ϕ then Γ,Γ′ ` ϕ (infinitary cut)

W if Γ ` ϕ then Γ,∆ ` ϕ (weakening)

Ded if Γ, ϕ ` ψ, then Γ ` ϕ→ ψ (deduction)

Name if Γ, i ` ϕ, then Γ ` ϕ, provided i 6∈nom(Γ, ϕ)

(name)

Paste if Γ,@i3j,@jϕ ` ψ, then Γ,@i3ϕ ` ψ,provided j 6∈ nom(Γ, ϕ, ψ) ∪ {i}

(paste)

Figure 1: The axiom sequents and sequent rules of Khybω

5

216

Page 226: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 Strong completeness of Khybω plus countably manysequents

Take the basic system Khybω defined above, and add a denumerable set ofadditional axiom sequents:

AS = {Γn ` ϕn | n ∈ N}

What AS contains may depend on the language, i.e. the parameters P and I.E.g., AS may contain (or even consist of) sequents generated by substitutionfrom sequent schema’s. When we add all instances of a sequent Γ ` ϕ toAS that are obtained by arbitrary substitutions of formulas for propositionalvariables and nominals for nominals, then countability is only guaranteed ifΓ ` ϕ is parameter-finite, i.e. if Γ ` ϕ contains only finitely many nominalsand propositional variables.

In this section we provide a completeness proof for Khybω +AS. For thecompleteness proof we follow the completeness proofs for the infinitary logicLω1ω presented in [4] and hybrid logic presented in [1]. The completenessproof for hybrid logic in [1] is very general and also shows that if theirproof system is extended with extra pure axioms, then this extended proofsystem is automatically strongly complete with respect to the class of framesdefined by these pure axioms. However, it is a finitary proof system, and thecompleteness proof hinges on a Lindenbaum Lemma where compactness isassumed. So if we were to add an infinitary rule to their system, we wouldnot get a complete proof system. Therefore we follow the completenessproof of [4], which does not depend on compactness. Furthermore we showthat extensions of Khybω with extra pure axiom sequents with finitely manynominals are also complete for the class of frames defined by these rules.

Theorem 1 (Completeness)Every Khybω + AS-consistent set of formulas in language L is satisfiable ina countable named model. �

Proof We prove that if Γ 6` ⊥ (i.e. Γ ` ⊥ is not derivable in Khybω + AS),then there is a named model M satisfying AS with (M,w) |= Γ.

Assume Γ is consistent, i.e. Γ 6` ⊥. We extend the language L to L+

by adding the countable set of new nominals J. (Consequently AS maynow also be extended.) We claim Γ 6`L+ ⊥. For assume Γ `L+ ⊥ and letΠ+ be the L+-derivation of this sequent. We shall show that Π+ can betransformed into a L-derivation Π of Γ ` ⊥, contradicting our assumption.Let

c : (nom(Π+)− nom(Γ)) → (I− nom(Γ))

be an injection (such a c exists, for (I− nom(Γ)) is infinite, by the definitionof sequent). Now replace in Π+ all nominals i ∈ nom(Π+)− nom(Γ) by c(i):this yields the L-derivation Π.

6

217

Page 227: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A collection ∆ ⊆ L+ is called admissible if it is L+-consistent and J −nom(∆) is infinite. We shall define an increasing sequence ∆0 ⊆ ∆1 ⊆· · · ⊆ ∆n ⊆ . . . of admissible sets containing Γ, and use ∆ω =

⋃n ∆n

to construct the model. Let i ∈ J. We put ∆0 = Γ ∪ {i}, so ∆0 is indeedadmissible (by Name). Now let {ϕn | n ∈ ω} be an enumeration of L+ whereevery formula occurs infinitely often. This yields an infinite set of numbersNϕ = {n | ϕn = ϕ} for every formula ϕ ∈ L+. Let {Θn ` ψn | n ∈ N} be anenumeration of @AS = {@jΘ ` @jψ | (Θ ` ψ) ∈ AS, j ∈ J}; and let Mϕ ={m | (Θm ` ϕ) ∈ @AS} be a (possibly empty) set of numbers for everyformula ϕ. Now we define for each ϕ an injective function fϕ : Mϕ → Nϕ

such that fϕ(mk) = nk (i.e. the k-th number in Mϕ is mapped to the k-thnumber in Nϕ).

Now we define ∆n+1 in terms of ∆n. We distinguish between ∆n ` ¬ϕn

and ∆n ∪ {ϕn} consistent.If ∆n ` ¬ϕn, then

∆n+1 =

∆n ∪ {¬θ} if there is an m with fϕn(m) = n

(i.e. there exists a (Θm, ϕn) ∈ @AS)and θ is the smallest formula in Θm

such that ∆n ∪ {¬θ} is consistent.∆n otherwise

We claim that the definition is correct, i.e. that, in the first clause, such a θcan always be found: for if not, then we would have ∆n ` θ for all θ ∈ Θm

and hence ∆n ` ϕn: with ∆n ` ¬ϕn, this contradicts the consistency of ∆n.If ∆n ∪ {ϕn} is consistent, then

∆n+1 =

∆n ∪ {ϕn,@jk} if ϕn = k with k 6∈ Jwhere j ∈ J− nom(∆n)

∆n ∪ {ϕn, χk} if ϕn = χ0 ∨ χ1

where k is the least number in {0, 1}such that ∆n ∪ {χk} is consistent

∆n ∪ {ϕn,@k3j,@jψ} if ϕn = @k3ψ,where j ∈ J− nom(∆n, ϕn)

∆n ∪ {ϕn} otherwise

Again we claim that the definition is correct. For the first clause, this comesdown to showing that ∆n ∪ {k,@jk} is consistent. Assume ∆n, k,@jk ` ⊥,then (by propositional reasoning with the axioms on nominals) ∆n, k, j ` ⊥,so with the rule Name we get ∆n, k ` ⊥, contradicting the consistency of∆n ∪ {ϕn}. For the second clause, the reasoning is standard. For the thirdclause, the reasoning is as for the first, but now we use the rule Paste.

We observe that ∆ω satisfies the following properties:

• ∆ω is maximal: for all ϕ ∈ L+, either ϕ ∈ ∆ω or ¬ϕ ∈ ∆ω, but notboth;

7

218

Page 228: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

• ∆ω is finitely `-closed: if ∆′ ⊆ ∆ω finite and ∆′ ` ϕ, then ϕ ∈ ∆ω;

• ∆ω is closed under all rules (Θ, ϕ) ∈ @AS: if Θ ⊆ ∆ω, then ϕ ∈ ∆ω.

We shall use these properties frequently below, without explicit mention.Before we construct the model from ∆ω, we define ∼ on J:

j ∼ k iff @jk ∈ ∆ω

and observe that, by the three axioms on nominals, ∼ is an equivalencerelation on J. We define

[k] = {j ∈ J | @jk ∈ ∆ω}

so, for j ∈ J, we have [j] = [j]∼ ∈ J/∼, the ∼-equivalence class of j. By thedefinition of ∆n+1 and the axioms on nominals, we have [k] ∈ J/∼, also ifk 6∈ J.

Now we construct the model M = (W,R, V ) from ∆ω:

• W = {[j] | j ∈ J}(= J/∼),

• R = {([j], [k]) | @j3k ∈ ∆ω},

• V (p) = {[j] | @jp ∈ ∆ω}

• V (j) = {[j]}

We claim that for all ϕ ∈ L+ and all j ∈ J, the following analogue of theTruth Lemma holds:

@jϕ ∈ ∆ω ⇔ (M, [j]) |= ϕ

This is proved with formula induction. The base cases (for propositionalvariables and nominals) follow immediately from the definition of the model.The cases for the Boolean connectives follow straightforwardly from theinduction hypothesis (for the negation case, we use that ∆ω is maximal).For the 2 step, we use the property

@j2ϕ ∈ ∆ω ⇒ ∀k ∈ J(@j3k ∈ ∆ω ⇒ @kϕ ∈ ∆ω)

This property follows from the fact that ∆ω is finitely `-closed. Since@j2ϕ,@j3k ` @j3(k∧ϕ), and by Intr, Back and Agree in @j3k∧ϕ ` @kϕ.

As a consequence, we have (by the definition of ∆0):

ϕ ∈ ∆ω ⇔ (M, [i]) |= ϕ

Moreover, we have that all (Θ, ϕ) ∈ AS hold in M :

if (M, [j]) |= Θ, then (M, [j]) |= ϕ

8

219

Page 229: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

To see this observe that:

(M, [j]) |= Θ ≡ for all θ ∈ Θ, (M, [j]) |= θ≡ for all θ ∈ Θ,@jθ ∈ ∆ω

⇒ @jϕ ∈ ∆ω

≡ (M, [j]) |= ϕ

So if Γ 6` ⊥, then there is a named model M satisfying AS with (M, [i]) |= Γ.This ends the proof �

Although the model which is constructed in this proof satisfies Γ, it is notnecessarily the case that the underlying frame satisfies the additional se-quents, i.e. we do not show canonicity. However if we restrict the additionalaxiom sequents to those generated by pure sequents (where no propositionalvariables occur), we do get canonicity, due to the following lemma. Thus, fornamed models and pure formulas containing only finitely many nominals,truth in a model and validity in a frame coincide

Lemma 1Let M = (F, V ) be a named model and Γ ` ϕ be a pure sequent. Supposethat for all pure instances ∆ ` ψ of Γ ` ϕ, M |= ∆ implies M |= ψ.Then F |= Γ/ϕ, i.e. for all V ′, w we have that ((F, V ′), w) |= Γ implies((F, V ′), w) |= ϕ. �

Proof (sketch) V : I → {{w}|w ∈W} is surjective, so it has a right inverseV −1 : W → I with V (V −1(w)) = {w}. Define the nominal substitutionσ : I → I by σ(i) = V −1(V ′(i)). Now we can prove, with straightforwardformula induction, that for all pure formulas θ:

((F, V ), w) |= σ(θ) ⇔ ((F, V ′), w) |= θ

This implies the lemma. �

5 Application to non-compact modal logics

We provide a number of interesting instances of Theorem 1. These areexamples of cases where pure axioms do not suffice to obtain completenessfor the relevant class of models, but pure sequents do. Since pure axiomsare a special case of pure sequents, these are generalizations of Blackburnand Tzakova’s result. Thus, if AS contains only pure formulas and finitelymany nominals, not only the model provided by Theorem 1, but also theframe underlying it, validates AS. In this section by Khybω + AS we meanKhybω extended with all pure instances of AS.

9

220

Page 230: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

5.1 Hybrid ancestral logic

Ancestral logic is the modal logic with two modalities 2 and 2∗, wherethe accessibility relation associated with the latter is the reflexive transitiveclosure of the accessibility relation associated with the former. Ancestrallogic is non-compact, and in fact a counterexample to compactness givesthe inspiration for a suitable hybrid version of this logic, namely Khybω

extended with a countable set of pure sequents containing only finitely manynominals. Let 2nϕ stand for ϕ preceded by n 2-operators.

AS1 {@i2n¬j | n ∈ N} ` @i2

∗¬j

It is clear that AS1 is valid exactly in those frames in which the accessibilityrelation of 2∗ is the reflexive transitive closure for the accessibility relationof 2. Thus, by Theorem 1 and Lemma 1, Khybω + AS1 is strongly completewith respect to such frames.

5.2 Hybrid reachability logic

Let us define Hybrid reachability logic as the hybrid logic given by Khybω

+ AS2, as follows, where 3nϕ stands for ϕ preceded by n 3-operators:

AS2 {¬@i3nj | n ∈ N} ` ⊥

It is clear that AS2 is valid exactly in those frames in which the accessibilityrelation R is reachable, in the sense that for any two states i, j in the modeleither i = j or there is a sequence s0R . . . sn where s0 = i and sn = j,where n ≥ 1. Thus, by Theorem 1 and Lemma 1, Khybω + AS2 is stronglycomplete with respect to reachable frames.

5.3 Hybrid no-cycle logic

Let us define Hybrid no-cycle logic as the hybrid logic given by Khybω +AS3, as follows:

AS3 {¬@i3ni | n ∈ N, n ≥ 1} ` ⊥

It is clear that AS3 is valid exactly in those frames in which the accessibilityrelation R contains no cycles, in the sense that for any state i in the model,there is no sequence s0R . . . sn where s0 = i and sn = i, where n ≥ 1.Thus, by Theorem 1 and Lemma 1, Khybω + AS3 is strongly complete withrespect to frames without cycles.

10

221

Page 231: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

5.4 Hybrid BCC logic

BCC-logic is the logic of the bounded chain condition, as defined in [8]: forall states i, there is a bound n ∈ N such that for any j there are only chainsiR . . . j of length smaller than n from i to j.

Let us define Hybrid BCC logic as the hybrid logic given by Khybω +AS4, the infinitary rule of [8], which turns out to be pure and does notcontain any nominals:

AS4 {3n> | n ∈ N} ` ⊥

Thus, by Theorem 1 and Lemma 1, Khybω + AS4 is strongly complete withrespect to frames with the bounded chain condition.

Note that the BCC-condition is stronger than converse wellfoundedness(no infinite ascending chains), for which we did not find a characterizingcountable set of sequents containing only finitely many nominals.

6 Conclusion and further research

In this paper we provided a strongly complete infinitary proof system forhybrid logic. The completeness proof worked in such a way that we immedi-ately got completeness for logics that extend the proof system with countablymany axiom sequents. This allowed us to obtain strongly complete proofsystems for non-compact hybrid logics. If the additional axiom sequents arepure and contain only finitely many nominals, then we automatically havecanonicity.

In the future we hope to attain similar results for hybrid logic withuncountably many pure rules with countably many nominals. This wouldyield strongly complete proof systems for many more interesting classes offrames.

References

[1] P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic, volume 53 ofCambridge Tracts in Theoretical Computer Science. Cambridge Univer-sity Press, Cambridge, 2001.

[2] P. Blackburn and M. Tzakova. Hybrid languages and temporal logic.Logic Journal of the IGPL, 7(1):27–54, 1999.

[3] R. Goldblatt. Mathematics of Modality, volume 43 of CSLI LectureNotes. CSLI Publications, Stanford, California, 1993.

[4] H. J. Keisler. Model Theory for Infinitary Logic: Logic with CountableConjunctions and Finite Quantifiers. North-Holland Publishing Com-pany, Amsterdam, 1971.

11

222

Page 232: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[5] S. Passy and T. Tinchev. An essay in combinatory dynamic logic. In-formation and Computation, pages 263–332, 1991.

[6] A. N. Prior. Past, Present and Future. Oxford University Press, 1967.

[7] Gerard R. Renardel de Lavalette, Barteld P. Kooi, and Rineke Ver-brugge. Strong completeness for propositional dynamic logic. In PhilippeBalbiani, Nobu-Yuki Suzuki, and Frank Wolter, editors, AiML2002 —Advances in Modal Logic (conference proceedings), pages 377–393. Insti-tut de Recherche en Informatique de Toulouse IRIT, 2002.

[8] K. Segerberg. A model existence theorem in infinitary propositionalmodal logic. Journal of Philosophical Logic, 23:337–367, 1994.

12

223

Page 233: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A Lower Complexity Bound for Propositional

Dynamic Logic with Intersection

Martin Lange

Institut fur Informatik, University of Munich

July 13, 2004

Abstract

This paper shows that satisfiability for Propositional Dynamic Logicwith Intersection is EXPSPACE-hard. The proof uses a reduction fromthe word problem for alternating, exponential time bounded TuringMachines.

Keywords: PDL, intersection, satisfiability, complexity

1 Introduction

Propositional Dynamic Logic, PDL, was defined in [4] to reason about pro-gram behaviour but is nowadays mainly interesting because of its connec-tion to Description Logics (DL) [16]. It is an extension of multi-modal logicwhere modalities take as arguments elements of a Kleene Algebra with atest operator.

PDL enjoys nice algorithmic properties: its model checking problem isP-complete and solvable in linear running time [4]; its satisfiability problemis complete for EXPTIME [14, 15]. PDL is embeddable into infinitary multi-modal logic and thus has the tree model property. It also has the finite modelproperty [6]. It is finitely axiomatisable [17, 10]. However, it is rather weakin expressive power since it is strictly less expressive than the alternation-freefragment of Kozen’s modal µ-calculus [9].

Several variants of PDL have been studied since, for example restrictionsto deterministic atomic programs, etc. Most variants aim at extending theset of operators in the underlying Kleene Algebra in order to allow propertiesof more programs to be expressed. Examples of these are loop constructs,the converse operator [18], an interleaving operator [12], etc. In most casesaxiomatisations and decision procedures for these extension can be obtainedby extending PDL’s axiom system and its decision procedures.

One variant for which this approach fails entirely is PDL with Inter-section, IPDL [7]. Using the connection to Description Logics mentioned

1

224

Page 234: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

above, IPDL can be seen as a DL which can express intersection of roles.Although intersection looks like yet another regular operation it cannot bedefined using the operators of a Kleene Algebra. This is simply because ofthe lack of both tree and finite model property for IPDL which makes it agood candidate for an undecidable logic.

Nevertheless, IPDL is only undecidable if atomic programs are requiredto be deterministic [7]. If they are allowed to be non-deterministic then itssatisfiability problem is decidable [3]. Danecki even showed that it can bedecided in double exponential running time.1 This is proved by constructingBuchi tree automata for IPDL formulas. They do not work on the formula’smodels directly but on trees describing their models.

These are the only results about IPDL so far. It is not known whetherthere is a better decision procedure for IPDL or whether it is complete fordouble exponential time. Complexity issues for modal logics containing theintersection operator have been addressed in [11] for instance. However,there modalities take elements of a boolean algebra rather than a Kleenealgebra like PDL does.

Fragments of IPDL have been studied in [5, 13, 1] for instance, mainlyregarding the issue of axiomatisability. They also show that the presence ofthe intersection operator makes IPDL a “strange” logic compared to PDLfor example.

In this paper we show that the satisfiability problem for IPDL is hard forexponential space. The proof presents a reduction from the word problemfor alternating exponential time bounded Turing Machines. This is inspiredby [19] where satisfiability of the temporal logic CTL∗ is shown to be hardfor double exponential time. It remains to be seen whether the lower boundcan be pushed up match IPDL’s 2-EXPTIME upper bound.

2 Preliminaries

2.1 Propositional Dynamic Logic with Intersection

Let P = {p, q, . . .} be a finite set of propositional constants which includestt and ff. Let A = {a, b, . . .} be a finite set of atomic program names. AKripke structure is a triple (S, { a−→ | a ∈ A}, L) with S being a set of states,

a−→ for every a ∈ A is a binary relation on states, and L : S → 2P labels thestates with propositions, s.t. for all s ∈ S: tt ∈ L(s) and ff �∈ L(s).

Formulas ϕ and programs α of IPDL are defined as:

ϕ ::= q | ϕ ∨ ϕ | ¬ϕ | 〈α〉ϕα ::= a | α ∪ α | α ∩ α | α;α | α∗ | ϕ?

1However, this piece of information seems to have never made it into common knowl-edge. Many seem to believe that only decidability but no complexity results are known.

2

225

Page 235: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

where q ranges over P, and a ranges over A.We will use the standard abbreviations ϕ∧ψ := ¬(¬ϕ∨¬ψ), ϕ→ ψ :=

¬ϕ ∨ ψ, [α]ϕ := ¬〈α〉¬ϕ and α+ := α;α∗.IPDL formulas are interpreted over Kripke structures. The semantics of

an IPDL formula is explained using an extension of the accessibility relationsa−→ to full programs α.

sα;β−−−→ t iff ∃u ∈ S s.t. s α−→u and u β−→ t

sα∪β−−−→ t iff s α−→ t or s β−→ t

sα∩β−−−→ t iff s α−→ t and s β−→ t

s α∗−−→ t iff ∃n ∈ N, s αn−−→ t where

∀s, t ∈ S : s α0−−→ s, and s αn+1−−−−→ t iff sα;αn−−−−→ t

sϕ?−−→ s iff s |= ϕ

where the meaning of s |= ϕ is explained below.Assuming a Kripke structure T to be fixed we define the semantics of a

formula ϕ just as s |= ϕ instead of T , s |= ϕ.

s |= q iff q ∈ L(s)s |= ϕ ∨ ψ iff s |= ϕ or s |= ψs |= ¬ϕ iff s �|= ϕ

s |= 〈α〉ϕ iff ∃t ∈ S s.t. s α−→ t and t |= ϕ

2.2 Alternating Turing Machines.

We use the following model of an alternating Turing Machine, which differsslightly from the standard model [2] in the way that it either moves its heador it writes a symbol and branches existentially or universally. It is not hardto see that this model is equivalent to the standard one w.r.t. the standardtime and space complexity classes.

An alternating Turing Machine M is of the form M = (Q,Σ, q0, qacc , δ),where Q is the set of states, Σ is the alphabet which contains a blank symbol�, and q0, qacc ∈ Q – the starting and the accepting state.

The set Q of states is partitioned into Q = Q∃ ∪ Q∀ ∪ Qm, where wewrite Qb for Q∃ ∪ Q∀, these are the branching states. We assume Qm tocontain the only accepting state qacc in which M simply moves to the leftend of the tape whilst staying in state qacc . Having arrived there, it stayson this cell in state qacc.

The transition relation δ is of the form

δ ⊆ (Qb × Σ ×Q× Σ

) ∪ (Qm × Σ ×Q× {L,R}) .

We also write (q′, b) ∈ δ(q, a) to denote (q, a, q′, b) ∈ δ for given q and a.

3

226

Page 236: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

In a branching state q ∈ Qb, the machine can overwrite the symbol underthe head but not move the head. Furthermore, it can branch nondeterminis-tically or universally. In a state q ∈ Qm, the machine acts deterministicallyand moves its head, i.e., for each a ∈ Σ, there is exactly one transition(q, a, q′,D) ∈ δ, for q′ ∈ Q and D ∈ {L,R}, meaning that the head movesto the left (L) or right (R), and the machine enters state q′.

Every alternating, exponential time bounded Turing Machine can betransformed into such one that still decides its language in alternating ex-ponential time.

A configuration of a Turing Machine is a snapshot in time consisting ofthe actual state that the machine is in plus the current content of the tape.The initial configuration consists of q0 and the input word written on thetape followed by blank symbols.

The Turing Machine accepts the input if its initial configuration is ac-cepting. The configurations’ acceptance depends on the kind of state:

• If the state is qacc then the configuration is accepting.

• If the state is in Qm \ {qacc}, then the configuration is accepting iff itsunique successor is accepting.

• If the state is in Q∃, then the configuration is accepting iff at least oneof its successors is accepting.

• If the state is in Q∀, then the configuration is accepting iff all of itssuccessors are accepting.

The entire computation is accepting if the initial configuration is. Notethat a witness for an accepting computation can be represented as a tree ofconfigurations with the starting configuration as its root, and where everyexistential and deterministic configuration has exactly one successor whereasall possible successors of a universal configuration are preserved in the tree.2

2.3 Complexity classes

We will quickly recall the definitions of the complexity classes used in thispaper. Let DTIME(f(n)), resp. DSPACE(f(n)), be the classes of problemsthat can be decided by a deterministic Turing Machine in time f(n), resp.with space f(n), where n is the size of the input to the machine. The classeswe refer to in this paper are those of exponential time, space and double

2If computations of alternating Turing Machines are regarded as a game then such awitness is nothing more than a winning strategy for the existential player.

4

227

Page 237: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

exponential time. They are defined as

EXPTIME :=⋃

DTIME(2p(n))

EXPSPACE :=⋃

DSPACE(2p(n))

2−EXPTIME :=⋃

DTIME(22p(n))

where p(n) ranges over all polynomials in n. These classes also have charac-terisations via alternating Turing Machines [2]. The following hold:EXPSPACE = AEXPTIME and 2-EXPTIME = AEXPSPACE.

3 The Reduction

Theorem 1 Satisfiability of IPDL is EXPSPACE-hard.

Proof According to [2], there is an alternating, exponential time boundedTuring Machine whose word problem is EXPSPACE-hard. Suppose M =(Q,Σ, q0, qa, δ) is such a Turing Machine that has been tailored so that itobeys the restrictions laid out in the previous section. Let w = a0 . . . an−1 ∈Σ∗ be an input for M. W.l.o.g. we assume the space needed by M on inputw to be bounded by 2p(n) for some polynomial p. Let N := 2p(n) − 1 be themaximal index of a tape cell.

In the following we will construct an IPDL formula ϕM,w over a singletonset A = {x} of atomic programs s.t. w ∈ L(M) iff ϕM,w is satisfiable. Theletter x indicates “moving to the next state”. Informally, a witness for anaccepting computation of M on w will serve as a model for ϕM,w.

The following propositions are needed.

P = Q ∪ Σ ∪ {c0, . . . , cp(n)−1} ∪ {d0, . . . , dp(n)−1}

• q ∈ Q is true in a state of the model iff the head of M is on thecorresponding tape cell in the corresponding configuration while themachine is in state q. The formula h :=

∨q∈Q q says that the tape

head is on the current cell.

• a ∈ Σ is true iff a is the symbol on the corresponding tape cell.

• cp(n)−1, . . . , c0 represent a counter C in binary coding. The countervalue will be 0 in the leftmost and N in the rightmost tape cell forinstance. Let P := p(n)−1 be the index of the most significant counterbit.

• dp(n)−1, . . . , d0 represent a counter D. The value of this counter will bethe index of the actual configuration (i.e. the distance to the startingconfiguration in M’s computation tree).

5

228

Page 238: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The encoding of M’s configurations is very similar to the ones presented in[19] or [8]. A configuration of the form

a0 a1 a2 . . . aN

with the tape head on, say, the third cell from the left and the machine instate q is modelled by a sequence of states of the form

cP

q...

c0 c1 c0a0 a1 a2 aN

• x−→ • x−→ • x−→ . . .x−→•

Here, the counter bits for D are left out to avoid clutter. Successive config-urations are modelled by concatenating these sequences.

For every fixed m ∈ {0, . . . , N} we can write a formula χC=m, resp.χD=m, which says that the counter value of C or D is m in the currentstate, e.g.

χC=0 :=P∧

i=0

¬ci , χC=1 := c0 ∧P∧

i=1

¬ci and χC=N :=P∧

i=0

ci

for the leftmost (m = 0), second (m = 1) and rightmost (m = N) positionin a configuration.

For the next part we need auxiliary formulas ϕCinc , ϕ

Dinc that increase the

value of C, resp. D, as long as they do not equal N .

ϕCinc :=

P∨i=0

( ¬ci ∧ [x]ci ∧∧j<i

(cj ∧ [x]¬cj) ∧∧j>i

(cj → [x]cj) ∧ (¬cj → [x]¬cj) )

This is just the standard formula for incrementing a binary value: there is abit which changes from 0 to 1, all higher bits remain the same and all lowerbits change from 1 to 0. We also need to be able to say that the value of Dremains unchanged.

ϕDremain :=

P∧i=0

( di → [x]di ) ∧ ( ¬di → [x]¬di )

Then we can write down a formula which requires the counter C to beincreased by one modulo N + 1 in every move from a state to a successor.

6

229

Page 239: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

At the same time, counter D is incremented iff the value of C changes fromN back to 0.

ϕcount := [x∗]( ( χC=N ∧ [x]χC=0 ∧ ϕDinc ) ∨

( ¬χC=N ∧ ϕCinc ∧ ϕD

remain ) )

We form programs

• αrest which goes from any state to the beginning of the next configu-ration, i.e. it traverses the rest of the current configuration,

αrest := (¬χC=N?;x)∗;χC=N?

• α2h, α0h which do an arbitrary amount of x-actions whilst seeing atleast two, resp. no tape heads.

α2h := x∗;h?;x+;h?;x∗

α0h := (¬h?;x)∗;¬h?

Then we can formalise the general requirements on a Turing Machine: everytape cell is marked with exactly one symbol from Σ and never with twodifferent states; no configuration has more or less than one cell marked withthe tape head. Finally, as long as the counters both do not have the valueN there is still a successor.

ϕgen := [x∗]( (∨a∈Σ

a ) ∧∧

a,b∈Σ,b�=a

¬(a ∧ b) ∧∧

q,q′∈Q,q �=q′¬(q ∧ q′)

∧ ( χC=0 → [αrest ∩ (α2h ∪ α0h)]ff )

∧ ( ¬χC=N ∧ ¬χD=N → 〈x〉tt ) )

At the beginning, the input word w = a0 . . . an−1 is written on the tape,followed by blank symbols � until a state with counter value 0 is reachedagain.

ϕstart := χC=0 ∧ q0 ∧ a0 ∧[x]( a1 ∧

[x]( a2 ∧. . . ∧ . . .

[x]( an−1 ∧[(x;¬χC=0?)+]� ) . . .))

Next we give a formula which expresses the fact that M’s computation isaccepting. Note that we assumed M to move its head to the very left, gointo state qa , and leave the head there once its computation is finished.

ϕacc := [x∗;χC=0?;χD=N?]qacc

7

230

Page 240: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Before we can encode M’s transition function δ, we need to write programsthat relate a tape cell in one configuration to itself or its neighbours in thenext configuration. Program α0! runs the atomic program x arbitrarily oftenwhilst seeing the counter value C = 0 only once.

α0! := (x;¬χC=0?)∗;x;χC=0?; (x;¬χC=0?)∗

Using this program and the usual trick of incrementing, resp. decrementinga binary counter we can write a program α−1 that turns a tape cell into itsleft neighbour in the following configuration.

α−1 := ¬χC=0?;α0! ∩P⋃

i=0

( ci?;x+;¬ci? ∩⋂j<i

¬cj?;x+; cj?

∩⋂j>i

( cj?;x+; cj? ∪ ¬cj?;x+;¬cj? ) )

Equally, α= and α+1 turn it into itself, resp. its right neighbour.

α= := α0! ∩P⋂

i=0

( ci?;x+; ci? ∪ ¬ci?;x+;¬ci? )

α+1 := ¬χC=N?;α=;x

Now we are able to formalise δ’s transitions in IPDL. Again, there are dif-ferent requirements on the model depending on the nature of a machine’sstate.

In an existential state, there is one transition that determines all suc-cessors. In a universal state, all successors behave according to one of thepossible transitions and every possible transition is present in the computa-tion tree. In a moving state the machine acts deterministically, hence, allsuccessor configurations must be the same. Finally, the label of any tapecell which is not under the tape head remains the same in the followingconfiguration.

ϕδ := [x∗](∧

q∈Q∃,a∈Σ

( q ∧ a →∨

(p,b)∈δ(q,a)

[α=](p ∧ b) )

∧∧

q∈Q∀,a∈Σ

( q ∧ a → ( [α=]∨

(p,b)∈δ(q,a)

(p ∧ b) ) ∧∧

(p,b)∈δ(q,a)

〈α=〉(p ∧ b) ) )

∧∧

(q,a,q′,L)∈δ

( ¬χC=0 ∧ q ∧ a → [α−1]q′ )

8

231

Page 241: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

∧∧

(q,a,q′,R)∈δ

( ¬χC=N ∧ q ∧ a → [α+1]q′ )

∧∧a∈Σ

( ¬h ∧ a → [α=]a ) )

Altogether, the machine’s behaviour is described by the formula

ϕM,w := ϕcount ∧ ϕgen ∧ ϕstart ∧ ϕacc ∧ ϕδ

Then, a model for ϕM,w bears a witness for a successful computation ofM on w. Conversely, each successful computation can be transformed intoa model for ϕM,w by removing the nondeterminism from the model andkeeping the universal branches. Finally, |ϕM,w| is polynomial in |M| and|w|. �

Corollary 2 Satisfiability of IPDL over a singleton set of atomic programsand tests restricted to atomic propositions is already EXPSPACE-hard.

This is not surprising since every IPDL formula ϕ with complex tests canbe transformed into a ϕ′ over additional propositions, s.t. ϕ′ only featuresatomic tests. Moreover, ϕ′ is satisfiable iff ϕ is satisfiable. On the otherhand, the reduction in the proof of Theorem 1 can easily be rewritten s.t.all tests are atomic using for instance the equivalence

(∨q∈Q

q)? ≡⋃q∈Q

q?

Corollary 3 Satisfiability of IPDL is 2-EXPTIME-hard under EXPTIME-reductions.

Proof According to [2], there is an alternating, exponential space boundedTuring Machine whose word problem is hard for double exponential time andpolynomial time reductions. If the reduction is allowed to take exponentialtime then one can use exponentially many counter bits for D and makethe reduction go through for an AEXPSPACE machine. Note that withexponentially many counter bits one can count up to 22p(n)

which – withthe right choice of p – is the maximal number of different configurations anAEXPSPACE machine can be in. �

Corollary 4 Satisfiability for IPDL is already EXPSPACE-hard over theclass of trees.

Proof Take M, w and ϕM,w from the proof of Theorem 1. Note that therepresentation of a witness for an accepting run of M on w is a tree sinceonly one atomic program is used. Hence, formula ϕM,w has the tree modelproperty which is not true for arbitrary IPDL formulas. �

9

232

Page 242: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 Conclusions

This is the first step towards closing the complexity gap for satisfiability ofIPDL. It remains to be seen whether this lower bound can be improved inorder to achieve 2-EXPTIME-completeness or Danecki’s upper bound canbe improved in order to obtain EXPSPACE-completeness.

The other main question that remains open is the exact complexity (i.e.both upper and lower bound) for test-free IPDL. The proof of the lowerbound presented here relies heavily on the presence of the test operator. Asimilar reduction might work for test-free IPDL, but then the encoding of amodel would have to be altered.

Acknowledgments I would like to thank Carsten Lutz for valuable sug-gestions concerning the improvement of this paper. For example, he pointedout Corollary 4. He also has found a simpler reduction using nondeterminis-tic, exponential space bounded Turing Machines. I would also like to thankthe two anonymous referees as well as Jan Johannsen and Stephane Remrifor their comments and suggestions.

References

[1] P. Balbiani and D. Vakarelov. Iteration-free PDL with intersection:a complete axiomatization. Fundamenta Informaticae, 45(3):173–194,February 2001.

[2] A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer. Alternation. Jour-nal of the ACM, 28(1):114–133, January 1981.

[3] S. Danecki. Nondeterministic propositional dynamic logic with inter-section is decidable. In A. Skowron, editor, Proc. 5th Symp. on Com-putation Theory, volume 208 of LNCS, pages 34–53, Zaborow, Poland,December 1984. Springer.

[4] M. J. Fischer and R. E. Ladner. Propositional dynamic logic of regularprograms. Journal of Computer and System Sciences, 18(2):194–211,April 1979.

[5] R. I. Goldblatt and S. K. Thomason. Axiomatic classes in propositionalmodal logic. In J. N. Crossley, editor, Algebra and Logic: Papers 14thSummer Research Inst. of the Australian Math. Soc., volume 450 ofLecture Notes in Mathematics, pages 163–173. Springer, 1975.

[6] D. Harel, D. Kozen, and J. Tiuryn. Dynamic Logic. MIT Press, 2000.

10

233

Page 243: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[7] David Harel. Recurring dominoes: Making the highly undecidablehighly understandable. Annals of Discrete Mathematics, 24:51–72,1985.

[8] J. Johannsen and M. Lange. CTL+ is complete for double exponentialtime. In J. C. M. Baeten, J. K. Lenstra, J. Parrow, and G. J. Woeginger,editors, Proc. 30th Int. Coll. on Automata, Logics and Programming,ICALP’03, volume 2719 of LNCS, pages 767 – 775, Eindhoven, NL,June 2003. Springer.

[9] D. Kozen. Results on the propositional µ-calculus. TCS, 27:333–354,December 1983.

[10] D. Kozen and R. Parikh. An elementary proof of the completeness ofPDL (note). TCS, 14:113 – 118, 1981.

[11] C. Lutz and U. Sattler. The complexity of reasoning with booleanmodal logic. In Advances in Modal Logic 2000 (AiML 2000), Leipzig,Germany, 2000. Final version appeared in Advances in Modal LogicVolume 3, 2001.

[12] A. J. Mayer and L. J. Stockmeyer. The complexity of PDL with inter-leaving. TCS, 161(1–2):109–122, 15 July 1996.

[13] S. Passy and T. Tinchev. An essay in combinatory dynamic logic.Information and Computation, 93(2):263–332, 1991.

[14] V. R. Pratt. A practical decision method for propositional dynamiclogic. In Proc. 10th Symp. on Theory of Computing, STOC’78, pages326–337, San Diego, California, May 1978.

[15] V. R. Pratt. Models of program logics. In Proc. 20th Symp. on Foun-dations of Computer Science, FOCS’79, pages 115–122. IEEE, 1979.

[16] K. Schild. A correspondence theory for terminological logics: Prelimi-nary report. In Proc. 12th Int. Joint Conf. on Artificial Intelligence, IJ-CAI’91, pages 466–471, Sydney, Australia, August 1991. Morgan Kauf-mann.

[17] K. Segerberg. A completeness theorem in the modal logic of programs.Notices of the AMS, 24(6):A–552, October 1977.

[18] R. S. Streett. Propositional dynamic logic of looping and converse iselementarily decidable. Information and Control, 54(1/2):121–141, July1982.

[19] M. Y. Vardi and L. Stockmeyer. Improved upper and lower bounds formodal logics of programs. In Proc. 17th Symp. on Theory of Computing,STOC’85, pages 240–251, Baltimore, USA, May 1985. ACM.

11

234

Page 244: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� ������� �� ���������� �� ��� �� � ����� ����������

������� ���

�� ��� �� ���������� �������� �����

�������� �� �� ����� �������� �����

���������������

��������

��� ��� �� ��� ��� � �� ���� � � ���������� ������ �� �� �� ������������ �� ��� ������� ��� ���� ��� ��� �� ������ �� ����� ��������� � ��� ������� ��� ����� ���� ������� ������ ������������� �������� ��������� �������� ��� �� ����� ��� �� �� �� ���������������� ���� ���������� �� ��� ����� �� ���������� ������ �� ��������� ���� ����� ��������� ����� ��� ������ ������� �������� ��������� � ����� �� � ��� ��� ��� ���� ��� �������� ��������� ������ � ���� ������������� �� ������� ���� �������� � ! � ����� ���� ����� ����� �� ��� ��� ������ ���� ��� ������ �� ���� ��������� ����������� ���� �� ���

������� "#$% � �� ������� ���� ������ ������� ����� ����&����� ������������� ���� ��� ���� ������ �� ��� �� �� �� ����&������

��������� ����� �� ��� ����� ���� ����� �' � ���� �� ��� �������� �� � ������� �� ���������� � ����� �� ���������� � ����� ���������� ���� ����� �� ������ ��� � ���� ������������� ���� ����� �� ������ ���� ��� ���������� ���� ����� �� ������� ��� �� ������ ��� ������� ���� �� �� ���� � ��� ��� ����

��������� ����������� � ���� ����� ������ ������ ������ �������������� �������������������������� ����� ������� �������� ���� ��������� ������� ����� �����������

� ����������

�� �� ���������� ���� ������������� ������ ���� ������ ���� ������� ���� � ��������� ������ ���������� ��� ������� ���� ������� �� ��������� ������� �������� ���� ��������� ������ ����� ��� ���������� ����� �� �� ���� ����� ���� ����� ��� ����������������� ������ �� �� ������ ��� ������������� �� ������ � �������������� ����� ��� ������������ ���� � ����� ���� !"����� �� #������� �� ����� ���� ������ ����� ��������� �� �������� ������ �� ���������� ������ ���� ������������� ����������� #��� ��� ������ ��� ����� � ������ ������ �� ������ ����� ������� ����� $%�� �� #������� &������� �� �������

'������ ��������� ����������� ������� ������ ����� �� �������� �� ����� � �������� ����������� �� ������������ �� ������ �� ����������� ������ ��� �� ��� ����������� �� ����������������� ��� ���� ������� ������ �� ��� ������ ���������� �������� �� �������� ()* ���� ������ �������� ��� �� ��������� ���� ������ � ������������ ��������������� ��� (+* ��� ������� ���������������� ����� �� ������ ���� ������ ��������� �� �������� �� ������ ���� ��� ��������������������� ������� �����

'� �� �� ����� ����� $%�� �� ��� ����� � #������� ��� ������ � �����,����� ���� ������ ����� ��� ����������� ���������� ���������� #��� �� �� ��� ��������� ���� ���� �������������� ������������ ����� ����������� �� ��� ��������� �������� � ���������� �����,����� ��������� #� ����������� ���������� ��� �� �� (-.* �������� ������ � ���� ��������� ����������� �������� ������������������ �����,����� ���� ������ ����� �� �����,����� �������� ���� ������� ���� ����� �� /�������� ���� �����,����� �������� �� ������� ������������ �� ���� ������� �� �����0������ ������

-

235

Page 245: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

���� ���� � ��� ���������� ����� ��� �������� ���� ������ ���� �����,����� ��������� #��� ����� ������� � ��� ������ � ������������ ��� �������1 ������ ���� ������ ������ �����,����� ��� ����� ����������� 2�� ��������� ���� ��������� ����������� ��� �,����� ���� ����������������� ������ � ���������� �������������� 3���� ��� ������ � ���� �����,����� ������ ���� �� ����� �� ������ �� ��� ��� � ��� �� �� (-.* ���� �� �� � ����� �������� �������������������� #��� ����� ������ �� ����������� �� ������� ����� ����4 ���� ������ (5* ����� ������� ��� ��� �� ��� �� �� (-.* ��������� � ����� ����� �� 0������ ��������6���� ������� ����� ������������������ 7����������� ��� ���� �������� � ���� ���������� �����,����� �� ����������� ��� ��� ���� �� ���� ���� ������������

�� ���� ����� �� ��� ����� �� ����������� ������� � ������������ �� �����,����� ������ ��������� ��������� ���� ������ ������ ����� ��� �� ��������� ������������ 8�� �������� ����� ��� ������ ,�������1 �� ������ ������������ �� ������������ ���� ������� �� �������� ������ ������������ ������ ���� ���� ���� ������ � �� ��� ���� � ����� ���������9 �� ��� ����������� ������ �������������� �� ������������ �������������� ����� �� ��� ���������� ������������ ����� ����������9 �� ���� �� ����� ���� �� ���� ��� ��� ������� ������� �������� ������� ������� ����� ������ ������������� ���� �� �� � ��������� ��������� /� ��� ����� �� ����������� ������� ������������� ���� ������� �� �������� ����� ��� ������ ������ �������� ���������� ������ �� ������������ ���� ���� ������ � ������������ �������� ���� �������������� � ������� ����� �����������4�� ���� �� ���� ��������� �� ������� ������������ � ����� �����������

#�� ��������� � ��� ������� ���� �� �� ������� ������ : ��������� �� ���������6�� ����������������� ������� �� ������ �����,����� ���������� ������ ) ������ ��� ��������� � ���������� ���������� ������ ��� ���������� � ����� ������������ ���� ������� �� ������ ����4 ���� ��������� ���� �������� �� ��� 0�� ���� ������ ) �� �������� ������ 5 ������ ���� ����0���� ����� ������� ������ ���� ��� ������������ ���� ������� �� �������� ����� ;������ ������ + �������� ���������������� ������� �� ���� ����� ���� �������� �� ���������� ���� ��� �� ���� (-.* �����

#�� ������ ����� ���� ��� ���� ������� ��� ��������� �� ���������� � ��������� ������ ���������� �� ���� ������

� ����� ����

#��� ������� ���� �� ���� �������� ������ �������� ������������ ����� �������� �������� � �������������� ��������� ������� ����������� �� � ������ � ���� ��������� �������� �� ����� �������������� <���� � ��������� ��� �0�� �� ��� ������ ���� /� ���� ������ ������ ���� ��� ���������� ������ 0����4 ��� ����� ����� ��������� ���� �������� � ��� ����� ���� ��������� �� � ������� ���������� �� ���� �������� ��� ��� � ������� ����� ���� ������������ =��� >����� ��������������� ���������� ��� ������ � ��������� ����� ����

���� ��� ���� ����

�� ����� � �� ��� ���������� ����� �� � ����� ���� ��� ����� ���������� ����� �� ���� ��������� ��������� ����� ����� � � � �� � �� ������� ��� ������ � ��������� �������� �� � �������������� ����1 ����� ����� ���� �������� ��,����� �� ����� ��������� �� � ������� ���������������� ��� ����� ���������� ����� �� ���� �������� � ��������� ����� ���� � ������� ��������� �0���� ����� �� ����� ������ �� ������� �� ��� � ������ � �������� ���� �� ����� ��� ���� �������� ��� ����� � ��� ������� �� ����� �������� �� � 0���� �� ���� ���������� �� � ����������� ���� �������� ������������ � �������� ������ ���������� ������ ���� ?����������� � ����������� ���� ����� ���@�������� � ���������� ����������� ;�� �� ���� � ��� ���� ����� ��� ������ ��� ����� � ��

:

236

Page 246: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� ������������ ����� ������������ �����

����� � �������� ������' ������ ������� �� ����� ������� ��� ���� ��� �� � �� � � ����

�����' ������ ������� �� ����� ������� ��� ���� ��� � � � �� �� ����

�������� � ����� ��� � ������ �'������� ( ����������� �� ( ��� �) � �

��� � ����� �'������� ( ���� ���� �� ����� � � ��� �������� � ��� � � � �

���

��������� ������� � ���� �� ��� ��� � �) �� �������� � ��

������������ ������ � ���� �� ���� ����� � � ��� �) � � � �����

��� ���������� ��� �������������� ��������

������ ���� �������������

����� ���� ������� ������

������ � ����� ����������� #�� ���� ���������� ������������ ����� ���� �� ��� (+*�

A�� �� �������� ��� �������� �������������� ������� ����� ������ ������ ���� �� � ���� ����������� ����� ���

������� ��� � ������ �� � ��� �� �� ��� �� ��������� �� ������ �� ���������� ������� ��

���� ������ ���� ����� ���� � �������� ��� ���������� �� ��

������ ;������ ��� ����(-$ #������ +�-*� �

/� ���� ���� �� �������� ��� ���� ������ �������� ���� �������� ��������� '� �� �� ��������� �� ������� �������� ��������� � ������� ��������� �B �� ��������� � ��������� ���������� #� ����� ������� �� ��� ��������� �� � ��� � �� �� �������� ���������� � ������� ��������� � ������ ���� �� � �� ��� � � �B � � � ��� � �� ���� � ��� � ��������� ��������� � ���� �� ������ � �� ��� � � � � �B � � � ��� � �� ���� ���������� ��� ���� ������ �� ���� ����� �� �,������� ��� � � � ����4 �� �� ������� ����� (C*� '� �������� � ������ � ���@����� ������ � � ��������� ���� � ���@����� �� ������ �� �������� ����� ����� � ����� �� ���� ������� A�� �� �0�� ��� ������ � ����� �� ��� ����� � ����� �������� ����� ����� �������� ��� � ���@�����D������� ���������� ������ � ��������� ����� ����� �� ������� ���� ��� �� ������ ���� �� ���� � ������� ��������� �� ���������� /� ����� ������ ��� ��������� �����1 �� ���� ��,���� �������� �� �� ������0������ #��� ����� �B������ ������� ��� � ����� �� ��������� ���� � ����� (C*4 � ����� ��� ���� ����� ��������������

���� ��� ���� � ��� �� � ���� � �������� ������� ��� ������� ���� �����! � �� � ������ �� �� � �� ����

#�� ����� ��������� �� @����0� �� ��� ����������� ���� �� ��� �������� � �� � ���������� ���� �� ����� � �� ��� �0�� ��� ���@����� �� ��

���� ����� � � � �����

�� ���������� ��� ������� �������� � ������ ����� ��� �� ������ ;�� � ����� �� ��� ����� ��������������� ������� ������� �� :�-1 �� ������ ������� �� � � ���� �B �� ��� �� ������ �� � ����� �� ������� ������� ���������� ��� ��������� ������� � � ���� �� ����� ����� �������� ��� ��� ��������E ���� � ����� �����

#�� ������ ��� �� ��� ������ �� ���������� ��� �������� � �������� �������� �� ���� ����� ����� �� � ������������� �� ���� ��� ������ �� ���� ���� ���������� �� ���������� ��� ��������

)

237

Page 247: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ����� � � ���� �� � 0���� ���� �� ����������� �� � ������������ ����� �"���� � �������� ���� ����

� � � � �� ����� �������� �� ������������� �� ������� ������� �� �� ��� �������� �� ����������� ����� ���� ��� ��������� ����� �������� �� � ����� ����� � ���� ���� �������� �������� ����� ������ � ������ ;���������� ��� ��������� ������� ���� �������� ��� ������0����� �� �� �� ���� �������� �� ���� � ��������� ��� �������� �� ��� ������� ���� �� � ������ 7������ ���� ���� ��� ���������� ����� �� ������4 ������������� ��������� ������ ������� � ��� ����� �� ������ �������� ����������

' ���� � �0��� ��� ���@����� � �� �������� � �� ��� �������� �� ��� ������� � �B ��� ��� ����������� �,������� ��� �� �1

�� � ������ � �� �� � ���� � �� �-�

3���� ��� ���� ������������� ����� �� ���� ��� �������� �� ��� ������� � ����� �� �� � ������� �������� � �� ��� ���������� ���� � ��� �������� � �� ������ � ��� ��� -� #��� ���������� ����� �� ������ ������� ����� �"��������4 ������� ����� ��������� � � ����� � �� ����� �� ����#��� �� ������ �� ��� ���� ��������� ������ �� ��������� �������� �� � �������������4 �� ����� ����� �������������� � ������� ����� ���������� �� ��� ������ ����� ���� ����� �� �������� ���� ��������� � ����� � �������� ����� ��� ������ � � ������ A�� �� ����� ���� �� ��� ��������

���� ��� ����� �� � �������� �� ��� ������� ����� �"������� �� ������������

7�� �� ��� ���� �� ������ ��� ������� �������� ������ ������������ �� �� � ������ � ��������������� ���� ������� �� � ����� � ������ �� �,���������� �� �����4 �� �� �� ��������������� � �������� �������� ������������ ������������ �� ����������� ������������ �� ���������������� � ��������

��� ��� ����� ����� �������� ������������ � � � 9 /�� ���� � ������ ��� �� �� ���� ����������9 �� �� �������� �� �� ������������ �� �����������9 /��� ���� �� ����� �� ���� � �������������9

/� ���� ���� �� ����� ����� ,�������� �� � ���������� ���� ;�� � E ��������� ������������ � � � E ��������� ��� � ���� ������� �� � 0�� �������� � E � ������ �� ��� ������������ �� �0�� � �� � �B � ���� �� ��� � ����� �� ��� ������� � ���� ������������ �� �� #��� � ����� ��������6� �� � �� � ����������� �B �� ���� � � �� � �� ��� � � ��

' ������������ ����������� ������ ������� �� ���� ��� �� � ��������������� ' ����� ��������6� �� ��� � ������������� �B �� �� ���������� �� ��� � �� � ���� ����� �� �� � ���� ������ �������� �� ���������� ��� �� ��� ������������ �������� 8 ������ � �������������� ������� � ���������������� ;����� � � � � �������������� ������� ��������������� �� � ��������������� ������� �����������������#��� ��� ������������ � �������� (-5* ��� �� ����� �� ��� 0��� ������� � � �� ������������������ �� ���� � �������������� ��� �� �� ����� ����� ����� �� �� �B������ ������� ����� �����������

�� �� ��� �������� �� ������ ��� ������� � �� ������������� ����� �� ��� ����� ���� ��������������� �� =��������� ������� ��� ���������� ������� � ���� ���� ��� ����� �������� �������� ������� ��� ����������� ������� ��� �� ������� ������� ��� ������������ (: ������� C�.$*��7����������� ��� ����� � �������� ������� �� #������� �� ������� �� ������ �� /�������� ���� ����� (-% ������� .*� ����� ��� �� ������� �������� � � ������������� ������ ���� � ����������� ���� �� ��� ����� ���� ���������� �����

F�����������6 ������� ����� ������� � ������������ ��� �����,������ ��� �� ����� �� ;����� -������ ��� ����� ����� ���������� ������ ��������� �� � ��� ���� ��� ��������� � � ��������������� ����� ��� ��������������

5

238

Page 248: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� �� ��

��� ��� ���

����

����

����

����

����

����

����

����

����

;����� -1 >����� ���������� ������� ������� ��������� �����,����� ���������� #��� �� ��� � ������� E������� @���� ��� ����� ���� �� �� ��������

'� ��� ������ �������� ��� ��� ���� ���� ��� ���������� ��� ������� #�� ���� ���������� ����� ������� ���� ���� �� �� ����� ����� ����1

�� � ��� � �� � ��� � �� � ��� �

#�� 0��� ��� ���� �� ������ �� ������ ) ��� ����� ��� �� ������ 5 ��� ���� ��� ���� �� ��������� ������ +�

� �������� ���� �� ����� �������

A�� �� ����� ���� ��� ���� ���� �� � ��� � /� ��������6� ���� � ������ � ���� (-C* ��� ��������� ����� ��� ������������� ������� /� ��� ����� �� ����� ���� ����� �� ����� ������ � �� ���������� ����� �� �������������� #��� ������ �� ����� ��� ���� �������� �� ��� 0��� #�� 0��� ��� ������� �� ��� ������� ������ �� 2� �������1 �������� �� ��� ��� ,������� ������� ����� ��������������� �� ������������ �������� ����� ������� ��� � ��� ��� � ������� ����� 2� F������ �� A�=���������� ���� �� ��� ���@��� �� ��� -G$%��� '������ ��� ��� ���� �� H� 2����� ������1 ������ ������ � ������� ���� �� ������ ������� ����� � � ����9

#�� �������� ������������ � ���� (-C* �� ������� �� ��� ���� �� ���� �� ���� �� �� ���� ��� ���������� �������������� � ��� �������� ��������� ������� �������� 3���� ������������������������ ���� � �������� �������� ���� ������������ 7����������� �� ��� �� ������� �������� ���������������� ���� ������������ �� ���� �� ��� ������ � �������� �� ������������ 8� ��� ����� ��� ��� ������� ������������ ����� �� ������� � �� �������� ������� ������� ���� E @��� � ������� ������ ������ �������� �� �� ������ �� ��� �������� �������

;�� ��������� ��� � ��� � ����� ��� �������� �� ��� E ��� ��� � ��� �������� ������������ �������� �� A�� � �� ��� ���� �������� ��� �� � � �� ��� ���� � E ��� ������� ������� ����� �������� ��� �� ��� ���������� ������� ���� ������ ��������������� ���� �E ��� ������� ������� � ����������� ���� ��� �4 ���� �� ���� � � � � �B � � ���������� ����� ��� �� �� ��� ������������ �������� �� ������� ���������� ���� ������� �� ������ I������� �������� � ��� ������ �� �� ���������� ���� ���� � �� � ��� �������� ��������� ;�� � � � � � � � ��� ���� �������� ���������� � �� � ���� �� ����� �� ��� �� ������� ��� �� ���� �� ������ �� ����� ����� /� ���� ����� ���������� ������ � ����� ��� ���� ������ �� ������� �� ��� ���������

+

239

Page 249: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

������� ��� #��� �"���� � ��������� �������� ��� ���������� �������� ��� � ���� ��� ������ ���

����� �� � $ ����% ��� ��� & �������� �� ������ �������

/� ���� ������ ��� � ������ � ������ ����0���� �������

�� �� ��� � � ���� �� ���� �"���� �� ���� ��������� ��� � ���� � �����

��� � ���% ���� �� ����

� � % � � �� � �� � � ��� �� ���� � �� �% ����� �� � �� � �� � ��� #�� ��� ���� � ��� � � �� ������ ������ ��� ��������� �������������� � � ��� ������� �� ���

;�� ��� � �

�� �� ������ � �� � ����� ��� �� ���� ���

8�� ���� ��@��� � �������� ���� �� ������� �� �� ��� ����� ���� �� ����� ������� ���� ������������������� '� ���� �� � ������ � ��� �������� ������� �������� �� ���

�� �� ��� �� �� � �������� ������� �������

7�� �� ��� ����� �� �0�� 0�� ����� ���������� �� ��� �������� � ��� �������� ��������� ���� �������� �� �� �� �������6� ��������� ���� ���� (-C* �������������

;���� �� ��� ����� �� �0�� ��� ��������� ��������1

� � � � � �� �� � � �� � � ���� ��

� � � � � ���

��

���� ��

� � � � �� �� � � ��� � ���� � ���� ��

� � � � � �� �� � � ��� � �� � ���� � � ��� � ��� � ����!� ��

<�0�� ��������� �� �� ������1

������� �� ����������

������� �� ���� ������

������ �� ���������

������� �� ����������

;������ ��� ���� �� ��� ����� ������� ������������ ������� �� ��� �� �� �������� ���� � ������������� ���� ���� �� �0�� �� ��� ��� � ��� ����� ���������� ��� ����� � �� �� ��� ���� ������������� /� ���� �� ����� ���� ����� ���������� ��������� ���� ��������� @����� �� �� ������� ����� ����� ������������ � �� �� �� ���� �� � ��� ��������� ����� �� ���� ���������� ����������� � ������ ���� ����� �������� �� �� '����� ��� �� ���� � ����� � ����� ���� � ������� � #���

.

240

Page 250: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

���

������ � ��� ��

���

����������� � �

���

���������

�� ���������

���

������ � ��� ��

���

����� ������ � �����

���

��������

3���� �� �� ������ �� ���������

���

����� � ���

���

���� �:�

��

���

������ � ���

���

���� �)�

8������ ���� �� ���� ����� ��� ���������� �� �������� #� ��������� ��� �������� �� : ������ ����������� � � ����

��� �

���

��� � ���� ��� � �� �� � ! �����

��� ���� �� ���� � � �� � ���� ����� �� � ������� � ��������� �� #��� �� �������

��� � ��������� � ���� � �������������

#� ����������

��

���

��

� ��� ��

���

��

���

�����

���� ��� " ���� �� ���� � � �

���

���� � "� 8�������� �� ���� � ����

�� ����� ������ � � ����

� ��� � 3���� �� ���� � ����

�� ��� ��

����

���� � " �� ���� �� ����

��

���

��

� ��� � " �����

����� �� )� �� ��� ���� � ���� � ��� � ��0���� ������������� ����� ���� �� ����� �������� �������4 ���� ������� ���� ������ ��� ����� (-:*� #���� �� ��� ��� �����

�� �� �� ��% ��% �% �� ��� ��� �� ���������� �������� ��������

/� ��� ���� �� �0�� ��� �������� �� ���������� ������ ��� � ���������� � ������� ���� �������� �������1 ���� �� �� ����� ��������� �� ������� ���� ��� ��� ��������� �0�� ����� ���� �� �� � �� �� ���� #���� ���� ���� �� ����� �� � ���� ���������� �����������

7�� @��� ���� �� ���� � ���� (-C* �� ����� ��� �������� � ������� �������� �� ������ ����������� ��� ���������� ���� � �4 ���� �� � ��������� �� �� � �� �� ���� #�� �������� ����������� ���� �� ����� �� #� A��

$ �� ���

��

��� �� �������� � �������

$

241

Page 251: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� �� ��� $� � �������� '����% �� ������� �� � �� ����� $� �� �� �� �� ��� ��� ���� �� �� ����� ��

�� �� ��� � ��� � �%

������ �� � ��� � � ��� � �������

�� �� ��� � ��� � �%

������ �� �� ������

�� �� � ��� �����

�� �� ��� � ��� � �%

������ ��� � �����

���

���� �� � �� � ���!���

#��� ����� �� ��� ��������

�� �� �� #��� �"���� � �������� ��� $ ��� �� ���� ��� �� �������� # ����

� � $ % � �5�

���

� � �� �� � � � $� � � ��� � ��� �+�

� � �� � ����������� ������% ����% ��� ������� ���� �������� ��������% ����� ��� ����� ��� ��

���������� �� �������!

� � ���$ � �� �.�

� � ���� ������� $� � ��� ��� � ����� � � ���� ����� ���� ��� �$�

#�� ����� ����� ������� ��� ���� �������� ;�� ������� ��������� �� ���� (-C* �� �� ������� �������� ������� � ���� � ������������ �� ����� � �������� (--* � �

� �������� ���� �� ������� �������

#��� ������� ����������� �� � ��� �� ���������� ���� ������������ ������� �������� � ��������������������� �� ��� G%�� �������� (G* �� �� � (:-* ����� ����������� ���� ����� ��� ���������������� ���� ������� ���������� ���������� � ���� E ��� ����� ����� � ������ ����� /� ��� ������� ���� ���� ����� ������� ���� ����� ����� �� ��� ������ ���� ����� �� ��� �������� �� � 0�������� �� ��� ���� ���� ������� �� ��� ����� � ���� �������� ���������� �� ����� ��� ��������������������� ��� ���� ��� �������� �� � ������ ��� ��� �� ��� ������������������� ������� 8� ��� �������� �� ��� �� ����� ���� ��� ������� ���������� ���������� � ���� ��� ������� �� ��� �����0�������������� �� 7>���������� =������� ������ ��� � ���� ���� ��� ��� 0������ ��������6����� '� ����� �� ���� ���� ���� �� �� ����� �� ����� ����������

��!"����" �� (� � ����������� ����� �� ���� ��������� ��� ����� ����� ��������� �� � ���������� ������ �� �������� ������� �������% ����

� &� ' ( � �% ) ��� �� ��� ( �&� '�% ���� ) � ��%

C

242

Page 252: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� � � � � ���� � � �Æ

� �� ��� ��

�� �� � � �� ����

& ) '

;����� :1 ;���� ������� 7����� ���� � ������ & �� ' �� �,��� �� ( ) � � �� ����� �� �� ��� �� ��������

� �� �� ��� �� &� � �*� � * )� � �+� �+ '�%

� ������ �� ���� �+� � �� &�+ '� � ���� �+� � ���+ &� � � +� � ���� �+� � ���+ '� � % +� � ���� � *��� �*�� *��� �*��+� � �� &�+ '� * )�%

� ������ �� �� ������� ��� � (� � �� ������� ��������� ��� � (��

7��� ���� �� ���� ��� �0������ ���� ������� �� ������ ��� 7������ ���������� � ������������� �������4 ����� ���� � � ���� ;����� ����J����� ����� � ������ + ��� �� ���� ���������� �������� �� ��� �� � ���� � � � � +�

�� ������ �0��

��� �� � ���

����� � ��� �� ������ �� �� ��

� ����� �������� 1

�� � �� �� � � � ��� ��

������� ���� �� ���� � ����� ������ � ������ ���� ���� ������� ����� �� ��������� ��������� ����� ������������ �� ���������� ���������� �������� ����� ��,������ ��� ��� �� ���� ��������������

� ��� ��,������ � ������������ ������� ����� ���� ����� �� ����� �� ������1

�� �� ��� � ������ � �� �� ��� � �����

� � �� (��

������� �� ����� ����� ���������� ���� �� ��"���� ���������� �� �� �� � ����������� ������

������ ?� �������� (G* �� �� � (:-*� �

#��� �� ���� ��� ��� ����� ����������� ����� �� ��"���� ���� ����� ���������������� #������������ �� � ���� � ��� ��� ������ ����� ������ & �� ' ��� ����� �� ����� �,��� �� ( ���� ������� ��)���� ��� ������ E ������ �� )�����

;��� ��� �� ���� ������� �� 0� ��� �������� �� ���� ����������� �� ����������� ������ ����& � ( ��� ����� ��� ��������� �������� �� ����� ���� 0���� & �� ' � ( E �� �� ������ �� ������������������ '� �� ��� ��0���� ���� ) �� ������ �,��� �� % ������ � �������� ������ �� ���� ����������� �����

G

243

Page 253: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

���� �� #�� ��������� ���������� ������� �� ������� �� ���� ����!

�� �� �� (�� �C�

�� � �� �� �� �� (� � % ��� �G�

*� ��������%

�� � ���

�-%�

�� � ������ �� ���� �� ���� � �� ��

������ ��������� C �� G ��� ���� �� ��� ����� � ��� ���� ��� ���� ������ �� �� ���� ������� ����� ����� ������ �� �� ���� �� ��� ����� � ����� �� ����� �� � ����� �������� �

��� ����� � % ��

�������� -% ������ ��� ��� ��� ���� ��� ����� ���@���� � �� �� � �������� � ��� ��� ���@���� ��� �� �� ���� �� �� ��

� � �� � ������� � ����4 ��� ���� �� ������ �� ������ ��������� �

���� �� ���� �������� �� ���� (��������� ��� �������� ��� ���������� ������� �� ���������� ��� ���� &�6�����6�� ����� � ��� ��� � ���� �������� ����% ��� ������

��� �� ��� � ��

� �� � �� ��� � �� �����

������ #��� ��� �������� ��� � �������� ��� ������������ ������� ���� �� ����� A�� �� �������� ������� � ������������ �� �� �� ��� �� ��� � (� 7�� �0�� ��� ��,������ � �������� ���� �������1

�� �� ���� �� (�

��

�� �� ��� ��� �� (��

�� ��������� -% � A���� 5�: ��� � �� � ��������� �� � �� �� ������ ��������

�� � �����

��

������

�� � �����

��� �� ����� G

�� � �� �� � ��

���

��

�� �� ������ ��������

�� � ������

��4 �� � ������� ��� ��� ����������

�� � ������

��� 3���� ����

� ��������� �0�� ��

��� ���

���

��

��� ����� � ��� ���� �� �,��� �� ���� ���� ����� � ��� ��������� ������ �� ��

� �� ��� ����������� C � A���� 5�: ��� �� � ����� ���� �� �� �

������� � #��� �� �� �������� ��� �� ��� ������ ����������� �� �� ��)���� ��"���� ����������

-%

244

Page 254: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

������ �� A���� 5�) �� �� ������ �� ����� ���� ��� ��� � ���� �������� ����� �� �� ���� � ������ �� ����� �� ��� ���� � ��� ��� ����� '����� ����� �� � ����� �� ����� ������ ����� ����� ���� ���� ��������� �� �� ��� ����� � ���� ����� ����� ���� � ����� �� ��������� �� �����4 �� ��� ����� � �� ����� ���� �� � ����� �� ��

���� �� ��������� �� ���� ��� �� ��� ���������� �� �� ��0���� ��� ���� ��0���� ���������� �� ���� ������

���� ����� �� � ������������� 3���� ��� ��� � ���� �������� ������� �� ����� ������� ���������� ���� ����� �� ��� �� �� ����� �� ����������� ������� �������� ��� ������������ ������� ���� �� � ����� ��0���� ������� ���� ������ �

8�� ��� ����� ���� ��� ����������� ������ ��� ������� �� ����� �����0������� ������� �� 7>���������� =������� ��� � ���� ���� ��� ��������� � ��� ��� �������� �� ���� ��� 0������ ���������6����4 ��� ��������6����� � ���� �� ������ �� ��� �������� ��� � ������1

���

� �� ��� �� ��� �� �

�� ��+� �� � ���� � � ����

� �� � �� ��� �

��� ��� ��� � � ��� �� � �� ���� �� ��� � � ����

#���� ������� ����� ��� �� � (:-* �� � ���������� ����� �� �� !� �� (-G*� �� ���� ������ ��� ����� ������ ��� ��� �������� ����� ���� ��� ���������� � ���� ��� � ����������#��� ��� ��������� �� ��� ������� � ��� ���������� � ���� �� ������ ��������1 �������� ���������������� � ������ ���������� �� ������ ��� ���� ������� ��� ���� ����������� � ��� �������� �� �������������������� � ���� ������������ �� ������ ������ ������ ) ����� ���� �� ������� �� �� ��� �������� �� ������ ���� ���@����� ��������� �� ��������� �������� >��� � ���� ������ ���� ��,������� ���� ������,��� ������ ��� ����� � ��� ������� ����� #������� �� ���� �� ������ �� � ������������� �������� ���� � ���� ������� �������� � �������������� �������� �� ����� ������ � ������ �����

� �������� ��� � �� ��� ����� ��������

#��� ������� ���� ���� � ��������������� ���� ������������������ � ������� ����� ����������� �� �������� ����� ���� ������ ���� ���������������� ������� ����� ���������� ���� � �����4 �� ��� ����������� ��������6����� � ������ �������������� ������� ��� �� ��� �� ������������ ��"(::*1

������� ��� � ��� �������� ����� ����� �� ��� � �������������� �� ��� ������� �� ����� �"��������

�� �% ���� �"���� � ��������� �� ������ ����� ���� ��� ���� ������� ����� �"�������� �����% ������� ��

� ���� �� ���� ���������� �������� �� �������

������ ������� ��� �� ������ �� ��������� ������� ���� ��� �� ��� ���� � #������ +�- ������� �� � ���� �������� �� ��� �� �� (-.*� 3� �0�� � ��������� � ���������� � �������������� ���� ������� �������� ������� ���� �� ��� ������ ! �� !� �� ! ��� ��J����� �� �������� ������� ������� !� � ��� ����J����� �� ��� ������� ��� !� #�� ������� � ��������� ��������������� � ��0���� ������� ���������� ! �� ����� ������������ 8������ ���� ��� �������� ����1

���� ��� ���� ������� ��� �� �� ���"���

������ '����� � � ���"�� � � �� ����� ������ ���@����� � ���� ����������� ������0������� ���� �������� ��� �� ����� ���� � �� ���� � ���� �� ������� ��� ��� ������ �������� <�0��� �� ����� � � ��� ������� � ��� ��� ���� � � ������� 8�� ���� �� �� ���� ����

� � �� �--�

--

245

Page 255: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��

� � ��� �-:�

'� �� ���������� � � ���� �� ���� ����� ����

�� � � � ����� ������ " ��� � ����� �-)�

����� ������� ��� �������� -) ������ ��� -: �� -- ������� -: ������� ���� � ���� � � ������� � � ����"���� �� -- ������ ��� ��� ��� ���� �� ��� � � ����� � � ������ '� � � �����

�� ���� �� �� ����� � � � ����� �� ��� ���� � � ��� ����� ����� �� -:� �

�� � (:%* ��� ����� ���� ������� ������ �� ��� ��� ��� �� ��� ������� �� ������ � ��������������� ������� A�� �� 0� ��� ������������ �����0���� , �� ��� ��� � ��� ������� ��������'�������� ������� � ��� ���� ��� �0�� �� ����� ����� ������� ! �� ����� ������������ ���� ������ � ������� ������� ������� �� , �� ����������� � ���� ����� � �� ��� ����� ���� ���������� �� ��� ��� �� ����� ���� ��� ����� �������� �� � �� ��� ���� �� ��� ����� �������� �� ������������ � ��������� ��� ������ ��� 0������ ��������6� �� ������ (5*� �� ���� ������������ � ������ � �� � �� ��� ��� " ��� #��� ��������� ��� ���� ���� �� � ��� ������ ��� ���������� �� ;����� - ��� ������� '������� �� �� ��� �������� �������� �� ��� ����� �� �������������� ������1

������� ��� #��� �"���� � ���������� )������ �"������+���� ������� ����� ���� �� ��������� � ����� ��� ���������� ������ ����� ������������ ������� ����� �"���������

������ ' ��������� ������� ��� �� ��� ���� �� 8 ������ ��� ���� ���� ������� ����0�������������� #��� �� ��� ���������� ������� �� ��� ��������� ������� ���"�� �� �� 7�� �������� ��������� � A���� +�:� #��� ��� � ������ � �� � �� �� ������� � ��� ��������6�� ������������� ������ �

8������ ���� �� ��� ����� ����

���� �� ���� ������� ��� �� �� ���"��

������ #��� ���� � �� �0�� �� �� ��������� ���� ����� ��� � �� �� �� ���� �� '�� � �

��� � � ���� �� � �� ���������� ������ ����� ���� ����� �� ���� � ���� � � �� ��� ���

���� ��� ��� ���� � �� �� ������ � � � �� ���� �� ���� ���� -- �� -: �� ���� ����� �� -) �� ������� ������� �� ��� ���� � ����� +�:� � -: ��� ��� ��� ���� �� � �� �� ���� � � ��� ������ ����� �� -) ������� �

#��� �� # �� � ���� #�� ��� ������� ���� ,������� ��� �� �������� ������� ������� � ����� ������� ��� �� ����� � ������� �������� ���������� ��� ���������� � � ����� �����K����������� �� �� ��� �� ��� ����� ��� �� ���� ���������� ��� ����� ������������ �� � ���� ������� � � �� �� ����� �� � ���� � ��������� � � ����������� ������ '�������� �� ��� ���� �A���� +�5 �� ��������� � �������� �� ��� ���� � A���� +�: ��� ��� �� � ������� ���� #��� ��������� ����� ���� �������� �������� ����� �� ��� � ���� ��������� ��������� ��� ����� �� ����� ������������� ���� ��� �� ��� ����� ��� ��������� �������� � ��������� �� �������� ���� �����,�������������� ��� ��� ��������������� �� �� (-.* �������� � ������ ���� ��������� ����� $� �� �������� #��#� ����������

������� �� � ������ � �������� �������� ������� �������� �� ��� ���������� ���� � � ����� ������������ #� $� ��� ��������6� ���� � ������ ������ � ������ �� 0�������� ����� ���� ���������� ��� ��� ������ �� I������� �������� ���� � ���������� ����� ��� � �-!� � ����� - ��

-:

246

Page 256: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ������ ���� �� �������� ,�����0���� �� ��� ������� ���� ���� ������ �� �������� ���� ������� ��3����� ����� ����� ���� #���0��������� ?������ ���� #���0�������� ����� ���� �� ��� ������- ������� ���������� ,�����0��� ��� ��,����� � ��������� ������� �� �� ��� ��,����� � ���������. ��� ��� � �������� � ��������� -��� �.� ��� �� ���������� ��� � ���� �� ������ � ������� ���� ���� �� �0����� �� ��� ������ ���� =��� ������� ��� �� ��������� �� ��������� � #� ��� ������������� ���������� ��/ � #��� �� ���� ������� � $� � ���� �� ��� �� � ��/ �-� �- �� $� ��/ ����#�� �������� ����1

������� ��� ���� ���������� ���� '����� ������ ������ ���� #���)�������� ��� � �������� � �����

������ #������ :�- ������� ���� 3����� ����� ����� ���� #���0�������� ��� � ������������� �� ��� ���������� ��� �0����� �� ������� ������� ���� ��� ���������� ��� �������� ���#���0����� �� ���� E � �� �� �0�� �� ������ � ���� ���� ��� �� �0����� �� ������ � ����� �

#���

�� � ��� � $� � ���� � �-5�

'� ��� � ��� ��� ������ � ���� ���������� �����,����� �������� �� ��� ������ �� ��������� �����,����� �������� ���� ������������ ������ 7����������� ��� ,������� ��� ������ ������� ���� ��� ���������� �� -5 ��� ������� ��� �� �� (-.* ����� ����

���� ������� ��� �� $� ��� "���

'� �� ������ ���� ���� ����������� �� ������ ���� ������ � A����� +�: �� +�5 ��� �� ����� ��A���� +�5 �� �� ������� ������� �� �� �� ������ ������������� � ��� ��������� �������

A�� �� 0���� ���� ���� � ��0�� ������� � ;����� -� �� �� ���� ��������� ��� ������� ����������������������

�� �� �

��� ��� ���

�� ��

��

����

������

������������

������

������

�����

� � � � � � � � � � � � � �

������

������

������

�����

�����

;����� )1 <���� ����� ����� ���������� �� ����� ���� �� �� ������� ������� ���� ��� ������ �� ����

-)

247

Page 257: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

#$%"�����&���"��� #�� ������ ������ �� ����� ��� ���������� 3�������� 8�� �� ��� ����������� ��������� ������ �� ������������� � ��� ����� H� 2����� �� #����6 �������� �� ��������������������� �� ���������� ��� ��� ���@��� �� ����� ������� � ����� �� 2������� ������� �� ���������������� !���� ��� ������� �� ��� ��� ����������� �� ������ �� ��� ��������� ������ �������� ��������� #����� ��� ���� �� �� ��� ������� � ��� 8�� A��������� �� !'� # �� ���������� ����������� � ����� ����������� �� �����

������ ��

"#% � ��� ������ �� �� ����� ��� �� ������� ��� ��� �� *����� �� ����� �� ����������� *������� +������ ,-� .//#�

".% ���� ������ ��� ���� ��������� ���� ��� ��� �� *����� �� 0��� 1���� � #223�

"-% ������������ �� ��� ������ � ������������ � ����������� � ������ ����� ��� �� ������� ���� ��� � �� *���������� *+45� #226� � $-73.�

"8% �� � ����!���� �� � ������� �� ����� ����� ��� � ������ �� ����� � ��� �� 82',./7,.3� #268�

",% "� #$��� ������ �������� ����� �������� ��� ������������ ������ ��� � ��� ��� 86'.#27.-8�#262�

"$% ��%� &����� ��� ������� � � ��� ������������ ������ � �� ����� ��� �� ������ �� ����� ���� �� 8/'#8#7#86� #23,�

"3% ��%� &����� �� �������� � �� ������� �� ��� ������������ ������ � ��� � ������� �� ������������ ������ �� ��� � ��� ��� -8'---7-8.� #23,�

"6% � �'���� �� ������� ������� � ������� �������� ��� ���������� ������ � ��� �� 9����� *�����0���������� ���� .6� ����� #22-�

"2% (� "!������ ������� � ����� ��������� ������� �� �������� �� ��� �� -. �#226 � � ,-72,�

"#/% �� "�� ��� ���� �� ����� ���� � ��� ��� �� +�� �� �� 4���� �� ��� :��� ����� �� ����������� ���� #8.� ;������� #222�

"##% �� "�� ��� (� "!������ ���� � ����������� �� ��� �� ��� ��� ��� ��� ��� ������� ���� � ����� ���� �� �� ��� �� -/'3#736� .//#�

"#.% "� "����!���� �� ���!���� ��� ������ � �� �� �������� �� �� !���� �� �� ��� �������+�� �� �� ����� �� ��� ���� ����� �� ���������� �� 6$� ;������� #23$�

"#-% (� ������ ����� ��� � �� � ��� ������ ��������� �� ����� ��� �� � ������� �� ��� ��� ������������� �� �������� �� ��� �� ������������

"#8% %�"� (������ ������ ������� � ����� ��� �� ������ �� ����� � ��� �� -3'#,/7#,6� #23.�

"#,% %�"� (������ �������� � ����� �� ����� ��� � ������ �� ����� � ��� �� 8/'8-2788.� #23,�

"#$% �)���"� ��� �������� ����� � ���� �� � ����� � ����������� ��������� ����� �� 8,'$-733� #232�

"#3% �� ������� ���� ���� ������ ������� � ��� ��� � �� *���������� *+45� #226� � $-73.�

"#6% �� ������� �������� �������� ������ �� ����� � ��� �� $6'$/37$#8� .//-�

"#2% )� *����� (� ������ !����� ������"���� ����� ��� � � ����� ��� ��� #$% �������� �� ����������

"./% )� *����� ���� ��� �� ��� ��� ��� 0�< ����� #22-�

".#% )� *����� ����� ��� ������ ����� ������������������ �� ��� � "�������� 8. �#22$ � #8,7#3#�

"..% �� ��������� ���� )� *���� ��� �� ������� ����� �� ����� ��� � "��'% #�����$ �� %� �&���� �� ��� �' (� ) � ��� ���� -� ������� .//#� � 6-7.$$�

-5

248

Page 258: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

����� ����� � ����������� �������

������� ���

�������� � ����������� �������� �������

�� �������� �������

���������� ��������������

����� �����

���������� � �������� �������

���!������ � ��!������� �"

�����������!������

��������

�� �������� ��� � �� ��� ���� ��� �� ����������� �� ������ ��������� ��������� �� ���� ��� ����� �� ����� � ��� �� � ��� ��� ����� ���� ��� ������� � �������� ����������� �� ��� ������� �������������������� ����� ������� ����������� �� ����� ���� �� �� ��� �������� � ����� �� ��������� ��� ����� �� ��� ���� ������� ������� �� �������! ���� �� ������������ ������ ��� ��� ���� ��� � ��������� "��������� ���� �� �� ��� � ��������� �� ����� ��� � �� ������ � � ������� ������ �� �� �� �� �� ��������� ��� ����� �� ��� �� �� �"����� ������������ �� �� ������ ����� #� ����������� �� ������ ���� ��������� �� ����� � �� ��� �� ���� ������� �� �����"�� � ������ �������� ��� ���� �� �� $ ���� �� %���� �� ��� �� ��� ���� �������� ��� �� ����� ������ ��� � �� ��� �� �� ������������ �� �� ����� ���������� �� �� � ��� ��� ���� ���� �� ��� ��� �� ������� ����� ������ ���� ���� ��� ��� ���� �� ������

� �����������

��������� ��� ������� �� �������� ������ ������� �� ��� � �� �� ������� �� ��� �� �� ���������� ��� ���������� ������ � ����� ����� ������ ��������� �� �� ��� ����������� ��������� �������� ��������� ��� ��������� �� ���������� ������ ������ ������ � �� ��� ���� �� �� �����!�� � ������ � �� ���� �� ������� ��������"

# ������� !��$����� ������� �� ������� �� �������� ������ ������� ��� ����� �� �� ������������� %&' ()' (&*� ������ �������� %(+' ,-* ���� ���� �� �������� .�������� �� %/' ,0*'

# ����� �1������� ������� �������� �� ������� ������ ��� ������ �.�� 2����� �1����.���� ����� ���2��� ��� �� ������� �� �������� ������ ������� %3' (/' 0+' ,-' ()*

# 4���� ���� � �� ��� � ��� �������� ��������� �� �� ����� ��� ������� ������� �� ����������� ������� �� ��� � ��� ����������� ��������� �� ����� �� �� ������� �� ��� ����� � � � %0/' +' 0*

� ������� �����! ���� �� �� ���� ��� 5������� ���������" ������� !��$����� ������� %,*�������� ������� ������ %(' (6* ��� ����� ������� ���� � ��7� �����8� ���� ���� �� 959� ���

249

Page 259: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� �� �

� �� �

� ��� �

� ��� �

� �� � � �� �

��

� �� �

� �

� �� �

:���� 0" ;1������ ��� �� ���� �������� �� �� �����

�59 %0(* <���.��� ��� �� �� ��� ������� ��� ��=����� ������ ��� �� ������� ������������ �� �� ���� � ���� ��.������ ������� �� �� ������ ��������� ������ � ������" <��������� ������8� 4���� 9��� �� ����.��� %0&*� �� ��� � ����������� ��� ��������� �� ��� ������.��� ����� �� ��� � ��� ��� ��� ��� ������� � � ���� ����.��� �� ������� �� ������������� ��������� �� �����8� 0, �������� ������ ����.��� ������ �� ��� �������������.��� ���� ������� ����� �� �.�� � � �� ��� ��������7 ���.�� �1������ ����� �����=����� �� ������� ���������� ��� � � %,(' 6' 03*

5� ��.���� �� �>���� ������� ��� ���� ����� ������� ��� ��������� ��� ������� ���������� ������ �������� �� !�� ��.� � ���� � �� �� ���� �������� �� �� �������� �������������� � ������ ���� ����� � �� ������� �� �� ���� ����$��������� ��� � ��� ���� 7������ ?;��������$:�������@$�������� ��� ��.� ���� ������������ ����� �� �� %()* ��� %0-*� ���.��� ����� ��� �������A��� ���� � ������ �� ��� � ��� � �� .�������� ��� � � %(+'3* 5��� �� ��� ����� �� ���� ������� ����� ���� � ��� ���� ����� �������� ��������� ���� ���� ����$��������� ��� ����� ������� ��� ��������� �� ��� �� ������� ����� �� ��� ���� ��� ��� 5��� ����� ��������7 ��� ��������� ��� ������� ��� ���� ������� �� �� ���������� �� ���� %)* ��� ����� ��������� �� %,,* <���.��� � ���.�� ��Æ � � ������� �� ��������� ����.��� �� � � ���� � ���� ������ ��.���� �B���� � �� ��� �� �� 7���������� ����� ��.� ���� ������� �� ���

5� ����� ��� ������ � � ���.��� ��� ������� ���7 �� !��$����� ������� �� ������� %()'(&' (+' ,-*� � �� ������� � �����.� �� �� ����� ���� �� ������� �� � ������� �� !��$��������� ��� �� ���� ������ ����$�������� ��� ��������� ��� ���� ����� ���� 4��� ��� �������� ���� ���� �� �� ���� ��� �1� �� �� ���� �1������.� ����� �� �� ��$.������� ������� ����� :C ���� A������ �� ���� �� �1���������� ���� � �� ��� � ��� !��$����� ��������� ������� ���� ���������� ���� .�������� � �� ���� ����� ���� ���� �1������.� ����� ���� ��������� � �� ��� �� �� ����� ���� � D� ���� �� �� �.��������� �� ���� ����� ����� �������� ��� � ���� ����� �� ����� ���� �����" �� ����� � ��� ������ ���� ����� ������� ����������� �� �������� � ������ � ���� ���� ���� ��������� � � �� ����� � ���� ��� ������ � ���� ���� � �� ��$������� ������ �� � ���� �� ���� ��� � ������ �������� �� ������� ���� � �� �� ��� ������� �������� � � �� ?���� ��� �� �� �.��������� ��������� ��� ��� ������@ ��� ?�.��� ��.�� �� ���� �� � �� � ��� �� � ��7�@

5�� ������ �� ��� ����� �� � ������ � ���� ���� � �� �������� �� ��������� �� � �������� ���� ������ �� ������� ������������� �� ����� �������������� ��� �������������� ��� ������� ����� ���������� �������� 4��� ��� ������ ��� ����� �� ��������� �� �������" �� �� ��� (� ������� � ������ ��� ��� ��� � ���� �� ������ �� ����� ��� �� ���� � 5�� ����� ������� ������� �� �� �� ��� ,� ��� � ����� �������� �� �� �1������.����� �� ��������� �� �� ��� 6� ��������� � ����� �� ����� ���� � ��� �� �� ��B���� ������ �� ������ �� ���� ��� ����������� ����������� �� �� ��� +� �� ��� ���.� �� ���� ���� �� ��� ����� ������� �� ��������� � �� ������� �� �������� ��� ����� ��� ������ D� ���� ���� �� ��� �������� ���� �� �.�� ��� �� ������� ���� !����� �� ���������� ���� ������� 5���� ����� ������������� �� �� ��� & ����� �� ���.� �� ���� ���� � �� ������� �� �������� ��� �.��E��$���� :������� �� �� ��� ) �� ��.� ��� �������� ����� ��� ����� ���� � ������� ���� ��

���� �� �� ��������

(

250

Page 260: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Æ �� �� ��� ���� �� ���� �����

������ ������ ������ �������� � ������� ������� �� ������� ������� ��

���� ���� ���� ����

������ ������ ������ ������ ����� �������� ������� ���� ����� ������� ���� ��

����� ������� ���� ���� ����

������ ������ ��������� �� ����� �������� ������� ������� ���� ��������

������� ���� �����

������ ������ ������ ��� ������� �������� ����� ���� ���������� ����� ���� �����

����� ����� ���� ����� ����

������ ������ ��� ������ ��� �������� �������� �������� ���� �������� � ���� ��������

����� ����� ���� ����� ���� �����

������ ���������� �� �� ���� ������� ������� ���� �

���� ����

������ ��� ��� ��� ��� ���������� �������� ����� ����� ����� ����� ��������� �����

����� ����� ����� ����� ��������

:���� (" 5�� ���� ��������� ����

���������

D� ��� � ������ ��� ������ ����� ������� ����� �� ������� �� ��� ������ �� �� ��������$�������� �� �F��� ���� ��8�� �� �F�1������� ���� ��8�� ��� �F�������� ������ ���8����� �F��.���� �� �������� ������ ���8�� � �F������ �.�����8�� �� �F�>��8�� ���� �F���$�������������� ���8�� ��� ���� �F��.���� �� ���$�������� ������ ���8� :���� 0 ��.�� �1������ �� ������ �������� �� �� ���� ����� ��� ����� ������� ��� �� ������ ��B���� ������ �����������.� ���� � ��B���� ������ �� ������� ���� ��������� ��B���� ������ �� ������ ����� �����!������ �� ������� ���.���� �� �� �������� ����.��� ��.� �� ����� �� �� ������������� ��� ������ ��� ���� � G ��� ���� ���� � � ��� ����� � �� � ���$���� �� ��� ����������� �� � ��� ������ �������� �� � �� ��� ����� ���2��� �� � � � � �� G �� ��� �G ���2����� �1����.� �� � � �� ���� �� ��� � �� � �� �� ��� ������ �� ���������"

� �� �� ��������� �� �� ������ �� � � ��� ��� ��� � ��� ������� � ��� ��� ��� ����

��� �� ��.���� �������� �� ��� ��� ����� ����� �.���'

� �� ���� �� �� ���� ��������� ���� �:���� (� ��� ����!�� �� �� ����� ��� ��� ������� ��� � � � � �� �� ��� � ��� ���� �� �� !��$����� ����� �

�������� ���� �� � ���� ��� ������ �� � � � ����� ���

�� .���� �� �� �� ���2� ��� �.�� ��� ����$���������

����� �� ���� �� ��� ������ �� ��� �� � ������ �� ��!��� ������ �� � ���� ������ ������ ����������� �� ������� ������ ������� �� ��� �� ���� ���� ����� �� ��!�������� ������ �� ��� ��� ��������� C� �� ��� ����� ��� �� ������� ����� !��$����� �������� �� ������ �� ���� ���� ���������� �� �� ���$����� %()* ������ ���������� ��� ������� ���� ������ �� ��� �� ��� ��� �� �� � ������ ��� ������� �� ��� �� �� ����� � ������� �� ��� � �� � ���� � G ��� ��� ����� � �� � �� ��� � �� �� �������� �������� ��� � � � � ��� ��� �� � � � � �� ��.�

���� G �� ����� �� ����� ���� G ���� ��� ��� ����� G �����

,

251

Page 261: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

5�� ����� � ��� �� � �� ��� � ��� G � � ���� � ���� C� ���� ��� ������ ��� $�����������; ������ ��� �� �� ����� �� �������� ���� � �� �� ���� ���� ��� �� ������� ���������� �� �� �� ; ������ ����

��������� �� �� ����� ���� ������� ��B���� ��!������ �� ������� �� ������� �� ��� ����.� ���� ����� �� � ����� ������� ����� �����!�� ������� ��� ���$����� ������ ��������� � � ���$���� ����� � � � � � �� � ���� G � D� ���� �� � ����� �� �� �� ���$����� ������ ����� ����� �� �� ������� �� ��� � � H����� ���� ����� �.� ��!������ ��������� ��� ������� �� �� ; ������ ��� �� ��� � � �

� �� �� ������ �� ���$���� ��.�1 ������ ����� ����� �� ��'

� �� �� ����� �� ����� �����$�� ������ ����� �� ��� � � � ������� �� �� ����

��

��� ������� �� � � � � � ��� ���$�������� ����� ����.��� �� �

�� ��� ���� �� ����� ������� �������� �� ���� ��� �� <���.��� �� ����� ��� �� ��� ��� �� ����� ���.�� �� ��� ����� ���� ���� �� �� ������� ������ �������� ����

��.�� � ������� �� ��� � � ��� � �� �� ������� �� �� � �� ����� �� ���.�� �� ����� ������� �� �� ���� ��� G ���� ��

�� � � �� �� ����"

��� �� � ��� �B � � � G ���� �� � ��� �B ����� ���� G � � � � � �G ���� �� � �� �B ����� ���� �G � � � � � �G � � � � � �G ���� �� � ��� �B � G �

��� �� � ���� �B � � � G � � � � ���� �G �

��� �� � ����� �B � � ���� G ���� �� � ���� �B ��� �� � ���

��� �� � ����� �B ��� �� � ����

���� ��� �� ����� �� ������ �� �� ��� �� �� ��� ��� � �� ���� �� ������ � .����� ���� �������� �� ������ �� ��� ��� ����!�� D� �� � �� G �������� � � ������� �� ��� � ��

� ��������

5�� ����� ������� �� � �1���� ������������ ���� ��� ������ ���� .�������� ��� ��� � � ���� �� I������ ���� �.�� � ��� � �� ����� �� �� ����� �������� %��*� %��*� � ������� �� � ���� �������� � ������ ���� � G ��� ��� � ��� � � � �� ��� �� � ������ �� � �������� �� � G ��� ���� � � �� ��� �� ����������� ��� �� �� .�������� �� �� � �� ����� �� � � ������ � �� ����� �� � � � � ������ �� � �G �� �� ����� � � ������ �� � ��G ��� ����� �.� ��!����� ����� �� �������"

0 �� � �� � ���� .�������� ��� �� � �G � �B � � ��

( �� � �G �� �B �� � ��G �

, �� � �G �� � �� �B � �G �� ��� � �G ��

6 �� � �G % *� �B� ��� ��� � �� � ��� �� � � ������� �� � �G �

D� �� �� ��� �����.������" � ��� �� ��� � �� ��� �% *�� 5�� ��� ����� �� �� �1������.� ����� �� �� ���� ���� ��� ���� ������ �1������ :����

�� ���� �������� ��� ��� �� ��.����� �� �1������� ���

�� �% *�� �� ���� �� �� �1������� � ������ � ����� �� ��� ����� ��� ������ ��� � ������� ��

����� G ���� ��

�� ������

% *����

6

252

Page 262: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

����� ��� G ����� 5�� ��!������� �� �������� ������ �� ���� ���� �� �� �� �1���������$ �������� %(/* �� �� �������" 2� �����.� �� �������� �� ��� ����� �� �� ����$�������� ��������� � �� �������� ��� � � ����� ������� ������� ������ ���� ��.������� ��.��� �������� �� �� �� �� ����� � ����� ��� �������' � � � �� ������� �������� �������������� ��� �� ?����@ ��� ���� �� � ��.��� ��� ?�������@ �� � ����� � �1� �� ��������� 5����� � �� ���� � ��!�� �������� %��* ��� %��* �� �����.������"

%��*� G %���*� � %����*� %��*� G %���*� � %����*��

�� �� �� ������� ��� %0&* ��� ��������� ���� %)*� �� �� ������� ����������� � � ������ �������

� ��� ��� ����������� � � ��� ���� �������� ������ ���������" ���� %��*��

� ��� ��� ���$����������� � � ��� ���� ���� �� ������� ����� �������� ��� ��������2���" ���� �%��*�� � %�*���

����� �� �� ���$���������� ����������� ��� ?�����@ ��� ? ���@� ����� ?� �������������@ ������������ 5�� ��������� ��� ���� �1����� ������� �� �� ���� ������ ��� ��� ������ ���� �1���� � �� ��� ��������"

���������� ��� � � ��� � ������������� � ���������������������� ������� ����������� ���������������� �������� � %�*������ ����������������� %��*�������

:��� ���� �������� � ������� �� ������� ��� � ������ �� �� ������ .�� �� � �� ��.��;���

D� ��� ����� �� �1������.� ����� �� �� ����� ������� �� � � �� �1������.� ������� !��$����� �������� �.�� ������ �� ��� ��� � ������ !��$����� ������� ��� ������������� �� !��$����� �������� �>�.���� � ��� � ��� ���� ������ �������� ��� �� ������������ ��� �� ���� ����� ��� %(+' (&' ,-' ()*� �� ���� ��� � �� � � � � �������� � ������ ����� �� ���.���� �� �� �����.���� �� ��� � �� �� ������ �.�� �� ������ ��� � �������� �� �� ������ �� �� ��� �� � �� ��� ���� � �� �� ������ !�� ����� ����� ����� ���� ����� �� �� ��� + �� ���� �� �� � �� �� �.�� � � �.�� �� ��� � 5��� �������� !��$����� ������� � ������ �� � ��� �� �� ������ �1������ ���� � �� ��� �

�� �� ������� �� ������ ���� ����� ��� ��� ��� � � � I� ����$7���� ����� ���� ����� ����������� � ������ ��� ����� ������ � �� �� ������������ �������� ��� �� �>�.���������� �� �� ���� � ��� ���� �� .�������� � � ��� ��� ��� ������ ������ ��� ��� ���������� � �G � �B � �G ��%�*� 4��� ������������ �� ��.���� ����� �� ����" ��� ������� ���� �� ������� �� %(-* ��� � �� ���� �������� ��� ����� �1 ���.� ��� 2����� �1����.�

������� �� ��� ����� ���� ������� ���� ���� ���� �������� � ���� ��� ���� ��� ������������� �� ��� ������ ����� � � �� ������� �� �� ������ �� ��� ����������� �� ��� ������ ������ � � ����� ��� ����� ������ ���� � ��� ��� ������ �� �� � �G �� �� � �G ��%�*�

� ����� �7� � �� �� ���� �� %(0* 5����� �� ���� ���� ��� �� � � ���� �� ;�������H����� ��� D��7� %00*� ���� �1�� ��������� �� �� �� �������� �1���������� ���� � �� �� �� ��$.������� ������� �� ���� � ��� �� �� �

� ����

�� ��� �� ���� �� ������� �� ���� �� ������� ��B���� ��������� ������ �� ������ �� ��� �� ��� ���� �� �� ��� (� �� ��� ������� � � ������ ��� ��� �� �� ������� �� ��� ��

+

253

Page 263: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

���������������� & ���

������������ �

���������

����

����� � ���������

����

���� �

���������

����

����� � ���������

����

���� �

����������� & ���

������ ��� & ���������

����

����� � �

��������� & �������� ��� & ��

����������

�����

������� ����

�����������

����

:���� ," �� ������ ������ ���� �

� ������ � �� ����� �� � ���� �� ������� �� ��� � �� � �� �� �� ��� ����� �� ��� ����������� �� ������ �� ��� ���� � D� �� �� ���� � ����� �� ���� �� �� ���� �� � � �� �� �� ��� �� �$������� .���� �� � �� � G ����� ���� ��� ���� ������� �� ��� � � ���������� ��� ��� �� ���� �� ���� ��� ������ �� �� �����

5�� ���� ���� �� ������� �� �� �� ��� �� ���� �� ��� ������ �� ��� C� ��������������� �� ��� ��� �� ��.������ �� ��� ���� �� ���� �� ��� ������ �� ��� ��� �� ��������� �� ��� �� �� ��� � ������� ��� ���$���� ������ ����� ��� C� ��� � 0� ����.������ �� ������ �� ���� � �� ���

�� ���� ����� ��� � �� � �

����� �� ���� ���� �� ���

��.� �� G ������

�� ���� ����� ������ � ���� �� ���� ����� � ����� � �� ������ � �� ��� ������� ������� ���� ����� � ����� ���$���� ��� ������ ������ �� �.��� ����� ������� ��� ����� ��� ���� ���� ������� �� ��� ?����.��@ ��� �� ����� ���� 5��� ��.�� � ���� � �������� �� ���� �� ��� �������� �� �� ������ ���� �� ��� ���� ��� �� �� ��� �� � 5��� �� ���� ��

� ���� �� �����.����� �� �� ������� ����� ��� ��� ������ ��� ����.�� ������ �� �� �� �� ����.�� ������� �� !��� �� �������� 5�� �� �� ������ ������ ��� !��� ���� ��� �� �� ��� �� � ��� ���� ���

� ���� ��� �� ��������� � �� ����� � ��7 ��� �� ����������� ������ �� ���� � 2� ����� �� D� ��� ���

�� �1������" !��� ���� �� �� ���� ��� ���� ���� �� ��� �� ��� !����� ���� �������� ��B���

���� �� �� ��� �� ��� ��� ��� �� ���� � �� ����� ��� ��� �

%��*�%��*� �� %��*�

�� .���� �� ���� �� �� ���� �� ���� ���� �� �1�� �� ��!��� �� ������ ��$ ����� �� ������ ���� �� �� �� ��B��� ���� �� ��� ��� ��� �� ���� � �� ���

�� ��� ��� �

���� � ������ �������� � ������

�� �� .���� �� � �� ���� �� ��� �� ��� ���� �� ������� ��� ������ ���� �� � ������� 5���� ��� ���� ���� ������������ ��� ��������� �� :���� , ������� ��� ���������

�� �� �� �� ���� �� �� ��� �� �� �� �� �� �������� �� ���� � �� ������� �� ��� ��� ���

�.�� �� ��"

������� �� ��� � -��� ���

� �� �� G ���� ��� ��� G ���

� �������

��

��� �� �� �� G ��� ��� ��� G ��

� �������

���

&

254

Page 264: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

' ( )

*+

,

� ��

� ��

� ��

� ��

� ��

� ��

� ��

:���� 6" ;�������� ��� ��������

� ����� �� �� ������ ��� � 2��! ���� �� �� �� ������ �� :���� , �� �� ���� �� %(0*

� �����������

�� ��� �� ���� �� ���.� �� ��� �������� �� �� ���� � ����� �� ���.����� 5�� ����� ������������ �� �� ��������� ������ D� ��� ��� ����������� � >�� ������� ��� ������������

������� � !�� ����� �� � � � � ���� ����� � � � ��� ��� � -" #��� �� ���� ��

��� ������" #�� ��� ���� � �� ���� ��� ��� ���� ��� ��� ������� ��� � ��� �� �� � ���

���������� �������

������ ��� ������������ ���� � -"

5�� ����� �� �� ��� ��� �� �� ������ ������� �� ��>���� ����� �� �� !�� >����� �� ������� � �� ����!������ �������

��� � �� �� 9� � G ����� � � �� � ����� ������ ����� � �� � !��� �� �� ���� ����� ������ � � �� �������� �� ��������� ��� .��� �� �� ���� �������� D� ��� �� � ����� ������� $������ �� ��� ����� �B ���� �1��� � ������� � " �� � � � ��� ��� ��� ��� �� � ��"

� �� ���� �� G � ��� ��� J 0� �� G ��� ��� ��� ��� � �

� �� ���� �� G � ��� ���� � J 0� G ��� ��� ��� ��� � �

� � � ������� � �� ����� � ������� ��� � �

:�� ��� ��� ��� ������ ������� � ����!������ �� �� � ���� �� �� !1 �� ��������� �� ����� ��� �������� �� �� !�� >����� �� �� ����� �� ���� ��� �� :���� 6 5�� �� ��� �

�7�� �����.� ������� � ���$��������� � � ��0� G �-� -�� ��(� G �0� -�� ��,� G �0� 0�� �

C� ����� ������ �� �������� �� %(,' (3* 9� � G ����� � � �� � ������ ����� �� ����� ���� �� �� �� ��������� ������������ �����"

� ��� �� � ��� ��� � � � � � ���� ��'

� ������������ ����� �� �� ��� � �� ��� ��� � ���7 ������ ������� �������'

� ������������ ����� ���� ��� � �� ��� ��� � ������� ������� ������������ � ������� �������� ���� �� ��� �-� �� ��� �� �-�� ����� �.���

5�� ��� ��� ������ �� �� ��!��� ��

� � � � ���� � � � %����*�� � ����

����� � �� �� ��2� ��� �� � ����� �� ������� D� ��� ���� ������� ������ ��� ��������.� �1���������"

)

255

Page 265: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� � �

� � �

� � �

� � �

�� � �

� � ��

� � �

� � ��

� � �

���� �

���� �

���� �

:���� +" 9��" � ��� ��� �������� �� �� ����� ����" �� ?����� ����@ �������

0 ����� �� �� ������� �� � � ��� � �G �� ��� ������� �� �� ������� �� �� � �� ������ ��� ��� �����"

� %��*�� � %��*�� � %�*�� �0�

( ����� � �� �� ������� �� � � � �G � � �� ��� ��� ������ ������� �� ���� 5���� ����������� ��������� � �������� �� �� ���� �� ����� � ����� ��� �������� �� �� ��>�� ��� ��������� � � � ��� � � �� ������� �� ����� �� �� ��� ��� �� :���� +

� � � ������� � ��� �(�

� � �� ������� � �� �,�

� � �� %���*�� �� �6�

� � � %���*�� ��� �+�

�� �� ��� � �� ��� ������� �� �� � ��� �� ������ ������������ � �� ��1 ���� ��������� � �� !1�� ��������� ��� � �� ���.��� ���� ������� ����

����� G ������� � �� � ������� � � � ���

����� G ������� � �� � ������� � � � ����

, �� ���� � ��� � ?�� ����@ �� �� ���� 5� ��� ���� �� ����� � ��������� ���������������� � �� ��������� �� �� ���� ��� �� :���� + :�� �1������ ���� ��� ( �� ��!��� �� ���� �� ���� ��� 0� ��� ���� ��� 6 �� ���� �� ���� ��� (

� � � ������ �&�

� ������� � �� �)�

� %��*�� � %��*�� � %�*�� � %���*�� � %���*�� �/�

D� �� �� � �� ���� ��� ���� ������ ���

����� G ������� � ������� � � � ���

���� G �

���������

���������� �� �� �� � �� ��� ��� ����"

���� G ������� � ������� � � � ���

����� G �

��������

/

256

Page 266: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

����������� :������ �&� � �/�� � �� �� �� 7�� �� ����� � �� ���� �� � ���������� ��2� �.� ��� �� ��� ���� %(0*�

6 �1������� �� ����.��� �� ���� �� �� =��� ��� �� �� ���� � ����� � �� �� ?����� ��� ����@ ������� ������ � � �� �1�� �� �������"

%����*�� ���� � ������ �3�

���� ��� �0-�

���� � ������ �00�

%����*�� ����� ��������� �0(�

��������� �0,�

����� �! �06�

+ !������ �� ����� � �� �����"

�����

���� � ���� �0+�

���������

�� ������ �0&�

���������

�� ����� �0)�

5�� ���� ������ �� �� ��� ��� �� �� � ��>���� ���� .��� ������ �����>����� ��������� ���� ���� �� ����!������ �� �� �� ��� ������ ����� ������� �� � ��� � ������ 5���� ���.� ��� �������� �� ���� ���� �� ����� � �Æ �� � ���� �� �� �� ����!���� �� � ��� ��� � ������ 5��� �� �� ���� ��� �� � ������ ��� � ����� �� ��� ��

��� � � ��� � -"

����� �� !�� � �� � ����� �����" #����

�� �� ��� ����� �� �� ���������� �� � ������ ����� ���� ��� ����� ����� � ��� � �������%

��� �� ��� ����� ����� � ��� � �������� ���� ��� ����� �� �� ���������� �� � ������ ��������� �� ����� ��� ��� �� � � - ��� �� � � ���� ��

��� � � "

5������ , �� �� �������� ����>�� � �� ��� ������ ��� � �� ���.�� �� %(0*

������ 5������ , �.��� �� ��� �� ���� �� ���� �� � ���� �� �.�� �� ��� ��!��� ������ ��� ��� �� ����� !��� ���� ��� I� ���� � ��B���� .����� �� �� �������������� ����.��� �� �� ���� �������� � >�� ������� ��� �������� ���� ��� ��� ���

������� �� &� ���������������� � � � ���� �� ��� ��� � 0� ���� �� ���� �� ��� ���

����" #��� ��� ��������� ���� � ��� ��� ������ ��� �� � � 0����� �� ��� ���

� ��� ���� ���� ���

������� �

��� ���

���������� ��� ���

� �������

����"

5�� �������� .����� �� �� ������ ������� �� �� �������" ��� � � �� �� ���������� �� �� ������ �� � �J � " �� � �� 5�� ��7 �� �� ��� ������ ������� �� � ��� �� �������� �$��������� � �� � � �� �� ������� �-� -� �� � ���� ��� � ����������� ��� �� ��� ���� ������� ��� ���� ��� � ����������� ��� �� � ����� �� ��� ������ ������� �� ��� ������ ��� �������� �� �� ����� �� 5������ & �� �� ���� �� %(0*

3

257

Page 267: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ������������

�� ��� �� ���� �� ���� �� ���� �� �� ����� �� ���� � ��� E��$����� �� ������ ��� ������

��� �� �.�� �� ���.��� ��������� D� ��� ��� ���� ���� ?�����.�@ ����� ��� ��� ���.�� ������� ?�����.�@ ���� :���� �� ������ �� ������ �� �� ��������� ����>�� � �� ����������� �� �� � ��� ���� �"

������ � �� �� &� � ���� � �� ������ ��� ���� �� ���� ������� �� � ����� ��� �� ����� ������ �� ���� �� ���� �� �� ������� ������������"

�� ��� �� � ��� ��!��� �� !��$����� �1���� <�� �� �� �� �� ��� ��� �� ���� ��� �� !��$����� ��!����� �� ���� �� � ��� �� ���.��� ��������� � ����� ���� ������� ������� ����� ���� � ��� ����� %06* ��� �� �� �� � �� �1�������� �� ��.����� !��$���������� ��� � �� �� ��Æ � � ���.��� � !����� �1���������� �� �� �� �� ���� ���$����������� I� 5������ (� �� ����� �1����������� ��� ��

� ��� ��� ��� �.��� ��� ���

�������

� - D� ��� �������� � ���$�1������������ ���� �� ������� � ���� ���� � �� ���� �����

���� �� ������ ��� �� � �� ��� �� �� � ���� �� ������� �� ��� ��"

������� �� #�� ��������� ���� � ��� E�������� �� ��� ��� ��� �� ���

�� ��� ���� �� ����

����������

����� ��� � 0"

5� ���.� ��� ����� �� ������ ������� �� ��!����� 6 �� ����!�� �� ��>����� ��� ���������� � ����������� ��� �� � � � �� ��!����� ���� �� �� !�� ���� �� �� ���� ���� ���� ����� �� %0)* �� ��� .����� �� �� ������ ������� �� K�

�$���� ��� � �� ��� � �� ����!������� ��� ������ � E�

�$�������� ���� ��� .������

�� � !�� ��� ����� ��� ��� ��� ������� .����� �� �� ������ �������� �� �1��� �� ����� ��������� ��2� "

��

��������� � � � ���� � ���� � %����*

��� � � � ���� � ���� �������� � � � ���� � ����

���0/�

<���.��� ��� �� �� �� �Æ ���" �� ������ �� ��� �� �� ��.� �� ���� ��� ��� ��� ���������� �� � �������� � ����� ���� ?�� 7��@ � � ��������� �� � ��� �B� � �� ���.� ���������� �� ����� � �� �� ���� �������� ����� �� ���� ��� ��� ��� � �� �� ����� � ������?����� � �� ���� �������@� ��� ��!����� ���� � ���� �� �� ��

� �� �� ��.��� �� �� ��.� � ���.�� �� 7�� ��������� 5��� �� ���� �� ����� ����� ���� �� ���� ��� ?���� ������@� � � ���� ��� ������ ������ ��� �� �� ��!��� ��>�� � ���$������� �� �� ���� D� ��� �� ��������� ������ � ��"

��

�%���*���� ��� � %���*�� � %����*���

��03�

9� ��� �� �� ������� �1������ �� �� 5�� ������ �� ������ ��� �� � ��� � �� �1�������� ��� ������� �� ���� ����� �� ��� ��� �� �������� ��� D� ���� �� ��������� �������"

��� � �� � ������� ����� ��� �� �� ����� ������ �� � G ��� ���� ���� � � �� �� ������� ��� � 5��� � �� ����� ����� ���� ������� ����� �� � G ���� ��� �� � ������ ��� ���� �� �� � ������� �� ��� � �� ���� ������������ � ����!�� �� ��������� �������" ��� �����>�� � ��� ��� � � � � � � � �� �� ���� �� ���� �� � � � � �� ��.� � ��

���� ���� � � �

D� �� ��� ������� �� !�� ��� �� ���� ���� ��� �� �1����� ��� ��� 5�� ������ ����� ��� �� ��������� ����� �� �� ���� �� %(0*

����� � � !�� ���� ��� G ��� ���� ���� � � �� �� � ������ ��� � ���� �� ����� ���� ������������ � � ���� ��� ������� �� �� ��� ������ �����" #��� ��� ����� ��� �� ���������� �� ������� ���� ����� �� � ���� �� ��� ����� ����� � ��� � ������� ���� �� � ����� �������������� �� ��� ����"

0-

258

Page 268: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

:�� �� �� ��� ��� �� ���� ����� �� ����� ������� ������ ��� �� ����� �� ��� ����� � �

����� ��� &� ��� ����� ����� � ��� � ������� ���� �� � ����� ��������� ����� �� �������� ���� ��� ����� ��� �� ���������� �� � ������ ���� ����� �� ����� ��� ��� �� � � 0��� �� � � ���� ��

��� � � � ���"

L�� �� �� ������ ��� �� ������������ �������

����� ��� ����������� ��� ����� ���� ��!$

��� ����� ��� � ����� � �

����� � �

��� 9����� 0- ��� 00 ���������� ����� 5������ /

� �� ���� ����� �� ���� ��� � ����� �� ��������� ������ ��� �� � ��� � ��� ���������� ���� �� ����� ��������� �1������ ��� �� ������ ��� � ����� �� ������ ���� ��������� �� �� %,-* ��� �� ��� � �� �������� �� �� %(&* ��� � ���� ��� �� ��� �� ����� ������!��� ������ �� ���.� ����� ���� �� ���� �� ���$�1������������ �� �� ��� �� ���� � D� ��2� ��� ����.��� �� ����� ����! ����� �� �� ����� ����� �� ���� �� �� ��� ����.� ���� E�

�$�������� �� ���� :������� �� ������� �� �� ���.� ����������� �� ���� � �� !��� ������ ��� ��" ��.������

��� �������� �� � ���� �� !��� ������ ��� �� ������� �� � �� �� �� ���.��� ��������� �� � ������� �� � ���� �� ������ �� ��� ������ ��� � !��$����� ��!����� ���� � �� �� ���������� !��� ������� 5��� 5������� & ��� 5������ ( ��.� � �� ���������"

��������! ��� #�� ��������� ���� � ��� ��� �"�"� ��� �� � � 0� ���� �� ��� ���

� ��� �������� ���

������"

� ��� ���� �� � !������

:�� ��.���� ����� ������ �� ���� �������� ��� ���7���� ��� � �� �� ���� + �������� ��������� ��� ����� �������� ������� �� ��������� %0+' /* 5��� �� ���� �� 7������ �� ��������� ��� � � ��������� �0� �� ��� ��� ���� �������� ��� � ��� ?������$��� ��@ ���������' �(� �� ��� ��� ���� �������� ��� � ��� ?��� ������$���@ ������� ��' ��� �,� �� ��

��� �� �������� ��� � ��� ���2������� ������� � 5�� ����� ������� �� � ��� ������������ ����$���� ������ �� ��� � G ��� ���� � � �� �� �1���� ������������ ���� ��� ���������� % *� ����� ������ �.�� �� !.� ����$�������� 5��� ��� ��������� �� �� ������������ ��� ���

� � � G ��� # ���'

� ��� G ���� # �����'

� ��� G ���� # �����

��.�� � ���� � �� ������ �� ���� �� ����� �� �� ���� �� �� �� �� �$������� ��� ���� .���� �� ��� ������� �� � 5�� ��� ��

� ���� ��� ���� ���� ��� ��!��� ���������� � ��

���� ��� � ����� �� ����� ���� �� ��.�������� �� �� � ��.� ��.��� �������� ��� �� �" :����

�� �� ���� ����� �� ���� � ��� ��� ��� ��� ���

� ��� ��� �� ����� �� � ��������� ����"

����� �� � �� ���� �� ��� �� ��� � G ��� � �� ���� ���� ���� ���� ����� � �� ���$���� ��� �� � ��� ����� �1 ���.� ��� 2����� �1����.� ������ �������� �� � � � ���0� �� �� ��������� �� �� ������ ������� �� � � �(� � ��� � ��� ������� � �,� �� �� ����.���� �� �� ��� �6� �� ���� �� �� ����$ ��������� ���� �:���� &� ��� .���� �� ���� ��� �������� � ���.� �� ������� �� 5������ (� � � ��� ��� � 0� �� ��.�

��� ���� �� �

� G ���� ��� ��� G ���

� �������

��

���� �� �� �� G ��

� ��� ��� G ��� ���

�������

00

259

Page 269: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Æ � � �� ��

� � � ����� � ����� �

� � ������ � ���� � ������� � � ����� �� ��� � ������ ���� ���������� ��

:���� &" 5�� ���� ��������� ����

5����� �� ������ ������� �� � ��� �� ���� �1������.� ����� �� �� ��$.������� ������� ������ �� � � �� !��$����� ������� ��� �� !.� ������ ����$������� ������� ��� ��!��������� ���� ����� ���

D� ��� ��.������ �� ��������� ��������� �� ���� � ����� �� �� � ����������� �� ���� ���� �� ��� ����� ���� � ��� ��� ������ ����� �� ���� ��� ������������ �� ���� ������� ��� �� ��� ��� ���� 4��� ��� ������ �� ��.� � ����� �����.��� ������ �� ��� ��� ����� ���������" ����� �� �� �� ���� �� ��� ������ �� ���� G ��� ���� � � �� � � ��� ��� ��� �� �� �� ��������� �� �� ����� ���� �1��� � ��>������� ���� � � � ��

� � �� ���� � �� � �� ���� � ��� �� � � � '

� ��� �.��� ������ � � � ��� � �� � ��� �� � � � � �� ��.� ���� � �� � �� ���� � �� ��

� ��� �.��� ������ � � � ��� � � � ��� �� � � � � �� ��.� � � ���� �

� �� ���� � .����� �� � �� � �� ��� ��������� � �

� ��� �� � � -

������� �� '����� ��������� � � � �

�� ��� ��� � 0" #��� �� ���� �� ��� �������" #��� ��� ��������� ���� � ��� ��� ������� ��� �� � � 0� �� ��� ��� ��� �� ���

������"

5�� ����� �� �� ��� ��� �� �� ����!������ ������� ��� �� ��� ������ ����� ���� ��� ����%((* ��� �� �������� ����� �� �� �������� ����� D� �� �� ����� ������ �� %0,*� �� ���� � ���������� �� ����� �� ������ � %0,* �� � %(0* ��� � ������ ��!����� �� ���� ��� 2��� ��� ���� �� �� ������ �� ��� �� � ���� �� ��� ���� ��� �� ���� ��� ���� ������������� ��� ��������� � ������ �� ��� ����� �� � �� ���� ���� � ��� ��� �������

D�� �.��� ���$������ �� �� ���� ��� � �� �$������

��� � ! � �� ���

� � �� � �� ���$����!���� �B ��� � ! � �� �� ����!���� �� � ����� ���� � �� ���� � �� �� ��2� ��� �� �� ��������� �������"

0 ;� � ��� �� �� ���$������ �� ������� �� �� �� �� �� ��� � �G ��� 5��� �� ����$� � ����� .�������� ��� � G 0� (� ,� ��� ���

�� �

�������

�%��*��� � %��*��� � %�*���� �(-�

�� ���� �� ���� �� ���� �(0��

�������

���� �((�

0(

260

Page 270: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

( �� �� �� ��� ��� �� ������� �� � ����� .������� !� �� �� ���

!� ��

�������

������� � �������

�������

������� �(,�

, �� ��� �� ��� � 0 " � � � " , ��� ������� �� ����� .�������� !�� � �� �� ���

!�� � �������

������� � ������������

�������� �(6�

L��� �� ��!�� �� ��� �.��� ��

��� "G ��

����� "G ! � ���

�� � �� "G �� � �

������ "G �����!�� � �����! � ���������� "G �����!�� � �����! � ���������� "G �����!�� � �����! � ����

5�� ��������� 9���� ���������� ������ 5������ 0, ��� �� ���.�� �� %(0*

����� ��� '����� ��������� � � � �

�� ��� ��� � 0" #��� �� ��������� � ��

���������� �� �� �������� �� ��� � ! � �� �� ���������� �� �"

" #����� ��

��.���� ���� >������ ��� ��� ������ � ������ 5�� ���� �������� �� � �1���� � �� ��$���� ��� ���� ���� .����� �� �� ���� � �������� �� ��� ����� ������� �� ��� ������$��� ������� �� �� �� ��� ��� � ��� ���� ��� �� ���� �� ���� � ��

� ��� ��� ������ ���

������� ��� � ��� � �� ����� �� 5������ 0, ���� �� ����� C��� �������� ���

�� ������� �� ��������� �� �� �� ��������� � � ��.��� � ���� �� ��� � ��� ��� �1��������� ����� �� �������� � ���� ����� �� � ���� ��������� ��� ��������� ����� ����� :������� � �� �� ���� ������� ������ �� ��� ��� ��� �� ���

������ ��� �� ���.��� ��$

������� 9� � ���� ����� ����� ��� ��� ���� ��� �� ����� � <������ ��� ������8� ����� ��

����.�� ������� ���� %0&*" 5������� ,� /� ��� & ����� � ���� � ��� �� �� �� ������ ��� ������������ ��� � �� ������ �� ����.�� �� �� ������������ �� ��� ����.�� �� �� ������� ��� ������� ��� ���� �� <������ ��� ������ �� �� ����� �" !��� �� ���� ������ ���� ��������� ��� � �� �� .����� �� � ? ���������@ �� �� ����� ����.�� �������� ����� <������ ��� ������ �� ��� ��� ���� ������������ �� 5������ , �� ��.� ���� ���.����� �������� �� �� ����� ��� ���� ��

� ����������� ��� � �� � ����� � � � ���7��.����� �� �� ��� �����.�� �������� ���� �� ����������� ��� �� ����� �� <������ ���������8� ��� �������� �����

!� ������

%0* 4 ������ ��� M .�� I����� � ����� ���7 ����� ��� � (����� �� )������ *���+����� �� ���� � ����� ,03 # ,&,� (--(

%(* M ����� 4��������� 7�������� ��� ������� ����.��� +��� ������ �� ��� )+,�(&�00�� 03/,

0,

261

Page 271: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

%,* M : ����� 5������ � ������� ����� �� � ��� ��� ��� )���� ��� &��������� �� (,�(�"0(,#0+6� 03/6

%6* � ����� ��� ; :��� ��� � ������� ��� ������ ���� ��� ��������� ��� � ���� �������� (����� �� )���� ��� &��������� � -����� � �()&- � 3"6&,#+-&� 033/

%+* I I���� 4���� ���� � ��� >�����.� ������ ��������� .������ �� ��� &������� /����� 0�� ��� )������ !��� �&/0! � 6�0�"(,#6+� 033&

%&* I ����7� ����.����� ��� ����� *���� ��� (����� �� ����� !��� � (&"&0#)+� 03/+

%)* � � ���� 4���� ��� ��� ����� >�����.� ������ ���� � �� : � ������ < 4 ������� ��� M .�� I������ ������� 0�� " �� ��� &(+)&�12 3��4���� �� '������ ���#������ -��������� 033,

%/* � �N�� �� < D���� ��� � 4 ����7�� � ������� �������� ������ � � �� ����������� ��� ��� �� #������� �� +����� ' ��� �� (++"&,#/,� (--0

%3* 4 ;�������� ����.��� �� ��������� �� ������ ������� �� �������� (����� �� 5����!������� ��� +������� + �(�"0,,#063� 0336

%0-* 4 ;�������� ��� � :������� ����$�� ������� �� ������ �������� &������������ (������� /�������� �� &��������� '������ + �(�"0&0#0)6� 0330

%00* O ;������� 4 P H����� ��� 5� D��7� :���$����� ���� ��� �� .�������� ��� ����������� ���� &����" ��� +��������� 0)3�(�"()3#(3+� (--(

%0(* � 4 ������� � 4 <��7������ ��� 4 � ��������� ������ #������ !��� � ,�������� �� ���������� ��� +���������� )��� ��� 5��� 6 C1���� Q��.����� ������ 9��� ����� (/� 0336

%0,* � ������� � O� �� : D����� ��� 4 R�7������ ��. ,�������������� ,���� !��� ��#����� ��� )���� ������ ;���.���� (--,

%06* � �����. ��� H �����7� 4���� ���� ��� ����� (����� �� 0��������� �� !��� �(("&-)#&,&� 033,

%0+* 4 ������� � ��������� ��� � ����������� 5������� �� ������� � �� 0�� " �� ��� 67��&��" (���� +���" �� )& �&(+)& � ����� 3-0#3-& 4����� O������� 033+

%0&* M <������ ��� P ������ � ������������ ����� ���� �� ��� ����.��� (����� �� ���)+,� ,/�6�"3,+#3&(� 0330

%0)* � <���� �� ����� ��������" 4�7��� �� ������ ��� ������ ������ ������������ )����� �� ��� ���� ,������� �� (6"+0#)(� 03/+

%0/* M 4 O����� ��� � 5���7� 5�� ������� �� ������� )����� �� ,������� �� �����060#030� 0366

%03* � 9� ��������� ����.��$����� ������� ��������� ��� ������� ��1�� )���� ���&��������� �� (--, � ������

%(-* � 9�� Q ������ ��� : D���� 4���� ���� ��� �� ��$.������� ������� �� +�����' ��� � !��� � ����� (06( �� 9L��� ����� (6)#(&0 ��������$H������ (--0

%(0* � 9� ��� : D���� 4���� ���� � �� ������� �� �������� 95��$����� -6$-+� 5� ��� �� Q��.����� �������� (--6 �.������� ���� ��"SS�� ��� $������� ��S������ �S������ ���

06

262

Page 272: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

%((* � 4���1 5�� �>������ ����� �� ��� �� ��� ������ (����� �� '����� !��� � 6+",00# ,0+� 03/-

%(,* 4 4��1 ��� 4 �������� Q��� �������� �� ������ ���� (����� �� !��� ��� +���������� 3�&�� 0333

%(6* I L���� ��� < IN� 7�� ��������� ��� ������� ��������" � ��1���� �� ������ ���� �� �����8� ����.�� ������� (����� �� ��� )+,� 6(�0�"6,#&&� 033+

%(+* � ������������ � � �� ��� H H��� 5������� �� >����� �� ������ �������� (������� +����� ��� '���� ' ��� ��� +/"(3#+,� 0333

%(&* � ��� ��� � � ���� � ������ �1��� ����� ��� ��������� ������������ �� �� ��������� (����� �� 0��������� �� !��� � ()"&(0#&+/� 033/

%()* � �������� R ��� ��� � ���� � ������ ���� ����� �� ������� ��� ���� ��� �� 0�� "2�� &��" +���" �� 8-9-� ����� 0&+#0)&� ��� 4���� 033( 4����� O������

%(/* M ���� ��� I L���� C� �� �����1�� �� >�����.� ������ ���������" � ��1������ ���� ������� �� �� ������ ���� ��� �� �� )���� ��� &��������� �� 0-/�0$(�"&3#0(,� 0333

%(3* �������� ��� R�7������ ��. C� �� ���� � �� ������ ����� ���� � (����� �� !��� ���+��������� 00"3-3#3,0� (--0

%,-* 4 � ������ ��� � �����7�.� �� �������� �� ����� ������ �� 0�� " �� ��� ����������������� )+, ������ �� #����� �� +������� ����� (60#(6& ��4 ������ (--0

%,0* M � ���� I������ ���� ��� ��������" � ��� ������ � � �� ������$ ���� ��� �� �� )���� ��� &��������� �� 0(("000#0,&� (---

%,(* P H����� ;1������.����� ��� ���������� �� �� ����.�� ���� ���� *���� ��� (������� ����� !��� � ,0�6�"+(3#+6)� 033-

%,,* 4 D����� C��� ��� �� �� ��� � >�����.� ������ ��������� ��� ��� ������ ���� �"���� ��� �������� ����� �� 0�� " �� ��� &��" 3��4���� �� ��� ������� !��� � :;;6��!:;;6 � ����� 63 �� �;Q�$D� ���"SS ��$�� ���S�� ����� 3&#0-+� (--0

0+

263

Page 273: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Normal modal logics containing KTBwith some finiteness conditions

YUTAKA [email protected]

Meme Media Laboratory, Hokkaido UniversityN13 W8 Kita-ku Sapporo, 060-8628 Japan

Key words: KTB, finiteness, tabularity, splitting

1 Introduction

1.1 Background

The semantical study of propositional modal logics using Kripke type semantics has broughtus a great success, especially to the investigation of the class of logics containing K4. This classof modal logics corresponds to the class of transitive frames. In order to analyze this class oflogics, many algebraic and frame-theoretic techniques have been developed, and by using them,several general results have been established, which give us a perspective of the lattice of normalmodal logics characterized by transitive frames (for example, [2],[5],[6],[14],[15]).However, there is no effective way of clarifying the lattice structure of modal logics of non-

transitive logics, and almost all techniques for the logics of transitive frames are not valid forthe logics of non-transitive frames. Therefore the study of logics of non-transitive frames hasnot yet grown to be so fruitful as that of logics of transitive frames.Here we discuss the structure of the lattice of logics containing KTB, whose frames are reflexive

and symmetric. The logic KTB is also known as Brouwerian system. The lattice of extensionsof KTB seems rather complicated, and so very little is known about the structure of this lattice.In this paper, we will focus on a little smaller class of the whole extensions of KTB, that is,

a class of logics containing KTB that are characterized by modal algebras, or general frameswith some finiteness conditions. Such classes sit on at the upper part of the lattice of extensionsof KTB. We will prove the following facts:

(1) We characterize subdirectly irreducible members and simple members of KTB Kripke al-gebras, and give an example of KTB Kripke algebras that is subdirectly irreducible butnot simple.

(2) We completely characterize all tabular and locally tabular members in Next(KTB). Inparticular, S5 is the only logic which is not tabular, but locally tabular.

(3) We characterize KTB-logics of frames with finite ’diameter’, and find that every finitealgebra for such a logic splits the lattice above the logic.

(4) We investigate the structure of the upper part of the lattice Next(KTB), and in partic-ular, we show that the logic determined by a frame of two reflexive points jointed with asymmetric relation is the third greatest logic of all Kripke complete KTB-logics.

1

264

Page 274: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1.2 Preliminaries

We will follow notions and nomenclature of modal logics from [4]. But here, we want to recallthe basic terminology of syntax and semantics of modal logics that we will use in this paper.First of all, we fix our language and introduce propositional modal logics. Our language con-

sists of: (1) a denumerable set of propositional variables {p0, p1, . . .}, (2) a propositional constant⊥, (3) conjunction ∧, (4) negation ¬, (5) box �, and (6) a pair of parentheses (, ). The connec-tives ∨ (disjunction), ⊃ (implication), and � (diamond) can be treated as auxiliary connectives(abbreviations). The set Φ of formulas on our language is defined as usual.L ⊆ Φ is a normal modal logic (a logic for short) if: (1) L contains the classical logic, (2)

�(p ∧ q) ⊃ (�p ∧ �q) ∈ L, and (3) L is closed under modus ponens, substitution, and the ruleof necessitation. The smallest normal modal logic is denoted by K. For a set of formulas Γ, thesmallest normal modal logic that includes both K and Γ is denoted by K ⊕ Γ. For a normalmodal logic L, the class of normal extensions of L is denoted by Next(L).In this paper, we concentrate on the class Next(KTB), where T := �p ⊃ p and B := p ⊃

��p, and KTB = K ⊕ T ⊕ B. Each member of Next(KTB) is called a KTB-logic for short.Next we review semantics for propositional modal logics. The first one is the algebraic seman-

tics. A structure A = 〈A,∩,∪,−, I, 0, 1〉 is a modal algebra if: (1) 〈A,∩,∪,−0, 1〉 is a Booleanalgebra, and (2) I is a unary operator that satisfies (i)I(1) = 1 and (ii)I(a∩ b) = I(a)∩ I(b) fora, b ∈ A. Formulas are interpreted in a modal algebra by a valuation in the standard way andit is also defined in the standard way that a formula A is valid in a modal algebra A (A |= A).For a class K of modal algebras, L(K) := {A ∈ Φ | A |= A for any A ∈ K} is called the normalmodal logic determined by K. A modal algebra A is a KTB-algebra if it satisfies that I(x) ≤ xand that x ≤ I(−I(−x)) for any x ∈ A.Another semantical tool is so called frames. A structure F = 〈W,R,P 〉 is a (general) frame if:

(1) W is not an empty set, (2) R is a binary relation on W , and (3) P ⊆ P(W ) contains ∅,W andclosed under ∩,−, IR, where IR(X) := {x ∈W | ∀y ∈W (xRy implies y ∈ X)} for X ∈ P(W ). Aframe 〈W,R〉 := 〈W,R,P(W )〉 is called a Kripke frame. The way how we interpret each formulain a frame is in the standard way. A class C of frames also determines a normal modal logic as:L(C) := {A ∈ Φ | A |= A for any A ∈ C}. As is well known, the axioms T and B correspondto reflexivity and symmetry of Kripke frames respectively. That is, the following holds: for anyKripke frame F , F |= T iff F |≡ ∀x(xRx), and F |= B iff F |≡ ∀x, y(xRy implies yRx). Here,the symbol |≡ represents the relation that the frame F is a model of the first order conditionon the right hand side. A reflexive and symmetric general frame is called a KTB-frame.It is well known as the representation theory that there exists a close relation between modal

algebras and general frames. For any modal algebra A, we define A∗ = 〈WA, RA, PA〉 as follows:(1) WA is the set of all prime filters in A, (2) RA is the binary relation on WA defined as: for anyF,G ∈WA, FRAG iff for all a ∈ A, I(a) ∈ F implies a ∈ G, and (3) PA := {θ(a) | a ∈ A}, whereθ(a) := {F ∈WA |a ∈ F}. Then, it is easily shown that A∗ is a general frame and that A and A∗validates the same set of formulas. Conversely, for any given general frame F = 〈W,R,P 〉, thecorresponding modal algebra F∗ can be defined as: F∗ = 〈P,∩,∪,−, IR, ∅,W 〉. Here, ∩,∪,− arethe set theoretic operations, whereas IR is the same one defined above. It is also easy to provethat both F and F∗ validate the same set of formulas. Moreover, about two transformation (·)∗and (·)∗, it is known that (1): A ∼= (A∗)∗ for any modal algebra A and that (2): F ∼= (F∗)∗ iffthe frame F is descriptive. We call an algebra A a Kripke algebra if there exists a Kripke frameF such that A ∼= F∗.

2

265

Page 275: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 Subdirectly irreducible and simple members of KTB-algebras

The first result in this paper is about subdirectly irreducible, and simple members of KTB-algebras. An algebra A is a subdirect product of an indexed family {Ai|i ∈ I} of the same type ifthere exists a one-to-one homomorphism f : A →

i∈I

Ai such that for any i ∈ I, πi ◦ f : A → Ai

is onto, where πi is a projection map to i-th coordinate. A non-trivial algebra A is subdirectlyirreducible (s.i. for short), if for any subdirect representation f : A →

i∈I

Ai of A, there exists

i ∈ I such that πi ◦ f : A → Ai is one-to-one.The notion of subdirectly irreducible algebras is important for logicians, because the following

holds.Proposition 2.1 Let A be a modal algebra that has a subdirect representation f : A →

i∈I

Ai.

Then L(A) =⋂

i∈I

L(Ai). �

Due to this proposition, when we consider a logic that is determined by a class of algebras,we have only to take logics determined by s.i. members of them into account, and then takeintersection of such logics. It is well known that A is s.i. iff A has the smallest non-trivialcongruence. An algebra A is simple if the congruence lattice of A consists of only two elements.Subdirectly irreducible, and simple members of modal algebras are well understood by the

following characterization.

Proposition 2.2 Let A = 〈A,∩,∪,−, I, 0, 1〉 be a modal algebra.

(1) A is subdirectly irreducible if and only if there exists d ∈ A (d �= 1) such that for any a ∈ A(a �= 1), there exists n ∈ ω, a ∩ I(a) ∩ · · · ∩ In(a) ≤ d.

(2) A is simple if and only if for any a ∈ A (a �= 1), there exists n ∈ ω, a∩I(a)∩· · ·∩In(a) = 0.

On the above proposition, (1) is by Rautenberg ([13]), and (2) is by the fact that for any non-empty subset X ⊆ A, the open filter F generated by X is given by: F = {a ∈ A | ∃x1, . . . , xk ∈X, and ∃n1, . . . , nk ∈ ω, In1(x1) ∩ · · · ∩ Ink(xk) ≤ a }. Here, in a modal algebra A, an openfilter F is a filter which satisfies that a ∈ F implies I(a) ∈ F for any a ∈ A.Now we prove the following characterization of s.i. and simple members of KTB Kripke

algebras. Below, Rn is defined as: xR0y iff x = y, and xRn+1y iff there is z ∈ W such thatxRnz and zRy.

Theorem 2.3 Let F = 〈W,R〉 be a KTB Kripke frame.

(1) F∗ is subdirectly irreducible if and only if F |≡ ∀x, y,∃n ∈ ω(xRny).

(2) F∗ is simple if and only if F |≡ ∃n ∈ ω,∀x, y(xRny).

Proof : (1): Suppose that F is s.i. Then, by Proposition 2.2 and the reflexivity of R, thereexists V ∈ P(W ) such that V �= W , and for any U ⊂ W , for some n ∈ ω, In

R(U) ⊆ V . Here wecan pick up some w ∈ W − V . For an arbitrary x ∈ W , put U0 := W − {x}. Then for this U0,there exists n ∈ ω such that In

R(U0) ⊆ V , which implies that {w} ⊆ −V ⊆ −InR(U0). This shows

in particular that wRnx. The same argument can also show that wRmy for some m ∈ ω. Thuswe have xRn+my because of symmetry of R. Conversely, suppose F |≡ ∀x, y,∃n ∈ ω(xRny).

3

266

Page 276: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Fix an arbitrary x0 ∈W . Then, by our condition, for any A ⊂W , there exists n ∈ ω such thatInR(A) ⊆ W − {x0}. Consider the smallest open filter F in F∗ that contains W − {x0}. Then

the previous fact implies that F is the smallest non-trivial open filter in F∗, which correspondsto the smallest non-trivial congruence.(2): Suppose that F is simple. Then, by Proposition 2.2 and the reflexivity of R, for anyU ⊂ W , there exists n ∈ ω such that In

R(U) = ∅. For any x ∈ W , put U0 := W − {x}. Thenthe previous fact means that for any y, y ∈ W = −In

R(U0). Thus xRny. Conversely, supposeF |≡ ∃n ∈ ω,∀x, y(xRny). Take an arbitrary non-trivial open filter F in F∗, where non-trivialmeans that F �= {W}. Then there exists X ⊂ W such that X ∈ F . By our condition it iseasily seen that In

R(X) = ∅. Since F is an open filter, ∅ ∈ F , which implies that F = P(W ).Therefore, there are only two open filters in F∗. Hence F∗ is simple. �

For a KTB-frame F = 〈W,R,P 〉, and n ∈ ω, F is n-connected if F |≡ ∀x, y(xRny). F is(finitely) connected if F is n-connected for some n ∈ ω. By the above theorem, as long as KTBKripke frames are considered, F is connected iff F∗ is simple. As seen in an example below, anyKTB Kripke frame that produces an s.i., but not simple algebra must be an infinite frame.

Example 2.4Let a Kripke frame H := 〈ω,R〉, where the relation R is defined as: mRn iff |m − n| ≤ 1 form,n ∈ ω. Then, clearly this R is reflexive and symmetric, and so, H is a KTB Kripke frame.By Theorem 2.3, it is also easy to prove that H∗ is s.i. but not simple.To close this section, we make one more comment on the decomposition of a KTB Kripke alge-

bra into s.i. members. According to Blok’s result ([2]), any Kripke algebra can be decomposedinto s.i. Kripke algebras in a following way.

Theorem 2.5 Let F = 〈W,R〉 be a Kripke frame. For each x ∈ W , define a Kripke frameFx = 〈Wx, Rx〉, where Wx := {a ∈ W | ∃n ∈ ωxRna}, and Rx = R ∩ (Wx ×Wx). Then, foreach x ∈W , F∗

x is subdirectly irreducible, and F∗ can be represented as a subdirect product of{F∗

x |x ∈W}, where the subdirect representation f : F∗ →∏

x∈W

F∗x is determined by (f(X))x :=

X ∩Wx for X ∈ P(W ). �

For a KTB Kripke frame F = 〈W,R〉, define a binary relation ≈ on W as: x ≈ y if there existsn ∈ ω such that xRny. Then, because of symmetry of R, this ≈ turns out to be an equivalencerelation, and so, x ≈ y iff Wx = Wy for any x, y ∈ W . Therefore, in employing the abovetheorem to decompose a given KTB Kripke algebra F = 〈W,R〉, we divide W by the relation≈ first, and we collect one representative xλ from every equivalence class Wλ, and then we canobtain our desirable subdirect representation of F from {Fxλ

∗ | xλ ∈Wλ}.In this way, every KTB Kripke algebra is decomposed into s.i. algebras. Therefore, when we

consider a KTB-logic which is Kripke complete, we may assume that a class of KTB Kripkeframes, each member F of which satisfies F |≡ ∀x, y,∃n ∈ ω(xRny).

3 Tabularity and local tabularity of KTB-logics

The second result in this paper is about tabular and locally tabular members in Next(KTB).A logic L is tabular if there exists a single finite frame F such that L = L(F) = {A ∈ Φ|F |= A}.A logic L is locally tabular if for any n ∈ ω, L contains only a finite number of pairwisenon-equivalent formulas built from variables {p1, p2, . . . , pn}, where a formula A is said to beequivalent to a formula B in L if (A ⊃ B) ∧ (B ⊃ A) ∈ L.

4

267

Page 277: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

In order to characterize tabularity and local tabularity of KTB-logics, we introduce well-knownaxiom schemata, Alt(n) and Tra(m,n).

Alt(n) := �p0 ∨ �(p0 ⊃ p1) ∨ · · · ∨ �((p0 ∧ p1 ∧ · · · ∧ pn−1) ⊃ pn) for n ≥ 0

Tra(m,n) := �mp ⊃ �np for n > m ≥ 0

Note that in Tra(m,n), �0p := p and �n+1p := �(�np). It is well known that these axiomschemata correspond to the following first order conditions on a frame F .

F |= Alt(n) iff F |≡ ∀z, x0, x1, . . . , xn[&ni=0zRxi implies �1≤i�=j≤n(xi = xj)]

F |= Tra(m,n) iff F |≡ ∀x, y(xRny implies xRmy)

In the above conditions, � is a logical connective of “big or”. In other words, every point ina frame which validates Alt(n) has at most n successors (n ≥ 0). Apparently Tra(m,n) is ageneralization of transitivity axiom (�p ⊃ ��p), and says that for any point x, if x is accessibleto y by n steps through R, then so by only m steps (n > m ≥ 0).For all normal modal logics, there is a following characterization of tabular logics using Alt(n)

and Tra(m,n).

Proposition 3.1 (See [4]) For any L ∈ Next(K), L is tabular if and only if Alt(n)∧Tra(n, n+1) ∈ L for some n ∈ ω. �

On the other hand, we can show the following fact about locally tabular members in Next(K).

Proposition 3.2 For any L ∈ Next(K), if L is locally tabular, then Tra(m,n) ∈ L for somem,n ∈ ω (0 ≤ m < n).Proof : Suppose that Tra(m,n) �∈ L for any m,n ∈ ω (0 ≤ m < n). Then, all the formulas of�ip, (i = 0, 1, . . .) are not pairwise equivalent in L. Thus L is not locally tabular. �

Note that for L ∈ Next(KT), Tra(m,n) ∈ L if and only if Tra(m,m + 1) ∈ L for somem,n(0 ≤ m < n). We need one more to deduce our result about locally tabular KTB-logics,that is, the theorem by Michael Byrd ([3]).

Theorem 3.3 There exists a KTB-frame F that satisfies the following properties.

(1) F |= Tra(2, 3).

(2) F �|= Alt(�) for any � ∈ ω.

(3) There is an infinite sequence of formulas, all members of which are constructed from onlytwo variables and that are not pairwise logically equivalent in L(F). �

By this theorem the next corollary is easily verified.

Corollary 3.4 Let L be a KTB-logic such that Tra(m,n) ∈ L for some m,n(2 ≤ m < n). IfL is locally tabular, then Alt(�) ∈ L for some � ∈ ω. �

By using all these, we can reach to the following theorem. Below, • represents a frame of singlereflexive point, whereas ◦ a frame of single irreflexive point.

Theorem 3.5 Let L ∈ Next(KTB) be locally tabular. Then one of the following conditionsholds.

5

268

Page 278: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(1) L = L(•).(2) L = KTB4 = S5.

(3) Alt(n) ∧ Tra(n, n+ 1) ∈ L for some n ≥ 2. �

Since L is really tabular in the cases (1) and (3) above, it is easily seen that S5 is the onlyexample in Next(KTB) which is not tabular but locally tabular.It has to be mentioned here that the above discussion can also be applied to locally finite

members in Next(KB). First, we point out that for L ∈ Next(KB), if �np ⊃ �n+k+2p ∈ L,then �np ⊃ �n+kp ∈ L due to the axiom B. Then on the axiom Tra(m,n), we have thefollowing:For n ≥ 1,

KB⊕ p ⊃ �np ={

KB⊕ p ⊃ �1p (n : odd)KB⊕ p ⊃ �2p (n : even)

Similarly, for n ≥ 2,

KB⊕ �p ⊃ �np ={

KB⊕ �p ⊃ �2p (n : even)KB⊕ �p ⊃ �3p (n : odd)

Then, Theorem 3.5 can be extended into the following form:

Theorem 3.6 Let L ∈ Next(KB) be locally tabular. Then one of the following conditionsholds.

(0) L is a locally tabular member in Next(KTB). (This case is covered by Theorem 3.5.)

(1) L = KB ⊕ p ⊃ �p = L(•, ◦).(2) L = KB ⊕ p ⊃ �2p = L(•, ◦, ◦−◦).(3) L = KB ⊕ �p ⊃ �2p = KB4 = L(◦) ∩ S5.

(4) L = KB ⊕ �p ⊃ �3p

(5) Alt(n) ∧ Tra(n,m) ∈ L for some m > n ≥ 2. �

In the case (2) above, ◦−◦ is a frame of two irreflexive points jointed with symmetric relationR. Clearly, in cases (1), (2), and (5), L is tabular, whereas (3) is classified into the case of nottabular but locally tabular. It is unknown whether L = KB⊕�p ⊃ �3p is really locally tabularor not.

4 KTB-logics determined by frames with finite diameter

In the previous section, tabular and locally tabular members in Next(KTB) are completelydetermined. In this section, we will consider another finiteness condition on frames and KTB-logics characterized by such frames.

6

269

Page 279: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4.1 Logics of frames with finite diameter

Let F = 〈W,R,P 〉 be a frame. A path (of length n) in F is a finite sequence {xi}ni=0 ⊆ W of

distinct points which satisfies xiRxi+1 for i = 0, 1, . . . , n−1. The diameter of F is the maximumlength of a path that F contains. It may happen that the diameter of a frame is infinite.

Definition 4.1 A frame F is n-bounded if the diameter of F is at most n− 1, specifically Fis a model of the following first order sentence δ(n):

δ(n) := ∀x0, x1, . . . , xn[x0Rx1Rx2 · · · xn−1Rxn implies �i�=j(xi = xj)]

F is bounded if it is n-bounded for some n ∈ ω.

For each n, n-bounded Kripke frames can be characterized by a modal formula. For thispurpose, first we introduce the notion of exclusive sequence of formulas. An exclusive sequenceof formulas is a sequence {Fi}n

i=0 (n ≥ 1) of classical formulas ( i.e. formulas on our languagewithout �,� ) such that (1) each Fi is satisfiable in one-point frame, or in the classical logic,and that (2) every pair of formulas Fi, Fj (i �= j), Fi ∧ Fj is not satisfiable at any one point ina frame. Of course, we can easily construct an exclusive sequence of formulas for any n ≥ 1.Now, we define a formula Dn by employing an exclusive sequence of formulas {Fi}n

i=0.

Dn := ¬{F0 ∧ �(F1 ∧ �(· · · ∧ �(Fn+1 ∧ �Fn) · · · ))}= F0 ⊃ �{F1 ⊃ �(· · · ⊃ �(Fn−1 ⊃ �¬Fn) · · · )}

Then the following holds.

Lemma 4.2 Let F = 〈W,R〉 be a Kripke frame and n ≥ 1. Then F |= Dn if and only ifF |≡ δ(n)Proof : Suppose F�|≡ δ(n). Then there exists a sequence of distinct points x0, x1, . . . , xn ∈ Wsuch that x0Rx1R · · ·Rxn. Since {Fi}n

i=0 is exclusive, we can define a valuation V on F as:for each i, xi |= Fi. Thus we have F �|= Dn. Conversely, suppose F |≡ δ(n), and consider anyvaluation V on F . Take arbitrary points x0, . . . , xn ∈W such that x0 |= F0, xi−1Rxi and xi |= Fi

for 1 ≤ i ≤ n − 1, and xn−1Rxn. Then, because of δ(n), there exists at least one pair (xi, xj)such that xi = xj . But since {Fi}n

i=0 is exclusive, we can see that for some i (0 ≤ i ≤ n − 1),xi = xn. Therefore, xi = xn |= Fi, which means that xn �|= Fn. Thus, we have x0 |= Dn, and soF |= Dn. �

By this lemma, it is obvious that the logic K ⊕ Dn does not depend on what its exclusivesequence of formulas {Fi}n

i=0 really looks like. To be more important, this lemma also says thatfor n ≥ 1, the logic K⊕Dn is elementary. This fact implies that these logics have the followingnice property. Below, a logic L is d-persistent if for every descriptive frame F = 〈W,R,P 〉,F |= L implies κF |= L, where κF = 〈W,R〉 (underlying Kripke frame of F).

Theorem 4.3 For every n ≥ 1,

(1) K ⊕ Dn is d-persistent, in particular canonical.

(2) K ⊕ Dn is Kripke complete.

(3) K ⊕ Dn has the finite model property. �

7

270

Page 280: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

For the proof of (3) above, we can appeal to the filtration method. Since the axioms T andB will not cause any trouble along the proof, we can apply the same argument to the logicKTB⊕ Dn to show the following theorem on it.

Theorem 4.4 For every n ≥ 1,

(1) KTB ⊕Dn is d-persistent, in particular canonical.

(2) KTB ⊕Dn is Kripke complete.

(3) KTB ⊕Dn has the finite model property. �

4.2 All finite frames split Next(KTBDn)

First we recall the notion of splitting of a lattice, that is often used for investigating the latticeof modal logics.

Definition 4.5 (splitting) Let L = 〈L,∧,∨〉 be a complete lattice and a ∈ L. Then a splitsL if there exists b ∈ L such that for any x ∈ L, either x ≤ a or b ≤ x, but not both. Such a pair(a, b) is called a splitting pair of the lattice L.

When we think about the splitting of a lattice of logics, if a logic L(A) (or L(F)) splits thelattice, then we say that the algebra A (or the frame F) splits it.Although we want to take splitting members for the lattice Next(KTB) into consideration,

it seems rather difficult. Instead, we discuss here the splitting of the lattice Next(KTBDn)(n ≥ 1), where KTBDn := KTB⊕ Dn.It is well known that when a logic L has the finite model property, if an algebra A splits

Next(L), then there exists a finite s.i. algebra A′such that L(A) = L(A

′) ([13]). Since KTBDn

has the finite model property, we can take only finite algebras for this logic into consideration inorder to find some splitting algebras of the lattice Next(KTBDn). Every finite KTB-algebraA is a Kripke algebra, and so, by Theorem 2.3, A is simple. Moreover, if A∗ |= Dn, then A∗ isn-bounded, and so, it is (n − 1)-connected. With the help of these facts together, we can showthat every finite algebra for KTBDn splits Next(KTBDn), that is, the following holds.

Theorem 4.6 Let A be an arbitrary finite KTB-algebra such that A |= D(n+1) (n ∈ ω). ThenA splits Next(KTBD(n+1)).Proof : Since A |= D(n+1), we have A∗ |≡ ∀x, y(xRny), and so, A |= Tra(n, n + 1). For eachelement a ∈ A, prepare propositional variable pa, and let A be the conjunction of the followingformulas. For any a, b ∈ A,

pa ∧ pb ↔ pa∩b, pa ∨ pb ↔ pa∪b, ¬pa ↔ p−a, �pa ↔ pIa

Here C ↔ D := (C ⊃ D) ∧ (D ⊃ C). Put B := �nA ⊃ p0. Then it can be verified that for anyalgebra B that validates D(n+1), the following three conditions are equivalent.

(1) A ∈ SH(B). (This means that A is a subalgebra of a homomorphic image of A.)

(2) L(B) ⊆ L(A).

(3) B �∈ L(B).

8

271

Page 281: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

It is easy to show that (1) implies (2), and that (2) implies (3). We will show here that (3)implies (1). By (3), there is a valuation h : Φ → B such that h(B) �= 1, that is, h(�nA) =Inh(A) := e �≤ h(p0). Here F := {x ∈ B | e ≤ x} is an open filter since B |= D(n+1). Take thecongruence relation ≡ on B that is induced by F , namely ≡:= {〈x, y〉 ∈ | x ↔ y ∈ F}. Due tothe construction of A, we have, for a, b ∈ A,

h(pa∩b) ≡ h(pa) ∩ h(pb), h(pa∪b) ≡ h(pa) ∪ h(pb), h(p−a) ≡ −h(pa), h(pIa) ≡ Ih(pa)

Put C := B/ ≡. Then C is a homomorphic image of B. Define a map f : A → C as: f(a) :=h(pa)/ ≡, then it is easily seen that f is a homomorphism. Moreover, we can prove that f isone to one. Since e �≤ h(p0), we have f(0) = h(p0)/ ≡ �= 1/ ≡ in C. Besides, C |= D(n+1),so C∗ |≡ ∀x, y(xRny), therefore the algebra C is simple. Thus f(0) �= 1/≡ implies that f is atrivial map. Finally we have that A is a subalgebra of C, and so, A ∈ SH(B).Now Let L∗ := KTBD(n+1)⊕B. Then for any L ∈ Next(KTBD(n+1)), there exists an algebra

B such that L = L(B). By the above three equivalent conditions, we have L = L(B) ⊆ L(A)if and only if B �∈ L(B) if and only if L∗ �⊆ L(B). Therefore (L(A),L∗) is a splitting pair. �

This proof is based on the proof of Rautenberg’s splitting theorem on m-transitive logics ([13]),which says that for L0 ∈ Next(K ⊕ tm), every finite s.i. algebra A for L0 splits Next(L0).Here tm := p ∧ �p ∧ · · · ∧ �mp ⊃ �(m+1)p. In order to apply the same technique, we cannothelp bringing some axiom like m-transitivity into our consideration. Therefore we have treatedhere the splitting of the lattice Next(KTBDn). Up to now, we have little knowledge about thesplitting members of Next(KTB), except the logic L(•). This logic obviously splits the lattice,because it is proved that this logic sits on top of all consistent normal modal logics containingT ([10]).It is easily proved that for any finite KTB algebra A, if A∗ |= Dn for some n ∈ ω, then A∗ |= Dm

for any m(m > n). Therefore, Theorem 4.6 says that any given finite KTB algebra splits thelattice Next(KTBD�) for every sufficiently large � ∈ ω. From the reverse viewpoint, this meansthat if there exists a KTB-logic that prevents some KTB-algebra from being a splitting algebraof Next(KTB), it must be the logic determined by infinite s.i. algebras such as H∗ in Example2.4, and that logic must be located in the bottom part of the lattice Next(KTB).

5 The upper part of the lattice Next(KTB)

As we mentioned above, the logic L(•) is the second greatest element in Next(KTB). Thenwhat kind of logics follow L(•)? A plausible candidate is the logic L(•−•). In fact, by Jonsson’slemma ([8]) we can show that above it are only L(•) and the inconsistent logic. Moreover, itcan be proved by the following argument that the logic L(•−•) is the third greatest of all Kripkecomplete logics in Next(KTB).First, we prepare the notion of p-morphism between two Kripke frames. Let F = 〈W,R〉 and

G = 〈V, S〉 be two Kripke frames. Then a map f : W → V is called p-morphism from F to G,if (1): f is onto, (2): for any x, y ∈W , xRy implies f(x)Sf(y), and (3): for x ∈W and a ∈ V ,f(x)Sa implies that there exists z ∈ W such that xRz and a = f(z). It is crucial that if thereexists a p-morphism from F to G, then L(F) ⊆ L(G) holds between two logics. We denote theframe •−• by G2, that is, G2 = 〈V, S〉, where V := {0, 1} and S := {(0, 0), (1, 1), (0, 1), (1, 0)}.Then, we can show the following.

Lemma 5.1 Let F = 〈W,R〉 be a KTB Kripke frame, where F∗ is subdirectly irreducibleand |W | ≥ 2. Then L(F) ⊆ L(G2)

9

272

Page 282: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Proof : Since F∗ is s.i., we have F |≡ ∀x, y∃n ∈ ω(xRny) by Theorem 2.3. We define a partition{W (i)}∞i=0 of W in the following: For i = 0, pick up an arbitrary x0 ∈W and W (0) := {x0}. For

i = n+ 1, W (n+1) := {y ∈W | xRy for some x ∈W (n)} −n⋃

i=0

W (i). Then by this construction,

|W | ≥ 2, and our first order condition for F , we have W (0) �= ∅, W (1) �= ∅, and∞⋃

i=0

W (i) = W .

Here a map f : W → {0, 1} is defined as: for a ∈W ,

f(a) ={

0 (if a ∈W (2k) for some k ∈ ω)1 (if a ∈W (2k+1) for some k ∈ ω)

Then about this f , first, because W (0) �= ∅ and W (1) �= ∅, we can say that f is onto. SupposeaRb for a, b ∈W . Then by our construction of W (i), we can see that one of the following casesoccurs, that is, (1): for some m, a ∈ W (m) and b ∈ W (m+1), (2): for some m, a ∈ W (m+1)

and b ∈ W (m), (3): for some m, a, b ∈ W (m). In case (1) and (2), f(a)Sf(b) holds due to thedefinition of f . In (3), f(a)Sf(b) also holds since S is reflexive. Last, suppose f(a)Sz for a ∈Wand z ∈ {0, 1}. Then, because W (0),W (1) �= ∅ and R is reflexive, it is obvious that there existsb ∈W such that aRb and f(b) = z. Thus we have checked that this f is a p-morphism from Fto G2, and so, L(F) ⊆ L(G2) holds. �

From a graph theoretical point of view, this lemma can be translated into the following way.Namely, every countable, non-directed, and connected graph (here connected means the graphtheoretical sense), whose number of nodes are greater than one, can be colored in two colors,say, black and white, in such a way that any black node can see at least a white node, and vicea versa. This fact suggests that the existence of a p-morphism from a KTB Kripke frame Fto a KTB Kripke frame G has something to do with the possibility of coloring the undirectedgraph F in some coloring pattern determined by the frame G. By the above lemma, we have thefollowing.

Theorem 5.2 The KTB-logic L(•−•) is the third greatest logics of all Kripke complete KTB-logics. �

In fact, by using Jonsson’s lemma for some finite frames, we can draw a picture of the upperpart of the lattice Next(KTB). Figure 1 in the next page shows how the lattice structure ofthe upper part of Next(KTB), which consists of connected frames of, at most, four reflexivepoints looks like. Each of ten circles represents the logic determined by the Kripke frame in it.On the other hand, it is not known at present whether there exist other predecessors of L(•)

than the logic L(•−•). We cannot remove the possibility that there exist other predecessors, inparticular, logics of some infinite frames might sit on just below L(•). The situation is the samefor other tabular logics. For a particular tabular logic L, it is easy to find which logics do existabove L by using Jonsson’s lemma. However, up to now we do not have any idea how to checkwhat exists just below L.In Blok’s paper ([1]), the following result is established on the lattice Next(KT).

Theorem 5.3 For any consistent logic L ∈ Next(KT), there exists a continuum of logics inNext(KT) that are covered by L. �

Of course this theorem can be applied to the logic L(•) to show that there are uncountablymany predecessors of this logic in Next(KT). When the axiom B comes in their sight, it is notclear how many logics can survive of all predecessors of L(•) in Next(KT). But there might be

10

273

Page 283: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

a possibility that Theorem 5.3 can be transformed, in some way, into the fact about the classNext(KTB) and that the fact will show there is no third greatest logic in Next(KTB).

The inconsistent logic

Figure 1: The upper part of Next(KTB)

6 Remarks and questions

The author cannot help saying that the study of the lattice Next(KTB) here is very farfrom complete at present, because this paper treats only a very restricted class of the wholeKTB-logics. But the approach in this study of assuming some conditions in order to make theobject logics tractable seems to be reasonable, in the face of the situation that there is very fewknown facts about this class of logics.The logics in Next(KTB) are extremely difficult to investigate by nature, which is mentioned

in several texts and papers in modal logics (for instance,[4],[7],[9]). In fact, there are only fewfacts known about this class of logics at present. Namely,

(1) There exists a continuum in Next(KTB) ([12]).

(2) The cardinality of pretabular members in Next(KTB) is at least countably infinite ([11]).

(3) The universal frame of rank 2 for KTB⊕ �2p ⊃ �3p is infinite ([3]).

To finish off this paper, we will list up some open questions about the KTB-logics and thestructure of the lattice Next(KTB). We expect that many researchers are getting interestedin this area of modal logics, and they will find answers to the following questions. Moreover, wehope that the investigation of this class of logics will lead to a discovery of some new techniquesand new perspectives for modal logics of broader classes rather than logics of transitive frames.

11

274

Page 284: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� The structure of the lattice Next(KTB).

(Q1): Is the logic L(•−•) the third greatest element of Next(KTB)?

(Q2): What is the degree of incompleteness of L(•) with respect to KTB?

See [1], [4] to find information about degree of incompleteness of a logic.

� The existence of peculiar members in Next(KTB).

(Q3): Is there any KTB-logic that is finitely axiomatizable but not Kripke complete?

(Q4): Is there any KTB-logic that is decidable but without the finite model property? ([7])

� General properties of members in Next(KTB).

(Q5): Is it possible to show that for any n ∈ ω, every member in Next(KTBDn) has the finitemodel property?

(Q6): Does there exist a continuum of pretabular members in Next(KTB)? ([11])

References

[1] Blok W.J., On the degree of incompleteness in modal logics and the covering relation in thelattice of modal logics, Technical Report 78-07, Department of Mathematics, University ofAmsterdam, 1978.

[2] Blok W.J., The lattice of modal logics : an algebraic investigation, Journal of SymbolicLogic, 45, 221–236 (1980).

[3] Byrd M.,On the addition of weakened L-reduction axioms to the Brouwer system, Zeitschriftfur Mathematische Logik und Grundlagen der Mathematik, 24, 405–408 (1978).

[4] Chagrov A., Zakharyaschev M., Modal logic, Oxford University Press, 1997.

[5] Fine K., Logics containing K4, part I, Journal of Symbolic Logic, 39, 229–237 (1974).

[6] Fine K., Logics containing K4, part II, Journal of Symbolic Logic, 50, 619–651 (1985).

[7] Gabbay D.M.,On decidable, finitely axiomatizable modal and tense logics without the finitemodel property II, Israel Journal of Mathematics, 10, 496–503 (1971).

[8] Jonsson B., Algebras whose congruence lattices are distributive, Mathematica Scandinavica,21, 110–121 (1967).

[9] Makinson D.C., A normal modal calculus between T and S4 without the finite model prop-erty, Journal of Symbolic Logic, 34, 35–39 (1969).

[10] Makinson D.C., Some embedding theorems for modal logic, Notre Dame Journal of FormalLogic, 12, 252–254 (1971).

[11] Meskhi V.Yu., Critical modal logics containing the Brouwer axiom, Mathematical Notes,33, 65–69 (1983).

12

275

Page 285: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[12] Miyazaki Y., Binary logics, orthologics and their relations to normal modal logics, in:Advances in Modal Logic, 4, eds. by P. Balbiani et al, King’s College Publications (2003),313–333.

[13] Rautenberg W., Splitting lattices of logics, Archiv fur Mathematische logik, 20, 155–159(1980).

[14] Zakharyaschev M., Canonical formulas for K4, part I:Basic results, Journal of SymbolicLogic, 57, 1377–1402 (1992).

[15] Zakharyaschev M., Canonical formulas for K4, part II:Cofinal subframe logics, Journal ofSymbolic Logic, 61, 421–449 (1996).

13

276

Page 286: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

On the formal structure of continuous action

Thomas MullerPhilosophisches Seminar, LFB III, Lennestr. 39, 53113 Bonn, Germany

[email protected]: +49 228 73-3786; Fax: +49 228 73-5066

Keywords: agency; indeterminism; free will; stit theory; imperfective aspect

Abstract

Analytical investigations of agency are mostly concerned with a descrip-tion ex post acto. However, continuous action (being doing something) needsto be considered as well. The paper shows that (1) the modal-logical treat-ment of agency in branching time-based stit theory in its present form isunable to handle continuous action, but (2) nonetheless the stit frameworkcan be extended such as to handle these cases as well. A new operator,istit,provides for an adequate expression of the notion of being doing something.In the extended framework, agency, ability and refraining are linked to anagent’s current strategy.

1

277

Page 287: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

On the formal structure of continuous action

1 Introduction

Since the 1980ies, a number of agency-related concepts have been explored usingthe resources of modal logic. The key idea, sometimes dubbed the “Anselmianapproach” since there is textual evidence for the analysis in some writings ofSt. Anselm’s, is that acting is best described in terms of an agent’s bringing aboutsome state of affairs. Thus, the concept of agency is seen to give rise to a familyof (agent-indexed) modal operators. A natural reading for these modalities is “αsees to it thatφ”, abbreviated as “α stit : φ”. The stit-modalities have been givena formally rigorous semantics in the framework of branching time; the approach islaid out and well argued for in the recent book,Facing the Future(Belnap et al.2001).

The present paper is an attempt at extending that formally rigorous modal-logical treatment of agency to cover the case of continuous action. Thus, we willtry to establish two claims: (1) Present stit theory is unable to handle cases ofcontinuous action in an adequate manner; (2) nonetheless, the framework can beextended in a formally rigorous way such as to cover continuous action as well.

In order to establish claim (1), we first need to explicate the phenomenon ofcontinuous action.1 Together with some other truths of descriptive metaphysics,the phenomenology of continuous action sets conditions of adequacy for any suc-cessful analysis. These conditions will be spelt out in section 2. In section 3 wewill then show why present stit theory, despite its variety of stit operators, does notmeet these conditions. Recalcitrant cases are, e.g., ones in which an agent is reallydoing something, but (as can turn out only later) does not finish the task.

Having established so much, in the rest of the paper (section 4) we will try toestablish claim (2) by adding yet another modality to the growing zoo of stit oper-ators. The new operator,istit (“. . . is seeing to it that”), is based on the frameworkof branching time, but it involves a new index of evaluation for the formal lan-guage. This new index,s (for “strategy”) provides a link between formal modelingon the one hand and our everyday mentalistic vocabulary on the other hand.

2 Continuous action: From the phenomenologyto criteria for an adequate theory

2.1 A metaphysical presupposition

Before considering the phenomenon of continuous action, it is important to spellout a metaphysical presupposition of this paper. In full accordance with the frame-

1The main example will be apple peeling.—Besides having the advantage of not involvingmanslaughter in the way most action-theoretic examples (including St. Anselm’s!) do, apples havea venerable tradition in practical philosophy, as witnessed, e.g., by Kant’sMetaphysics of Morals(1797) and, more recently, by Segerberg (1989) and Xu (1996).

2

278

Page 288: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

work of stit theory, we assume that agency presupposes indeterminism. I.e., therecan be no agency if the future is not open, if it does not contain more than one pos-sible course of events. To some this may seem like a bold assumption, taking sidesin the endless debate about compatibilism. A full defence of the assumption is cer-tainly out of place here. Suffice it to say that from the point of view of descriptivemetaphysics the assumption is inevitable.

2.2 Two kinds of examples of agency

In analyses of agency, the examples used point to two differing approaches.2 Con-sider the following examples from Davidson’s “The logical form of action sen-tences” (1967):

• Jones buttered the toast in the bathroom with a knife at midnight. (107)

• The doctor removed the patient’s appendix. (111)

• Amundsen flew to the north pole. (115)

In Intention(1963), Anscombe mostly uses examples of the following kind:

• I’m pumping. (38)

• He is replenishing the water-supply. (39)

• She is making tea. (40)

The first difference that one may notice is one of tense: Davisons’s examplesare in the past tense, whereas Anscombe’s are in the present tense. Still more im-portantly, the examples differ with respect to aspect: Davidson considers actions inthe perfective aspect (from a point of view after the action is finished). Anscombeuses the imperfective aspect (from a point of view while the action is occurring),as marked by the present continuous.3 Analytical investigations of agency havemostly been concerned with Davidson-type examples. Stit theory is no exceptionin this respect. Certainly these examples are important, but a full account of agencyneeds to consider Anscombe-type examples of continuous action, too. The twoclasses of examples differ with respect to what may be called their “phenomenol-ogy”.

Davidson When an action is finished, the following account will be adequate formany purposes: First a certain initial state of affairs obtained. Due to the agent’saction, some outcome state of affairs obtained later. The agent saw to it that atransition from initial to outcome occurred. Consider apple peeling.Ex post acto,

2This point is made in Thompson (2004).3In English, the difference between the two aspects is not as marked as in other languages—e.g.,

in Slavic languages there are two different verb forms for the two aspects.

3

279

Page 289: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

the situation seems to be simple enough: First the apple wasn’t peeled, then it waspeeled, and the agent did it.4 Stit theory is able to handle many cases of that kindsmoothly, e.g., via thedstit operator to be discussed in section 3.1.

Anscombe While an agent is acting, the phenomenology is different. E.g., theongoing action of apple peeling may be described like this: Some time in the past,the agent decided to peel, and from then until the present moment, she chose (whenshe had a choice) in such a way as to continue peeling. Before the peeling will befinished, the agent will usually have more choices. In particular, if the agent isacting freely, we need to allow for “dropping out”, i.e., not finishing. The laterchoices need to be made when they are due; they cannot be made now (“no choicebefore its time”). But still, when the agent is really peeling, there will be “defaults”for the later choices. We may say that the agent is now committed to the futuredefaults, but she still cannotchoosethese defaults now.

2.3 Conditions of adequacy

If the given description is correct, the following type of situation has to be possible:At 9:00 an agent is peeling an apple in such a way that as far as things are plannedout, she should be finished by 9:04—but as it turns out, at 9:02 the agent drops theapple and walks away; not compelled by any outside force, but simply “of her ownfree will”. She was peeling, but she didn’t finish in the end.5

In summary, continuous action has the following marks:

1. If an agent is doing something, she has defaults set for her future choices.

2. The future choices are real choices nonetheless: When the time comes, thedefault is not forced upon the agent.

3. An agent may be truly said to be doing something even though it may turnout later that she didn’t finish.

These characteristic marks impose conditions of adequacy on any theory of con-tinuous action: The theory must be able to save these phenomena.

3 Why current stit theory fails the test of adequacy

In this section I will argue that current stit theory is unable to account for thephenomenon of continuous action. The presentation will be limited to the case ofa single agent for the sake of simplicity.

4Certainly a transition from apple not peeled to peeled involves an agent’s continuous action inany case. However,ex post actoit is often feasible to ignore this as an unimportant detail.

5It may be that such examples sound the more plausible the longer the envisaged action is goingto take. E.g., the following may be more convincing: “I was walking to the office, but then I didn’twalk there”. Or consider this: “I was writing a thesis on Hegel, but then I never finished”. Theimportant point is that such actions arestructurallypossible; the duration should not matter as far asthe structure is concerned.

4

280

Page 290: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3.1 Outline of stit theory

Stit theory as laid down in Belnap et al. (2001)6 is based on indeterministic branch-ing time models with agents and choices,M = 〈M,≤, Agents, Choice〉, whereM is a nonempty set of moments partially ordered by≤. The partial ordering≤additionally satisfies the axiom of “no backward branching”:

∀x∀y∀z ((x < z ∧ y < z) → (x ≤ y ∨ y ≤ x)),

and any two moments have a common lower bound:

∀x∀y∃z (z ≤ x ∧ z ≤ y).

Further postulates are discussed in Belnap et al. (2001), Chap. 7. The setAgents ={α} for simplicity, andChoice is a function determining the choices for agentα atany moment. Maximal linear subsets ofM are calledhistories. In M, branchingoccurs where histories diverge. At each pointm ∈ M , the setH(m) of historiescontainingm is partitioned via the equivalence relation≡m of being undivided atm, whereh1 ≡m h2 iff m ∈ h1 ∩ h2 and there ism′ ∈ h1 ∩ h2 such thatm < m′.Πm is the respective partition ofH(m). If Πm has more than one element, then atm there is (indeterministic) splitting of histories.

The metaphysical basis for agency is not just indeterminism, but agent-relatedindeterminism. This concept is formally captured viaChoiceα

m, which is the setof choices open for agentα at momentm. The partitionChoiceα

m may be morecoarse-grained, but not more fine-grained, than the partitionΠm: An agent hasat most as fine a “control” over what happens as nature’s indeterminism allows. IfChoiceα

m has only the one elementH(m), this means that at momentm agentα hasno choice. As usually, a modelM is taken to be a formal picture of the ontology.This means that the partitionsChoiceα

m are given by nature and cannot be changedby the agent.7

In accord with standard two-dimensional semantics, sentences are evaluated atan index of evaluation, which usually consists of a context of utterance and somemore indexes. In one-agent stit theory, the context is taken to be a moment of utter-ancemc, and formulae are evaluated additionally at a moment-history pairm,h,wherem ∈ h, to allow for standard Prior-Thomason tense operatorsP (“it wasthe case that”) andF (“it will be the case that”).8 Two stit operators are normallyconsidered: the “deliberative stit”,dstit, and the “achievement stit”,astit.

dstit To start with the simpler of the two, the semantics fordstit is as follows:

M,mc,m, h |= α dstit : φ iff

6For further uses of the framework cf. the extensive bibliography in that work.7If you wish, you may read an existentialist note into this: According to stit theory, our freedom

is not just given, but forced upon us by the way the world is.8The semantics of the Occamist future tense operatorF is the following: M, m, h |= Fφ iff

there ism′ ∈ h with m < m′ such thatM, m′, h |= φ. The past tense operatorP employs the(unproblematic) mirror image of this, exchanging “m′ < m” for “ m < m′”. —For a full outline cf.Belnap et al. (2001), Chap. 8.

5

281

Page 291: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. for all h′ ∈ Choiceαm(h) we haveM,mc,m, h′ |= φ and

2. there is a “counter”h′ ∈ H(m) such thatM,mc,m, h′ 6|= φ.

The first clause is positive: It states that the current choice ofα (singled outfrom Choiceα

m through the historyh in the index of evaluation) secures the truthof φ. The negative second clause, on the other hand, excludes suchφ that are trueunder any circumstances: nobody sees to it that2 + 2 = 4.

astit Thedstit operator considers a current choice that brings aboutφ. The otherstit operator,astit, can only be assessedex post acto—α astit : φ is only true ifφis currently true. This already violates requirement (3) from the list at the end of thelast section, so we are spared the task of explaining the (somewhat more intricate)semantics ofastit. Thus, from among the currently available stit operators, onlydstit might be a candidate for a “continuous action” operator.9

3.2 Why dstit cannot handle continuous action

Consider Figure 1. Suppose that at momentm it is true to say that agentα is peel-ing an apple (A): She decided to peel a while ago, and she has already completedpart of the task. In the unrealistically simple picture given,α faces only two morechoices, atm and atm′, before the apple will be peeled, and by deciding to peel shehas fixed default choices leading to the desired result; these choices are indicatedby bold lines.10 We want to be able to say that atm, the agent is peeling—eventhough the outcome is not guaranteed yet. (The agent is facing real choices in thefuture, atm′, and she will be free to choose.) However, from the picture it is clearthat nothing that the agent can choose now (atm) will be enough to secure thedesired outcome. By being restricted to the current choices forα (i.e., the partitionChoiceα

m), dstit is unable to single out the “good” histories, i.e., the ones in whichthe current task will be finished. Accordingly,α dstit : A (whereA is for applepeeling) will be evaluated false atm. Thus,dstit is unable to capture the conceptof defaults for future choices, which is central to continuous agency.

4 A proposed solution

How should a positive formal account of continuous action look like? In theprevious section it was argued that the stit framework so far does not have thenecessary resources for the task. Thus there are two options—either to abandonthat framework or to extend it. In my view, the second option is to be preferred,

9The cstit operator that is used, e.g., in Horty (2001), consists ofdstit without the negativeclause. Thus the following discussion ofdstit holds forcstit as well.

10We make no assumptions about the psychological status of these choices, e.g., whether theyare conscious or subconscious. For the metaphysics of agency the important part is the branchingstructure.

6

282

Page 292: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A

m’

m

Figure 1: A crucial test case fordstit: default choices in a continuous action.At m and atm′, the agent must choose one of two available bundles of histories,indicated by the boxes.A will be true only if the agent chooses the rightmost optionboth atm and atm′.

since the branching time basis of stit theory gives a formally perspicious and well-understood picture of the metaphysical basis of agency. It would be a good thingto stay within that framework. In what follows I argue that it can be done.

4.1 Theistit operator

The target of analysis is a modal operator that I will callistit (“is seeing to it that”).It needs to do justice to the requirements laid down at the end of section 2.

The discussion of Figure 1 shows that for “is doing” we need a way of singlingout sets of histories (the “default” ones, given what the agent is doing) in a way thatis more fine-grained than the partitionChoiceα

m at momentm. On the other hand,selecting a single history is not an option, since usually an agent cannot secure asingle history: there are other sources of indeterminism besides the agent. On thebasis ofM and the indexmc,m, h, the required set of histories cannot be defined.More resources are needed.

The proposed solution is to add a new parameters to the index of evaluation.That new index is to stand for the agent’s strategy with respect to which a sentenceis to be evaluated.11 The concept of a strategy is firmly grounded in the frameworkof branching time. For a detailed introduction that describes various concepts ofstrategies, cf. Belnap et al. (2001), Chap. 13. In our context we will be using

11In the case of multiple agents, there will be a strategy parametersα for each agentα, and therewill be a “strategic future tense” operatorF α

S for each agent, too.

7

283

Page 293: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

so-called simple strategies. Roughly, a simple strategy is a partial function frommoments to the available choices at those moments:s(m) ∈ Choiceα

m, ands(m)may be interpreted as “what the strategy advises to choose atm”.12 The concept ofa strategy thus incorporates the required notion of “default choice” without alteringthe setsChoiceα

m (which, as we assume, are given by nature). A strategys admitsa historyh iff by following the advice ofs, historyh may be reached.13 The set ofadmitted histories is denotedAdmh(s). Given an indexm,h, s with m ∈ h andsa simple strategy, we can define a new (Peircean) “strategic future tense” operator,FS , as follows:

M,mc,m, h, s |= FSφ iff

1. s is defined atm and

2. on allh ∈ Admh(s) ∩H(m), φ will be settled true.14

Thus, it will strategically (as far asα’s current plans are concerned) be the case thatφ iff for all histories that are admitted byα’s current strategy,φ will be the case.

Theistit operator is based on this future operator, adding a negative clause toprevent trivial cases:

M,mc,m, h, s |= α istit : φ iff

1. (positive)M,mc,m, h, s |= FSφ and

2. (negative) there is a “counter”h′ ∈ H(m) such thatM,mc,m, h′, s 6|= Fφ.

This operator fulfills the formal requirements that the other stit operators wereunable to handle:

1. The strategy parameter,s, incorporates the notion of “defaults for futurechoices”.

2. The strategys leavesChoiceα unaltered: The status of future choices as realchoices is not changed.

3. As an example for doing and not finishing, consider the picture of Figure 1.If α at m′ chooses the left branch (which she is free to do), then it is true tosay that atm, α was peeling an apple, but she did not finish.

We conclude that while current stit theory is unable to handle the phenomenon ofcontinuous action in an adequate way, theistit operator presented here providesfor an adequate extension of stit theory.

12The formal definition is given in Belnap et al. (2001), Chap. 13C. For our purposes, all thatis important is that a simple strategy always gives advice of the most specific kind: It specifies amember ofChoiceα

m, while a general strategy may specify a subset ofChoiceαm.

13As above, normally a strategy will not be able to guarantee a single history—apart from theindeterminism over whichα has control, there is usually also other indeterminism in a model.

14Settled truth means truth with respect to all histories that contain the moment in question—historical necessity, if you wish. Cf. note 15 below.

8

284

Page 294: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4.2 Status and initialization of the strategy parameters

In the usual treatment of Prior-Thomason tense logic, there is a difference betweenthe indexesm andh: If a stand-alone sentence is to be evaluated,m gets an initialvalue from the context via the moment of utterancemc, but there is no context-initialization for the history parameter,h: Assuming a “history of the context”would mean falling prey to the myth of “the real future”, a notion decisively criti-cized (under the name of “the thin red line”) in Belnap et al. (2001), Chap. 6. Whatis the status of thes parameter?

In the istit picture, s functions as an interface between our everyday men-talistic vocabulary that describes an agent’s plans and intentions and the formalbranching time framework. We talk about an agent’s current plans or intentionsas something objective (something that may be relevant, e.g., in a court case), andthis is whats should capture. Thus,s is an initialized parameter likem, not anuninitialized one likeh. Paralleling the treatment ofm means that we should adda “current strategy” indexsc to the context.

4.3 Further extensions

So far we have introduced the bare minimum for handling continuous action: Themodal operatoristit is adequate for the task. In the extended framework that in-cludes the strategy index, a number of other modalities can be defined, showing theflexibility of the framework. In closing we mention three such extensions.

Strategic modalities In two-dimensional semantics, to each index that is not partof the context there corresponds a number of one-place modal operators for shiftingthat parameter. E.g., the (weak) tense operatorsP andF (and their strong dualsHandG) shift the moment parameterm backward or forward in time. For the historyparameterh there are also corresponding “historical” modalities〈h〉 (weak) anddually [h] (strong).15 Along the same lines one can define “strategic” modalities〈s〉 and[s], where the clause for the weak operator〈s〉 is as follows:

M,mc,m, h, s |= 〈s〉φ iff M,mc,m, h, s′ |= φ for some simple strategys′

that agrees withs everywhere in the past ofm.

Ability The weak strategic modality〈s〉 allows one to express the concept ofability: If 〈s〉FSφ is true atm,h, s, then this means that the agentα could changeher current strategys such as to guarantee outcomeφ. If φ is not guaranteedanyway,〈s〉α istit : φ is true, meaning that the agent could beφ-ing.

15Formally, the notion of “settled truth” that was used above in the definition of the semanticsof FS (cf. note 14) may be expressed as follows:φ is settled true atm, h iff [h]φ is true atm, h,meaning thatφ is true atm, h′ for all h′ ∈ H(m).

9

285

Page 295: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Refraining Refraining means not doing something while one could do it. Thus,we may say thatα refrains fromφ-ing iff α is not φ-ing (¬α istit : φ) and stillcould beφ-ing (〈s〉α istit : φ).16

5 Conclusion

The phenomenon of continuous action poses a challenge for a formal theory ofagency. The branching time-based stit framework in its present form, comprisingthe operatorsastit anddstit, is unable to capture that phenomenon. However, byincorporating an agent’s current strategy as a new index of evaluation it is possi-ble to extend the stit framework such as to capture continuous action. The newoperator,istit, forms the basis of a rich structure of agency-related modalities.

References

Anscombe, G.E.M. (1963).Intention, 2nd ed. Oxford: Blackwell.

Belnap, N., Perloff, M., and Xu, M. (2001).Facing the Future. Oxford: OxfordUniversity Press.

Davidson, D. (1967). The Logical Form of Action Sentences. In hisEssays onActions and Events, Oxford: Oxford University Press 1980, 105–122.

Horty, J.F. (2001).Agency and Deontic Logic. Oxford: Oxford University Press.

Kant, I. (1797).Metaphysische Anfangsgrunde der Rechtslehre. Ed. B. Ludwig.Hamburg21998.

Segerberg, K. (1989). Bringing it about.Journal of Philosophical Logic18:327–347.

Thompson, M. (2004).Life and Action. Cambridge, MA: Harvard University Press.To appear.

Xu, M. (1996).An Investigation into the Logics of Seeing-to-it-that. UnpublishedDissertation, University of Pittsburgh, Pittsburgh, PA.

16In contrast to the standard treatment of refraining in stit theory, this notion of refraining is notagentive, i.e., it cannot be paraphrased as anistit clause. Further work will be required to find outwhich of the two approaches (taking refrainings to be doings, as in standard stit theory, or not, ashere) is more in line with our actual usage of the notion of refraining.

10

286

Page 296: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Utilitarian deontic logic

Yuko [email protected]

Department of PhilosophyIndiana UniversityBloomington, USA

Keywords and phrases: deontic logic, stit logic of agency, game-theoretical logic

287

Page 297: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC

YUKO MURAKAMI

1. Introduction

A lesson from the obstacles for deontic logic proposed before 1980s, in-cluding standard deontic logic, is that conceptual analysis of action1 shouldprecede that of deontic notions. It actually fits to our intuition, for a majoreveryday usage of “ought” concerns evaluation of human actions, not thatof situations. In particular, if a deontic operator is motivated by game the-ory (for example, if it is designed to reflect logical properties of the bestchoice toward an action among available options), their truth conditionsshould involve evaluation of options by all possible means in a semantic ap-proach. Syntactical characterization, on the other hand, of such operatorsshould take an action sentence as its argument, since the deontic judgmentconcerns with actions.

See-to-it-that (stit) theory of logic of agency (Belnap et al. [1] ) satis-fies the requisites. With philosophical foundations in free will arguments,the theory proposes semantics of agency. The game-theoretically motivatednotion of choice is cast against branching semantics of indeterminist time(Thomason [8]); at each moment, each agent is assigned a set of availableoptions.

Several kinds of agency operators have been proposed in the stit the-ory: so-called achievement stit operators and deliberative stit operators.Achievement stit operators are supposed to reflect the intuition that an ac-tion is evaluated at the moment of its achievement, which is preceded by themoment of the agent’s choice toward the action. On the other hand, deliber-ative stit (dstit, in short) operators aims to capture agency where the actionis evaluated at the moment of an agent’s choice. It is shown in [1] that bothdeliberative stit logic and achievement stit logic are axiomatizable, and thatthe latter is a sublogic of the former.

Horty [5] emphasizes game-theoretical motivations in the stit theory. Heintroduces to stit semantics a utilitarian value assignment over branches,which represent possibilities at a moment, so as to define preference anddominance orderings among options at each moment. He proposes several

Key words and phrases. deontic logic, stit theory of agency, game-theoretical logic.I am particularly grateful to Professor Ming Xu for his immeasurable help and encour-

agement. I also appreciate Professors Larry Moss, Nuel Belnap, John Horty, and RyoKashima, who have kindly offered opportunities of discussion. The author is responsiblefor any mistakes and unclarity, however.

1Since deontic notions concern human actions and agency, logic of action in the currentcontext, which should found any analysis of deontic notions, should reflect properties ofagency and human actions. On the other hand, dynamic logic, often considered as logicof action, is intended to represent actions of machines, thus seems rather inappropriatefor applications with deontic notions.

1

288

Page 298: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 YUKO MURAKAMI

deontic modalities including operators according to properties of value as-signment; “· · · holds in every dominant choice” and “· · · holds in everyoptimal choice,” for example.

Figure 1 illustrates Horty’s utilitarian frame for a single agent. The treestructure represents a branching structure for the indeterministic time. Itis open to the future, where each branch (or history) represents a possiblegrowth of the world. There are three moments where the agent can make achoice among two options Ci0 and Ci1 available at each mi. A real numberis assigned to each history in the structure to represent its utilitarian value,based on which the dominance ordering and the optimal ordering are defined.While readers should consult the following section for the exact definitions,it should also fit to a game-theoretical intuition that C01 is dominant overC00 at m0 and C21 over C20 at m2, while neither C01 nor C11 is dominantover each other at m1.

Figure 1. Utilitarian frame for a single agent

h1 h2 h3 h4 h5

0 1 −1 2 3

m0

m1 m2

C00 C01

C10 C11 C20 C21

This paper will show that deontic logic based on dominance orderingsin Horty’s semantics is not only axiomatizable but also has finite modelproperty, and thus is decidable. Section 2 describes a formal language andutilitarian deontic semantics on the background of the stit theory. Withpreference and dominance orders, optimal choices are defined according toHorty’s conceptual analysis to sort out three notions of obligations: domi-nant ought, optimal ought, and two-valued ought. Completeness and decid-ability of such logics are simultaneously proved via construction of a finitemodel in Sections 3 and 4. Section 5 suggests further investigations, basedon the consideration that the result implies that a single logical system char-acterizes not only all the three notions considered here but also the otherutilitarian deontic operators proposed by Horty, plain utilitarian ought andstandard ought and its individualized version. Discussion involves branchingtime (BT) semantics for stit logics, although our main text relies on sim-plified modal semantics through this paper, since validity in BT semanticscoincide with that in modal semantics with no temporal vocabulary.

289

Page 299: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 3

An open question is how various operators for deontic notions behaveand interact in a temporal structure, while the current discussion involvesno temporal references.

Readers should refer to [1] and [5] for details of BT semantics for delib-erative stit logic and deontic notions.

2. Preliminary

2.1. Language. The language L contains denumerably many propositionalvariables p0, p1, . . ., denumerably many terms for agents α0, α1, . . ., an iden-tity symbol =, truth-functional operators

∑and ∧, and non-truth-functional

operators ¤, [ cstit : ] and ¯. Formulas are defined as usual, except α = β,¤A, [α cstit : A], ¯[α cstit : A] are formulas whenever α and β are terms foragents and A is a formula. Abbreviations such as →, ↔, ∨, >, ⊥ and 3 areintroduced as usual. The following abbreviations are also defined:

α 6= β =df ∼(α = β),4αA =df [α cstit : A],ªαA =df ¯[α cstit : A].2

An additional abbreviation diff (β0, . . . , βn) (where n > 0) is defined asfollows:

diff (β0, β1) =df β0 6= β1,diff (β0, β1, . . . , βn+1) =df diff(β0, . . . , βn) ∧ (

∧06k6n βk 6= βn+1).

A stit formula is a formula whose major operator is 4α for some α, and anought formula is a formula whose major operator is ªα for some α.

2.2. Branching-time semantics. Horty [5] proposes semantics for stitand dominant ought based on the branching semantics for indeterministtime proposed by Prior and Thomason (see [7], [8] and [10])3.

A tree structure is F = 〈T, <〉, where T is a nonempty set, whose membersm,m′ etc. are called moments; < is a tree relation on T , i.e. a partial orderon T being (1) linear to the past and (2) connected. A history in F is amaximal linear subset of T . A history h is said to go through a moment mwhen m ∈ h. A moment-history pair is a pair of a moment m and a historywhich goes through m. Hm denotes the set of histories which go through m;HF denotes the set of all the histories in F; and Moment-History denotesthe set of all moment-history pair in F.

A stit frame is a tree structure together with a set of agents and a choicefunction for each agents, i.e. a quadruple F = 〈T, <, Agent, choice〉 satisfyingthe following. Agent is an nonempty set of agents; choice is a function that

2This paper follows Horty who uses S5-like cstit operators for modalities of agency.The dstit operators [ ] and the dstit-based dominant ought operator ⊕α can be defined

as follows: [α]A =df 4αA∧∼˜A, and ⊕αA =df ªαA∧∼˜A. [ ] can be taken primitive,and let 4αA =df [α]A ∨˜A, ⊕αA =df ¯[α]A and ªαA =df ⊕αA ∨˜A.

3We will closely follow the notation in [5] except in two cases. First, we use ασ as anagent, the interpretation of α in the model M = 〈F, σ〉, while in [5], α is used both as aterm for agent in the language and as the interpretation of the term in the model. Second,[5] uses |A|Mm for a “proposition”, the set of all histories h passing through m such thatM, m/h › A, while we use ‖A‖σ for the counterpart notion of the set of possible worldsand use |. . .| for the notion of cardinality.

290

Page 300: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 YUKO MURAKAMI

assigns to each a ∈ Agent and m ∈ T a partition choicema of Hm satisfying

the following conditions:Independence of Agents: For each function f that assigns toeach pair of a ∈ Agent and m ∈ T a member of choicem

a ,⋂

a∈Agent

f(a) 6= ∅

.In other words: Let selectm be the set of all functions f on Agent

such that f(a) ∈ choicema for every a ∈ Agent; For each proper

subset A of Agent, let StatemA = {⋂a∈Agent−A f(a) | f ∈ selectm};

Then, for any m ∈ T , ∅ /∈ Statem? .

No choice between undivided histories: For any h, h′ ∈ HF,m0 ∈ h∩h′, and α ∈ Agent, if there is m ∈ h∩h′ such that m0 < m,h ∈ choicem0

α if and only if h′ ∈ choicem0α .

A utilitarian stit frame is a quintuple F = 〈T, <,Agent, choice, value〉,where 〈T, <,Agent, choice〉 is a stit frame, and value is a function from HF

to the set of real numbers.Let F = 〈T, <, Agent, choice, value〉 be a utilitarian stit frame. We will use

K, K ′ etc. to range over subsets of HF. For each a ∈ Agent, m ∈ T, andh ∈ Hm, we use choicem

a (h) for the member of choicema to which h belongs.

For each m ∈ T, h0, h1 ∈ Hm, let Rma h0h1 iff choicem

a (h0) = choicema (h1).

2.3. States, preference, and dominance. The notion of states plays themain role in preference and dominance. Let Γ ⊆ Agent and m ∈ T. DefineStatem

Γ (h) =⋂

a 6∈Γ choicema (h). Let Statem

Γ = {StatemΓ (h) : h ∈ Hm}.

For each a ∈ Agent and m ∈ T, we will write Statema for Statem

{a}. LetK, K ′ ⊆ Hm. K ′ is weakly preferred to K, written K ≤ K ′, iff for eachh0 ∈ K and each h1 ∈ K ′, value(h0) ≤ value(h1). K ′ is strongly preferredto K, written K < K ′, iff K ≤ K ′ and not K ′ ≤ K. Let a ∈ Agent andlet K, K ′ ∈ choicem

a . K ′ weakly dominates K, written K ¹ K ′, iff for eachS ∈ statem

a , K ∩ S ≤ K ′ ∩ S. K ′ strongly dominates K, written K ≺ K ′, iffK ¹ K ′ and not K ′ ¹ K.

The following, established in Horty [5] may not all be necessary for ourproofs, but it should aid the reader’s intuition.

Proposition 2.1. Let 〈T, <,Agent, choice, value〉 be a utilitarian stit frame,let m ∈ T, a ∈ Agent, and let Statem

a be as defined above. Then the followinghold.

(i) X < Y iff value(h′) ≤ value(h) for each h′ ∈ X and h ∈ Y , andvalue(h2) < value(h1) for some h2 ∈ X and h1 ∈ Y .

(ii) The weak preference ordering ≤ is transitive.(iii) The strong preference ordering < is a strict partial ordering.(iv) X < Y and Y ≤ Z only if X < Z.(v) X ≺ Y iff X ∩S ≤ Y ∩S for every S ∈ Statem

a , and X ∩S < Y ∩Sfor some S ∈ Statem

a .(vi) The weak dominance ordering ¹ is transitive.(vii) The strong dominance ordering ≺ is a strict partial ordering.(viii) X ≺ Y and Y ¹ Z only if X ≺ Z.

291

Page 301: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 5

2.4. Utilitarian stit model, truth conditions, and semantic notions.Let F = 〈T, <,Agent, choice, value〉 be a utilitarian stit frame. A valuation τon F is a function that assigns each agent term a member of Agent and eachatomic formula a subset of Moment-History. A utilitarian stit model is apair M = 〈F, τ〉 where F is a utilitarian stit frame and τ is a valuation onF, which assigns each agent term α a member ατ ∈ Agent, and assigns eachatomic formula a subset of Moment-History. That A is true in M = 〈F, τ〉at m/h, written M,m/h ² A, is defined recursively as follows, where p isany atomic formula and α is any agent term.

M, m/h ² p iff m/h ∈ τ(p);M, m/h ² α = β iff ατ = βτ ;M, m/h ² ∼A iff M, m/h 2 A (i.e., not M,m/h ² A);M, m/h ² A ∧B iff M, m/h ² A and M, m/h ² B;M, m/h ² ¤A iff M, m/h′ ² A for every h′ ∈ Hm;M, m/h ² 4αA iff M, m/h′ ² A for every h′ ∈ choicem

ατ (h);M, m/h ² ªαA iff for each K ∈ choicem

ατ such that K * ‖A‖τm,

there is a K ′ ∈ choicemατ such that (i) K ≺ K ′,

(ii) K ′ ⊆ ‖A‖τm, and (iii) K ′′ ⊆ ‖A‖τ

m for everyK ′′ ∈ choicem

ατ with K ′ ¹ K ′′.

where ‖A‖τm = {h ∈ Hm | M,m/h ² A}. Notions of validity and satisfiabil-

ity are defined the same way as in modal logic. A is valid in M if M,m/h ² Afor every m/h ∈ Moment-History, and A is valid in F if A is valid in everymodel on F. A is valid if A is valid in every utilitarian stit frame. Φ is sat-isfiable in M if for some m/h in Moment-History, M,m/h ² A for everyA ∈ Φ, and Φ is satisfiable if Φ is satisfiable in a utilitarian stit model.

2.5. Classification of utilitarian stit frames. F = 〈T, <, Agent, choice, value〉is two-valued if the range of value is {0, 1}. Counterpart notions on modelsare defined as usual.

Let F = 〈T, <, Agent, choice, value〉 be a utilitarian stit frame. For eacha ∈ Agent and m ∈ T, we define

optimalma = {K ∈ choicema | ∼∃K ′(K ′ ∈ choicea ∧K ≺ K ′)}.

F is optimal at m if for each a ∈ Agent, K ∈ choicema − optimalma only if

K ≺ K ′ for some K ′ ∈ optimalma . A utilitarian stit model M = 〈F, τ〉 isoptimal if F is. It is easy to see that F is optimal only if optimalma 6= ∅ forevery m ∈ T and every a ∈ Agent. It has been shown in Horty [5] that F isa finite-choice utilitarian frame only if F is optimal. It is also easy to verifythe following.

Proposition 2.2. Let F = 〈T, <,Agent, choice, value〉 be any optimal util-itarian stit frame, and let M = 〈F, σ〉 be any model on F. Then for eachm,h in F with m ∈ h and each formula ªαA,

(1) M,m/h ² ªαA iff X ⊆ ‖A‖σm for every X ∈ optimalmασ

4 .

4When taking the dstit-based dominant operator primitive, we have the following. If M

is optimal, then for each m, h in M with m ∈ h, and for each formula ⊕αA, M, m/h › ⊕αAiff Hm * ‖A‖σ

m and X ⊆ ‖A‖σm for each X ∈ optimalmασ .

292

Page 302: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 YUKO MURAKAMI

Proof. Suppose first that M,m/h ² ªαA. Then, if X * ‖A‖σm for any X ∈

choicemασ , we know by the truth-definition of ª that there is an Y ∈ choicem

ασ

such that X ≺ Y , and hence X /∈ optimalmασ . It follows that X ⊆ ‖A‖σm

for every X ∈ optimalmασ . Suppose next that M,m/h 2 ªαA. Then bythe truth-definition of ª, there is an X ∈ choicem

ασ such that X * ‖A‖σm.

Suppose for reductio that

(2) X ′ ⊆ ‖A‖σm for every X ′ ∈ optimalmασ .

Then, since M is optimal, there is an Y ∈ optimalmασ such that X ≺ Y ,and then Y ⊆ ‖A‖σ

m. Applying the truth-definition of ª again, there is aZ ∈ choicem

ασ such that

(3) Z * ‖A‖σm

and Y ¹ Z. If Z /∈ optimalmασ , there is an X0 ∈ Choicemασ such that Z ≺

X0, and then, since Y ¹ Z, Y ≺ X0 by Proposition 2.1(viii), and henceY /∈ optimalmασ , a contradiction. It follows that Z ∈ optimalmασ , and henceZ ⊆ ‖A‖σ

m by (2), contrary to (3). From this reductio we conclude thatX ′ * ‖A‖σ

m for some X ′ ∈ optimalmασ . ¤

2.6. Axiomatic systems. The logic L0 takes as axioms all substitutions oftruth-functional tautologies and all formulas of the following forms, wherein A8, A(α/β) is any formula obtained from A by replacing one or moreoccurrences of β with occurrences of α, and AIA is a scheme of schemeswhich holds for all n > 0:

A1 ¤(A → B) → (¤A → ¤B), ¤A → A, 3A → ¤3AA2 4α(A → B) → (4αA →4αB),4αA → A, ∼4αA →4α∼4αAA3 ªα(A → B) → (ªαA → ªαB)A4 ¤A →4αA ∧ ªαAA5 ªαA → 34αAA6 ¤ªαA ∨¤∼ªαAA7 ¤(4αA →4αB) → (ªαA → ªαB)A8 α = α, α = β → (A → A(α/β))

AIA diff (β0, . . . , βn) ∧ (∧

06k6n 34βkBk) → 3(∧

06k6n4βkBk)and takes as rules of inference modus ponens and necessitation, i.e.

RN from A to infer ¤A.5

It has been shown in Horty [5] that for each agent term α, ªα is a normalmodal operator. It is easy to show that A4–A8 are all valid in all utilitarianstit frames. Xu [11] shows that each instance of AIA is valid for all stitframes.

By ordinary modal logic we know that the following are derivable in eachlogic mentioned above:

DR1 A → B / ¤A → ¤B, 4αA →4αB, ªαA → ªαBDR2 A / 4αA, ªαA

T1 α = β → ¤(α = β)T2 α 6= β → ¤(α 6= β)T3 ªαA ↔ ªα4αA (A2, RN, A7, DR1)T4 ªαA → ∼ªα∼A (A3, A5, A2, DR1)

5When taking [ ] primitive, we should replace A4 by, e.g., [α]A ∨ ⊕αA → ∼˜A.

293

Page 303: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 7

3. Canonical Modal Frames

3.1. Modal utilitarian stit frames. The semantics presented in Section2.2 is based on BT + AC structures in Belnap et al. [1]. Since the formallanguage in this paper contains no operators whose interpretation involvestemporal reference (such as tense operators or achievement stit operators),and thus from a technical point of view, the temporal relation in utilitarianstit frames can be eliminated when stit formulas and ought formulas are inquestion.

For each utilitarian stit model M = 〈T, <,Agent, choice, value, τ〉 and anarbitrary moment m ∈ T , let M(m/h) = 〈Hm,Agent, choicem, valuem, τm〉,where choicem is the restriction of choice on m, valuem the restriction ofvalue on m, and τm is the function which assigns {h ∈ Hm : m/h ∈ τ(p)}to each propositional variable p. Counterpart notions of states, preference,and dominance on M(m/h) are to be defined by taking a history instead ofthe corresponding moment-history pair. Then, by induction:

Observation 3.1. For each formula A of L and for each moment-historypair m/h, M,m/h |= A iff M(m/h), h |= A.

Let us call F = 〈W,Agent, {Ra}a∈Agent, v, V 〉 a modal utilitarian stitframe, where 〈W,Agent, {Ra}a∈Agent, v〉 be an arbitrary multi-S5 frame withAgent being the (non-empty) index set for modalities, and V be a functionfrom W to real numbers (a value assignment). A modal utilitarian stit modelis 〈F, τ〉 where τ is a truth-value assignment in the usual sense.

For an arbitrary modal utilitarian stit model M = 〈W,Agent, R, v, τ〉 anequivalent utilitarian stit model can be easily constructed. That is, withr 6∈ W , Mr = 〈W ∪ {r}, Agent, choicer, vr, τr〉 is a utilitarian stit model,where choicer is a function which assigns to each member a of Agent thepartition generated from Ra.

Notions such as truth and validity on modal utilitarian stit frames andmodels are defined in the same way to the branching-time (BT) version.

Thus, the construction in the next subsection aims a modal utilitarianstit model to satisfy a consistent formula.

3.2. Construction of a modal frame. In the rest of this paper, when weuse L without subscript, we presuppose that all axioms of L0 are theoremsof L, and modus ponens and RN hold in L. In particular, L could be any Ln

with n > 0.Let Φ be any L-MCS. We use WΦ to denote the set of all L-MCSs w such

that {¤A | ¤A ∈ Φ} ⊆ w. It is easy to verify, by ordinary modal logic(applying A1, A8 and A4), that the following holds:

Remark 3.2. Let Φ be any L-MCS. Then

(i) for each w ∈ WΦ, {A | ¤A ∈ w} = {¤A | ¤A ∈ w} = {¤A | ¤A ∈Φ};

(ii) for each w ∈ WΦ, ¤A ∈ w iff A ∈ w′ for every w′ ∈ WΦ;(iii) for each α and β, α = β ∈ Φ iff α = β ∈ w for every w ∈ WΦ;(iv) for each w ∈ WΦ and each α, {A | ¤A ∈ w} ⊆ {A | 4αA ∈ w}.

294

Page 304: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

8 YUKO MURAKAMI

For each set Ξ of formulas, we use Sub(Ξ) for the closure of Ξ undersubformulas. A set Ξ of formulas is suitable if the following conditions hold:

(i) Ξ = Sub(Ξ)(ii) α and β occur in Ξ only if α = β ∈ Ξ,(iii) ªβB ∈ Ξ only if 4βB ∈ Ξ.6

Note that each set of formulas has a smallest suitable extension. For eachsuitable set Ξ of formulas, we fix Ξ+ for Ξ ∪ {∼A | A ∈ Ξ}.

Let Φ be any L-MCS, and let Ξ be any non-empty set of formulas. Ξis coherent with Φ if for each A, and for each α and β with α = β ∈ Φ,4αA ∈ Ξ iff 4βA ∈ Ξ, and ªαA ∈ Ξ iff ªβA ∈ Ξ.

Let Φ be any L-MCS, and let Ξ be any suitable set coherent with Φ. Wedefine the “regular modal frame for L with respect to Φ and Ξ” as follows.

First, when Ξ contains no modal formula, consider the model 〈{Φ}, {a}, {=}, vΦ, VΦ〉, where vΦ(p) = {Φ} if p ∈ Φ and vΦ = ∅ otherwise, and VΦ(Φ) = 1.The argument in Section 4 will work for such a case.

Now, the following construction in this section assumes that Ξ containsmodal formulas. Let W = {w∩Ξ+ | w ∈ WΦ}. We will use x, y etc. to rangeover members of W.

For each α and β occurring in Ξ, α ∼=Φ,Ξ β iff α = β ∈ Φ∩Ξ+. Let us use〈α〉Φ,Ξ for the ∼=Φ,Ξ-equivalence class to which α belongs, and let Agent bethe set of all ∼=Φ,Ξ-equivalence classes of terms for agents. We will use a,betc. to range over members of Agent.

For each a ∈ Agent, let Ra be the relation on W such that for eachx, y ∈ W, Raxy iff {4αA ∈ x | α ∈ a} = {4αA ∈ y | α ∈ a}. Obviously,each Ra is an equivalence relation. We use choice to denote the functionon Agent that assigns each a ∈ Agent the set choicea of all Ra-equivalenceclasses, and use [x]a for the Ra-equivalence class to which x belongs, i.e.,[x]a = choicea(x).

Finally, 〈W, Agent, choice〉 is the regular modal frame for L w.r.t. Φ andΞ if W, Agent and choice are as specified above. Let, for each a ∈ Agent,Σa = {4αA | ªαA ∈ Φ ∩ Ξ+ ∧ α ∈ a}, and let Σ =

⋃a∈Agent Σa. It is

easy to see that Σ = {4αA | ªαA ∈ Φ ∩ Ξ+}. The canonical modal framefor L w.r.t. Φ and Ξ is 〈W, Agent, choice, value〉, where 〈W, Agent, choice〉is the regular modal frame for L w.r.t. Φ and Ξ, and value is the functionon W such that for each x ∈ W, value(x) = 1 if Σ ⊆ x, and value(x) = 0otherwise. When Φ and Ξ are given explicit in the context, we will dropthe references to Φ or Ξ as much as we can, and thus use 〈α〉 for 〈α〉Φ,Ξ,and [x]〈α〉 for [x]〈α〉Φ,Ξ

etc. From now on, when we speak of “the regular(canonical) modal frame for L w.r.t. Φ and Ξ”, we presuppose that Φ is anL-MCS and Ξ is a suitable set of formulas coherent with Φ. We say that F isa regular (canonical) modal frame for L if it is a regular (canonical) modalframe for L w.r.t. some Φ and Ξ.

We want to consider both the compactness and the finite model propertyat the same time, and thus, consider the ordinary canonical frames andtheir filtrations at the same time—this way we can avoid going over similar

6When taking [ ] primitive, the last condition should be replaced by (a) ⊕βB ∈ Ξ onlyif 4βB,ªβB ∈ Ξ, and (b) [β]B ∈ Ξ only if 4βB ∈ Ξ.

295

Page 305: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 9

arguments twice. It is easy to see that when 〈W,Agent, choice, value〉 is thecanonical modal frame for L w.r.t. Φ and Ξ, if Ξ is the set of all formulas,W = WΦ, and thus we have a subframe (generated by Φ) of the ordinarycanonical frame; and if Ξ is (“logically”) finite, we have a filtration.

The following remark is an immediate consequence of A8, our definitionof coherence and ordinary modal logic:

Remark 3.3. Let 〈W, Agent, choice, value〉 be the regular modal frame forL w.r.t. Φ and Ξ. Then

(i) for each x ∈ W and each A ∈ Ξ, A ∈ x or ∼A ∈ x;(ii) for each x, y ∈ W, {4αA | 4αA ∈ x} = {4αA | 4αA ∈ y} only if

R〈α〉xy;(iii) for each x ∈ W, {4αA | ªαA ∈ Φ ∩ Ξ+} ⊆ x only if Σ〈α〉 ⊆ x.

The lemma below handles the desired properties of ¤ and 4, on the basisof which we can start dealing with independence of agents and the conditionfor the operator of dominant ought.

Lemma 3.4. Let 〈W,Agent, choice〉 be the regular modal frame for L w.r.t. Φand Ξ, and let x ∈ W. Then

(i) for each ¤A ∈ Ξ, ¤A ∈ x iff A ∈ y for every y ∈ W;(ii) for each α and β occurring in Ξ, α = β ∈ x iff 〈α〉 = 〈β〉;(iii) for each 4αA ∈ Ξ, 4αA ∈ x iff A ∈ y for every y ∈ [x]〈α〉.

7

The following lemma shows that for each L, all regular modal framessatisfy the version of independence of agents.

Lemma 3.5. Let 〈W,Agent, choice〉 be the regular modal frame for L w.r.t. Φand Ξ. Then for each function f on Agent such that f(a) ∈ choicea for everya ∈ Agent,

⋂a∈Agent f(a) 6= ∅.

The theorem below follows immediately from our definition and Lemma 3.5.

Theorem 3.6. Each regular modal frame for L is a stit frame, each canon-ical modal frame for L is a two-valued utilitarian stit frame.

The lemma below gives us a property all regular modal frames have,which will be used to deal with the semantic condition for the dominantought operators.

Lemma 3.7. Let 〈W,Agent, choice〉 be the regular modal frame for L w.r.t. Φand Ξ. Then for each ªαA ∈ Ξ and each x ∈ W, ªαA ∈ x iff 4αA ∈ y forevery y ∈ W such that Σ〈α〉 ⊆ y.

From now on, for each regular modal frame F = 〈W,Agent, choice〉 for L,we use DF for the set {y ∈ W | Σ ⊆ y}. If F = 〈W, Agent, choice, value〉 is acanonical modal frame for L, let DF = DF′ , where F′ = 〈W, Agent, choice〉.

7When taking [ ] primitive, we need one more clause, i.e., that for each [α]A ∈ Ξ,[α]A ∈ x iff A ∈ y for every y ∈ [x]〈α〉 and ∼A ∈ z for some z ∈ W , which followsimmediately from (i) and (iii) above and a theorem [α]A ↔ 4αA ∧ ∼˜A, which in turnfollows from a replacement of A4 mentioned in an earlier note.

296

Page 306: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

10 YUKO MURAKAMI

Note that for each canonical modal frame F for L, DF = {y ∈ W | value(y) =1} by definition.

Lemma 3.8. Let 〈W, Agent, choice〉 be the regular modal frame for L w.r.t.Φ and Ξ. Then for each ªαA ∈ Ξ and each x ∈ W, ªαA ∈ x iff 4αA ∈ yfor every y ∈ DF.

Applying Lemma 3.8, Lemma 3.7 and T4, one can easily verify the fol-lowing.

Corollary 3.9. Let 〈W,Agent, choice〉 be the regular modal frame for Lw.r.t. Φ and Ξ. Then DF 6= ∅.

Because the canonical modal frame F = 〈W, Agent, choice, value〉 for Lw.r.t. Φ and Ξ is a utilitarian stit frame, optimala is defined for every a ∈Agent (recall that optimala = {K ∈ choicea | ∼∃K ′(K ′ ∈ choicea ∧ K ≺K ′)}). Our next goal is to show that each canonical modal frame is optimal,but first we need to show the following key lemma.

Lemma 3.10. Let F = 〈W,Agent, choice, value〉 be the canonical modalframe for L w.r.t. Φ and Ξ. Then for each x ∈ W and each a ∈ Agent,Σa ⊆ x iff [x]a ∈ optimala.

Proof. Let x ∈ W and a ∈ Agent. First suppose that Σa ⊆ x. We showbelow that [x]a ∈ optimala, i.e., [x]a ≺ K for no K ∈ choicea. To that end,we observe that by definitions of Σa and Ra,

(4) Σa ⊆ y for every y ∈ [x]a.

Consider any K ∈ choicea. We first show that for each S ∈ statea,

(5) S ∩K ≤ S ∩ [x]a.

By definition of value, it is sufficient to show that

if value(y) = 1 for some y ∈ S ∩K, value(z) = 1 for all z ∈ S ∩ [x]a.

Let S ∈ statea, K ∈ choicea and x0 ∈ S∩K with value(x0) = 1. We know bydefinition of statea that S =

⋂b∈Agent−{a}[x0]b. For each b ∈ Agent − {a},

since value(x0) = 1, Σb ⊆ Σ ⊆ x0, and then by definitions of Σb and Rb,Σb ⊆ y for every y ∈ [x0]b. It follows from (4) that Σ ⊆ z for everyz ∈ [x]a∩ (

⋂b∈Agent−{a}[x0]b), and hence value(z) = 1 for every z ∈ S∩ [x]a.

It follows that (5) holds. Now if [x]a ≺ K, K � [x]a by definition, i.e.,there is an S′ ∈ statea such that S′ ∩K � S′ ∩ [x]a, contrary to (5). Hence[x]a ⊀ K. It follows that [x]a ∈ optimala.

Next suppose that Σa * x. Then there are α ∈ a and 4αA ∈ Σa suchthat 4αA /∈ x. By definition of Ra, 4αA /∈ y for each y ∈ [x]a, and thusΣa * y for each y ∈ [x]a, and hence

(6) [x]a ∩ DF = ∅.

There is, by Corollary 3.9, a z ∈ DF. Since Σ ⊆ z, an argument similar tothe one given above will show that for each b ∈ Agent, Σb ⊆ y for everyy ∈ [z]b. Letting Sz =

⋂b∈Agent−{a}[z]b, we know that Sz ∈ statea and

z ∈ Sz ∩ [z]a. Since value(z) = 1 and Sz ∩ [x]a 6= ∅ (by Theorem 3.6),Sz ∩ [z]a � Sz ∩ [x]a by (6), and hence [z]a � [x]a. It follows, also from

297

Page 307: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 11

(6), that for each S ∈ statea, S ∩ [x]a ≤ S ∩ [z]a, i.e., [x]a ¹ [z]a. Hence[x]a ≺ [z]a, and therefore [x]a /∈ optimala. ¤

Theorem 3.11. Each canonical modal frame for L is optimal.

Proof. Let a ∈ Agent, and let K ∈ choicea − optimala. We show as followsthat there is a K ′ ∈ optimala such that K ≺ K ′, which is sufficient. Letx ∈ K. We know that K = [x]a and, by Lemma 3.10, that Σa * x. Anargument similar to that in Lemma 3.10 shows that there is a z ∈ DF suchthat [x]a ≺ [z]a. Since Σa ⊆ Σ ⊆ z, [z]a ∈ optimala by Lemma 3.10. ¤

For each n > 0, we know that the characteristic axiom of Ln, i.e., APCn,corresponds to the semantic condition that the stit frames are at-most-n-ary. The following lemma shows that all regular (canonical) modal framesfor Ln are at-most-n-ary.

Lemma 3.12. Let n > 0, and let 〈W, Agent, choice〉 be the regular modalframe for Ln w.r.t. Φ and Ξ. Then for each a ∈ Agent, |choicea| 6 n.

Proof. Let a ∈ Agent. Suppose for reductio that |choicea| > n. Thenthere are U0, . . . , Un ∈ choicea such that U0, . . . , Un are all different. Selectα ∈ a, select x0 ∈ U0, . . . , xn ∈ Un, and select w0, . . . , wn ∈ WΦ such thatxk = wk ∩ Ξ+ for every k with 0 6 k 6 n. Consider any i and k suchthat 1 6 k 6 n, 0 6 i 6 n and i 6= k. By definition of Ra, there isan A such that either ∼4αA ∈ xk and 4αA ∈ xi, or 4αA ∈ xk and∼4αA ∈ xi; and then by A2, either 4α∼4αA ∈ wk and ∼4α∼4αA ∈wi, or 4αA ∈ wk and ∼4αA ∈ wi; and hence there is an A′ such that4αA′ ∈ wk and ∼4αA′ ∈ wi. It follows that for each k with 1 6 k 6 n,there are 4αAk,0, . . . ,4αAk,n ∈ wk such that ∼4αAk,i ∈ wi for every iwith 0 6 i 6 n and i 6= k. For each k with 1 6 k 6 n, letting Bk =4αAk,0 ∧ . . . ∧4αAk,n, it is easy to see by ordinary modal logic (applyingA2, RN and A4) that 4αBk ∈ wk and ∼Bk ∈ wi for every i with 0 6i 6 n and i 6= k. It follows that 4αB1 ∈ w1, ∼B1 ∧ 4αB2 ∈ w2, . . . ,∼B1∧. . .∧∼Bn−1∧4αBn ∈ wn, and then by Remark 3.2(ii), 34αB1 ∈ w0,3(∼B1 ∧4αB2) ∈ w0, . . . ,3(∼B1 ∧ . . .∧∼Bn−1 ∧4αBn) ∈ w0, and henceby APCn, B1∨. . .∨Bn ∈ w0. But, since ∼Bi ∈ w0 for every i with 0 6 i 6 n,∼(B1 ∨ . . . ∨ Bn) ∈ w0, contrary to the assumption of L-consistency on w0.We conclude from this reductio that |choicea| 6 n. ¤

4. Main Results

Let F = 〈W, Agent, choice, value〉 be the canonical frame for L w.r.t. Φand Ξ. M = 〈F, τ〉 is the canonical model for L w.r.t. Φ and Ξ if τ is thevaluation on F such that for each agent term α, ατ = 〈α〉, and for eachatomic formula p ∈ Ξ, τ(p) = {x ∈ W | p ∈ x}.Theorem 4.1. Let M = 〈F, τ〉 be the canonical model for L with respect toΦ and Ξ, where F = 〈W, Agent, choice, value〉. Then for each A ∈ Ξ and foreach x ∈ W, M, x ² A iff A ∈ x.

Proof. By a routine induction we can show that for each A ∈ Ξ and x ∈ W,

(7) M, x ² A iff A ∈ x.

298

Page 308: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

12 YUKO MURAKAMI

It is straightforward that (7) holds when A is an atomic formula, or ∼B orB ∧ C. We know by Lemma 3.4(i)–(iii). that (7) holds when A is α = β,¤B or 4αB. Now let A be ªαB ∈ Ξ. Consider any x ∈ W. By Lemma 3.7,ªαB ∈ x iff 4αB ∈ y for every y ∈ W such that Σ〈α〉 ⊆ y, and then byLemma 3.10,

(8) ªαB ∈ x iff 4αB ∈ y for every y ∈ W such that [y]〈α〉 ∈ optimal〈α〉.

Consider any y ∈ W such that [y]〈α〉 ∈ optimal〈α〉. We know by Lemma 3.4(iii)that 4αB ∈ y iff B ∈ z for every z ∈ [y]〈α〉, and then by induction hy-pothesis, 4αB ∈ y iff [y]〈α〉 ⊆ ‖B‖τ . It follows from (8) that ªαB ∈ x

iff [y]〈α〉 ⊆ ‖B‖τ for every y ∈ W such that [y]〈α〉 ∈ optimal〈α〉. Sinceeach K ∈ optimal〈α〉 is [y]〈α〉 for some y ∈ W, we know that ªαB ∈ x

iff K ⊆ ‖B‖τ for every K ∈ optimal〈α〉. Hence by Theorem 3.11 and 2.5,ªαB ∈ x iff M, x ² ªαB. ¤

Theorem 4.2. L0 is complete and compact.

Proof. For each L0-consistent set Ψ of formulas, let Φ be any L0-MCS includ-ing Ψ, let Ξ be the set of all formulas, and let M = 〈W,Agent, choice, value, τ〉be the canonical model for L0 with respect to Φ and Ξ. It is easy to see thatΦ ∈ W. Then by Theorem 4.1, M, Φ ² Ψ. ¤

By Theorem 3.11, each canonical modal frame for L0 is an optimal utili-tarian stit frame, and hence the only formulas valid in all optimal utilitarianstit frames are theorems of L0. It follows that the theorem below holds, whichshows a certain limitation of the language.

Theorem 4.3. There is no set Θ of formulas such that for each utilitarianstit frame F, F is optimal iff all members of Θ are valid in F; and if L isthe set of all formulas valid in all optimal utilitarian stit frames, L = L0

(identifying L0 with the set of all its theorems).

Similarly, because canonical modal frames for L0 are all two-valued, thefollowing holds.

Theorem 4.4. There is no set Θ of formulas such that for each utilitarianstit frame F, F is two-valued iff all members of Θ are valid in F; and if L isthe set of all formulas valid in all two-valued utilitarian stit frames, L = L0

(identifying L0 with the set of all its theorems).

The theorem below follows from Theorem 4.1 and Lemma 3.12, whoseproof is similar to that of Theorem 4.2, with an application of Lemma 3.12.

Theorem 4.5. Let n > 0. Then the following hold:(i) Ln is complete; and(ii) for each set Θ of formulas, if every finite subset of Θ has an at-

most-n-ary utilitarian stit model, so does Θ.

In the following, we prove the decidability of all Ln with n > 0 by way offinite model property. For each L-consistent formula A, let Φ be an L-MCScontaining A, and let

Π0 = {α = β | α and β occur in A},

299

Page 309: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 13

Π1 = {4βB,ªβB | ªαB ∈ Sub({A}), α = β ∈ Φ, β occurs in A},Π2 = {4βB | 4αB ∈ Sub({A}), α = β ∈ Φ, β occurs in A}, andΞΦ

A = Sub({A}) ∪Π0 ∪Π1 ∪Π2.

It is easy to verify that for each A, ΞΦA is finite, suitable, and coherent with

Φ.

Theorem 4.6. For each n > 0, Ln has the finite model property, and henceis decidable.

Proof. Let n > 0 and let A be any Ln-consistent formula. Select an Ln-MCS Φ containing A, let Ξ = ΞΦ

A, and let M = 〈W,Agent, choice, value, τ〉be the canonical model for Ln w.r.t. Φ and Ξ. Since Ξ is finite, so is W.Because A ∈ Φ ∩ Ξ+ ∈ W, we know by Theorem 4.1 that M, x ² A, wherex = Φ ∩ Ξ+. When n > 0, we apply Lemma 3.12 to conclude that M isat-most-n-ary. ¤

5. Implications of the main results

Let us call the notion of ought defined by (2.4) in terms of BT utilitar-ian stit models dominant ought, and call the notion of ought defined by(2.5) in terms of optimal BT utilitarian stit models optimal ought, and callthe notion of ought defined by (2.5) in terms of two-valued BT utilitarianstit models two-valued dominant ought. By Theorem 4.2, Theorem 4.3 andTheorem 4.4, L0 characterizes the dominant ought, the optimal ought andthe two-valued dominant ought. Here we consider some other notions ofought and present a simple result of the discussion.

Let F = 〈T, <,Agent, choice, value〉 be any BT utilitarian stit frame, andlet M = 〈F, σ〉 be any model on F. The following semantic interpretation isdiscussed in Horty [5], where m ∈ h in F:8

(9)M,m/h ² ªαA iff for some h′ ∈ H(m), choicem

ασ(h′) ⊆ ‖A‖σm, and

choicemασ(h′′) ⊆ ‖A‖σ

m for every h′′ ∈ H(m) suchthat Value(h′) 6 Value(h′′).

Let us call the notion of ought defined by (9) in terms of BT utilitarianstit models plain utilitarian ought. It is easy to verify that all axioms of L0

are valid when taking ªαA to be the plain utilitarian ought. In order toshow that L0 is complete, with ªαA to be the plain utilitarian ought, weapply Lemma 3.8 in a routine induction, as we did in Theorem 4.1. HenceL0 characterizes not only the dominant ought and the optimal ought, butalso the plain utilitarian ought.

F is a standard deontic stit frame if F = 〈T, <, Agent, choice, ought〉, whereT, <, Agent and choice are as specified in a BT stit frame, and ought isa function on T such that for each m ∈ T, ought(m) ⊆ Hm. A standarddeontic stit model is an M = 〈F, σ〉, where F is a standard deontic stit frame,and σ is just like a valuation on a BT utilitarian stit frame. Let M be astandard deontic stit model with m,h in M such that m ∈ h. The following

8This formulation is actually derived from a general interpretation of an ought-to-beoperator combined with the interpretation of a doing operator, i.e., cstit.

300

Page 310: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

14 YUKO MURAKAMI

is a semantic interpretation of ªαA discussed in both Horty-Belnap [6] andHorty [5]:9

(10) M,m/h ² ªαA iff choicemασ(h′) ⊆ ‖A‖σ

m for every h′ ∈ ought(m).

Let us call the notion of ought defined by (10) in terms of standard deonticstit models standard ought. This notion is the result of building the notionof ought in traditional deontic logic into the context of branching time andstit. It is easy to see that each axiom of L0 is valid when ªαA is takento be the standard ought. It is also easy to see that from each two-valuedBT utilitarian stit frame M = 〈T, <, Agent, choice, value, σ〉, we can easilyget a standard deontic stit model sd(M) = 〈T, <, Agent, choice, ought, σ〉,where for each m ∈ T, ought(m) = {h ∈ Hm | value(h) = 1}. It is easy toverify that for each A, and for each m and h in a two-valued BT utilitarianstit model M with m ∈ h, M,m/h ² A iff sd(M),m/h ² A. It follows,from the fact that L0 characterizes the two-valued dominant ought, that L0

characterizes the standard ought.Thomason [9] proposed the idea that each different agent should have his

own “ought set”. Let us see what happens in our current context if we applythat idea. F is an indexed deontic stit frame if F = 〈T, <,Agent, choice, ought〉,where T, <, Agent and Choice are as specified in a BT utilitarian stitframe, and ought is a function on T × Agent such that for each m ∈ Tand each a ∈ Agent, ought(m, a) ⊆ Hm. M is an indexed deontic stit modelif M = 〈F, σ〉 where F is an indexed deontic stit frame and σ is a valuationon F, which is just like a valuation on a BT utilitarian stit frame. Let Mbe an indexed deontic stit model with m,h in M such that m ∈ h. Thefollowing semantic interpretation, discussed in Horty-Belnap [6], seems toresemble Thomason’s idea:10

(11)M,m/h ² ªαA iff choicem

ασ(h′) ⊆ ‖A‖σm for every h′ ∈ ought(m, a).

Let us call the notion of ought defined by (11) in terms of indexed deontic stitmodels individual ought. It is routine to verify that each axiom of L0 is validwhen ªαA is taken to be the individual ought. Let 〈T, <, Agent, choice〉be any regular modal frame for L0 w.r.t. Φ and Ξ. We define ought tobe the function on Agent such that for each a ∈ Agent, ought(a) = {x ∈Moment-History | Σa ⊆ x}. Let F = 〈T, <,Agent, choice, ought〉 and letM = 〈F, τ〉, where τ is as defined in the canonical model for L0 w.r.t. Φand Ξ. Applying Lemma 3.7 and a routine induction, we can show that foreach A ∈ Ξ and each x ∈ Moment-History, M, x ² A iff A ∈ x. Then wecan convert M to an indexed deontic stit model the same way as we convertutilitarian stit models to BT utilitarian stit models, and finally we arrive atthe completeness of L0 with ªαA to be interpreted as the individual ought.Hence L0 also characterizes the individual ought.

References

[1] Nuel Belnap, Michael Perloff, and Ming Xu. Facing the Future: Agents and Choicesin Our Indeterminist World. Oxford Univerity Press, Oxford University Press, 2001.

9In Horty-Belnap [6], the corresponding formulation uses dstit instead of cstit.10Here we replace the dstit notion in Horty-Belnap [6] with cstit.

301

Page 311: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

UTILITARIAN DEONTIC LOGIC 15

[2] Dov Gabbay and F. Guenthner, editors. Handbook of Philosophical Logic, volume 2.D. Reidel Publishing Company, Dordrecht, 1984.

[3] Dov M. Gabbay and Hans J. Ohlbach, editors. Temporal Logic, First InternationalConference, ICTL’94, Bonn, Germany, Proceedings, volume 827. Springer-Verlag,1994.

[4] Risto Hilpinen, editor. New Studies in Deontic Logic. D. Reidel Publishing Company,Dordrecht, 1981.

[5] John F. Horty. Agency and Deontic Logic. Oxford University Press, Oxford, 2000.[6] John F. Horty and Nuel Belnap. The deliberative stit : a study of action, omission,

ability, and obligation. Journal of Philosophical Logic, 24:583–644, 1995.[7] Arthur Prior. Past, Present and Future. Oxford Univerity Press, Oxford, 1967.[8] Richmond H. Thomason. Indeterminist time and truth-value gaps. Theoria, 36:264–

281, 1970.[9] Richmond H. Thomason. Deontic logic and the role of freedom in moral deliberation.

In Hilpinen [4], pages 177–186.[10] Richmond H. Thomason. Combination of tense and modality. In Gabbay and Guen-

thner [2], pages 135–165.[11] Ming Xu. Decidability of deliberative stit theories with multiple agents. In Gabbay

and Ohlbach [3], pages 332–348.

302

Page 312: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING

C. NALON, C. DIXON, AND M. FISHER

ABSTRACT. We present a clausal resolution-based method for temporal logics of knowledge withsynchrony and no learning. This and related logics admit axioms which include operators fromboth the temporal and epistemic logics, which enable the description of how knowledge evolvesover time. Instead of proposing new resolution rules, further information is added to the set ofclauses in order to deal with this particular interaction. We give outline of the proofs of soundness,termination, and completeness.

1. INTRODUCTION

Logics have been used in Computer Science for many years as a natural way for describingproperties of complex systems. More recently, there has been an increasing interest in combinedmodal logics, as different logical languages are more suitable to specify different properties withina system. Typical examples are the specification and verification of distributed systems [5] and thespecification of agent-based systems [11, 12]. Given such logical characterisation of a system, itis then desirable to have the appropriate tools in order to verify whether a particular property holdsfor this system. By verifying that a property holds, we mean to prove that the property is a logicalconsequence of the specification.

There is a wide range of logics that could be chosen in order to model and characterise suchcomplex systems. Moreover, there is a variety of ways of combining the chosen logics. In thefollowing, we concentrate on a particular combination that has been proved useful in modellingdistributed and agent-based systems, namely, we are looking at Propositional Temporal Logics ofKnowledge (KL(n), for short). In such logics, the dynamic component is described by a proposi-tional linear temporal logic and the informational component is described by a propositional logicof knowledge. When this combination is given by the union of the axiomatic systems of both log-ics, where the underlying logics are then said to be independent, proof methods can be relativelyeasily obtained by taking the union of proof methods for the logics considered alone and makingsure that enough information is passed to each component.

Proof methods for combined logics cannot be obtained in a straightforward way, however, whenthe logics interact, that is, when, besides the characterisation of both logics (given by their ax-iomatic systems), further axioms, including operators of both logics, are needed in order to modela specific situation. Contexts where we are particularly interested in how the knowledge of anagent evolves over time is a typical example where interaction axioms are required. Interactionsoften increase the complexity of the validity problem for the logical language and proof methodsfor such logics are, to our knowledge, rare.

In this paper, we introduce a proof method for a particular interaction between time and knowl-edge: synchrony and no learning. This property was firstly discussed in the context of blind-fold games [9]. Recently, a similar characterisation has proved useful in the description of non-decreasing domains [8]. In such systems, once two situations are indistinguishable to an agent, theagent will never acquire any knowledge that would allow her to distinguish between such situa-tions. Although the complexity of the interacting logic is high (non-elementary, for the multi-agentcase), the axiom that expresses this property has a simple form, which allowed us to investigate indetail the requirements for its proof method.

Key words and phrases. Proof theory, interacting modal logics, temporal and epistemic reasoning.

1

303

Page 313: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 C. NALON, C. DIXON, AND M. FISHER

The structure of the paper is as follows. In Section 2, we review the syntax, semantics andthe resolution method for the combined logic of time and knowledge. In Section 3, we present aresolution method for synchronous systems with no learning. The resolution method introducesadditional information into the set of clauses, instead of introducing new (possibly complicated)resolution inference rules. The multi-agent case is non-trivial, therefore we discuss the issuesrelated to the granularity of the information that needs to be provided in order to obtain complete-ness for these systems. We discuss the correctness results of the method for the multi-agent casein Section 4. In Section 5, we discuss our results and other directions for future research.

2. TEMPORAL LOGICS OF KNOWLEDGE

The syntax of KL(n) comprises a set of modal operators and a set of temporal operators. For-mulae are constructed from a denumerable set, P = {p, q, p′, q′, . . .}, of propositional symbols;nullary connectives, true and false; propositional connectives, ¬, ∧, ∨, ⇒, and ⇔; temporal con-nectives, ♦, , �, U , and W ; and a set of unary modal operators Ki , for all i ∈ A, whereA = {1, . . . , n} is the set of agents.

Definition 1. The set of well-formed formulae WFF is defined recursively as follows:

• the nullary connectives and propositional symbols are in WFF;• if φ and ϕ are in WFF , then so are ¬ϕ, (ϕ ∧ φ), (ϕ ∨ φ), (ϕ ⇒ φ), (ϕ ⇔ φ), ♦ϕ, ϕ, �ϕ,(ϕU φ), (ϕW φ) and Ki ϕ, ∀i ∈ A.

A literal is either a proposition or its negation; a modal literal is either Ki l or ¬Ki l, where l is aliteral and i ∈ A; and an eventuality is in the form ♦l, where l is a literal.

The semantics of KL(n) interprets formulae over a set of temporal lines, each of which corre-sponds to a discrete, linear model of time with finite past and infinite future, together with theagents’ accessibility relations Ki. We define a timeline t as an infinitely long, linear, discrete se-quence of states, indexed by the natural numbers. Let TLines be the set of all timelines. A pointq is a pair q = (t, u), where t ∈ TLines and u ∈ N is a temporal index to t. Let Points be theset of all points. A model is a structure M = 〈TL,K1, . . .Kn, π〉 where TL ⊆ TLines is a setof timelines with a distinguished timeline t0; Ki, for all i ∈ A, is the accessibility relation overpoints, i.e., Ki ⊆ Points× Points, where each Ki is an equivalence relation; and π is a functionπ : Points× P → {true, false}.

Definition 2. Truth of a formula is given as follows:

• 〈M, (t, u)〉 |= true• 〈M, (t, u)〉 |= p iff π(t, u)(p) = true, where p ∈ P• 〈M, (t, u)〉 |= ¬ϕ iff 〈M, (t, u)〉 �|= ϕ• 〈M, (t, u)〉 |= (ϕ ∧ φ) iff 〈M, (t, u)〉 |= ϕ and 〈M, (t, u)〉 |= φ• 〈M, (t, u)〉 |= �ϕ iff 〈M, (t, u + 1)〉 |= ϕ• 〈M, (t, u)〉 |= ϕUφ iff ∃k∈N, k≥ u, 〈M, (t, k)〉 |= φ and ∀j∈ N, u≤ j< k, 〈M, (t, j)〉 |= ϕ• 〈M, (t, u)〉 |= Ki ϕ iff ∀t′, u′, such that ((t, u), (t′, u′)) ∈ Ki, 〈M, (t′, u′)〉 |= ϕ.

The semantics of the other connectives are given by false ≡ ¬true, (ϕ ∨ ψ) ≡ ¬(¬ϕ ∧ ¬ψ),(ϕ ⇒ ψ) ≡ (¬ϕ ∨ ψ), (ϕ ⇔ ψ) ≡ ((ϕ ⇒ ψ) ∧ (ψ ⇒ ϕ)), ♦ϕ ≡ (trueUϕ), ϕ ≡ ¬♦¬ϕ,and (ϕW ψ) ≡ ( ¬ϕ ∨ ϕUψ). We write (t, u) ∼i (t′, u′), if ((t, u), (t′, u′)) ∈ Ki. A formulaϕ is said to be satisfiable if there is a model M such that 〈M, (t0, 0)〉 |= ϕ; a formula ϕ is valid if〈M, (t0, 0)〉 |= ϕ, for every model M .

The resolution-based proof method for KL(n) in [3] combines the inference rules for temporaland multi-modal knowledge logics when considered alone. A formula is first translated into anormal form, called Separated Normal Form for Logics of Knowledge (SNFK ). A nullary con-nective, start, which intuitively represents the beginning of time, is introduced. Formally, we have〈M, (t, u)〉 |= start if, and only if, t = t0 and u = 0, where M is a model and (t, u) is a point.Formulae are represented by a conjunction of clauses, which are true in all states, that is, they have

304

Page 314: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 3

the general form ∗ ∧iAi, where the universal operator is defined as ∗ϕ⇔ ±(ϕ∧C ∗ϕ)

(with ±ϕ ⇔ ϕ ∧ −ϕ and 〈M, (t, u)〉 |= −ϕ if, and only if, ∀k, k ∈ N, if 0 ≤ k ≤ u,then 〈M, (t, k)〉 |= ϕ), C is the common knowledge operator (i.e. Cϕ ⇔ E(ϕ ∧ Cϕ), whereEϕ ⇔ ∧

i∈A Ki ϕ), and Ai is a clause which is in one of the following forms, where l, li, ki areliterals, mij are literals or modal literals in the form Ki l or ¬Ki l:

Initial clause: start ⇒r∨lb

b=1

Sometime clause:g∧ka

a=1

⇒ ♦ l

Step clause:g∧ka

a=1

⇒r

�∨lb

b=1

Ki -clause: true ⇒r∨mib

b=1

Literal clause: true ⇒r∨lb

b=1

Transformation into the SNFK, whose satisfiability preserving transformation rules are given in[4] and [3], depends on three main operations: the renaming of complex subformulae by introduc-ing new variables whose truth value is linked to the formula they replace in all states; the removalof temporal operators; and classical style rewrite operations. Once a formula has been transformedinto SNFK , the resolution method can be applied. The method consists of two main procedures:the first performs initial, modal and step resolution; the second performs temporal resolution. Eachprocedure is performed until a contradiction (either true ⇒ false or start ⇒ false) is generated orno new clauses can be generated. In the following l, li are literals; mi are literals or modal literals;D, D′ are disjunctions of literals; M , M′ are disjunction of literals or modal literals; and C , C′are conjunctions of literals.

Initial Resolution is applied to clauses that hold at the beginning of time:

[IRES1] true ⇒ (D ∨ l)start ⇒ (D′ ∨ ¬l)start ⇒ (D ∨D′)

[IRES2] start ⇒ (D ∨ l)start ⇒ (D′ ∨ ¬l)start ⇒ (D ∨D′)

Modal Resolution is applied between clauses referring to the same agent (i.e. two Ki -clauses; aliteral and a Ki -clause; or two literal clauses):

[MRES1] true ⇒ (M ∨mi)true ⇒ (M ′ ∨ ¬mi)true ⇒ (M ∨M ′)

[MRES2] true ⇒ (M ∨ Ki l)true ⇒ (M ′ ∨ Ki ¬l)true ⇒ (M ∨M ′)

[MRES3] true ⇒ (M ∨ Ki l)true ⇒ (M ′ ∨ ¬l)true ⇒ (M ∨M ′)

[MRES5] true ⇒ (D ∨ Ki l1 ∨ Ki l2 ∨ . . .)true ⇒ (D ∨ l1 ∨ l2 ∨ . . .)

[MRES4] true ⇒ (M ∨ ¬Ki l)true ⇒ (M ′ ∨ l)true ⇒ (M ∨modi(M ′))

where modi(A ∨B) = modi(A) ∨modi(B)modi(Ki l) = Ki l

modi(¬Ki l) = ¬Ki lmodi(l) = ¬Ki¬ l

The first inference rule corresponds to classical resolution. MRES2 is justified by the axiom D,i.e. � Ki ϕ ⇒ ¬Ki¬ϕ, for any formula ϕ. The rules MRES3 and MRES5 by the axiom T,i.e. � Ki ϕ ⇒ ϕ, for any formula ϕ. The rule MRES4 is justified by the external commonknowledge operator surrounding each clause. Recall that each clause in SNFK is of the form

305

Page 315: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 C. NALON, C. DIXON, AND M. FISHER

∗ ∧iAi. By modal reasoning, this implies Ki

∧iAi, for each agent i ∈ A. A literal or Ki -

clause as true ⇒ (D′ ∨ l) is, then, an abbreviated form of Ki (true ⇒ (D′ ∨ l)). By propositionalreasoning, this formula can be rewritten as (Ki (¬D′ ⇒ l)). From the axiom K, the knowledgeoperator can be distributed over the implication, resulting in (Ki ¬D′ ⇒ Ki l). Rewriting thisformula back into the normal form results in true ⇒ ¬Ki¬D′ ∨ Ki l. The resolution inferencerule is, then, applied between the clauses containing the complementary modal literals ¬Ki l (fromthe first premise) and Ki l (from the transformation of the second premise). The function modigenerates the clausal form of ¬Ki¬D′. This function makes use of the axioms K (for distributingthe knowledge operator over D′), 4, and 5 (for modal simplification, that is, replacing formulae as¬Ki¬Ki l and ¬Ki Ki l, where l is a literal, by Ki l and ¬Ki l, respectively).

Step Resolution is applied to clauses that hold at the same moment in time:

[SRES1] C ⇒ �(D ∨ l)C′ ⇒ �(D′ ∨ ¬l)

(C ∧ C′) ⇒ �(D ∨D′)

[SRES2] true ⇒ (D ∨ l)C ⇒ �(D′ ∨ ¬l)C ⇒ �(D ∨D′)

together with the following simplification rule:

[SIMP1] C ⇒ �falsetrue ⇒ ¬C

Temporal Resolution is applied between an eventuality ♦l and a set of clauses which forces lalways to be false. Schematically, the temporal resolution rule is (where A is a conjunction offormulae, C is a conjunction of literals, and l is a literal):

[TRES] A ⇒ � ¬lC ⇒ ♦lC ⇒ (¬A)W l

The intuition behind this rule is that, once C is satisfied, then there must be no other clauses forcingl to always be false. Thus, A can only be satisfied after the right-hand side of the sometime clause,l, is satisfied. In detail, the temporal resolution rule is (where Aj is a conjunction of literals, Bj isa disjunction of literals, and C and l are as above):

[TRES] A0 ⇒ �B0

...An ⇒ �Bn

C ⇒ ♦lC ⇒

⎛⎝ n∧

(¬Ai)i=0

⎞⎠ W l

where∀i, 0 ≤ i ≤ n,� Bi ⇒ ¬l∀i, 0 ≤ i ≤ n,� Bi ⇒ ∨n

j=0Aj

The set of clauses that satisfy these side conditions are together known as a loop in ¬l. Algorithmsfor finding such a loop can be found in [1]. We note that each Aj ⇒ �Bj are step clauses inmerged SNFK , that is, they correspond to a conjunction of step clauses in SNFK . A translation ofthe resolvent into the normal form is given below (where t is a new proposition):

true ⇒ (¬C ∨ ¬Ai ∨ l)t ⇒ �(¬Ai ∨ l)

true ⇒ (¬C ∨ t ∨ l)t ⇒ �(t ∨ l)

Clauses are kept in their simplest form by performing classical style simplification. Classicalsubsumption is also applied and valid formulae, as false ⇒ �A andA⇒ �true, can be removedduring simplification as they cannot contribute to the generation of a contradiction.

306

Page 316: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 5

3. SYNCHRONOUS SYSTEMS WITH NO LEARNING

We now describe a clausal resolution system for temporal logics of knowledge in synchronoussystems with no learning (KL

sync,nl(n) ). A system is said to be synchronous if the agent has access

to a common external clock. Intuitively, if the system is synchronous, the agent knows the time.Moreover, this time is common to all agents. The agent is said to have the property of no learning,if her knowledge does not increase over time. Formally, in the class of models for synchronoussystem with no learning, if two points, (s,m) and (t, n), are in the accessibility relation of agenti, i.e. (s,m) ∼i (t, n), then, because of synchrony, they share the same time index (m = n)and, because of no learning, their successors are also indistinguishable to agent i, i.e. (s,m +1) ∼i (t, n + 1). Note that the number of states that an agent considers possible stays the sameor increases over time and that once the agent considers two timelines indistinguishable, thosetimelines will always be indistinguishable to her. Figure 1 shows an example of a synchronoussystem with no learning, where horizontal lines represent timelines, time increases from left toright, the dots represent points, and the dashed lines enclose points in the same equivalence class.

t0

t1

t2

t3

t4

t5

t6

timelines

t7time

0 1 2 3 4 5 6 7

FIGURE 1. Example of a Synchronous System with No Learning

The syntax and semantics for KLsync,nl(n) are the same as for KL(n). A complete axiomatisation

for KLsync,nl(n) comprises the set of axioms of PTL, the set of axioms of S5(n), together with the

axioms � �Ki ϕ ⇒ Ki�ϕ (SNL), for all agents i ∈ A [7]. The validity problem for such class

of systems is EXPSPACE for n = 1 and non-elementary space for n ≥ 2 [7].

3.1. Proof Method. The general approach for dealing with synchrony and no learning is as fol-lows. Given a set of clauses in SNFK, we first add some new clauses which ensure that theconstraints expressed by the SNL axiom are made explicit before applying the resolution rulespresented in the previous section. As making such constraints explicit is the essential part of theproof method, we explain in more detail its motivation.

In resolution-based proof methods, generally speaking, one has to identify complementary for-mulae (or sets of formulae) in order to apply the inference rules. This procedure can be relativelyeasy for basic logics. For instance, for propositional logics, there is only one resolution inferencerule, which is applied to clauses containing complementary literals, l and ¬l. However, for morecomplex logics, trying to identify complementary formulae can be non-trivial, costly, and oftenachieved by the introduction of new inference rules. This is the case, for instance, in the modalepistemic case, where several modal (resolution) inference rules are introduced in order to resolvea literal l with its possible complements, namely ¬l and Ki¬ l. For the temporal case, besides

307

Page 317: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 C. NALON, C. DIXON, AND M. FISHER

the (step resolution) rules to resolve complementary literals which are true at the same moment intime, the temporal resolution rule is introduced, which requires the identification of a set of clausesthat imply � ¬l to be resolved with an eventuality ♦l. For the combined logics of knowledgeand time, the proof method might seem, at first, rather simple, as it consists of the same inferencerules for the logics considered alone. Nevertheless, its apparent simplicity comes from the sep-aration of the different dimensions (via the normal form) and from making sure that all relevantinformation is made available to these different dimensions (through the propositional language,which is shared by all logics, via simplification rules). Thus, there is no need for new inferencerules: separation provides an elegant way to deal with the combined logic.

Although elegance and simplicity are desirable features for any proof method, this cannot beachieved in a straightforward way when dealing with interactions. In this case, by definition,different dimensions are not separated. We have chosen to adopt the same set of inference rulesof KL(n), as the proof method for the interacting logic must still comprise all the inference rulesfor the underlying languages, so that we are still able to provide refutations for formulae in thoselanguages. Having chosen that, some extra mechanism should be added to the proof methodin order to deal with the interactions. For synchrony and no learning, instead of adding rathercomplex inference rules or trying to identify two complementary sets of clauses, we have chosento add further information to the set of clauses.

We remark that we use the contrapositive form of the SNL axiom: � ¬Ki¬ �ϕ⇒ �¬Ki¬ϕ.A set of clauses satisfying its antecedent is written into the normal form as (at least) two clauses:a clause (or a set of clauses which imply) true ⇒ ψ ∨ ¬Ki¬ l and a step clause (or a set of stepclauses which imply) l ⇒ �ϕ, where ψ is a disjunction of literals or modal literals, and ϕ and lare literals. That is, those clauses together imply ¬ψ ⇒ ¬Ki¬ �ϕ. Because of the SNL axiom,those clauses also imply ¬ψ ⇒ �¬Ki¬ϕ. This is the extra information that we make availableby introducing the new clauses. Instead of looking for such a set of clauses, we introduce newclauses for every step clause. Recall that a step clause is in the general form

X ⇒ �Y

where X is a conjunction and Y is a disjunction of literals. The information we wish to makeexplicit is

¬Ki¬X ⇒ �¬Ki¬Y.

The general approach to generating the new clauses consists of taking the contrapositive form ofa step clause and distributing the knowledge operator Ki through this clause1. Then, we take thecontrapositive form of the resulting clause, exchange the knowledge and temporal operators, andrename the modal literals to keep the normal form. That is, if X is a conjunction, then we replaceX by a new propositional symbol newX , called ∧-proposition; then, the modal literals in thetemporal clause are renamed by new propositional symbols, nkni(X) and nkni(Y ), called SNLi

propositions. The resulting clause, with the new propositional symbols representing the modalliterals, is called a SNLi clause. The SNLi clauses and the clauses defining the new propositionalsymbols are those added to the set of clauses.

3.2. Generating New Clauses. Here we give formal definitions for the new literals and clausesinformally discussed in the previous section. First, we make the distinction between the alreadyexisting literals and the new ones to be added. Basic literals are any literals in the original formulaand any new literals introduced during translation into SNFK . We use the term literal alone, ifthere is no need to distinguish which type of literal we are referring.

We firstly rename conjunctions of basic literals, adding the corresponding definitions to theset of clauses. Subsequently, we can re-use these definitions for the renaming of conjunctions

1Note that we cannot distribute the knowledge operator ¬Ki¬ over an implication, as ¬Ki¬ (ϕ ⇒ ψ) ⇒(¬Ki¬ϕ ⇒ ¬Ki¬ψ) (for any formulae ϕ and ψ) is not a valid formula in S5(n)

308

Page 318: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 7

involving any kind of literals. To aid this process, we define a function, NEW, which takes aconjunction of literals, ϕ, as its argument and returns the new name for this conjunction, newϕ.Those new propositions are called ∧-propositions.

Definition 3. Let c1∧ . . .∧cn, d1∧ . . .∧dm, d′1∧ . . .∧d′m′ , n,m,m′ ≥ 2, be conjunctions of basicand/or SNLi literals in the language of KL

sync,nl(n) . Assume there is an order over the set of literals,

such that li < ¬li < lj < ¬lj , if i < j, for all positive literals li and lj . Let SIMP(ϕ) be the resultof applying simplification rules to ϕ and of ordering the conjuncts. We define the function NEW asfollows:

• NEW(false) = false• NEW(true) = true• NEW(l) = l, for any literal l• NEW(c1 ∧ . . . ∧ cn) = newSIMP(c1∧...∧cn)

• NEW(c1 ∧ . . . ∧ cn ∧ newd1∧...∧dm ∧ . . . ∧ newd′1∧...∧d′m′ ) = NEW(SIMP(c1 ∧ . . . ∧ cn ∧ d1 ∧

. . . ∧ dm ∧ d′1 ∧ . . . ∧ d′m′))

The new proposition is labelled by the simplified, ordered form of the conjunction it is renaming.Simplification is given by usual classical rules, i.e. by deleting repeated literals and/or true fromconjunctions, and by reducing contradictions to false and tautologies to true. Although the defini-tion is applied to all literals, we do not need to rename either a constant, a literal or conjuncts whichare SNLi-literals (as we shall see later). We also do not rename conjuncts which are ∧-propositionsas they are only names for conjunctions. For instance, NEW(a ∧ newb∧c) is newa∧b∧c, as newb∧c

is the name for b ∧ c.Definition 4. For each ∧-proposition, newc1∧...∧cn , n ≥ 2, we add true ⇒ newc1∧...∧cn ∨ ¬c1 ∨. . . ∨ ¬cn to the set of clauses. These new clauses are called ∧-clauses.

The added ∧-clauses correspond to the normal form of one direction of the double implicationnewc1∧...∧cn ⇔ c1 ∧ . . . ∧ cn, which defines the ∧-propositions. As we rename conjunctions onthe left-hand side of step clauses (i.e. formulae of negative polarity), we only need the equivalent(in SNFK) to c1 ∧ . . . ∧ cn ⇒ newc1∧...∧cn .

Once the ∧-propositions have been generated, we define the new names for modal literals,which are the result of distributing the knowledge operator through step clauses. Renaming isused here in order to retain the normal form. We define a set of renaming functions, RENi, one foreach agent i ∈ A, each of which takes as its argument a conjunction or a disjunction of literals, sayϕ, returning the new name for ¬Ki¬ϕ, that is, nkni(ϕ). Because conjunctions are firstly renamedand the knowledge operator can be distributed over disjunctions, these functions will only beapplied to literals. The new names, nkni(l), where l is a literal, are called SNLi-propositions. ASNLi-literal is a SNLi-proposition or its negation.

Definition 5. Let∨

j lj be a disjunction of literals and∧lb,

∧lsi ,

∧lsj , and

∧lV

kbe conjunctions

of basic, SNLi, SNLj (j �= i), and ∧-literals respectively.

• RENi(l) = nkni(l), if l is either a basic, SNLj (j �= i) or ∧-literal;• RENi(l) = l, if l is a SNLi-literal;• RENi(

∨j lj) =

∨j RENi(lj), for any literal lj;

• RENi(∧lsi ∧

∧lsj ∧

∧lb ∧

∧lV

k) =

∧lsi ∧RENi(NEW(

∧lsj ∧

∧lb ∧

∧lV

k)), where j �= i.

The last case says that we rename the conjunctions which involve SNLj literals, for j �= i bythe corresponding ∧-proposition before renaming the modal literal, but we do not need to re-name the SNLi literals in the conjunction (because � ¬Ki¬ (¬Ki¬ϕ ∧ ¬Ki¬ψ) ⇔ (¬Ki¬ϕ ∧¬Ki¬ψ), for any formulae ϕ and ψ). For instance, RENi(a ∧ newb∧c ∧ nkni(d) ∧ nknj(e)) =nkni(d) ∧ nkni(newa∧b∧c∧nknj(e)). We also remark that clauses are kept in their simplest form(e.g. RENi(a ∧ nkni(a)) = nkni(a)).

309

Page 319: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

8 C. NALON, C. DIXON, AND M. FISHER

Definition 6. Let l be a basic, a SNLj (j �= i) or a ∧-literal l. We add SNL⇒i (l) : true ⇒¬nkni(l) ∨ ¬Ki¬ l and SNL⇐

i (l) : true ⇒ nkni(l) ∨ Ki¬ l to the set of clauses. The new clausesare called SNLi definition clauses.

These clauses correspond to the definitions of the SNLi literals, i.e. the equivalence nkni(l) ⇔¬Ki¬ l for each literal l. We need both sides of the double implication, because SNLi literals canoccur with both negative and positive polarities in the set of clauses.

Definition 7. Given a step clause X ⇒ �Y , the corresponding SNLi-clause is

RENi(X) ⇒ �(RENi(Y ))

whereX is a conjunction and Y is a disjunction of literals, and RENi is the function defined above.

Note that the SNLi-clauses are defined for both the initial set of step clauses and those step clausesgenerated while performing resolution.

Note also that these definitions alone could lead to the generation of an infinite number of newliterals. If we consider only one agent, it is clear that this process terminates, because once the∧-propositions have been generated, due to simplification, we can determine all the SNL1 literalsthat need to be generated. However, when we consider multiple agents, it is not clear where wecould stop generating new literals. Suppose, for instance, that A = {1, 2} and the set of basicliterals is {a, b}. In this case, we generate (among others) the ∧-proposition newa∧b. Then, wegenerate (among others) the SNL1 and SNL2 literals, nkn1(newa∧b) and nkn2(newa∧b). We mightneed, now, to consider these new propositions as part of possible conjunctions and generate therespective SNLi literals, as simplification might not apply in this case. For instance, we might needto generate (among others) newa∧nkn1(newa∧b) and the SNL2 literal nkn2(newa∧nkn1(newa∧b)).

We can prove that the number of literals that need to be generated depends on the structure ofthe original formula that we are trying to refute. We define the nesting depth of a SNLi literal,|nkni(l)|snl, as being the number of times that different RENi renaming functions have been ap-plied to any literal: |l|snl = 0, if l is a basic literal; |newl1∧...∧ln |snl = max(|l1|snl, . . . , |ln|snl),if newl1∧...∧ln is a ∧-literal; and |nkni(l)|snl = 1 + |l|snl, otherwise. Note that we have to con-sider the SNLj-literals, j �= i, that are renamed by the ∧-propositions (e.g. |newa∧nkn1(b)|snl =max(|a|snl, |nkn1(b)|snl) = max(0, 1 + |b|snl) = max(0, 1) = 1). The maximum nesting depthof SNLi-literals needed in the resolution method is at most the same as the number of alternationsof distinct knowledge operators in the original formula, that is, the alternating modal depth of theformula. Given such an upper bound for the number of literals that need to generated, that is, byallowing only a finite number of literals, termination of the method is guaranteed.

We call SNFsnl the set of clauses resulting from the transformation of a formula into the SNFK,the SNLi, the ∧, and the SNLi definition clauses. The resolution method applied to a set of SNFsnl

clauses is essentially the same as that described in Section 2, except that we extend the functionmodi so that modi(l) = l, if l is a SNLi-literal.

Below, we illustrate the use of the method. The first example shows that �K1 K2 ϕ ⇒K1 K2

�ϕ is valid in KLsync,nl(n) . We start by negating the formula and then transforming it into its

normal form.

1. start ⇒ x2. x ⇒ �y3. true ⇒ ¬y ∨ K1 z4. true ⇒ ¬z ∨ K2 ϕ

5. true ⇒ ¬x ∨ ¬K1¬w6. true ⇒ ¬w ∨ ¬K2¬ t7. t ⇒ �¬ϕ

where clauses (1-4) correspond to �K1 K2 ϕ and the other clauses correspond to ¬K1 K2�ϕ.

We add the SNLi-clauses

8. nkn1(x) ⇒ �nkn1(y) [2, SNL1]9. nkn2(nkn1(x)) ⇒ �nkn2(nkn1(y)) [8, SNL2]

310

Page 320: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 9

and SNLi definition clauses that will be used in the refutation:

SNL⇐1 (x). true ⇒ nkn1(x) ∨ K1¬x

SNL⇐2 (nkn1(x)). true ⇒ nkn2(nkn1(x)) ∨ K2¬nkn1(x)

SNL⇒1 (y). true ⇒ ¬nkn1(y) ∨ ¬K1¬ y

SNL⇒2 (nkn1(y)). true ⇒ ¬nkn2(nkn1(y)) ∨ ¬K2¬nkn1(y)

The refutation proceeds as follows.

10. true ⇒ ¬nkn1(y) ∨ K1 z [3, SNL⇒1 (y),MRES4]

11. true ⇒ ¬nkn1(y) ∨ z [10,MRES5]12. true ⇒ ¬nkn2(nkn1(y)) ∨ ¬K2¬ z [11, SNL⇒

2 (nkn1(y)),MRES4]13. true ⇒ ¬nkn2(nkn1(y)) ∨ K2 ϕ [12, 4,MRES4]14. true ⇒ ¬nkn2(nkn1(y)) ∨ ϕ [13,MRES5]15. nkn2(nkn1(x)) ⇒ �ϕ [14, 9,SRES1]16. t ∧ nkn2(nkn1(x)) ⇒ �false [15, 7,SRES2]17. true ⇒ ¬t ∨ ¬nkn2(nkn1(x)) [16,SIMP1]18. true ⇒ ¬w ∨ ¬nkn2(nkn1(x)) [17, 6,MRES4]19. true ⇒ ¬w ∨ K2 ¬nkn1(x) [18, SNL⇐

2 (nkn1(x)),MRES1]20. true ⇒ ¬w ∨ ¬nkn1(x) [19,MRES5]21. true ⇒ ¬x ∨ ¬nkn1(x) [20, 5,MRES4]22. true ⇒ ¬x ∨ K1¬x [21, SNL⇐

1 (x),MRES1]23. true ⇒ ¬x [22,MRES5, SIMP]34. start ⇒ false [23, 1, IRES1]

The following example is the proof that K1 K2 ϕ ⇒ K1 K2 ϕ is valid in KLsync,nl(n) . We start

by transforming the negation of this formula into its normal form.

1. start ⇒ x2. true ⇒ ¬x ∨ y3. true ⇒ ¬x ∨ z4. z ⇒ �y5. z ⇒ �z

6. true ⇒ ¬y ∨ K1w7. true ⇒ ¬w ∨ K2 ϕ8. true ⇒ ¬x ∨ ¬K1¬ r9. true ⇒ ¬r ∨ ¬K2¬ s

10. s ⇒ ♦¬ϕThen, we add the new SNLi clauses:

11. nkn1(z) ⇒ �nkn1(y) [4, SNL1]12. nkn1(z) ⇒ �nkn1(z) [5, SNL1]13. nkn2(nkn1(z)) ⇒ �nkn2(nkn1(y)) [11, SNL2]14. nkn2(nkn1(z)) ⇒ �nkn2(nkn1(z)) [12, SNL2]

and also the SNLi definition clauses that will be needed in the refutation:

SNL⇒1 (y) : true ⇒ ¬nkn1(y) ∨ ¬K1¬ y

SNL⇐1 (z) : true ⇒ nkn1(z) ∨ K1¬ z

SNL⇒2 (nkn1(y)) : true ⇒ ¬nkn2(nkn1(y)) ∨ ¬K2¬nkn1(y)

SNL⇐2 (nkn1(z)) : true ⇒ nkn2(nkn1(z)) ∨ K2¬nkn1(z)

The refutation now proceeds as follows

15. true ⇒ ¬nkn1(y) ∨ K1w [6, SNL⇒1 (y),MRES4]

16. true ⇒ ¬nkn1(y) ∨ w [15,MRES5]17. true ⇒ ¬nkn2(nkn1(y)) ∨ ¬K2¬w [16, SNL⇒

2 (nkn1(y)),MRES4]18. true ⇒ ¬nkn2(nkn1(y)) ∨ K2 ϕ [17, 7,MRES4]

311

Page 321: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

10 C. NALON, C. DIXON, AND M. FISHER

19. true ⇒ ¬nkn2(nkn1(y)) ∨ ϕ [18,MRES5]20. nkn2(nkn1(z)) ⇒ �ϕ [13, 19,SRES2]21. s ⇒ ¬nkn2(nkn1(z))W ¬ϕ [20, 14, 10,TRES]22. true ⇒ ¬s ∨ ¬nkn2(nkn1(z)) ∨ ¬ϕ [21,SNF]23. true ⇒ ¬r ∨ ¬nkn2(nkn1(z)) ∨ ¬K2 ϕ [22, 9,MRES4]24. true ⇒ ¬r ∨ ¬nkn2(nkn1(z)) ∨ ¬w [23, 7,MRES1]25. true ⇒ ¬r ∨ K2 ¬nkn1(z) ∨ ¬w [24, SNL⇐

2 (nkn1(z)),MRES1]26. true ⇒ ¬r ∨ ¬nkn1(z) ∨ ¬w [25,MRES5]27. true ⇒ ¬x ∨ ¬nkn1(z) ∨ ¬K1w [26, 8,MRES4]28. true ⇒ ¬x ∨ ¬nkn1(z) ∨ ¬y [27, 6,MRES1]29. true ⇒ ¬x ∨ ¬nkn1(z) [28, 2,MRES1]30. true ⇒ ¬x ∨ K1¬ z [29, SNL⇐

1 (z),MRES1]31. true ⇒ ¬x [30, 3,MRES3]32. start ⇒ false [31, 1, IRES1]

4. CORRECTNESS

In this section, we give the results for soundness, termination, and completeness of the resolu-tion method for KLsync,nl

(n) . Full proofs can be found in [10].

Soundness. The soundness proof consists of showing that, given a formula ϕ in KLsync,nl(n) , its

transformation into SNFsnl is satisfiability preserving and that the application of the inferencerules to the set of clauses in the normal form is also satisfiability preserving.

Theorem 1. Let ϕ be a well-formed formula in KLsync,nl(n) and τ(ϕ) = ∗ ∧

i Ti be the transfor-mation of ϕ into SNFsnl, where every Ti is a clause. ϕ is satisfiable in KLsync,nl

(n) if, and only if,τ(ϕ) is satisfiable in KLsync,nl

(n) .

This result is obtained from correctness of transformation into SNFK (given in [3]) and fromLemmas 1.1, 1.2, and 1.3, which show that the addition of new literals and definition clauses aresatisfiability preserving.

Lemma 1.1. Let T be a set of clauses in SNFsnl. Let T ′ be the set of clauses augmented withthe definitions of ∧-propositions, that is, T′ = T ∪ {true ⇒ newl1∧...∧ln ∨ ¬l1 ∨ . . . ∨ ¬ln |for all literals li occurring in T}. T is satisfiable in KLsync,nl

(n) if, and only if, T ′ is satisfiableKLsync,nl

(n) .

Proof of Lemma 1.1 (outline) We show that given a model M for T , we can construct a modelM ′ for T ′. For every state (t, u) in M , let the corresponding state (t′, u′) in M ′ be exactly as(t, u), except that π(t′, u′)(newl1∧...∧ln) = true, if 〈M, (t, u)〉 |= l1 ∧ . . . ∧ ln for all possibleconjunctions of literals. Temporal and equivalence relations are kept as in the original model.Obviously, M ′ satisfies all clauses in T ; also, it follows from the construction of M′, that such amodel satisfies all definition clauses for the ∧-literals. For the “only if part”, a model M for T canbe obtained from M by ignoring the values of the ∧-literals. �

Lemma 1.2. Let T be a set of clauses in SNFsnl. Let T ′ be the set of clauses augmented with thedefinitions of SNLi-literals, that is, T ′ = T ∪ {SNL⇒

i (l) : true ⇒ ¬nkni(l) ∨ ¬Ki¬ l, SNL⇐i (l) :

true ⇒ nkni(l)∨Ki¬ l | for all literals l occurring in T}. T is satisfiable KLsync,nl(n) if, and only

if, T ′ is satisfiable KLsync,nl(n) .

Proof of Lemma 1.2 (outline) This proof can be obtained in a similar way that the proof forLemma 1.1, but where for every state (t, u) in M , the corresponding state (t′, u′) in M ′ is exactlyas (t, u), except that π(t′, u′)(nkni(l)) = true if, and only if, 〈M, (t, u)〉 |= ¬Ki¬ l for all agentsi ∈ A and literals l. �

312

Page 322: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 11

Lemma 1.3. Let T be a set of clauses in SNFsnl. Let T ′ be the set of clauses augmented withthe SNLi-clauses, i.e. T ′ = T ∪ {RENi(X) ⇒ �RENi(Y ) | for all ∗(X ⇒ �Y ) ∈ T},where RENi is the renaming function, X is a conjunction and Y is a disjunction of literals . T issatisfiable KL

sync,nl(n) if, and only if, T ′ is satisfiable KL

sync,nl(n) .

Proof of Lemma 1.3 (outline) If X is a conjunction, by Lemma 1.1, there is a ∧-proposition,newX , such that, if 〈M, (t, u)〉 |= X, then 〈M, (t, u)〉 |= newX . Thus, without loss of generality,we assume that X is a proposition. This proof can be, then, obtained from the semantics of theuniversal operator, ∗, which surrounds all clauses, propositional reasoning and applications ofthe axioms K and SNL. The “only if” part is obvious: if there is a model that satisfies all clausesin T ′, T ⊂ T ′, the model satisfies all clauses in T . �

The set of inference rules is the same as that described in [3], except that we add to the definitionof the functionmodi thatmodi(l) = l, if l is a SNLi-literal. We remark that redefining this functionis not essential, but it saves some steps in the refutation.

Theorem 2. Let T be a set of clauses in the SNFsnl. Let T ′ be the set of clauses augmented withany resolvents obtained by applying the inference rules IRES1-2, MRES1-5 (where modi(l) = l,if l is a SNLi-literal), SRES1-2, SIMP1, and TRES to T . T is satisfiable in KLsync,nl

(n) if, and onlyif, T ′ is satisfiable in KL

sync,nl(n) .

Proof of Theorem 2 This result follows from soundness of the resolution method for KL(n) [3],that is, that all inference rules, including MRES4 without modification, are sound, and from theobservation that the same resolvent from the modified MRES4 can also be obtained from suc-cessive applications of the original MRES4 and MRES1 to the original resolvent and the SNLidefinition clauses. �

Termination. The algorithm for the method presented here is based on that for KL(n). The dif-ference is that SNLi and ∧-literals, together with their corresponding definition clauses, are intro-duced before starting the application of the resolution method. Also, SNLi-clauses, correspondingto existing or newly generated step clauses, are introduced when performing the algorithm. It hasbeen shown in [3] that the method for KL(n) terminates, i.e. given a finite number of clauses onlya finite number of clauses (modulo order and simplification) can be generated, so at some point ei-ther false is generated or no new clauses are generated. In order to transfer the termination resultsfrom KL(n) to KLsync,nl

(n) , we have to show that all propositional symbols that might be needed forthe refutation can be defined before the resolution rules are applied.

Firstly, it has been shown in [6] that the new propositional symbols required for translating theresolvent obtained by an application of the temporal resolution rule can be added at the beginningof the proof. No other inference rule requires the introduction of new symbols. As there is afinite number of symbols, due to simplification, only a finite number of clauses is generated.Secondly, simplification is applied when generating the ∧-literals and SNLi-literals, so the numberof definition clauses for these literals is also finite. Thirdly, the number of step clauses is (atany point) finite, and so it is the number of SNLi clauses. Finally, given the alternating modaldepth of the original formula, the maximum nesting depth of SNLi-literals can be determined, andso the number of new literals that might be needed in the refutation is finite. Given that only afinite number of new symbols and definition clauses is introduced, there is only a finite numberof clauses that can be defined. Thus, as the resolution method applied to the set of clauses in theSNFsnl is the same as the method for KL(n), termination follows from the results in [3].

Completeness. This proof is based on that given in [6], where a graph is built from a set of clauses.The construction of the graph is given in more detail in [10]. The proof consists in showing that anempty graph corresponds to an unsatisfiable set of clauses and that, in this case, there is a refutationby the resolution method presented here.

Let T be a set of clauses into SNFsnl. We construct a finite direct graph G = 〈N,E〉 for T ,where N is a set of nodes and E is a set of labelled edges, as follows. A node η = (V, Y ) is a pair,

313

Page 323: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

12 C. NALON, C. DIXON, AND M. FISHER

where V is a maximal consistent set of basic literals, modal literals, SNLi literals, and ∧-literals;and Y is a subset of basic literals occurring on the right hand side of a sometime clause. Intuitively,V corresponds to states and Y is a set of witnesses for eventualities, that is, it corresponds toeventualities that have not been satisfied by the predecessors of V . There are n+1 types of edges:one for the temporal dimension plus one for each agent in A = {1, . . . , n}. For every consistentset V , we construct nodes η = (V, Y ), where Y is any of the possible subsets of literals occurringon the right-hand side of sometime clauses in the set of clauses. We delete any nodes that do notimmediately satisfy the literal and modal clauses, including the definition clauses, in T .

Given a non-empty set of nodes, we construct the set of labelled edges by firstly building theequivalence classes for each agent i ∈ A. There is an i-edge between two nodes η = (V, Y ) andη′ = (V ′, Y ′), if, and only if, V and V ′ contain the same set of modal literals for that agent. Wesay that a node η′ is i-reachable from η, if there is an i-edge between η and η′. We say that anode η′ is {i0, . . . , im}-reachable from η, if there is a sequence of nodes η0, . . . , ηm+1, such thatη0 = η, ηm+1 = η′, and there is a ij-edge between every two nodes ηj and ηj+1, for 0 ≤ j ≤ m.We define [η]i as the set of nodes that are i-reachable from η. Obviously, every [η]i defines anequivalence relation over the set of nodes.

Then, we construct the temporal edges. We start with a full (temporal) graph, i.e. there is at-edge linking every two nodes in the graph (because true ⇒ �true). We say that a node η′is t-reachable from η, if there is a sequence of nodes η0, . . . , ηm, such that η0 = η, ηm = η′,and there is a t-edge between every two nodes ηj and ηj+1, for 0 ≤ j < m. We say that η is apredecessor of η′, if η′ is t-reachable from η.

For every step clause (∧l ⇒ �

∨l′) ∈ T , we delete a t-edge between η = (V, Y ) and

η′ = (V ′, Y ′), if η |= ∧l and η′ �|= ∨

l′. For each sometime clause, ϕ ⇒ ♦l, in the set ofclauses, we also delete a node η = (V, Y ), if V |= ϕ and l �∈ Y ; and a t-edge from η = (V, Y )to η′ = (V ′, Y ′), if (a) l ∈ Y , l �∈ Y ′, and V �|= l; or (b) l ∈ Y ′, V |= l, and V ′ �|= ϕ . Thisensures that (a) eventualities not satisfied by a predecessor are not “forgotten” and (b) there willbe no edge from a node that satisfies an eventuality to another node that is “waiting” for the sameeventuality to be satisfied, unless this successor satisfies the left-hand side of a sometime clausethat also says that this eventuality should hold at some moment in the future. In other words, weare only waiting for eventualities to occur that originally came from satisfying the left-hand sideof a sometime clause.

A node η = (V, Y ) is an initial node if, and only if, V satisfies all initial clauses in T and, foreach sometime clause ϕ ⇒ ♦l, such that V |= ϕ, if, and only if, l ∈ Y . We say that a nodeη′ is reachable from a node η, if, and only if, there is a sequence of nodes η0, . . . , ηm, such that,η0 = η, ηm = η′, and either ηj+1 is t-reachable or i-reachable from ηj , for 0 ≤ j < m and i ∈ A.

We further delete any nodes that are not predecessors of any node which is reachable from aninitial node. This reduces the graph to (possibly disjoint) connected components which include atleast one initial node. The resulting graph is called a behaviour graph for T . Given a behaviourgraph for T , we recursively delete any nodes (V, Y ) (and edges to it) that have no temporal suc-cessors (no infinite temporal line can be constructed from the node, so it is not part of any model);satisfy the left-hand side of a sometime clause, but there is no node satisfying the eventuality thatis t-reachable from this node; and/or satisfy a formula as ¬Ki¬ p but there is no modal i-edge to anode which satisfies p. The resulting graph is called reduced behaviour graph. Deletions of eithernodes or edges are repeated non-deterministically until the graph is empty or no other deletioncan be done. This procedure corresponds, respectively, to application of temporal simplification(SIMP1), temporal resolution (TRES), and modal resolution to the set of clauses. If the graph isempty, then the set of clauses is not satisfiable in KL

sync,nl(n) .

If the reduced behaviour graph is not empty, we show that given two nodes η and µ in the sameequivalence class, i.e. [η]i = [µ]i, for agent i ∈ A, such that η′ is a successor of η, then µ has alsoa successor in [η′]i.

314

Page 324: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 13

Theorem 3. Let G be a non-empty reduced behaviour graph for a set of clauses T in SNFsnl. Letη, µ, η′, µ′, be nodes in G, such that [η]i = [µ]i and there is a t-edge from η to η′. Then there is anode µ′, [µ′]i = [η′]i, such that there is a t-edge from µ to µ′.Proof of Theorem 3 We show, by contradiction, that if the graph is not empty and we have atemporal edge from a node η to a node η′, then all nodes in [η]i have successors in [η′]i.

Suppose that there is a temporal edge from η to η′ and µ ∈ [η]i has no successors in [η′]i. Thus,we can identify a set of step clauses that together imply ψ ⇒ �χ, such that µ satisfies ψ and, forall µ′ ∈ [η′]i, µ′ satisfies ¬χ. If we could not identify this set of clauses, the temporal edges wouldnot have been removed.

If all µ′ ∈ [η′]i satisfies ¬χ, then, by the semantics of the knowledge operator, µ′ |= Ki¬χ. Theaddition of the SNLi-clause corresponding to ψ ⇒ �χ, i.e. RENi(ψ) ⇒ �RENi(χ), ensures thatevery node must satisfy ¬Ki¬ψ ⇒ �¬Ki¬χ. We have that µ satisfies ψ and, by the semanticsof the knowledge operator, it also satisfies ¬Ki¬ψ (or the corresponding ∧-proposition, newψ ,and the formula ¬Ki¬newψ, in the case where ψ is a conjunction). In fact, every node in [η]isatisfies ¬Ki¬ψ. If all nodes in [η′]i satisfy Ki¬χ, because of the SNLi clause, then there mustbe no temporal edge from η to η′. This contradicts with our initial assumptions, so there must bea temporal edge between µ and µ′ or none of the nodes in [η]i have successors in [η′]i.�Theorem 3 shows that every two nodes in the same equivalence class, say [η]i, have successorsin the same equivalence class, say [η′]i. However, there might be nodes in [η′]i that have nopredecessors in [η]i. In order to be able to construct a model, we also have to show that every nodehas a predecessor.

Theorem 4. Let G be a non-empty reduced behaviour graph for a set of clauses T in SNFsnl. Letη be a node in G. Then, there is a node η′ such that η′ is a predecessor of η.

Proof of Theorem 4 (outline) If η has no predecessor, we introduce a node η′ = (V ′, E′) (anda temporal edge from η′ to η) such that V ′ |= ¬start and V ′ |= ¬p for all propositional symbolsoccurring in the set of SNFsnl clauses. Note that such a node satisfies trivially all clauses in theset of SNFsnl clauses, either by falsifying the left-hand side of a (initial, step, or sometime clause)or by satisfying the right-hand side of a modal or literal clause (as, from translation, the right-handside of such a clause has at least one negative literal occurring as a disjunct).�

Hence, if the reduced behaviour graph is not empty, we can inductively construct a model in which,if two nodes, (s,m) and (t,m), are in the same equivalence class, i.e. (s,m) ∼i (t,m), then theirsuccessors are also in the same equivalence class, i.e. (s,m+ 1) ∼i (t,m+ 1), which suffices toprove that the model is in the class for synchronous systems with no learning [7]. In order to buildthe first timeline for the agent i, we choose an initial node η0 from the graph, which corresponds tothe point (t0, 0). Let the nodes of [η0]i be the initial points of other timelines for agent i. Note thatonly the first point at the first timeline needs to be an initial node in the graph. Then, we choose η1from the immediate successors (i.e. t-reachable by one t-edge) of η0. We build the next point ineach timeline, by choosing the immediate successors of each node in [η0]i from the nodes in [η1]i.From Theorem 3, this is always possible. From Theorem 4, for points at time greater than zero,we can construct the predecessors of a node back to initial point of every timeline.

We note that the introduction of the SNLi-clauses not only delete edges which are not part ofa model for KLsync,nl

(n) , but also contribute to the temporal resolution procedure. Temporal resolu-tion between a sometime clause, say l ⇒ ♦p, and a set of clauses that together imply � ¬p,corresponds to removing from the graph subcomponents which satisfy l, but never satisfy the even-tuality p. As we are considering the combined logics of knowledge and time, the sometime clausesmay be preceded by a chain of knowledge operators. For instance, the set of clauses (taken fromthe second example on Page 9):

8. true ⇒ ¬x ∨ ¬K1¬ r9. true ⇒ ¬r ∨ ¬K2¬ s

10. s ⇒ ♦¬ϕ

315

Page 325: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

14 C. NALON, C. DIXON, AND M. FISHER

t

t

1

2

1

2

x, y, z

nkn1(z)

nkn2(nkn1(z))

t y, z, ϕ

w, nkn1(z), ϕ

nk2(nkn1(z)), ϕ

FIGURE 2. Some temporal loops related to the second example.

shows that the left-hand side of the sometime clause is preceded by the knowledge operators ¬K1¬and ¬K2¬ . In systems with synchrony and no learning, ¬K1 K2¬♦¬ϕ implies ♦¬K1 K2¬ϕ,which should be resolved with a set of clauses that together imply � K1 K2 ϕ. Note that becauseof the axiom T, the same loop implies � ϕ. Although this loop could be resolved with thesometime clause, the resolvents generated by temporal resolution would not contribute to removingthe chain of knowledge operators preceding the eventuality. The sometime clause occurs in atimeline that is {1, 2}-reachable from the initial timeline. Therefore, we are interested in a loopthat is also {1, 2}-reachable from the initial timeline. In Figure 2, we show this loop, where thestates are represented by circles; each state is labelled by the propositional symbols which hold atthat state; and the edges are labelled either by t (indicating the temporal successor) or by the agentindex (indicating that those states are in the same equivalence class). We note that we could haveintroduced SNLi-literals in sometime clauses in the same way that those literals where introducedin step clauses, but this is not necessary for completeness. The next theorem ensures that theappropriate loop is found, if a chain of knowledge operators precede the sometime clause.

Theorem 5. Let C be the set of SNFsnl clauses that together imply ¬Ki1 Ki2 . . .Kim¬♦¬ϕwhere ϕ is a basic literal and every two consecutive knowledge operators differ in their index.If there is a set of clauses that imply � Ki1 Ki2 . . .Kim ϕ, then there is a set of nested SNLi

clauses C′ that imply � ϕ.

Specifically, if χ is the conjunction of literals on the left-hand side of the step clauses which satisfythe loop conditions, then we can prove that there is a loop given by nknim...1(χ) ⇒ �ϕ (wherenknim...1(l) is an abbreviation for successively applying the renaming functions REN1, . . . ,RENm

to a literal l). The temporal resolvents, the set of clauses that represent the chain of knowledge op-erators preceding the sometime clause, and the (nested) SNLi definitions of χ can be successivelyresolved by applying the modal inference rules. Intuitively, the nesting of SNLi literals transformsthe interaction between knowledge and time into two separated problems: a temporal and a modalproblem. Firstly, by introducing the SNLi propositions, the interaction is represented as a (propo-sitional) temporal problem. Once the temporal problem is solved, modal resolution takes place,“translating” back those SNLi propositions into their corresponding modal literals and performingresolution in the modal portion of the language.

In the proof, we assume that subsequent knowledge operators in the chain preceding a temporalformula had different indices. This corresponds to the alternating modal depth of the knowledgeoperators in the original formula and gives an upper bound for the maximum nesting of SNLiliterals. Note that this assumption can be made without loss of generality, as chains of modal

316

Page 326: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

RESOLUTION FOR SYNCHRONY AND NO LEARNING 15

clauses with same index can be successively resolved together, resulting in one modal clause ofthat index.

5. CONCLUSIONS

We have shown how to extend the method presented in [3] to deal with synchronous systemswith no learning. The addition of new information to the set of clauses, together with the use ofrenaming, provides an intuitive way of dealing with this particular interaction. Although the intro-duction of new clauses is costly, we avoid the introduction of new inference rules and, potentially,expensive searches over the set of clauses. As there is no change in the set of inference rules,implementation is relatively easily obtained by adapting and re-using existing theorem-provers.Correctness for the multi-agent case has been established. Results for the multi-agent case of themethod for synchrony and perfect recall given in [2] can be established in a similar way and it isongoing work.

Synchronous systems with either perfect recall or no learning have complete axiomatisationsgiven by axioms with simple structures. We have been investigating how the ideas behind our proofmethod could be applied to logics that include axioms of the (simple) form Ki

�pϕ⇒ �qKi ϕ or�pKi ϕ⇒ Ki

�qϕwhere p and q are integers (p, q > 0), and the iterated next operator is definedas �0ϕ = ϕ, and �pϕ = �p−1 �ϕ, for p > 0, where ϕ is a formula. Although the generationof new clauses would not be straightforward, we have also been investigating other interactionsexpressed by finite axiomatisations which include more complex axioms, as, for instance, the nolearning axiom, � (Ki ϕ1 U Ki ϕ2) ⇒ Ki (Ki ϕ1 U Ki ϕ2), when synchrony is not required.

REFERENCES

[1] C. Dixon. Temporal Resolution using a Breadth-First Search Algorithm. Annals of Mathematics and ArtificialIntelligence, 22:87–115, 1998.

[2] C. Dixon and M. Fisher. Clausal Resolution for Logics of Time and Knowledge with Synchrony and PerfectRecall. In Proceedings of ICTL 2000, Leipzig, Germany, 2000.

[3] C. Dixon and M. Fisher. Resolution-Based Proof for Multi-Modal Temporal Logics of Knowledge. In S. Good-win and A. Trudel, editors, Proceedings of the Seventh International Workshop on Temporal Representation andreasoning (TIME’00), pages 69–78, Cape Breton, Nova Scotia, Canada, July 2000. IEEE Computer Society Press.

[4] C. Dixon, M. Fisher, and M. Wooldridge. Resolution for Temporal Logics of Knowledge. Journal of Logic andComputation, 8(3):345–372, 1998.

[5] R. Fagin, J. Y. Halpern, Y. Moses, and M. Y. Vardi. Reasoning About Knowledge. MIT Press, 1995.[6] M. Fisher, C. Dixon, and M. Peim. Clausal Temporal Resolution. ACM Transactions on Computational Logic,

2(1), January 2001.[7] J. Y. Halpern, R. van der Meyden, and M. Y. Vardi. Complete Axiomatizations for Reasoning About Knowledge

and Time. SIAM Journal on Computing, 33(3):674–703, 2004.[8] B. Heinemann. Linear Tense Logics of Increasing sets. Journal of Logic and Computation, 12(4):583–606, 2002.[9] R. E. Ladner and J. H. Reif. The logic of distributed protocols (preliminary report). In Joseph Y. Halpern, editor,

Theoretical Aspects of Reasoning about Knowledge: Proceedings of the First Conference, pages 207–222, LosAltos, California, 1986. Morgan Kaufmann Publishers, Inc.

[10] C. Nalon. Resolution for Synchrony and No Learning. PhD thesis, University of Liverpool, March 2004.[11] A. S. Rao and M. P. Georgeff. Modeling Rational Agents within a BDI-Architecture. In R. Fikes and E. Sande-

wall, editors, Proceedings of Knowledge Representation and Reasoning (KR&R-91), pages 473–484. Morgan-Kaufmann, April 1991.

[12] B. van Linder, W. van der Hoek, and J. J. Ch. Meyer. Formalising Motivational Attitudes of Agents: On Pref-erences, Goals and Commitmentes. In M. Wooldridge, J. P.Puller, and M. Tambe, editors, Intelligent Agents II,volume 1037 of Lecture Notes in Artificial Intelligence, pages 17–32. Springer-Verlag, 1996.

ACKNOWLEDGMENTS

This work was partially supported by CAPES (grant BEX 1158-99-6) and the EPSRC researchgrants GR/R45376 and GR/S63182.

DEPARTMENT OF COMPUTER SCIENCE,UNIVERSITY OF LIVERPOOL,LIVERPOOL, L69 7ZF, UK.E-mail address: {C.Nalon, C.Dixon, M.Fisher}@csc.liv.ac.uk

317

Page 327: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

On the Complexity of Fragments ofModal Logics

Linh Anh [email protected]

Institute of Informatics, University of Warsawul. Banacha 2, 02-097 Warsaw, Poland

Abstract. We study and give a summary of the complexity of 15 basicnormal modal logics under the restriction to the Horn fragment and/orbounded modal depth. As new results, we show that the satisfiabilityproblem of sets of Horn modal clauses with modal depth bounded byk ≥ 2 in the modal logics K4 and KD4 is PSPACE-complete, in Kis NP-complete. We also show that the satisfiability problem of modalformulas with modal depth bounded by 1 in K4, KD4, and S4 is NP-complete; the satisfiability problem of sets of Horn modal clauses withmodal depth bounded by 1 in K , K4, KD4, and S4 is PTIME-complete.

1 Introduction

In the field of modal logics, a lot of works are devoted to modal logics thatextend the modal logic K by some of the axioms D, T , B, 4, and 5. Thereason is not that those logics are useful in practice, but because they arebasic modal logics. Many useful multimodal logics, e.g. ones for reasoningabout knowledge and belief, are also formed using the mentioned axiomsand are extensions of some basic (mono)modal logics.

Decidability and complexity are important aspects of logics. In [8],Ladner proved that the complexity of the satisfiability problem in themodal logics K , T , B , and S4 is PSPACE-complete, and in S5 is NP-complete. This means that the satisfiability problem is NP-hard in all ofthose logics. In order to reduce the complexity to PTIME, one must focuson fragments of the considered logic. Such fragments are often specifiedby restrictions on the language. There are of course many kinds of re-strictions, but the obtained fragments may be useful or not. The Hornfragment is very useful in logic programming, and in many logics it sig-nificantly reduces the complexity of the problem. For modal logics, therestriction of bounded modal depth is also acceptable, because in practicemodal formulas often have small modal depth. We can also combine thesetwo restrictions. Given an “acceptable” restriction and a modal logic, one

318

Page 328: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

may want to study the complexity of the satisfiability problem in theobtained fragment of the logic. The result may be positive (PTIME) ornegative (NP-hard, PSPACE-hard, etc). Both of the cases are useful: thepositive case is good for the fragment itself, while the negative case im-plies that every multimodal logic containing the fragment is hard at leastas the fragment.

In this work, we study and give a summary of the complexity of thesatisfiability problem in the basic normal modal logics (which are obtainedfrom the logic K by adding an arbitrary combination of the axioms D, T ,B, 4, and 5) under the restriction to the Horn fragment and/or boundedmodal depth.

In [6], Halpern studied the effect of bounding modal depth on thecomplexity of modal logics and showed that the complexity of the satisfi-ability problem of formulas with modal depth bounded by k ≥ 2 in K andT is NP-complete, and in S4 is PSPACE-complete. His arguments for Kand T can also be applied for the logics KB , KDB , and B , to obtain theNP-completeness.

In [4], Farinas del Cerro and Penttonen showed that the satisfiabilityproblem of sets of Horn modal clauses in S5 is decidable in PTIME. In[3], Chen and Lin showed that the similar problem for a normal modallogic L being an extension of K5 (write K5 ≤ L) is also decidable inPTIME. Chen and Lin also proved that for a normal modal logic L suchthat K ≤ L ≤ S4 or K ≤ L ≤ B, the problem is PSPACE-hard. Theyalso made a comment that the problem is still PSPACE-hard for S4 evenwhen the modal depth is restricted to 2.

In [9], we showed that the complexity of the satisfiability problem ofsets of Horn modal clauses with finitely bounded modal depth in KD , T ,KB , KDB , and B is decidable in PTIME. These PTIME results can fur-ther be categorized as PTIME-complete, because the satisfiability prob-lem of sets of Horn clauses in the classical propositional logic is PTIME-complete, as proved by Jones and Laaser [7].

In this paper, we show that the satisfiability problem of sets of Hornmodal clauses with modal depth bounded by k ≥ 2 in the modal logicsK4 and KD4 is PSPACE-complete, and in K is NP-complete. We alsoshow that the satisfiability problem of modal formulas with modal depthbounded by 1 in K4, KD4, and S4 is NP-complete; the satisfiabilityproblem of sets of Horn modal clauses with modal depth bounded by 1in K , K4, KD4, and S4 is PTIME-complete.

In Table 1, we summarize the complexity of the basic modal logics un-der the mentioned restrictions. There, mdepth stands for “modal depth”;

319

Page 329: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

PS-cp, NP-cp, and PT-cp respectively stand for PSPACE-complete, NP-complete, and PTIME-complete. The marks [?] and [∗] indicate the resultsof this work, where [∗] involves with K4 and KD4.

KD , T K5, KD5, KB5K KB , KDB , B K4, KD4, S4 K45, KD45, S5

no restrictions PS-cp [8] PS-cp [8] PS-cp [8] NP-cp [8]

mdepth ≤ k, k ≥ 2 NP-cp [6] NP-cp [6] PS-cp [6] [∗] NP-cp [8]

mdepth = 1 NP-cp [6] NP-cp [6] NP-cp [?] NP-cp [8]

Horn PS-cp [3] PS-cp [3] PS-cp [3] PT-cp [4, 3]

Horn, mdepth ≤ k, k ≥ 2 NP-cp [?] PT-cp [9] PS-cp [3] [∗] PT-cp [4, 3]

Horn, mdepth = 1 PT-cp [?] PT-cp [9] PT-cp [?] PT-cp [4, 3]

Table 1. The complexity of the satisfiability problem for modal logics

2 Preliminaries

2.1 Syntax and Semantics of Propositional Modal Logics

A modal formula, hereafter simply called a formula, is any finite sequenceobtained by applying the following rules: any primitive proposition pi isa formula, and if ϕ and ψ are formulas then so are ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ,ϕ→ ψ, 2ϕ, and 3ϕ.

The symbols 2 and 3 can take various meanings but traditionallystand for “necessity” and “possibility”. We use letters p and q to denoteprimitive propositions, and Greek letters ϕ, ψ, ζ to denote formulas.

A Kripke frame is a triple 〈W, τ,R〉, where W is a nonempty set ofpossible worlds, τ ∈W is the actual world, and R is a binary relation onW , called the accessibility relation. If R(w, u) holds then we say that theworld u is accessible from the world w.

A Kripke model is a tuple 〈W, τ,R, h〉, where 〈W, τ,R〉 is a Kripkeframe and h is a function mapping worlds to sets of primitive propositions.For w ∈ W , h(w) is the set of primitive propositions which are “true”at w.

We call a model 〈W, τ,R, h〉 flat if W = {τ} and R = ∅.A model graph is a tuple 〈W, τ,R,H〉, where 〈W, τ,R〉 is a Kripke

frame and H is a function mapping worlds to formula sets. We some-times treat model graphs as models with H being restricted to the set ofprimitive propositions.

320

Page 330: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Given a Kripke model M = 〈W, τ,R, h〉 and a world w ∈ W , thesatisfaction relation � is defined as follows:

M,w � p iff p ∈ h(w);M,w � ¬ϕ iff M,w 2 ϕ;M,w � ϕ ∧ ψ iff M,w � ϕ and M,w � ψ;M,w � ϕ ∨ ψ iff M,w � ϕ or M,w � ψ;M,w � ϕ→ ψ iff M,w 2 ϕ or M,w � ψ;M,w � 2ϕ iff for all v ∈W s.t. R(w, v), M,v � ϕ;M,w � 3ϕ iff there exists v ∈W s.t. R(w, v) and M,v � ϕ.

We say that ϕ is satisfied at w in M if M,w � ϕ, and that ϕ is satisfiedin M , write M � ϕ and call M a model of ϕ, if M, τ � ϕ.

The size of a finite Kripke model 〈W, τ,R, h〉 is |W | + |R| +Σw∈W |h(w)|. The length of a formula ϕ is the number of occurrencesof connectives and primitive propositions in ϕ. The modal depth of a for-mula ϕ is the maximal nesting depth of modalities occurring in ϕ, e.g.mdepth(p ∧2(3q ∨ ¬3r)) = 2.

The following lemma is well-known and can be proved easily.

Lemma 1. Given a finite model M and a formula ϕ, the problem ofchecking whether M � ϕ is decidable in polynomial time (in the size ofM and the length of ϕ).

If as the class of admissible interpretations we take the class of allKripke models (with no restrictions on the accessibility relations) thenwe obtain a normal modal logic which has a standard Hilbert-style ax-iomatization denoted by K. Other normal modal logics are obtained byadding to K certain axioms. The most popular axioms used for extendingK are D, T , B, 4, and 5, whose schemata are listed below. These axiomsrespectively correspond to seriality, reflexiveness, symmetry, transitive-ness, and euclideaness of the accessibility relation. A modal logic L isserial if it contains the axiom D.

Axiom Schema Corresponding Condition on R

D 2ϕ→ 3ϕ ∀w ∃u R(w, u)T 2ϕ→ ϕ ∀w R(w,w)B ϕ→ 23ϕ ∀w, u R(w, u)→ R(u,w)4 2ϕ→ 22ϕ ∀w, u, v R(w, u) ∧R(u, v)→ R(w, v)5 3ϕ→ 23ϕ ∀w, u, v R(w, u) ∧R(w, v)→ R(u, v)

In this work, we consider all of the 15 basic modal logics that areobtained from K by adding an arbitrary combination of the above axioms,

321

Page 331: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

namely K, KD, T , KB, KDB, B, K4, KD4, S4, K5, KD5, K45, KD45,KB5, S5. The names of these logics often consist of K and the names ofthe added axioms, e.g. KDB is the logic which extends K with the axiomsD and B. The special cases are T , B, S4, and S5, which stand for KT ,KTB, KT4, and KT5, respectively. For a further reading about modallogics, see, e.g., [1, 2].

We refer to the properties of the accessibility relation of a modallogic L as the L-frame restrictions. We call a model M an L-model if theaccessibility relation of M satisfies all L-frame restrictions. We say that ϕis L-satisfiable if there exists an L-model of ϕ. A formula is L-valid if it issatisfied in every L-model. We write ϕ �L ψ to denote that ψ is satisfiedin every L-model of ϕ.

2.2 Modal Horn Formulas and Positive Modal LogicPrograms

We call formulas of the form p or ¬p, where p is a primitive proposition,classical literals and use letters a, b, c to denote them. We call formulasof the form a, 2a, or 3a atoms and use letters A, B, C to denote them.

A clause is a formula of the form 2s(A1∨ . . .∨An∨¬B1∨ . . .∨¬Bm),where s,m, n ≥ 0. The sequence 2s is called the modal context of theclause1. If s = 0 then the clause is called a simple clause. Note that themodal depth of a clause is not greater than the length of its modal contextplus 1.

A formula set is sometimes considered as the conjunction of its for-mulas, in particular when we are talking about length, modal depth, orsatisfiability.

A formula is in negative normal form if it does not contain the con-nective →, and the connective ¬ can occur only immediately before aprimitive proposition. Every formula can be transformed to the equiva-lent negative normal form in the usual way. A formula is called negativeif in its negative normal form every primitive proposition is prefixed bynegation. A formula is called non-negative if it is not negative, and positiveif its negation is a negative formula.

A formula ϕ is called a Horn formula iff one of the following conditionsholds:

– ϕ is a primitive proposition;– ϕ is a negative formula;

1 Assume that the modal context of 2s2p is 2s+1.

322

Page 332: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

– ϕ = 2ψ, or ϕ = 3ψ, or ϕ = (ψ∧ζ), where ψ and ζ are Horn formulas;– ϕ = (ψ → ζ), where ψ is a positive formula and ζ is a Horn formula;– ϕ is a disjunction of a negative formula and a Horn formula.

A clause is called a Horn clause if it is a Horn formula.Our definitions of Horn clauses/formulas are different than the one of

Chen and Lin [3]. A Horn clause by our definition is also a Horn clause bythe definition of Chen and Lin, and the latter is a Horn formula by ourdefinition, but not vice versa. These definitions, however, are equivalent.As stated by Lemma 2 given below, every Horn formula ϕ can be trans-lated to a set X of Horn clauses such that for any normal modal logic L,ϕ is L-satisfiable iff X is L-satisfiable.

A positive propositional modal logic program is a finite set of rules ofthe following form: 2s(B1 ∧ . . . ∧ Bk → A), where s ≥ 0, k ≥ 0, andA,B1, . . . , Bk are atoms of the form p, 2p, or 3p, where p is a primitiveproposition.

Formula sets X and Y are said to be equisatisfiable in a logic L (orL-equisatisfiable) iff (X is L-satisfiable iff Y is L-satisfiable).

Lemma 2. For any formula set X, there exists a clause set Y such that:

– X and Y are equisatisfiable in any normal modal logic.– If X is a set of Horn formulas, then Y is a set of Horn clauses.– The modal depth of Y is equal to the modal depth of X, and the length

of Y is of quadratic order in the length of X.

Moreover, if X is a set of Horn formulas and Y is divided into two groupsP and Q such that P contains only non-negative clauses and Q containsonly negative clauses, then P can be treated as a positive program, andX is L-satisfiable iff P 2L ¬Q, where L is any normal modal logic. Thetranslation from X to Y is computable in polynomial time.

The proof for the case when X is a set of Horn formulas can be foundin [9]. The proof for the other case is similar. The translation technique isbased on replacing a complicated formula by a fresh primitive propositionand “defining” that primitive proposition by the formula. For example,2s(3ϕ∨ψ), where s ≥ 0 and ϕ is not a primitive proposition, is replacedby 2s(3p∨ψ) and 2s+1(¬p∨ϕ), where p is a fresh primitive proposition.

2.3 Ordering Kripke Models

Let M = 〈W, τ,R, h〉 and N = 〈W ′, τ ′, R′, h′〉 be Kripke models. We saythat M is less than or equal to N w.r.t. a binary relation r ⊆ W ×W ′,and write M ≤ N w.r.t. r, if the following conditions hold:

323

Page 333: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. r(τ, τ ′)2. ∀x, x′, y R(x, y) ∧ r(x, x′)→ ∃y′ R′(x′, y′) ∧ r(y, y′)3. ∀x, x′, y′ R′(x′, y′) ∧ r(x, x′)→ ∃y R(x, y) ∧ r(y, y′)4. ∀x, x′ r(x, x′)→ (h(x) ⊆ h′(x′)).

The first three conditions state that r is a bisimulation of the framesof M and N . Intuitively, r(x, x′) states that the world x is less than orequal to x′.

We say that a model M is less than or equal 2 to N , and write M ≤ N ,if M ≤ N w.r.t. some r. This relation is a pre-order [9]. Also see [9] forthe proof of the following lemma.

Lemma 3. Suppose that M ≤ N . Then M � ϕ implies N � ϕ for everypositive formula ϕ.

Let P be a positive program in a normal modal logic L. We say thatM is the least L-model of P if M is an L-model of P and M is less thanor equal to every L-model of P . Observe that if P is a positive programin a normal modal logic L, and M is the least L-model of P , then for anypositive formula ϕ, M � ϕ iff P �L ϕ.

A model M is called the least flat model of a positive program P if itis a flat model of P and is less than or equal to any flat model of P . In[9], we showed that any positive modal logic program that has some flatmodel has the least flat model, which can be constructed in polynomialtime and has polynomial size.

3 New Results

We first consider the complexity of the satisfiability problem of sets ofHorn formulas with modal depth bounded by k ≥ 2 in the logics K4,KD4, and S4.

If X and Y are formula sets then we write X;Y to denote the unionof them. We write X;ϕ for X; {ϕ}. We need the two following auxiliarylemmas. The first one is used to reduce lengths of modal contexts ofclauses.

Lemma 4. In the following, let p and q be new primitive propositions(i.e. p and q occur only at the indicated positions) and ϕ a simple clause.Then the following pairs of formula sets are equisatisfiable in any normalmodal logic that is an extension of K4.2 This kind of “equality” is induced by the pre-order ≤. By Lemma 3, if M is equal

to N (i.e. M ≤ N and N ≤ M) then for every positive formula ϕ, M � ϕ iff N � ϕ.

324

Page 334: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(1) X; 22ϕ and X; 22p; 2(¬p ∨ ϕ)(2) X; 22kϕ and X; 2kq; 2(¬q ∨2kϕ) where k ≥ 2(3) X; 22k+1ϕ and X; 2k+1q; 2(¬q ∨2kϕ) where k ≥ 1(4) X; 2(a ∨22kϕ) and X; 2(a ∨2kq); 2(¬q ∨2kϕ) where k ≥ 1(5) X; 2(a ∨22k+1ϕ) and X; 2(a ∨2k+1q); 2(¬q ∨2kϕ) where k ≥ 0

Proof. →) Choose one of the pairs. Suppose that the LHS set is satisfiedin a model M = 〈W, τ,R, h〉. Let M ′ = 〈W, τ,R, h′〉 with x ∈ h′(u) iffx ∈ h(u) for x 6= p and x 6= q, p ∈ h′(u) iff M,u � ϕ, and q ∈ h′(u) iffM,u � 2kϕ, where p and q are the new primitive propositions. It is easilyseen that the RHS set is satisfied in M ′.←) Choose one of the pairs. We show that the RHS formula set implies

the LHS set in any normal modal logic that is an extension of K4. Theassertion holds for the pair (1) because the following formulas are K4-valid.

2(¬p ∨ ϕ)→ 22(¬p ∨ ϕ)

22p ∧22(¬p ∨ ϕ)→ 22ϕ

The assertion holds for the pair (2) because the following formulas areK4-valid.

2(¬q ∨2kϕ)→ 2k(¬q ∨2kϕ)

2kq ∧2k(¬q ∨2kϕ)→ 22kϕ

The assertion holds for the pair (4) because the following formulas areK4-valid.

2(¬q ∨2kϕ)→ 2k+1(¬q ∨2kϕ)

2(a ∨2kq) ∧2k+1(¬q ∨2kϕ)→ 2(a ∨22kϕ)

For similar reasons, the assertion holds for the pairs (3) and (5).

Lemma 5. Let L be a normal modal logic that is an extension of K4.Every formula set X can be translated to an L-equisatisfiable set Y ofclauses with modal depth bounded by 2. Furthermore, if X is a set ofHorn formulas then Y is a set of Horn clauses. The translation can bedone in polynomial time and the length of Y is bounded by a polynomialin the length of X.

Proof. By Lemma 2, we can translate X to a clause set Z such that:

– X and Z are L-equisatisfiable;– if X is a set of Horn formulas then Z is a set of Horn clauses;

325

Page 335: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

– the modal depth of Z is equal to the modal depth of X, and the lengthof Z is of quadratic order in the length of X;

– the translation is done in polynomial time.

We refer to the pairs of equisatisfiable formula sets given in Lemma 4as translation rules (with left to right direction of application). We thenapply3 these translation rules to Z. We apply the rule (1) only when themodal depth of ϕ is 1, and the rule (5) only when k ≥ 1, or k = 0 andϕ is not a classical literal. We apply the rules until no more changes canbe made to the set. Let Y be the resulting set. Observe that the modaldepth of Y is bounded by 2.

Observe that each of the applications decreases the modal depth ofsome formula of the set by a half (with an inaccuracy up to 2) and in-creases the length of the set by a constant number (of symbols). Hencethere exists a constant h such that we can decrease the modal depth ofthe set by a half (with an inaccuracy up to 2) while the length of theset increases not more than h times. Hence the process terminates inpolynomial time. It is easily seen that the length of Y is bounded by apolynomial in the size of Z, and Y is a set of Horn clauses if so is Z.

Therefore, the translation from X to Y (via Z) is done in polynomialtime, the length of Y is bounded by a polynomial in the length of X, andY is a set of Horn clauses if X is a set of Horn formulas.

As a consequence we have the following result:

Theorem 1. The complexity of the satisfiability problem of sets of Hornformulas with modal depth bounded by k ≥ 2 in the logics K4, KD4, andS4 is PSPACE-complete.

This theorem follows from the above lemma and the reason that thesimilar problem without bounding modal depth is PSPACE-complete [3].The assertion for S4 has been previously proved by Chen and Lin [3].

By this theorem, the complexity of the satisfiability problem of for-mula sets (without the Horn restriction) with modal depth bounded byk ≥ 2 in K4, KD4, and S4 is PSPACE-complete (the upper bound followsfrom [8]).

We now study the complexity of the same problem for the modallogic K .

Theorem 2. The complexity of the satisfiability problem of sets of Hornformulas with modal depth bounded by k ≥ 2 in the logic K is NP-complete.3 Each application of a rule is done for the whole formula set but not a fragment.

326

Page 336: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Proof. The upper bound follows from Halpern [6]. For the lower bound,we use a reduction from the 3SAT problem, which is known to be NP-hard. The 3SAT problem is to check satisfiability of a clause set X ={C1, . . . , Cn}, where Ci = ci1 ∨ ci2 ∨ ci3 and ci1, ci2, ci3 are classicalliterals. Given such a set X, we construct in polynomial time a set Y ofHorn formulas with modal depth bounded by 2 such that X is satisfiableiff Y is K -satisfiable.

Let t and f be new propositions, which informally stand for “true” and“false”. The presence of the formula 2f (resp. 3t) at a world w informallysays that there are no worlds (resp. there is some world) accessible fromw. Let Y be the set consisting of the following formulas:

3pi,3qi,¬3(pi ∧ qi),

3(pi ∧2f) ∧3(qi ∧2f)→ ci1,

3(pi ∧3t)→ ci2,

3(qi ∧3t)→ ci3,

¬32f,22t,

for 1 ≤ i ≤ n, and pi and qi are new propositions. Note that Y containsonly Horn formulas with modal depth bounded by 2.

Suppose that X is satisfied by a variable assignment V . We show thatY is K -satisfiable. Let M = 〈W, τ,R, h〉 be a model defined as follows:

– W = {τ, w1p, w1q, . . . , wnp, wnq, u},– R = {(τ, wip), (τ, wiq) | 1 ≤ i ≤ n} ∪

{(wip, u) | 1 ≤ i ≤ n and V (ci2)} ∪{(wiq, u) | 1 ≤ i ≤ n and V (ci3)},

– h(τ) = {p |V (p)}, h(u) = {t},and h(wip) = {pi}, h(wiq) = {qi}, for 1 ≤ i ≤ n.

It is easy to verify that M � Y . Therefore Y is K -satisfiable.Now suppose that Y is K -satisfiable. We show that X is satisfiable.

Let M be a model of Y . Let wip, wiq be worlds accessible from τ suchthat M,wip � pi and M,wiq � qi, for 1 ≤ i ≤ n. If there exists a world ac-cessible from wip, then M, τ � 3(pi∧3t), and hence M, τ � ci2. Similarly,if there exists a world accessible from wiq, then M, τ � ci3. If there are noworlds accessible from wip or wiq, then M, τ � 3(pi ∧ 2f) ∧3(qi ∧ 2f),and hence M, τ � ci1. Consequently, M, τ � Ci, for 1 ≤ i ≤ n. ThereforeM, τ � X, and X is satisfiable.

327

Page 337: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

In the remainder of this section, we study the satisfiability problemof modal formulas with modal depth bounded by 1. The problem is NP-complete, and for the Horn fragment it is PTIME-complete, for all ofthe modal logics considered in this work. Some parts of these resultsimmediately follow from known ones. We complete the picture by thetwo following theorems.

Theorem 3. The complexity of the satisfiability problem of formulas withmodal depth bounded by 1 in the logics K4, KD4, and S4 is NP-complete.

Proof. The lower bound NP-hard follows from the fact that the satisfia-bility problem in the classical propositional logic is NP-complete. For theupper bound, let L be one of the logics K4, KD4, S4, and let X be anyL-satisfiable formula set with modal depth bounded by 1.

It can be proved that X has an L-model M = 〈W, τ,R, h〉 such thatfor any u and v different to τ , if R(τ, u) and R(u, v) hold, then u = v.In fact, if M ′ = 〈W, τ,R′, h〉 is an L-model of X, then by deleting edges(u, v) with u 6= τ from R′ and adding edges (u, u) for u 6= τ to the frame,we obtain such a mentioned L-model M of X.

An L-model M of X with the mentioned frame restriction can be non-deterministically constructed in polynomial time by building an L-modelgraph for X (see, e.g., [11, 5, 10] for the technique of building L-modelgraphs). Therefore the satisfiability problem of formulas with modal depthbounded by 1 in K4, KD4, and S4 belongs to the NP class.

Theorem 4. The complexity of the satisfiability problem of sets of Hornformulas with modal depth bounded by 1 in K, K4, KD4, and S4 isPTIME-complete.

Proof. The lower bound PTIME-hard follows from the result by Jonesand Laaser [7] that the complexity of the satisfiability problem of sets ofHorn formulas in the classical propositional logic is PTIME-complete.

By the result of [9], every positive modal logic program with modaldepth bounded by 1 has the least KD4-model and the least S4-model,which can be constructed in polynomial time and have polynomial size.Consequently, by Lemmas 2 and 1, the problem of checking satisfiabilityof sets of Horn formulas with modal depth bounded by 1 in KD4 and S4is decidable in PTIME.

It remains to show that the similar problem for the logics K and K4 isdecidable in PTIME. Let L denote K or K4, and P be any positive modallogic program with modal depth bounded by 1. Let M = 〈W, τ,R,H〉 bethe model graph constructed as follows.

328

Page 338: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. Let W = {τ, ρ}, R = {(τ, ρ)}, H(τ) = P , H(ρ) = ∅.2. For every w ∈W , and every ϕ ∈ H(w),

(a) Case ϕ = (B1 ∧ . . . ∧ Bk → A) : if M,w � Bi for all 1 ≤ i ≤ k,then add A to H(w);

(b) Case ϕ = 2ψ : add ψ to every world u accessible from w;(c) Case ϕ = 3p : if M,w 2 p then add a new world u with content{p} to W and connect w to u (i.e. let W = W ∪ {u}, H(u) = {p},R = R ∪ {(w, u)}).

3. While some change occurred, repeat step 2.

Observe that, for any w and u, R(w, u) holds only when w = τ (sincethe modal depth is bounded by 1). Hence, the above algorithm terminatesin polynomial time. It can be shown by induction on the structure of ϕthat for any w ∈W and any ϕ ∈ H(w), M,w � ϕ. Hence M is a K -modelof P . By the mentioned property of R, M is also a K4-model of P .

If N = 〈W ′, τ ′, R′, h′〉 is a model of P such that R′ 6= ∅ and for any x,y, R′(x, y) holds only when x = τ , then M ≤ N . This claim can be provedby showing that it is an invariant of the loop of the above algorithm thatthere exists a relation r ⊆W×W ′ such that the following assertions hold:

– r(τ, τ ′)– ∀x R(τ, x)→ ∃x′ R′(τ ′, x′) ∧ r(x, x′)– ∀x′ R′(τ ′, x′)→ ∃x R(τ, x) ∧ r(x, x′)– ∀x, x′ ∀ϕ ∈ H(x) r(x, x′)→ N,x′ � ϕ

Such relations r can be built as follows: After the execution of step 1,let r = {(τ, τ ′)} ∪ {(ρ,w′) | R′(τ ′, w′)}, and after each execution of step2c, let r = r ∪ {(u, u′) | R′(τ ′, u′) and p ∈ h′(u′)}.

If P has a flat model, then let M ′ be the least flat model of P , elselet M ′ = M . Both M and M ′ can be constructed in polynomial time andhave size bounded by a polynomial in the size of P .

We claim that for any positive formula ϕ with modal depth boundedby 1, P 2L ϕ iff M 2 ϕ or M ′ 2 ϕ. The “if” part clearly holds. For the“only if” part, suppose that P 2L ϕ, where ϕ is a positive formula withmodal depth bounded by 1. It follows that there exists an L-model N ofP such that N 2 ϕ. Let N|1 be the model obtained from N by deletingall edges not starting from τ . We have N|1 � P and N|1 2 ϕ, because themodal depths of P and ϕ are bounded by 1. If N|1 is a flat model, thenM ′ is the least flat model of P , and hence M ′ 2 ϕ. Otherwise, M ≤ N|1,and hence M 2 ϕ.

By Lemmas 2 and 1, we conclude that checking satisfiability of sets ofHorn formulas with modal depth bounded by 1 in K and K4 is decidablein PTIME.

329

Page 339: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4 Conclusions

We have summarized the complexity of the satisfiability problem in allof the 15 basic normal modal logics under the restriction to the Hornfragment and/or bounded modal depth. To fulfill the complexity table, wehave given some new results. Our Theorems 1 and 2 show that the modallogics K, K4, and KD4 are hard even under the mentioned restrictions.The restriction of modal depth to 1 is quite tight and the correspondingfragments are rather useless. However, our results for that case are stillinteresting from the theoretical point of view.

Acknowledgements

The author would like to thank professor Andrzej Sza las and the anony-mous reviewers for helpful comments.

References

1. P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge UniversityPress, 2002.

2. A. Chagrov and M. Zakharyaschev. Modal Logic. Clarendon Press, Oxford, OxfordLogic Guides 35, 1997.

3. C.C. Chen and I.P. Lin. The computational complexity of the satisfiability ofmodal Horn clauses for modal propositional logics. Theoretical Computer Science,129:95–121, 1994.

4. L. Farinas del Cerro and M. Penttonen. A note on the complexity of the satisfia-bility of modal Horn clauses. Logic Programming, 4:1–10, 1987.

5. R. Gore. Tableau methods for modal and temporal logics. In D’Agostino, Gab-bay, Hahnle, and Posegga, editors, Handbook of Tableau Methods, pages 297–396.Kluwer Academic Publishers, 1999.

6. J.Y. Halpern. The effect of bounding the number of primitive propositions andthe depth of nesting on the complexity of modal logic. Artificial Intelligence,75(2):361–372, 1995.

7. N.D. Jones and T.W. Laaser. Complete problems for deterministic polynomialtime. Theoretical Computer Science, 3:105–112, 1976.

8. R. Ladner. The computational complexity of provability in systems of modalpropositional logic. SIAM Journal of Computing, 6:467–480, 1977.

9. L.A. Nguyen. Constructing the least models for positive modal logic programs.Fundamenta Informaticae, 42(1):29–60, 2000.

10. L.A. Nguyen. Sequent-like tableau systems with the analytic superformula prop-erty for the modal logics KB, KDB, K5, KD5. In Roy Dyckhoff, editor, Proceedingsof TABLEAUX 2000, LNAI 1847, pages 341–351. Springer, 2000.

11. W. Rautenberg. Modal tableau calculi and interpolation. Journal of PhilosophicalLogic, 12:403–423, 1983.

330

Page 340: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

PSPACE decision procedure for some transitive

modal logics

I. Shapirovsky

Institute for Information Transmission ProblemsB.Karetny 19,Moscow, Russia, 101447E-mail: [email protected]

Keywords: modal logic, computational complexity, selective filtration

1

331

Page 341: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1 Introduction

In this paper we describe a new method for constructing PSPACE decisionprocedure for transitive modal logics.

The idea of our method is to reduce the satisfiability problem for a modallogic to the satisfiability problem on some tree-like frames using the selectivefiltration.

We show how this method works for two particular logics L1 and L2.These logics were introduced in the study of modalities in Minkowski space-time [4],[7]. In [7] the Kripke–Gabbay method of selective filtration (see e.g.[2]) in a combination with the method of maximal points [3] was used toprove the finite model property (FMP) of these to logics. This proof actuallyallows us to reduce the satisfiability problem for L1, L2 to the satisfiabilityproblem on certain frames. In spite of the exponential size of these frames(with respect to the length of a given formula), the algorithm works withina polynomial space.

We also show how to transfer our method to some other transitive logics.It is well known that the logic K4 and many of its extensions (for example,S4) are PSPACE-complete [6]. The proposed method provides a new proofof PSPACE-decidability, for example, for the logics S4, S4.2, S4.1.2.

For the logic L2 we also obtain the PSPACE-hardness by applying Lad-ner’s method of reduction of the QBF-validity problem to the modal satisfia-bility problem (the PSPACE-hardness for L1 follows from Ladner’s theorem[6]). So we obtain that L1, L2 are PSPACE-complete.

2 Preliminaries

In this paper by modal logic we always mean a normal monomodal logiccontaining K4. For a modal logic Λ and a modal formula A, the notationΛ+A denotes the smallest modal logic containing Λ∪{A}. We assume that♦, →, ⊥ are the basic connectives, and �, ¬, ∨, ∧, � are derived. PVdenotes the set of propositional variables. For a formula B, let Sub B be theset of all subformulas of B. Here are the names for some particular modalaxioms:

A4 := ♦♦p→ ♦p, AT := p→ ♦p,A1 := �♦p→ ♦�p, A2 := ♦�p→ �♦p,AD := ♦�, Ad := Ad1 = ♦p→ ♦♦p,Ad2 := ♦p1 ∧ ♦p2 → ♦(♦p1 ∧ ♦p2);

and the names for some modal logics:

K4 := K +A4, S4 := K4 +AT,S4.2 := S4 +A2, S4.1.2 := S4.2 +A1,L1 := K4 +AD +Ad2, L2 := L1 +A2.

2

332

Page 342: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

By (Kripke) frame we mean a transitive monomodal frame (W,R). A(Kripke) model is a Kripke frame with a valuation: M = (W,R, θ), whereθ : PV −→ 2W , 2W denotes the power set of W . For a model M = (W,R, θ)or a frame F = (W,R), the notation x ∈ M or x ∈ F means x ∈ W .As usual, for x ∈ W, V ⊆ W we denote R(x) := {y | xRy}, R(V ) :=⋃x∈V

R(x), R|V = R ∩ V × V . Also we put W x := {x} ∪ R(x), F x :=

(W x, R|W x). F x is called a cone in F . A model M1 = (W1, R1, θ1) is a(weak) submodel of M = (W,R, θ) (notation: M1 ⊆M) if W1 ⊆W , R1 ⊆ R,θ1(q) = θ(q) ∩ 2W1 for every q ∈ PV . If also R1 = R|W1 then M1 is calleda restriction of M to W1 and denoted by M |W1. We put Mx := M |W x.

The sign � denotes the truth at a point of a Kripke model and also thevalidity in a Kripke frame. For a class of frames F , L(F) denotes the set ofall formulas that are valid in all frames from F . For a single frame F , L(F )abbreviates L({F}).

Recall that a cluster in (W,R) is an equivalence class under the relation∼R:= (R∩R−1)∪ Id, where Id is the equality relation; a degenerate clusteris an irreflexive singleton; a simple cluster is an reflexive singleton. For apoint x, x denotes its cluster. C0 denotes a degenerate cluster, C1 denotesa simple cluster, Cn denotes an n-element cluster for n ≥ 2. For clustersC,D ∈W/ ∼R we put

C ≤R D := D ⊆ R(C), C <R D := C ≤R D and C = D.

Note that the relations ≤R, <R are transitive and antisymmetric, and <R

is irreflexive. For clusters C1, C2 we say that C2 is successor of C1 (C1 isancestor of C2), if C1 <R C2 and there is no cluster C, such that C1 ≤R Cand C ≤R C2. For x, y ∈ F we say that y is successor x (x is ancestor ofy), if y is successor of x. The frame F/ ∼R:= (W/ ∼R,≤R) is called theskeleton of F . A point x ∈ W is called maximal (minimal) if its cluster ismaximal (minimal) in F/ ∼R.

In this paper a frame (W,R) is called rooted if for some singleton cluster{x} W = R(x) ∪ {x}.

Consider a finite frame F = (W,R). For C ∈ W/ ∼R let v(C) be thenumber of all its successors. A singleton cluster C is called adherent if C hasa single successor D, and D is not a singleton. For a <R-chain Σ in W/ ∼R

h(Σ) denotes the number of all non-adherent clusters in Σ. We put:

v(F ) := max{v(C) | C ∈W/ ∼R},h(F ) := max{h(Σ) | Σ is a <R -chain in F/ ∼R},

m(F ) := max{|C| | C ∈W/ ∼R}.The disjoint union F1�F2 and the ordinal sum F1 +F2 of frames F1, F2

are defined in a standard way. The notation F1 � F2 means that thereexists a p-morphism from F1 onto F2.

One can easily check the following

3

333

Page 343: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

LEMMA 1 F � Ad2 iff F is 2-dense, i.e., the following holds:

∀x∀y1∀y2(xRy1 & xRy2 → ∃z(xRz & zRy1 & zRy2)).

Let us recall the definitions of the chronological future relation ≺ inMinkowski spacetime Rn, n > 1:

(x1, . . . , xn) ≺ (y1, . . . , yn) ⇔n−1∑i=1

(yi − xi)2 < (xn − yn)2 & xn < yn.

Here are the main completeness results for the logics L1 and L2 [7].

THEOREM 2 L(Rn,≺) = L2, n ≥ 2.

THEOREM 3 Let X be an open connected domain in R2 bounded by a closedsmooth curve. Then L(X,≺) = L1.

THEOREM 4 Let X be an open convex polygon in R2. Then L(X,≺) ∈{L1,L2}.

3 Strong FMP via selective filtration

Let us recall the notion of selective filtration.

DEFINITION 5 Let M be a Kripke model, Ψ a set of formulas closed undersubformulas. A submodel M1 ⊆M (with the relation R1) is called a selectivefiltration of M through Ψ if for any x ∈M1, for any formula A

♦A ∈ Ψ & M,x � ♦A⇒ ∃y ∈ R1(x) M,y � A.

The following Lemma is proved easily by induction on the length of aformula A.

LEMMA 6 If M1 is a selective filtration of M through Ψ, then for anyx ∈M1, for any A ∈ Ψ

M,x � A⇔M1, x � A.

Next, we formulate the maximality property of canonical model.

LEMMA 7 Let M be the canonical model of a modal logic Λ, and assumethat a formula B is satisfied at some x ∈ M. Consider the set of all thoseclusters in Mx, in which B is satisfied:

Γ := {C ⊆ Mx | ∃y ∈ C B ∈ y}.

Then M|⋃ Γ contains a maximal cluster (with respect to the relation ≤R).

4

334

Page 344: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The proof of this Lemma (in this formulation) was given in [7].Let F1 be the class of all finite rooted 2-dense serial frames, and let

F2 := {F + C | F ∈ F1, C is a finite non-degenerate cluster}

It was shown in [7] that the logics L1 and L2 have FMP. The proof givenin [7] actually yields the following strong finite model property (SFMP) ofL1 and L2.

LEMMA 8 Let B be a modal formula, n = |Sub B|.(i) B is L1-satisfiable ⇔ B is satisfiable in some F ∈ F1, such that h(F ) ≤n, v(F ) ≤ n, m(F ) ≤ n.

(ii) B is L2-satisfiable ⇔ B is satisfiable in some F ∈ F2, such that h(F ) ≤n+ 1, v(F ) ≤ n, m(F ) ≤ n.

Let FS4 be the class of all finite rooted reflexive transitive frames, andlet

FS4.2 := {F + C | F ∈ FS4, C is a finite non-degenerate cluster},

FS4.1.2 := {F + C | F ∈ FS4, C is a simple cluster}.Using Lemmas 6,7, it is possible to prove the SFMP of the logics S4, S4.2, S4.1.2

in the same way as for the logics L1, L2:

LEMMA 9 Let B be a modal formula, n = |Sub B|.(i) B is S4-satisfiable ⇔ B is satisfiable in some F ∈ FS4, such that h(F ) ≤n, v(F ) ≤ n, m(F ) ≤ n.

(ii) B is S4.2-satisfiable ⇔ B is satisfiable in some F ∈ FS4.2, such thath(F ) ≤ n+ 1, v(F ) ≤ n, m(F ) ≤ n.

(ii) B is S4.1.2-satisfiable ⇔ B is satisfiable in some F ∈ FS4.1.2, such thath(F ) ≤ n+ 1, v(F ) ≤ n, m(F ) ≤ n.

4 PSPACE upper bound for L1 and L2

In this section we show that the satisfiability problem for L1 and L2 is inPSPACE.

Let Tn,1 be the class of all frames isomorphic to C0 + Cn. We put

Tn,k+1 := {F + (F1 � . . . � Fn) | F ∈ Tn,1, F1, . . . , Fn ∈ Tn,k},

T+n,k := {F + F1 | F ∈ Tn,k, F1 ∈ Tn,1}.

Let Tn,k ∈ Tn,k, T+n,k ∈ T+

n,k (Figure 1).Using the standard unravelling technique, one can check the following

5

335

Page 345: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Figure 1: Tn,k+1, T+n,k+1

LEMMA 10

(i) Let F ∈ F1, h(F ) ≤ k, v(F ) ≤ n, m(F ) ≤ n. Then Tn,k � F .

(ii) Let F ∈ F2, h(F ) ≤ k + 1, v(F ) ≤ n, m(F ) ≤ n. Then T+n,k � F .

Proof. By induction on h(F ). �

The following Lemma allows us to reduce the satisfiability problem forthe logics L1, L2 to the satisfiability problem on the frames Tn,k, T

+n,k re-

spectively.

LEMMA 11 Let B be a modal formula, n = |Sub B|.(i) B is L1-satisfiable ⇔ B is satisfiable at the root of Tn,n.

(ii) B is L2-satisfiable ⇔ B is satisfiable at the root of T+n,n.

Proof.

(i) (⇒). By Lemma 8, there exists F ∈ F1 such that formula B is satisfiedat the root of F and h(F ) ≤ n, v(F ) ≤ n, m(G) ≤ n. By Lemma 10,Tn,n � F .(⇐). Note that L(Tn,n) ⊃ L1.

(ii) Similar to (i). �

Now our aim is to describe polynomial algorithms, deciding whether agiven formula is satisfiable at the root of Tn,n, T+

n,n.

6

336

Page 346: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Consider a formula B. Let A1, . . . , An be all subformulas of B, p1, . . . , pm

all variables of B. Let us order Sub B as follows: for i ≤ m we put Ai := pi,and if Ai is a subformula of Aj then i ≤ j. We can achieve this within n lognunits of space making an array of pointers to the beginnings of subformulas.Note that An = B.

For an arbitrary formula A we put A0 := ¬A, A1 := A. Let ξ =(ξ1, . . . , ξn) ∈ {0, 1}n. We put Bξ :=

∧iAξi

i .

A boolean vector ξ ∈ {0, 1}n is called B-consistent in a frame F at apoint x (notation: F, x �B ξ) if for some valuation θ we have F, θ, x � Bξ.Boolean vectors ξ1, . . . , ξl ∈ {0, 1}n are called simultaneously B-consistentin F on a tuple (y1, . . . , yl) ∈ W l (notation: F, (y1, . . . , yl) �B (ξ1, . . . , ξl))if for some valuation θ we have F, θ, yi � Bξi

for all i = 1, . . . , l.If x is the root of F , then the notation F �B ξ means F, x �B ξ. If also y

is the root of G then the notation F +G �B (ξ1, ξ2) means F +G, (x, y) �B

(ξ1, ξ2). From Lemma 11 we obtain

LEMMA 12

(i) B is L1-satisfiable ⇔ there exists ξ ∈ {0, 1}n such that Tn,n �B ξ andξn = 1.

(ii) B is L2-satisfiable ⇔ there exist ξ, ξmax ∈ {0, 1}n such that T+n,n �B

(ξ, ξmax) and ξn = 1.

Next, one can check the following

LEMMA 13 Let F0, F1, G1, . . . , Gl be rooted frames, y the root of F1, xi

the root of Gi, ξ, ξ1, . . . , ξl ∈ {0, 1}n. Then for the frames F = F0 + (G1 �. . . �Gl), F+ = F + F1 we have:

(i) F, (x1, . . . , xl) �B (ξ1, . . . , ξl) ⇔ Gi �B ξi, i = 1, . . . , l.

(ii) F+, (x1, . . . , xl, y) �B (ξ1, . . . , ξl, ξ) ⇔Gi + F1, (xi, y) �B (ξi, ξ), i = 1, . . . , l.

Consider the frame F = F0 +G, where G has exactly n minimal pointsx1, . . . , xn, F0 ∈ Tn,1, x is the root of F0 (Figure 2).

Note that the truth value of the formula B at x in a model over F is fullydetermined by the truth values of its variables in F0 and the truth valuesof its subformulas at x1, . . . , xn. In Appendix we describe the algorithmSATLocal, deciding whether F, (x, x1, . . . , xn) �B (ξ, ξ1, . . . , ξn), providedG, (x1, . . . , xn) �B (ξ1, . . . , ξn).

LEMMA 14 Let ξ, ξmax ∈ {0, 1}n.

(i) Tn,k+1 �B ξ ⇔ there exist ξ1, . . . , ξn ∈ {0, 1}n such that Tn,k �B ξi, i =1 . . . n and SATlocal(ξ, ξ1, . . . , ξn) =true.

7

337

Page 347: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Figure 2: F0 +G

(ii) T+n,k+1 �B (ξ, ξmax) ⇔ there exist ξ1, . . . , ξn ∈ {0, 1}n such that T+

n,k �B

(ξi, ξmax), i = 1 . . . n and SATlocal(ξ, ξ1, . . . , ξn) =true.

Proof. By definition Tn,k+1 = F + (G1 � . . . � Gn), T+n,k+1 = Tn,k+1 + F1,

where F, F1 ∈ Tn,1, G1, . . . , Gn ∈ Tn,k. Let x, x1, . . . , xn, y be the roots ofF,G1, . . . , Gn, F1 respectively.

(i) (⇒). For some θ we have: Tn,k+1, θ, x � Bξ. Obviously, for someξ1, . . . , ξn ∈ {0, 1} we have Tn,k+1, θ, xi � Bξi

, i = 1, . . . , n . It follows thatξ, ξ1, . . . , ξn are simultaneously B-consistent on the tuple (x, x1, . . . , xn), andthus SATlocal(ξ, ξ1, . . . , ξn)=true.(⇐). By Lemma 13, ξ1, . . . , ξn are simultaneously B-consistent on the tuple(x1, . . . , xn). Since SATlocal(ξ, ξ1, . . . , ξn)=true, we have that ξ, ξ1, . . . , ξn

are simultaneously B-consistent on the tuple (x, x1, . . . , xn), and thus ξ isB-consistent at x.

(ii) (⇒). Similar to (i).(⇐). By Lemma 13, ξ1, . . . , ξn, ξmax are simultaneously B-consistent onthe tuple (x1, . . . , xn, y). Since SATlocal(ξ, ξ1, . . . , ξn)=true, we have thatξ, ξ1, . . . , ξn are simultaneously B-consistent on the tuple (x, x1, . . . , xn). Wehave for some θ1, θ2:

T+n,k+1, θ1, x � Bξ, T+

n,k+1, θ1, xi � Bξi, i = 1, . . . , n;

T+n,k+1, θ2, y � Bξmax

, T+n,k+1, θ2, xi � Bξi

, i = 1, . . . , n.

We define the valuation θ on T+n,k+1 as follows:

θ(p) := {x | x ∈ F, x ∈ θ1(p)} ∪ {x | x ∈ F, x ∈ θ2(p)}One can see that T+

n,k+1, θ, x � Bξ and T+n,k+1, θ, y � Bξmax

. �

8

338

Page 348: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Now let us give a recursive description of the algorithms SATtree andSATtree+ determining whether Tn,k �B ξ and T+

n,k �B (ξ, ξmax) (for thebasic case k = 1 these algorithms - SAT1 and SAT+

1 are constructed inAppendix).

Function SATtree(ξ, k) returns booleanBegin

if k = 1 then return(SAT1(ξ));for all ξ1, . . . , ξn ∈ {0, 1}n:

if SATlocal(ξ, ξ1, . . . , ξn) thenif

∧i SATtree(ξ

i, k − 1) thenreturn(true);

return(false);End.

Function SATtree+(ξ, ξmax, k) returns booleanBegin

if k = 1 then return(SAT+1 (ξ, ξmax));

for all ξ1, . . . , ξn ∈ {0, 1}n:if SATlocal(ξ, ξ1, . . . , ξn) then

if∧

i SATtree+(ξi, ξmax, k − 1) then

return(true);return(false);

End.

By Lemma 14, we obtain

LEMMA 15

(i) Tn,k �B ξ ⇔ SATtree(ξ, k) =true.

(ii) T+n,k �B (ξ, ξmax) ⇔ SATtree(ξ, ξmax, k) =true.

Function SATL1(B) returns booleanBegin

for all ξ ∈ {0, 1}n, ξn = 1: if SATtree(ξ, n) return(true);return(false);

End.

Function SATL2(B) returns booleanBegin

for all ξ, ξmax ∈ {0, 1}n, ξn = 1: if SATtree+(ξ, ξmax, n) return(true);return(false);

End.

By Lemma 12 and Lemma 15, we obtain

9

339

Page 349: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

THEOREM 16 B is Li-satisfiable ⇔ SATLi(B)=true, i = 1, 2.

The space used by SATLocal, SAT1, SAT+1 is O(n2). The depth of

recursion of SATLi is n, and the total amount of space required is O(n3).

5 PSPACE lower bound for L1 and L2

In this section we show that the satisfiability problems for L1 and L2 isPSPACE-hard.

Note that K ⊆ L1 ⊆ S4. Since satisfiability problem for all modallogics between K and S4 is PSPACE-hard (Ladner’s Theorem, [6]), we ob-tain that L1 is PSPACE-hard. Since S4 � A2, the logic L2 ⊆ S4. How-ever, the following slight modification of Ladner’s construction [6] provesthe PSPACE-hardness for all modal logics between K4 and S4.1.2.

Let A be a propositional logic formula, p1, . . . , pn be the all variables ofA, and β = Q1p1 . . . QnpnA, where Q1, . . . , Qn ∈ {∀,∃}. We put

Sβ :=∧

0≤i≤n

(qi → ♦(¬qi ∧ qi+1)),

Qβ :=∧

{i|Qi=∀}(qi−1 → ♦(qi ∧ pi) ∧ ♦(qi ∧ ¬pi)),

Pβ :=∧

1≤i≤n

((qi ∧ pi → �(qn → pi)) ∧ (qi ∧ ¬pi → �(qn → ¬pi))),

f(β) := q0 ∧ �(qn → A) ∧ �(Sβ ∧Qβ ∧ Pβ).

A straightforward argument shows that

β is valid ⇒ f(β) is S4.1.2-satisfiable;

f(β) is K4-satisfiable ⇒ β is valid.

Since the validity problem for prenex quantified boolean formulas isPSPACE-complete [10], we obtain

THEOREM 17 Let L be a modal logic, K4 ⊆ L ⊆ S4.1.2. Then the satis-fiability problem for L is PSPACE-hard.

Note that K4 ⊂ L2 ⊂ S4.1.2. By Theorems 17, 16, we obtain

THEOREM 18 L1, L2 are PSPACE-complete.

10

340

Page 350: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

6 Conclusion

Is is easy to generalize the method from Section 4 for logics satisfying theSFMP similarly to those described in Lemma 11.

Let us explain it in more detail. Set the polynomials Pm, Qm, Pr, Qr,Ph, Pv. Consider the rooted frames Fn = (W,R), F ′

n = (W ′, R′), |W | =Pm(n), |W ′| = Qm(n). Assume that for every xi, xj ∈ W (yi, yj ∈ W ′) itis possible to decide whether xiRxj (yiR

′yj) using Pr(n) (Qr(n)) units ofspace. Let Fn,1 be the class of all frames isomorphic to Fn, F′

n the class ofall frames isomorphic to F ′

n,

Fn,k+1 := {F + (F1 � . . . � Fl) | F ∈ Fn,1, F1, . . . , Fl ∈ Fn,k, l = Pv(n)},

F+n,k := {F + F ′ | F ∈ Fn,k, F

′ ∈ F′n}.

Let Fn,k ∈ Fn,k, F+n,k ∈ F+

n,k

Consider the logics L, L′ with the following SFMP:

finF: FormulaB is L-satisfiable ⇔B is satisfiable at the root of Fn,Ph(n), n =|Sub B|;

finF+: FormulaB is L-satisfiable ⇔B is satisfiable at the root of F+n,Ph(n), n =

|Sub B|.Then L,L′ ∈ PSPACE. Indeed, under the above restrictions it is clear

that there exist algorithms, working in polynomial (of n) space, for theframes Fn,k+1, Fn, F

′n, analogous to SATLocal, SAT1, SAT+

1 To checkthe satisfiability of a formula we need to use the recursion on depth Ph(n)(similar to SATL1 SATL2), hence these logics are in PSPACE.

Note that the SFMP of many logics can be obtained in the form finF,finF+ using the method of selective filtration in a combination with themethod of maximal points.

For example, consider the logics S4.2, S4.1.2. It is possible to prove thePSPACE-decidability of these logics by Ladner’s method [6]. We can alsoobtain this result by our method.

Let TSn,1 be the class of all frames isomorphic to C1 + Cn. We put

TSn,k+1 := {F + (F1 � . . . � Fn) | F ∈ TSn,1, F1, . . . , Fn ∈ TSn,k},

TS+n,k := {F + F1 | F ∈ TSn,k, F1 ∈ TSn,1}.

TS◦n,k := {F + F1 | F ∈ TSn,k, F1 is a simple cluster}.

Let TSn,k ∈ TSn,k, TS+n,k ∈ TS+

n,k, TS◦n,k ∈ TS◦

n,k.Using Lemma 9, one can check the following analogue of Lemma 11.

LEMMA 19 Let B be a modal formula, n = |Sub B|.

11

341

Page 351: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(i) B is S4-satisfiable ⇔ B is satisfiable at the root of TSn,n.

(ii) B is S4.2-satisfiable ⇔ B is satisfiable at the root of TS+n,n.

(iii) B is S4.1.2-satisfiable ⇔ B is satisfiable at the root of TS◦n,n.

It follows that these logics are in PSPACE, and by Theorem 17, weobtain that S4.2,S4.1.2 are PSPACE-complete.

Finally let us make some remarks on the non-transitive logic K40 :=K + ♦♦p→ ♦p∨ p, axiomatizing derivation in arbitrary topological spaces.Using the filtration method from [7], one can prove the finite model propertyof K40. Moreover, using the ideas of the present paper, we can show thatthe satisfiability problem for K40 is in PSPACE, the proof will be publishedin the sequel.

7 Appendix

Function SATlocal(ξ, ξ1, . . . , ξn) returns booleanBegin

for all θ ∈ {0, 1}(n+1)×m:rm θi

j is the trues value of pj at xj, C0 = {x0}, Cn = {x1, . . . , xn}rm We construct η ∈ {0, 1}(n+1)×n: ηi

j is the trues value of Aj at xi

begin for j := 1 . . . n: rm Aj ∈ Sub B.begin

for i := 0 . . . n: rm xi ∈ C0 + Cn.begin

if Aj = ⊥ then ηij := 0;

if j ≤ m then ηij := θi

j ;if Aj = As → Al then rm Note that s, l < j.

if ηis = 0 or ηi

l = 1 then ηij := 1 else ηi

j := 0if Aj = ♦As then rm Note that s < j.

beginηi

j = 0;for l := 1 . . . n: if ηl

s = 1 then ηij := 1;

for l := 1 . . . n: if ξls = 1 or ξl

j = 1 then ηij := 1;

end;end;

end;if (η0

1, . . . , η0n) = ξ then return(true);

end;return(false);

End.

Function SAT1(ξ) returns booleanBegin return(SATLocal(ξ, 0, . . . , 0); End.

12

342

Page 352: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Function SAT+1 (ξ, ξmax) returns boolean

Begin return(SATLocal(ξ, ξmax, 0, . . . , 0)∧SAT1(ξmax)); End.

8 Acknowledgements

The author is grateful to prof. Valentin Shehtman for his help and also tothe anonymous referee for their useful comments.The work on this paper was supported by RFBR, project No.02-01-22003and by CNRS.

References

[1] P. Blackburn, M. de Rijke and Y. Venema. Modal Logic. CambridgeUniversity Press, 2001

[2] A. Chagrov, M. Zakharyaschev. Modal logic. Oxford University Press,1997.

[3] K. Fine. Logics containing K4. I. Journal of Symbolic Logic, v. 39(1974),31-42.

[4] R. Goldblatt. Diodorean modality in Minkowski spacetime. Studia Log-ica, v. 39 (1980), 219-236.

[5] R. Goldblatt. Logic of time and computation. CSLI Lecture Notes, No.7. Stanford, 1987.

[6] R. Ladner. The computational complexity of provability in systemsof modal propositional logic. SIAM Journal of Computing, 6:467-480,1977.

[7] I. Shapirovsky, V. Shehtman. Chronological future modality inMinkowski spacetime. Advances in Modal Logic, Volume 4, 437-459.King’s College Publications, 2003.

[8] V.B. Shehtman. Modal logics of domains on the real plane. Studia Log-ica, v. 42 (1983), 63-80.

[9] E. Spaan. Complexity of Modal Logics. PhD thesis, University of Am-sterdam, 1993.

[10] L. J. Stockmeyer and A. R. Meyer. Word problems requiring exponen-tial time: preliminary report. In Proc. 5th ACM Symp. on Theory ofComputing, pp.1-9, 1973.

13

343

Page 353: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� ��� ������� � �� � ����� ������

�������� �������

� �����������

��� ������� ����� � ��� ��� � �� ��� � � ������� ����� �

���� ���� ������� ��� � ���� ����� ��� � �������� � ��� ����

�� �� ������ !� "��� �� ��������� �� #� $����� �� �� ���������

������ �� %��� �%��� ������� �� ��&���� ���� � ��� ���� ���� �

'����� � ��� ���� � �������� ��� ���� ���� ��� ��������� ���

����� ����� � � ���� ��%��� � ������������ ��� � (���� ���������

�������� ��� ���� ��� �� � ���� �����)��� �%� ������ �������

����� ������ ������� %���� �����)��� �%� � *���� ����� ��� � ��

��� %��� �%� ���� ���� ��� ����� � +%� ����� ,� � ��� - ����� �

������ % �� � ��� ����� ������ ./0�

����� ��������� ������� ��� ����� ���� �� � ���� ����� ����

��� �� �����) ���� � ����� ( ��� ����� �� ��� � %�� � ��������

������ % � ������� �� ������� ��� �� �� � ���� �� ��� �� %��

1 �� �)����� �� ��� ���� ���� �� ������� � � � �� ��� �� �%�����

��������� � ��� ������

���� ���� �� �� ����� �� ������� �� � �� �� ������ ��� ��� ��������

������� ��� ������������ ��������� �� ������������ ��� �� !��"�

344

Page 354: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

"�� % ����2� �� ����� ��� ��� ����� 3 ������� � ��%�� � ����

������ ���� ����� � � �� � � �+%������ � � ���� ����� �����

����� ��� �� ��� ���� � ������� 4 �������� ������� ,������ 5 $���

���- �� 6����������� 7 ,"��� 5 ���������-� ��������� ������� �)�

������� ���� %����� ��� � �� �% �� ���� ������ �� �� �%� � .80

�� � ��� ����� ��������� ��� .�0�

3 ����������� ������� � ���� ���� �� �� 6������%�7 �+%�������

������� ( ��� ����� � �� � .90 ���� � �� � ���%�� : ��� �+%�������

�� � ����� � ��� ��%�� � ��� ��� ���%�� ��� :� (� : � ���� ��

������� ���� � ���� ���� �%� ��� �������� � ��� ������ ���� ��� ��

� �� � ���� ��� �� �� ��� � % � � ��� �+%������� �������

*%� �������� � �� �� ��� ������ ��� .;0 .<0� =� ���� ���

� ���� �� : �� � ������ ���� � > ,����� ���� ��� �- ���� ������ ���

��� ��� ������ � ��� ���� � � ��� �������� �� �+%������� ������

� � �� ���� ������ ��� �+%������� ��� � � 3 ���������� ����� � �

���� � ��� ���� ��� ������ ���� � ��� �������� ������

�� �)����� � .<0 ��� ����� �

������� � ��� ���� ����� ( ��� �� � ����� ��,�- � � ��% ���? �� �%�

� � � �& � �� � �� ��� ��� ������ � ��� ��� ���� � ��,�- �� ��,�-�

@%� %�� � ������ � �� �� �� � � � �+%������� ,���� �� ��� ����

� ��- ����% � � � � �%�� ���� ���� ��� ������ �� �� ���%��

� �� ��+%��� ���� ��� ���� ���� �������� ��� %����� ��� ����������

� ���� ������ ��� ����� � � �%�� �� � �� ����� ��� ����� � ���

��� � %����� � ��� �� � ��� � ��%���� �� ��� ���� ��� 1 �� ���

����� ������ ���� ��� ��� ������ � � ��� �� ��������� ��� ������ ��

�)����� ��� ����� �� ��� ���� ����� ����� ���� ��� ��� �� ��� .�0�

8

345

Page 355: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�� � ������� ��� ����

��� � ��� % ������ �� ������� ��������� *%� �������� �� ����

����� ���� .<0� �� ���� ��� �� � ��� ����� ���%�� �%��� ��� �

�%����� �� � > ��� �� � � � � � ��� ��� ������ ��� ���� �������

��� �� ���� ������� ��� � � � ���� "�� ���� �� ��� �� � ��� ���%��

� �� % �� �� ��� ��� ������ ��� ��� �� ��� > ��� �� � � � ���

���� �������� � � ���� ��� ��� ������ �

������ ������ � ����� � � ������ ���� 3 ����� �������

���� � � �%��� � > ,����� � � � � ��- ����� � > � � � �� � � ����

���� �� � � �� ��� ���� ������� ������ � 3 ����� ��� ��� � �

� ���� � > ,�� �-� ����� � 4 � � 8� � � �������� �

A��%��� ��� �)����� � ��� ���%�� � % %��4

�,�- > �� �, � �- > ,� �, -- � �,�-� �,�� - > �� � ��,�- � �, -��

��������� � ������� ����� ��� ���� � ��� ��� ������ � � > ,�� �-�

����� � 4 ��� � 8� ? � ��� �� � � � �)����� � ����� 3 ���%�� �

������ ��� �� � ���� � � � �� � �, - ,� ����� ����4 ��� � -�

3 ���%�� � ���� � � ���� � ,����4 � � - �� �� � ��%� ��

����� ���� � ����� ������ ���� ��� �� 3 �� � ���%�� B � ���� � �

,����4 � � B- �� ����� B � ������ ( ��� ������ �� � �� �� �� ����

� � � B������

���� ��� ��%�� � � ���%�� ����� �� ��� ��� ������ ��%����

� �� �� ��� �) � �� � %�� ���� ��� ������ ���� ��� ���� � ��� ���

������ �

������� �� �� � > ,����� � � � � ��- � � ����� �� � � � � � �� �

���� � ����� � ���� � �� � �� � ��� � ,��� ��� ��� � � � � ����� ��- ����

/

346

Page 356: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

���� � > ��� � > �� � ��� ��� � � �� ���������� � �� ���� ��� �

,�- �� �� ���� � ����� � ����� � �� � �� ����� ���� �� %������ � �

�������� �� � � ������ � �� �� �� ��������� �� � �� �� �� �� ��� � ����

���� ��� !���� � ���� ���� � �� �" ��������# � �������� ������ %�����

�� �� $ �

������� ��� � ���� ���� ���� � �� � ���� � ���� ���� � > � � � ���

��# � � ��� !���� � �� �� ���� ���� � �� �� %� � ��� �� ���� ����

�� ���� �� ������ �� � � �� �# �,�-� %� ����� �� � ��,� -� �� ��

��!���� �,�- ��� �� !������ �� � ��������

������� ��� &�� � 8������ �� � > ,����� � � � � ���-� �� ���� � >

,����� � � � � ��� ���

� � � � � � ���� -� ��� �� > ������

���� �� ���� �� ���������

���� ,���� ��� ������ �-� %� ����� �� ���� � � �� �� $ �� ��

����� �� ���� � ��

,������� �������� � ������� ���� � � ���� ���� � ����� �� � ���

���� ��� ��������-

������� ��� �� � > ,����� � � � � ��� �- � � ������ ����� ����

: � �� �� ������ �������� ���� � � ������������ &�� � � ��

:,�- 4> � : � ,���- � �� %�� ����� �� � � �� ���� :��+%������

� � � ������ ,���- �� ,���-� �� '��� � �� �- �� :,�- > :,�-�

������� ��� ���� ()*� + � �� ��������� � �� ,$ ���� -�)� �� � �

� ������ � ������ � � � �� �� �� �� ������� �� �� � �����

��� � � > ,� �� ���� ���� ���� �

�- �� ���� � ������� � � ���%�� :�� ��

��� � # �� � � � ��� � # ������� � � � � � �

��.� � � ��?

<

347

Page 357: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

��-� � � > �� �?

��/� ���� >� �������?

��)� ������� C ��� � �� C �� : >���� � ?

��0� ��� � � C � : � � >���� � ��

����� ��� �&�������� ������ �� � � � � $������� �� � ������� :���

%� ��� � # � �� ��� � # :

��� � �1 � �� �� � �

��� ��� .<0 .;0� �

����� ��� �� � > ,����� � � � � ��� �-� : � �� ��� �� � ,$ ����

-�)� � � ������ � ������ � � ���� ���� � � ��? � �� � � > �� ��

%� �� ��� � > ,� �� ��� � � � � ��� ��- ���� ����

� ��� � # �� � �� ������ �1 ��� �

� ��� �� ������"

� ��� � # � :� ��,�- > ��� ���� � ��

�� � $������� �� � ������� :�� ���� ��� � ���������

��� �������? �� .<0 .;0� �

����� ��� 2 �� ��������� � �� ���� -�3 ������ ���� �� ������ � ��

������# 4�������� �������# �� ���� �5 �� �� ���� ����� � �� ���� � ����

%� �� ��� � � > ,� �� ���� � � � � ���� �

�-� ��� � �� �� �� �� ��� ��

����� ��� > ,��-� �� �� �������� ����� ������ �� ��� ���� �� ������� �

�� ��� � � �� ���� ���� ���� � � ��� � ��� ��� ()*� %���� .-�.-� %�

� � �� � $������� �� � ������� :���

��� ��� .<0� �

D

348

Page 358: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

� �������� �� � �������

������� �� � �� ��%���� ��� ����� ���� � > ,����� ���� ��� �-�

� > ,�� ��� ���� ��� �- �� � ������ �� �� ���� �� ������� � ��������

� !�, - > � "

� !�, - > � "

� Æ �� � �� Æ ��� � � � � �"

� ���� Æ � Æ ���� ��� � � � � �"

� �� � �� �� ��� � # � ���� ��� � � �1 �� � � ��

4 ��� � �� � ��� ���� �� � ����������� ��� � � � ����

���� � �� %�� ����� � � � � � �� ���� �� ������ � ������

��� � �� �� �� ��� !���� � ����������� 4 ��� � �� ��

(� ���� �� ��� ��� ��� ������ ���� �� ��������� � � �+%������� �����

��� ��� ������ ������� � �������4

����� ��� �6���������� ����� 6���������� � ����� ����� ����� ��

��������� ���� ��� � �� � ������ ��� � �1 �� � � ��� � # �������

�����

��� E�� .�0 ������ 8�8�� �

������� ��� �� � > ,����� � � � � ��� ��� � � � � ��� �- � � ����� ����

: � �� �� �������� ���� � � ������������ &�� � � �� ���

��4> ,����� � � � � ��� ���

� � � � � � ���� � �-� �� ���� �� ������� �� �� 7��

� � � �1 ��� !���� 4 ��� � � ��� � ���� ���� ���

��� ��������� ��%� : ���� �� ���� � ��� � � � � ����

349

Page 359: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

(� � ����� ���� � � � �+%������� ������� @� ������ �� ���� � � ��

�� ��% �� �� � ���� ������� �� �� ��� �+%������� �������

����� ��� + � �� �� ���� � �� ,$ ���� /�/� ��� � # � � ��

� Æ�� > ��Æ ��

��� "�� % �� �� ���� � Æ�� � ��Æ �? ��� ���� � � ����� �

� ������ ���� 3 %�� � � ���"� ��� � � ��� �+%������ ��%� : ��

����� �)� � 4 � �� � � � �� � %�� ���� ���� � �, Æ ��-" �� ��%

�,�� Æ -" ���� ���� " �� �� �� ���� ��> �� �� � �> " � ��� ���

����� � � "� �

'��� �� ���� ������� ��� ����� �� ���%���� �������� �

����� ��� �� � � �� �����������# ����� : �� � ,$ ���� /�/� %�

��� �� ���� $������� �� � ������� :�� � ���

�.� �� Æ �� � �� Æ �� >� �� Æ �� � �� Æ ��?

�-� �� Æ �� � �� Æ�� >� �� Æ �� � �� Æ��?

�/� ���� Æ �� � �� Æ ���

� >� ���� Æ �� � �� Æ ���

� �

��� =� ���� �� ,�-? ��� ��� � ,8- ,/- ��� ������� � �������

��� ���� �� �������� ��� %� ����� �� �� 3 %�� � Æ � � � Æ �

��,� Æ �-��� ��� �� �� " ���"���� �� ���� �� ������ �

"���� 8�9 �� ���� �� � �� "� � "� � "� �� � � %�� ���� ���"�� "�����

� "�,� �-�� �� ��% �� "���� /�< ����� �)� � �� %�� ���� "����

�� � �� � �� F�)� � Æ � � � Æ � ������ ��,� Æ �-��� � ����� �)� � �

%�� ���� ������� �� ��% ��������� �

9

350

Page 360: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

����� ��� �� � � �� �����������# ����� : �� � ,$ ���� /�/� �

����� ���� �� ������ � �� �� ������� ������ �� ��� �� $������� ��

� ������� :�� ������ � �� ���� -�3 � � > ,� �� ���� � � � � ���� �

��� � � � � �

��-�

��� ��� �� �� ������� ����� ������ ��

��� ��� > ��� %� � ���

�.� �� Æ �� � �� Æ �� >� ��� Æ ��� � �

�� Æ �

��?

�-� �� Æ �� � �� Æ �� >� ��� Æ ��� � �

�� Æ �

��?

�/� ���� Æ �� � �� Æ���

� >� ����� Æ ��� � �

�� Æ �

�����

��� ��� ���� �� ��� ��� ��� �����% "����� ��� �� �� ��� ����

� �

���

����� ����� #� � ��� �������� �� %�� � ��� � ��� @� "���� /�D

,�- �� Æ �� � �� Æ �� ������ �� Æ �� � �� Æ �� �� ���� �� ��%���

�� Æ ����� � ����

� Æ �� ����� ������ �� Æ ��� � ��� Æ ��� ��������� �� ���

����� ,8- ,/-� �

� �� ��� �� ��� ���� ����� ��������

"�� % � ������ ��� ������ � ���%�� �� �������� �� ���%�� �

������� �� %� ���%�� �� ����� ����� � > ,����� � � � � ��-� � >

,�� ��� � � � � ��- �� �� ����

� �� > ,� � ������ � � � � ���� ���� � � � � ���-

���� ����

,�� �-���,��� ��- � ����

� C � > ��?

,�� �-���,��� ��- � � > �� C ����

��

� �������� �� ���%�� �� � � � �� � ��������# ������� �� � ���

;

351

Page 361: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

������� ��� � G�E����� �� � ���� ����� �!������� �# � $ �� �� ��

�������� �� �� ������� � �� �

� ��� ��� ,� � �-"

� ����� "

� ���� ���������

� G�E������ �� � ����� �� �� ���� ��� H ��� ��� �� �� � 8%��������

��� �� �� �� ���� ��������� ������

������� ��� � ���� ���%�� �� � ������ ����� �� � � ������

����� �� �� ����� ���� ���� ����� �� � �� �# �� � ��� ����������

�!���� �� �� �����

������� ������� ������� ������� ������� �������

I����� ���� ��� ���� ���� ����� ����� ���4

���� Æ ���� � ���� Æ���� � ���� Æ �� � �� Æ ����� �� Æ ���� � ���� Æ���

������ �� �� � � � ��� ������ �� �� � � 8%�������� ��� %�

� �� ���� �# ��� ����� �� �������9 ������� � � �� � ��� ���

�� �� � ����� �� � ���

��� ,������- @� ���������� ������� � � ��%��� ��� ./0 �����

���� ��� � �� ���� � � ������� �� ����+�� � ������� ���� � ����

����� � ���%��� � ���� ������� � �� ��� � � � �������� �����

� �� � � �� � �� ����� �� � � ������ ,�����- ���� �� � ��� '�

�� %�� ,���� ���� � ��� �)�� � ��� �� � �� ���� � #����� <�8-

� � ������ ����� � � ������� ��� ����� ���%�� �� ������������� ���%�

��� ���� �� ��� �� ��� ���%���� �)�� ��� #����� <�/ ��

352

Page 362: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

'� �)�� � ��� F�� ���� �� �� �)�� ��� ��2����� �� ������� � �

����� ��� %�� !��� $ ���# �� # ������ � ������ ���� ����� ��

�� ������ ��� � �� � ��� ���� ���� � � � �� $ �� ����� ���� �����

�� ��

��� ���� � ���� �� ������ ���� ��� � �+%�� ����� ��� � �� �

������ �,�- > �,�-� F� ��� ���%��� � �� ��%��� �,�-�

(� �,�- > � � � ��� ����� �� � � � �& ��� ��� ��� ���

������ ��� ��%� �� � �� �� � � ��� �� � ����� �)� � 8� ����� � ,������

���� � � ��� �)�� %���� � ��� ��� ������ -�

3 %�� ���� �,�- > $ �� ����� �)� � ! ����� � �� �,�- � $� @�

������ ��� � �� � �& ��� ��� ��� ��� ������ ��� ��%� �� � �� �

�� �� �� � � �"� � ���"� > �"� � ���"� ����� �� �� ��� ��� ������

� � � �� ��������� �� "� ���� ��� ����� � � � " ,� � � "-� �

����� �)� � ��� ��� 8� � 8�� ����� � � ��� ���� �,�- > $� �

����� ��� �� � � � � �� ��� �� � ,$ ���� /�/ � ����� ����

��# ������ � � �� � ���� �� ����� � $ ��� ��� $! $� %� �� ��

�� � �� $ ���

��� ( ���� � � � �� �� ��� � � � �� � �� �� ��� � � � �� � ����� �

� ��� ��� �� � � �� � �� � ��� %���� � ����� � � ���� �� "����

<�<� �

������ ��� :��# ����� � ���� %���� )�. ��� �� ����

��� @� ������ <�� ����� � � ���%��� � � ���� � ��� � ��

����� � � �� � �� ����� �� � � ������? ������ �� ��� � %�� ����

�� ,��� ��- � ����� �� � ��� �� � ���

��

353

Page 363: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

"�� %�,�- ���� ��� ���� ����� � � ���%�� � ���� �� ���� �

��� � � � ��� ,��� ��)���� %���� � � ��� �������� � ��� ����-� 3 %��

���� %�, - > !� �� � � ��� ��,�- �� ��� ������ � �� �� � �������

!�,�- � ���

"�� �� ,�� ��������� ��- �� ��� �� ������ � � ,�� ��������� � - �

��� �� �� � ��,�- � !�� ��� �� �� � �� �� �� ���%�� �

,�- �� ��,�- H %�,�- � ! ��� ��� � � �& ��� � � ��

��� � ����� �� ��%��� %�,�- ,��� "���� ���� � .<0-� ( ����

��� �� � %�,�- > � � �������� (� %�,�- & � ��� �� �������� �� � � ���

� > ��'� � � %�� ���� ,�- � ����� �� ' � � �? ��� � �� ���

��� � ' �� �� � ���,�-� ���� ����� ������ ��,�- > ��,�- � ��

���� %�,'- H %�,�- > %�,�- H %�,�- � ! �� �� ��%����� � %����

��� � � '� ��% ��� � � ��

(� � > ����' ��� ��� � � ��� �� ���,�- ��� � '� @%� �����

������ ��,�- > ��,�- �� ��% %�,'- H %�,�- � %�,�- H %�,�- � !� '���

��� � � ' �� ��%����� � %���� �� ��% ��� � � ��

��� ���� � ��� � � � >���� � � � ����� � ��� ��� ����

���� �,��- > � ��� ,�- �� ����4

,8- ��� ,��� ��- � �

=� �� ����

,/- �� � ��

( ���� �� ����� � �� ��� %������ �������� �� � ��� ��� ���

������� � � �� � �� ��� ��� ���� ��% �� � �� � ���

(� � �� ����� ���� �� ������ ��� ���%���� �������� � � � ��

�)����� � %�� ���� ��� Æ ��� � ��� Æ��� ��� � � � F� ��

,��� ��-���,��� ��-���,��� ��- � �� ��� �,��- � ! �� ,��� ��- ��?

��% ,��� ��-,��� Æ ���-,��� ��- � ���

��

354

Page 364: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

�������� �� �� ���� ��� ������� � �� � � "���� /��� ��� �� %���

�� ���� � � � ���� �%� � "���� <�D� �

��� �����% ������ �� �� �������J�� � ��� ������� �� ��

������ ��� �� � � � ��� ������ �� ��� � � 8%�������� ��� %�

� ��� �� ����

��� ���� � ��� ��� � ��� ��� �%� ������ �%�� �� �������� ����

������� ������ � '��� �� ���� ��� ��� �� �� ���%�� ���� 4

������� ��� %� ������ �������� �������� ��������� ����

���� ��� �� ����

������ � �� ��� ������ ��� ��� � ��� ���� ��� ,����� �� ��� �

�� +%� ��- �� �������� � � � ��� ��� ��� � ��� ������ �� %��

���� � .D0 � �%�� ��� �����������

8� =� �� ���� ��� ��� ��� �� � � ��� � & �� F�� ���� ���

������� ��� ���� � .<0 ������ � ��� ����� �� 3� �%�%�J ./0 �����

D�/�

/� ��� ��� � ���� �� ��� � �� +%� �� ./0 ��//��

���������

.�0 K� @�����%� L� �� I�M�� N� A����� L��� ����� E�������� O��

��� ��� K�� 8����

.80 #� $����� (� '��� L� I���� � ������� "��� �� �� *)���

O���� ��� K�� ���<�

./0 #� $����� 3� �%�%�J �� =���� L� P������� ����� L���

���� ��� ���� ���� � ����� �� ��������� � !� ����� 8��/�

�8

355

Page 365: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

.<0 #� $����� A� �������� K��%�� � ���� ���� (� ����� ;��� �� ��

�� 2<7� �4 ��� 9/��<� ���;�

.D0 #� $����� A� �������� K��%�� � ���� ���� � (((� K��%�� �

���� �� ������� ���� � ��%��� "���� �� 98 ,8��8- F� 8 �� �D9�

�;/�

.�0 #� $������� 3� �%�%�J L� P������� ����� K��%�� � Q��� ��

����R ���� ���� ����%� ��� ,�� �����- ���� ���� ������� 8��/�

����4SS������ ��������%�S ��&S�%��S�%��������

.90 ������������ #����������� � �<��� ������ ��/< ,���;- �� 9�8��

.;0 A��������� 3 ���� ���� ����� ��� �� � � #���� �� #���%�� 4

��%��� � K%�� �� 3������ (�� ��� "���� ,!�� �� L� #� I�M��-�

��%��� 3������� K%��� ��� ���/ �� 8DD�8;D�

.�0 A� ������� (� ������� ��� E�������� �%�%�� ������� �

L��� �� ��������� 3����� � L��� "��� �� < ��� </9�<D��

���R E����� K%������� 8��/�

�/

356

Page 366: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

A systematic proof theory for

several modal logics (extended abstract)

Charles Stewart ∗ Phiniki Stouppa

July 14th, 2004

Abstract

The family of normal propositional modal logic systems is given a very systematic organ-isation by their model theory. This model theory is generally given using frame semantics,and it is systematic in the sense that for the most important systems we have a clean, exactcorrespondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames.

By contrast, the usual structural proof theory of modal logic, as given in Gentzen sys-tems, is ad-hoc. While we can formulate several modal logics in the sequent calculus thatenjoy cut-elimination, their formalisation arises through system-by-system fine tuning toensure that the cut-elimination holds, and the correspondence to the formulation in theHilbert-Lewis systems becomes opaque.

This paper introduces a systematic presentation for the systems K, D, M, S4, andS5 in the calculus of structures, a structural proof theory that employs deep inference.Because of this, we are able to axiomatise the modal logics in a manner directly analogousto the Hilbert-Lewis axiomatisation. We show that the calculus possesses a cut-eliminationproperty directly analogous to cut-elimination for the sequent calculus for these systems,and we discuss the extension to several other modal logics.

Keywords: proof theory of modal logic, calculus of structures, deep inference, cut-elimination.

1 Introduction

Modal logic has been one of the most fundamental advances in logic sinces Frege’s invention ofthe quantifier, and indeed arguably the most important innovation of modern logic. We mightattribute this success to:

• The ability of modal notations to naturally accomodate key concepts needed in formaldescription, most fundamentally concepts of mode and tense. While many propositionalmodal logics can be encoded in first-order predicate logic in a straightforward way, formany applications the modal systems are simpler and more useful;

• Most of the widely used modal logics being decidable;

• The normal modal logics possession of an elementary model theory in the form of framesemantics (due to Kripke, Hintikka and Kanger, see [8]) that provides a systematic corre-spondence between their constitutive axioms as they are usually given in Hilbert style andconditions on the accessibility relation on frames1.

∗Corresponding author: [email protected], International Centre for Computational Logic, Technische Uni-versitat Dresden, D-01062 Dresden, GERMANY.

1See Garson [12], and Blackburn, de Rijke and Venema [3] for readable accounts of this relationship.

1

357

Page 367: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Let’s look at the toolkit available to the classical logician for propositional modal logic com-pared to that for propositional logic. In propositional logic we characterise consequence relationsusing Hilbert deductive systems, give our semantics using Boolean algebra valued models, pro-vide decision procedures using tableau methods, and provide techniques for proof analysis usingsequent calculi in the mould of Gentzen.

In modal logic, close substitutes for the first three of these tools can be found: Hilbert systemsare extended in a standard way with rules to provide what we might call Hilbert-Lewis deductivesystems; Kripke or Beth frame semantics provides a sufficiently rich universe to provide modelsfor almost all modal logics we care about; and tableau systems can be extended to accomodatemodal logics. However, when it comes to proof analysis the situation is more fraught. A relativelynatural extension of Gentzen-style sequent calculi provides cut-free characterisations of severalimportant modal logics; however it fails to provide characterisations of others, and even in thecases where sequent systems can be provided, these are arrived at in an ad-hoc manner2, andthe relationship to the Hilbert-Lewis calculus becomes opaque. It’s not an exaggeration to saythat, by and large, modal logicians have not found proof analysis to be worth the effort; this, nodoubt, is responsible for the omission of structural proof theory in texts such as Blackburn, deRijke and Venema [3].

In this paper we will provide a new proof theoretic basis for modal logic using the calculus ofstructures introduced by Guglielmi [14]. This proof calculus provides a structural proof theoryin the sense of Gentzen for logical systems because it possesses a notion of cut-free proof directlyanalogous to that for the sequent calculus. It possesses several properties that give proof theorycarried out in the calculus of structures important advantages over the traditional proof theorycarried out in structural proof theory, and which extend to the modal logic systems we cover. Itis our hope that these new possibilities for proof analysis can reinvigorate this neglected cornerof the modal logician’s toolkit.

The structure of the rest of this paper is as follows. In the next section we recall thecharacterisation of a much studied ‘cube’ of 15 normal modal logics in Hilbert-Lewis deductivesystems, and describe the sequent calculus for just one of these, the system M (also known assystem T). In section 3 we present the calculus of structures for classical logic, introduce itsextension to system M, and talk about some of the most important analytical properties thecalculus has in detail, including cut-elimination. In section 4 we discuss the generalisation ofthe structural proof theory to the other modal logics of the cube in both sequent calculus andthe calculus of structures. In section 5 we provide a critical and slightly philosophical discussioncomparing the achievements of this calculus to the rival calculus that is best placed to challengeus in providing a highly general approach to proof analysis, namely Belnap’s display logic asapplied to modal logic by Wansing, as well as the more conservative but nevertheless highlyinteresting approach of Pottinger’s hypersequent calculus as investigated by Avron. Finally,section six provides a brief, critical summary of what has been achieved in this paper.

2 Proof theory of system M

Let us begin by establishing some terminology. A logical system is defined over a formal languageby a deductive system that determines its theory, a subset of the sentences of the language. Eachformula in the theory of a logical system is called a theorem. Logical systems are schematic, thatis they allow schematic letters that may stand in for arbitrary propositions, so the theory of alogical system for a more restrictive language can naturally be projected to richer languages byapplication of substitution. Two logical systems are equivalent if their theories are the same,

2I do not mean by this that the designer of sequent systems has no methodology available other than trial anderror, rather I mean that this methodology consists of a complex set of heuristics of low generality, and so fallsshort of being systematic.

2

358

Page 368: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

they are distinct otherwise. If the theory of one logical system strictly contains the theory ofanother, then we call the former system the stronger system.

We begin by showing how propositional modal logic is defined using Hilbert-Lewis deduc-tive systems, as these proof calculi give the oldest and most common axiomatisations, and sowe may consider these axiomatisations as constituting a standard. The sentences of classicalpropositional logic contain an inexhaustible supply of schematic letters, letters for truth andfalsity, which we indicate tt and ff, and are closed under the binary operators ⊃, ↔, ∧ and ∨and the unary operator ¬. The Hilbert style axiomatisation of this logical system proceeds inthe standard way, where any substitution of propositions for schematic letters in the axiomsof the system may be used to start a derivation or add a new theorem to the derivation, andnon-axiomatic propositions may be obtained by modus ponens.

Hilbert-Lewis systems differ from Hilbert systems in that they possess (at least) a pair ofdual modalities, and characteristically are schematic not through the schematicity of the axioms,but via the rule uniform substitution, which allows the schematic letters to be replaced by anyproposition; the consequences of this difference are that deductions are normally a little longer,but it is slightly easier to define some metalogical apparatus; in any case the consequence relationshould be unaffected by this difference, and in this paper we will be casual about the difference,defining axioms as sentences, but applying them as if they were schematic. In addition, allof the Hilbert-Lewis systems in which we are here interested will have the single further ruleof necessitation: if ` p then ` �p, and will have the duality captured by the axiom DM:¬� p ↔ �¬p.

Then the weakest normal modal logic K is axiomatised as a Hilbert-Lewis system by ex-tending classical propositional logic with the axiom K: �(p ⊃ q) ⊃ (�p ⊃ �q). Naturally, thiscorresponds to the class of frame models3 where there are no conditions imposed on the frameaccessibility relation.

We then obtain the stronger modal logics we are interested in by adding further axioms. Allof the most studied systems of modal logic arise by adding some subset of the following axioms:

Definition 1

1. Axiom D: �p ⊃ �p

2. Axiom T: �p ⊃ p

3. Axiom 4: �p ⊃ � � p

4. Axiom B: p ⊃ � � p

5. Axiom 5: �p ⊃ � � p

which correspond to constraining the accessibility relation to be serial (ie. there are always nodesaccessible from any node), reflexive, transitive, symmetric and Euclidean respectively. There area total of fifteen distinct logical systems obtainable, which we may organise in a lattice accordingto the ‘stronger than’ relation, with the most important of these being:

1. K itself;

2. D, obtained by extending system K with rule D;

3. M, obtained by extending system K with rule T: note this system is often named T, anomenclature we avoid to prevent confusion with Godel’s system T.

3We avoid the terminology of possible worlds, since we consider model systems lacking the T axiom, that cannotreasonably be considered to be about possibility or contingency, and further we agree with Thomas Forster thatthe language of possible world semantics is intoxicating to the careless and is best avoided[11]. Hence we callHintikka/Kripke/Kanger style semantics, frame semantics; we avoid formalising these but will need to talk aboutthe frame accessibility relation, which is given over nodes.

3

359

Page 369: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

4. S4, obtained by extending system M by rule 4;

5. B, obtained by extending system M by rule B;

6. S5, obtained by extending system M by rule 5, or equivalently by extending system M byrules B and 4.

The strongest system from these modal logics that is perfectly straightforward to formulate ina sequent system and to prove cut-free is system G-M (for Gentzen system M): we formulate4

this using Schutte’s approach of one-sided sequents. We make use of a dualising function informulating the syntax of the calculus: φ is defined recursively to be the De Morgan dual of theformula φ:

¬A = A A ∧B = A ∨B tt = ff

p = ¬p A ∨B = A ∧B ff = tt

where p is a schematic letter, and A ⊃ B and A ↔ B are treated as derived connectives using thestandard encodings. The inferences of the system are given by trees generated by the followinginference rules, where the nodes are multisets of formulae (indicated by Γ,∆, with the notations�Γ, �∆ indicating the result of prefixing each formula of the multiset appropriately):

Axiom and cut:ax

` Γ, A, A

` Γ, A ` ∆, Acut` Γ,∆

Contraction and weakening:

` Γ, A, Acontr` Γ, A

` Γ wk` Γ, A

Truth:tt` tt

Logical connectives and modal operators:

` Γ, A, B∨` Γ, A ∨B

` Γ, A ` Γ, B∧` Γ, A ∧B

` Γ, A ¬` Γ,¬A

` Γ, A�1

` �Γ,�A

` Γ, A �1` Γ, �A

The theorems of this system then are given by the concluding formulae of inferences wherethe concluding sequent contains exactly one formula. It is easily shown that the sequent formu-lation contains all the theorems of the Hilbert-Lewis formulation, and with the lightest touch ofingenuity, modelling the multisets by disjunctions, the reverse containment can also be demon-strated. We can also show the following theorem, where an inference rule or set of inference rulesare admissible if all theorems of the system may be proven without their use:

Theorem 2 The cut rule is admissible in the sequent calculus formalisation of system M.4It is possible to show that, given very liberal constraints on the form of the sequent calculus, 2-sided (Gentzen

style) systems and 1-sided (Schutte style) systems characterise exactly the same consequence relations in thepresence of De Morgan dualities [20], and this demonstration generalises to hypersequents.

4

360

Page 370: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

This cut-elimination theorem is not really more complex to prove than that for its subsystemof classical logic, though there is a novelty: since the inference rules for the modal operatorsintroduce ‘side-effects’; the rule �1 adds � modalities to side assumptions, which can then beused in cuts. The new rule doesn’t interfere with permutability of cuts, however. These sideeffects are related to an important sense in which these ‘modal Gentzen calculi’ go beyondtraditional Gentzen calculi: with the traditional calculi only a formula in the consequence, andsome subset of its subformulae in the premisses, are ‘touched’ by the logical rule: we can say thatthe traditional calculus is strongly focussed; modal Gentzen calculi with rules such as �1 we callstrongly unfocussed, where we say a system is weakly unfocussed if it is not strongly focussed, andis weakly focussed if it is not strongly unfocussed5. Later we will encounter weakly unfocussedcalculi.

3 The calculus of structures

Now we introduce a characterisation of the system M in the calculus of structures. As forthe single-sided sequent calculus, we make use of De Morgan dualities between connectives inthe formulation of the system. By convention, the calculus has a rather different notation forformulae than is used in Hilbert-Lewis and Gentzen proof calculi: the calculus of structuresuses a bracket notation for the propositional logical connectives to suggest the associativityand commutativity properties: the calculus treats structures that are equivalence classes of thesyntactic representations quotiented over these relations, and also quotiented over rules for units.Conjunction is represented by ”(. . .)”, so (R,S) stands for the conjunction of R and S; likewisedisjunction is represented by ”[. . .]”. We also introduce a dualising operator in the same way asfor the single-sided sequent calculus.

So the formulae of the calculus of structures are built up from the units tt, ff , the schematicletters, where for each schematic letter a we admit the complement a as a formula, and wheneverR1, . . . , Rn are formulae, so are [R1, . . . , Rn] and (R1, . . . , Rn). A formula context S{−} isobtained from a formula by replacing any leaf (eg. a schematic letter) by ‘−’; then S{R} is theformula obtained by replacing this − by the formula R. Curly braces are ommited when theformula R is precisely of the form (R1, . . . , Rn) or [R1, . . . , Rn].

The duals of formulae are defined recursively, where the dual of each schematic letter is itscomplement and vica versa, and

(R1, . . . , Rn) = [R1, . . . , Rn]

[R1, . . . , Rn] = (R1, . . . , Rn)

tt = ff ff = tt

so for any formula R, R = R.The structures are the equivalence classes of formulae obtained by quotienting over

1. (Associativity)

[R1, . . . , Ri, [T1, . . . , Tj ], U1, . . . , Uk] = [R1, . . . , Ri, T1, . . . , Tj , U1, . . . , Uk]

and

(R1, . . . , Ri, (T1, . . . , Tj), U1, . . . , Uk) = (R1, . . . , Ri, T1, . . . , Tj , U1, . . . , Uk)

5The point of the terminology of focussing is to say that a calculus is focussed if each rule that deals with aconnective is only about that connective. As such this is related to the properties such as separation as definedby Wansing [26], but neither strong nor weak focussedness is expressible in terms of Wansings properties, andour interest here is technical, concerned with permutability of cut, rather that the meaning theoretic concerns ofWansing.

5

361

Page 371: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2. (Congruence) If R1 = R2 then S{R1} = S{R2}, for any formula context S{−};

3. (Commutativity) [R1, R2] = [R2, R1] and (R1, R2) = (R2, R1);

4. (Identities) R = [R,ff] and R = (R, tt).

The equations can be considered as being in the spirit of the equations Schutte introduced overpropositional formulae that ensure each formula is equal to its negation normal form, but theygo further, representing more complex, though still tractable, equivalences over formulae. Theirpresence makes the calculus rather more difficult to master than the sequent calculus (especiallythe identities present difficulties) but they make working with the system much easier once thisinitial hurdle is passed, and normally ensure that important theorems avoid the trivial issues ofsyntax that they would otherwise be burdened with.

Inferences are chains of applications of inference rules, where each inference rule has onepremise and one conclusion, which is a structure rather than a formula. All of the inference rulesare deep, which means that each rule is given by a pair of formulae specifying the premise andconclusion that are both given with the same formula context. The inference rules for classicallogic, the system SKSg are given:

Interaction and cut rules:

S{tt}i ↓

S[R,R]S(R,R)

i ↑S{ff}

The switch rule:S([R, T ], U)

sS[(R,U), T ]

The weakening rules:S{ff}

w ↓S{R}

S{R}w ↑

S{tt}And the contraction rules:

S[R,R]c ↓

S{R}S{R}

c ↑S(R,R)

The theorems of this system are the formulae belonging to the structures that occur asconclusions of inferences whose premise is the structure tt.

This system has many desirable properties:

1. The system is self dual: each rule labelled with ↑ takes form:

S{R}∗ ↑

S{T}

matching by De Morgan duality the form of the corresponding rule labelled with ↓ takesform:

S{T}∗ ↓

S{R}where rules not labelled with ↑ or ↓ are self-dual (so far we have seen just the switch rule).

2. The entire up-fragment, ie. the rules labelled with ↑, is admissible; that is, the full systemis equivalent to the system obtained by removing the whole up-fragment. This is shown byBruennler [6] and discussed in more detail below, by means of a translation from cut-freeproofs of the sequent calculus into proofs of system KSg, that is, system SKSg withoutthe rules of the up-fragment. Because the rule i ↑ closely models the cut rule, it is naturalto describe this admissibility result as cut-elimination for the calculus of structures.

6

362

Page 372: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3. We can restrict the interaction, cut and weakening rules to atoms, by which we mean thatthe applications of the rules using formulae R can be restricted to the case where R is anatom (ie. a or a) both for system SKSg and the cut-free system KSg. Furthermore, whenadding to system SKSg the medial rule

S[(R,U), (T, V )]m

S([R, T ], [U, V ])

we can also restrict the contraction rule to atoms. These restrictions are achieved by sim-ple local transformations on inferences, as opposed to the complex global transformationthat we associate with cut-elimination. The system with these restrictions we call atomicSKSg (analogously atomic KSg), or just SKS (analogously KS). An analogous restric-tion to atoms can be achieved for the sequent calculus by similarly local transformationsin the case of the axiom and weakening rules, but for fundamental reasons of the shallow-ness of inference cannot be achieved at all for contraction, and only by means of globaltransformations in the case of cut.

4. System SKS enjoys important computational properties: it is local, and so too is its sub-system KS, in the sense that looking at the inferences going either up or down, structure isrearranged, or atoms introduced, abandoned or duplicated, but arbitrarily large substruc-tures are never introduced, abandoned or duplicated. Brunnler also discusses an importantadvantage corollary to locality and atomicity of cut: using deep inference he gets a finitaryvariant of SKS by simple means (i.e. without cut elimination); this means that there areonly finitely different ways rules of the system can be applied to obtain a given conclusion,which in principle means the calculus can be used for proof search. However, it is fair topoint out that these benefits do carry a cost: by contrast to the sequent calculus there area great many possible rules that one may apply during proof search, so one cannot readoff a tableaux algorithm from KS in the way one can for cut-free LK. Devising efficientproof search algorithms that can take advantage of these properties is an important goalof the program of research in the calculus of structures.

A thorough mathematical and conceptual examination of these properties and their impor-tance is given in Bruennler’s [6]; a shorter discussion appears in [5].

We can extend this calculus to obtain the system M by allowing formulae of the form �R,and �R, extending the equivalences defining structures with tt = �tt and ff = �ff and extendingthe set of inferences as follows:

S{�[R, T ]}k ↓

S[�R, �T ]S{�R}

t ↓S{R}

We call the system that extends system KSg with the above down rules KSg-M, and thesymmetric system extending system SKSg with both the down and up rules SKSg-M.

We show the equivalence of system SKSg-M to M in two steps. Firstly we show we canmap inferences of SKSg-M onto inferences of M:

Definition 3 We define the map −s from formulae of M onto structures of SKSg-M recur-sively:

ps = p for p one of tt, ff, or a schematic letter;

(¬A)s = As (A ∧B)s = (As, Bs) (�A)s = �As

(A ∨B)s = [As, Bs] (�A)s = �As

Proposition 4

7

363

Page 373: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1. There is a map −h from structures of SKSg-M to formulae of M with the properties that−hs is the identity map on structures, and for all formulae A, B, if A is a subformula ofB then either Ash is a subformula of Bsh or (¬A)sh is a subformula of (¬B)sh.

2. If As = Bs then ` A ↔ B is a theorem of M;

3. If R = As and A is a theorem of M, then R is a theorem of SKSg-M;Proof Part 1 follows from defining the obvious recursive map from the formulae of SKSg-M,and observing that for any structure we may choose a lexically simplest representative formula(given some total order on atoms and their complements), to obtain the reverse map which iseasily verified to have the desired properties.

Part 2 follows easily from the observation that −s is bijective when restricted to disjunctivenormal forms.

To prove part 3, we must first prove that each axiom of M maps onto a theorem of SKSg-M,which is a straightforward exercise in the construction of inferences in the calculus of structures:for didactic reasons we recommend the reader works through at least two cases, for example(A ⊃ B ⊃ C) ⊃ (A ⊃ C) ⊃ B ⊃ C, and axiom K, and then show that inferences of the Hilbert-Lewis calculus map onto inferences of SKSg-M by considering the three inference rules. To seethat inferences making use of uniform substitutivity are modelled in SKSg-M we need simplyobserve that the inference rules of SKSg-M are schematic. In the other two rules proceed byan induction on the lenth of proofs. ��

^

Theorem 5 M and SKSg-M are equivalent.Proof For each inference rule of KSg-M, which takes the general form:

S{R}∗

S{T}

we show that Rh ⊃ Th is a theorem of M. The dual rules map onto these from the theorem ofclassical logic: (A ⊃ B) ↔ (¬B ⊃ ¬A). The following lemma is a straightforward exercise intheoremhood over K:

Lemma 6 If A ⊃ B is a theorem of M, then so are:

1. A ∧ C ⊃ B ∧ C;

2. A ∨ C ⊃ B ∨ C;

3. �A ⊃ �B;

4. �A ⊃ �B.

from which, by an induction on the makeup of formula contexts S{−}, we obtain for eachinference rule (S{R})h ⊃ (S{T})h. Then inferences of KSg-M can be mapped onto chains ofapplications of modus ponens, starting from the theorem tt. ��

^

Theorem 7 The up-fragment of SKSg-M is admissible.Proof We show this by mapping cut-free proofs of G-M onto proofs of KSg-M, followingjust the same technique as [5], that is, for each rule of G-M

Γ1 . . . Γn

we must find an inference of KSg-M with premise �i(Γs1, . . . ,Γ

sn) for some i and conclusion

∆s, where we extend the mapping −s from formulae to sequents by considering sequents to be

8

364

Page 374: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

disjunctions of their constituent formulae, and where the notation �iA indicates the formulaeobtained from A by prefixing � to it i times. These proofs are straightforward; the only casewhere we need i to be nonzero is mapping �1; the rule then maps onto a number of applicationsof the k ↓ rule equal to the number of formulae in Γ. We can then use these inferences toconstruct our map on cut-free proofs recursively. ��

^

Lastly if we add the following rules:

S[�R, �T ]j ↓

S{�[R, T ]}S{ff}

uw ↓S{�ff}

S[�R,�T ]l ↓

S{�[R, T ]}we can restrict cut, interaction, contraction and weakening to atoms, and so obtain a localsystem, which we call KS-M.

4 Systems K, D, S4, and S5

We can extend the account of the structural proof theory that we have given of M to the systemsK, D, and S4 quite analogously. First we consider the axiomatisation of these systems in thesequent calculus: define two new inference rules:

` �Γ, A�2

` �Γ,�A

` Γ, A �2` �Γ, �A

Then we obtain our axiomatisations as follows:

1. G-K is obtained by dropping rule �1 from G-M;

2. G-D is obtained from G-M by substituting rule �2 for �1;

3. G-S4 is obtained from G-M by substituting rule �2 for �1;

Theorem 8 Each of these systems is equivalent to the matching Hilbert system, and for each,the cut rule is admissible.Proof For G-K and G-D the proofs are quite analogous to that for G-M. G-S4 is similarlyshown to be equivalent to the matching system S4; the cut-elimination proof is more subtlebecause many of the usual permutations on cuts fail, a fact that is unfortunately rather glossedover in the literature. Cf. Ohnishi and Matsumoto, and Valenti [17, 23, 24]. ��

^

Our axiomatisation in the calculus of structures proceeds by defining the inference rules:

S{�R}d

S{�R}S{� �R}

4 ↓S{�R}

Then we obtain the equivalent systems:

1. KSg-K is KSg-M without the t ↓ rule;

2. KSg-D is KSg-K together with the d ↓ rule;

3. KSg-S4 is KSg-K and the t ↓ and 4 ↓ rules;

4. SKSg-K, SKSg-D, and SKSg-S4 are the symmetric versions of the above.

Note that inclusion of an axiom in the Hilbert system corresponds to inclusion of a downrule in the corresponding system in the calculus of structures.

9

365

Page 375: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Theorem 9 The analogs of Proposition 4 and Theorem 5 hold for each of SKSg-K, SKSg-D,and SKSg-S4.

The axiomatisation of systems S5 and B in the one-sided sequent calculus present seriousdifficulties, however. For example, it is possible to axiomatise S5 by replacing the rule �1 ofG-M by any one of the following three rules6:

` �Γ,�∆, A�3

` �Γ,�∆,�A

` Γ,�∆, A�4

` �Γ,�∆,�A

` �Γ,∆, A�5

` �Γ,�∆,�A

This system can be mapped onto a reasonable axiomatisation of KSg-S5 which we axiomatiseas KSg-S4 plus the following rule:

S{�� R}b ↓

S{R}We can show that the theories of the systems S5, the three possible formulations of G-S5 and

SKSg-S5 are equivalent, and that cut-free proofs of G-S5 map onto KSg-S5, but unfortunatelycut-elimination for all three variants of G-S5 fails:

Proposition 10

1. The axiomatisations of G-S5 formulated using rules �3 and �5 have no up-fragment freeproof of ¬A ∨� �A, a theorem of S5.

2. The axiomatisation of G-S5 formulated using rule �4 has no cut-free proof of ¬��A∨�A,a theorem of S5.

Proof The are cut-bearing proofs for part 1 with cut formula �A, and for part 2 with cutformula � �A, for which in each case all the side formulae have � as their main operator, whichin each case is the principal conclusion of the appropriate �i rule. ��

^

While cut-free sequent systems for S5 have appeared in the literature; for example [4], theseformulations do not take the simple form of the sequent systems treated in this paper, possessingeither sophisticated rules that expose proof structure (either explicitly as in Brauner’s connec-tions between formula, or tacitly by some form of extra-logical labelling), or introduce some kindof deep inference that involves judgements with more sophisticated structure than Gentzen’s se-quents. We have not heard of any truly cut-free sequent formulation of B in a Gentzen-stylesequent calculus.

We can axiomatise system B in the calculus of structure by adding b ↓ to KSg-M to obtainKSg-B, and likewise SKSg-B. Here our inference rules dealing with the modal operators findthemselves in a one-one relationship to four of the five axioms given in the lattice of Hilbertsystems described in Definition 1; we can extend this to rule 5 with the inference rule:

S{�� R}5 ↓

S{�R}

Conjecture 11 For all of the systems of the cube characterised by adding any subset of themodal down rules we have described above, cut is admissible.

6The first of these three alternatives, using �3 is the one-sided analogue of the two-sided sequent formulationof S5 due to Ohnishi and Matsumoto, [17].

10

366

Page 376: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Note that though there are fifteen systems of the cube, there are thirty-two possible ways ofcharacterising them; all of these characterisations are conjectured to allow cut elimination.

In particular we make an observation.

Theorem 12 KSg-S5 is cut-free.

We will discuss this result in section five, which provides the strongest of several pices ofevidence for the conjecture, when we treat hypersequents.

The approach to proofs we have so far outlined are proofs by translation which leverageknown cut elimination proofs for the sequent calculus. It is also possible to prove cut eliminationdirectly, although non-constructively, by semantic means. The advantage of this method is thatit is as systematic as the the formalisation in the calculus of structures; the disadvantages are,besides non-constructivity, that such methods give only limited proof-theoretic insight, and sofar are restricted to fewer axioms than are needed to describe the systems of the cube.

Let us introduce two new Hilbert-Lewis axioms:

Definition 13

1. Axiom W5: �� p ⊃ � � p

2. Axiom C4: � � p ⊃ �p

Then we have W5,C4 ` 5, 5 ` W5, and T ` C4; these new axioms then are weak versionsof 5 and T that combine to provide the inferential strength of axiom 5. Each of these can beincorporated into the calculus of structures as follows:

S{�� R}w5 ↓

S{� � R}S{� � R}

c4 ↓S{�R}

Conjecture 14 For all 128 possibilities of adding some subset of the seven rules d ↓, t ↓, 4 ↓,b ↓, c4 ↓, and w5 ↓ to KSg the up-fragment is admissible.

Obviously this conjecture is stronger than the first, and so far the evidence for it is weaker.At least a subsystem generated by four of the above seven rules is known to be cut-free by asemantic argument. It appears to be the case that the semantic argument works for systemsdescribed by axioms that correspond to rooted conditions on the frame accessibility relation; thatis, conditions that can be described by a Π0

2 formula where the universal quantifiers describe atree of arcs across nodes, and the existential quantifiers assert the existence of arcs each of whicheither originates from the root of the tree, or which originates from a node that is existentiallyasserted and distinct from any in the original tree. The rules that correspond to rooted conditionswe call rooted, these that don’t we call unrooted.

Theorem 15 For each of the 10 systems described by some subset of the four unrooted rulesd ↓, t ↓, 4 ↓, and c4 ↓, the up-fragment is admissible.

To repeat, this theorem is proven by semantic means that we believe extend to all systemsdescribed by rooted rules. Additionally the six systems that can be formalised without the c4 ↓rule can all be proven by means of translation from known systems in the sequent calculus (fourof which are the systems discussed in most depth at the beginning of this section).

11

367

Page 377: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

5 Display logic and hypersequents

As remarked before, the most obvious rival to our calculus is the application of display logic tomodal logic as investigated by Wansing [26]. Modal display logic is modular in the sense thatwe seek, and it has a syntactic cut elimination that ensures any properly presented system (thatis, a ‘properly displayable’ system) is cut-free, in the true sense of guaranteeing analytic normalforms. It embraces classes of calculi that we do not know how to express satisfactorily in thecalculus of structures, namely calculi without De Morgan duality, such as intuitionistic logic7.And indeed these strengths come from a property display logic possesses similar to a propertyof the calculus of structures, namely, the manipulations on structures allow the logical rules tobe applied at effectuvely unbounded depths, or in other words, display logic is also a calculus ofdeep inference.

Given all this, one might reasonably ask – why pursue another calculus of deep inference?Our answer is that we believe that for the purposes of proof analysis, the design of the calculus ofstructures will ultimately lead to technically better results, essentially for the following reason:display logic wishes to combine the sophisticated notion of proof structure one obtains with deepinference with the traditional approach to proof analysis based on the subformula property. Theresult is that display logic leads a double life, with the subformula property holding on formulae,the leaves of the structures display logic manipulates, but not able to say anything about the fullyfledged structures which is really where all the action happens. A second consequence is thatdisplay logic seeks to embrace constraints on its presentations, such as Dosen’s principle, thatappear to be necessary to get reasonable results from subformla-property-based proof analysis.Even so, the proof analysis does not seem to be as effective as in the propositional case: in thesequent calculus one can simply read of a tableu mechanism from a cut-free sequent calculus; indisplay logic, the procedure, or perhaps better heuristic, is more fraught. To be blunt, some ofthe technical advantages that flow from the subformula property in the context of the sequentcalculus do not flow from the property that display logicians call ‘subformula property’; perhaps,by analogy, the meaning-theoretic parallels depend upon a certain amount of optimistic charity.

By contrast the calculus of structures makes no distinction between logical and structuralrules, which brings simplicity; constraints such as Dosen’s principle, or some of the properties ofsystems described by Wansing like separation, symmetry and explicitness simply make no sensein this context; and seeks to substitute a wholly novel methodology of proof analysis, with somestriking properties that improve on Gentzen-family calculi, such as atomicity.

Additionally there is a technical disadvantage of display logic for our investigation into sys-tems of modal logic: display logic appears to be most naturally a tense logic, in that in intro-ducing a modality, one obtains not only introduction rules for that modality and its De Morgandual, but also the reverse modalities in the tense logical sense (so four modal operators, ratherthan the expected two). Hence, whereas when one has a proof in KS-S5 each structure occur-ing in the inference corresponds to a theorem of S58, in display logic the axiom dealing for 5is expressed in terms of tense logic (due to the conversion of modal axioms to rules involvingprimitive tense formula, following the results of Marcus Kracht [15]), and so the judgementsapprearing in the tree of a proof may be assertions not of S5 but of S5t, its tensed extension.By conservativity, we know that we have the right theorems, but conservativity of theoremhooddoes not map into conservativity of cut-free provability; there is still a problem unsolved. Inparticular, if consider these kind of results adequate, then the problem of cut elimination formodal logic becomes mostly a solved problem: we simply use calculi that embed modal logics insequent calculi for geometric theories, such as pioneered by Alex Simpson[?].

Certainly there is room for dispute over the relative merits of the two approaches to providinga structural proof theory for modal logic; however we think it is important to recognise firstly

7Though there are logics expressible in the calculus of structures which it appears that display logic cannotexpress at all, such as system NEL [13], due to the branching nature of proofs in display logic

8By means of the translation −h described in section three.

12

368

Page 378: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

that the ultimate test of health of a proof calculus will lie in its usefulness to logicians notinterested in proof theory for proof theory’s sake; secondly that in providing a toolkit for proofanalysis display logic and the calculus of structures are not necessarily rivals, but may insteadbe complementary formalisms.

We also wish to discuss hypersequents as they have been treated by Arnon Avron. Theseextend the sequent calculus, allowing several strands of inference to accur in parallel, and tocommunicate by means of certain ‘external’ structural rules (the familiar structural rules arecalled ‘internal’, we will call rules that act locally upon a sequent internal also). We note that forthe modal logics in which we are interested, the 2-sided hypersequents Avron treats are equivalentto 1-sided hypersequents; thus we may describe the general hypersequent ` Γ1| . . . | ` Γn by thestructure [�[Γs

1], . . . ,�[Γs1]]. Avron captures the system S5 in the hypersequent system HS5

by giving internal rules corresponding to the rules for S4, together with, in addition to externalcontraction and weakening, a modalised splitting rule (we give our 1-sided variant):

G| ` �Γ1, �Γ2,Γ3MS

G| ` �Γ1, �Γ2| ` Γ3

All internal rules map easily onto the rules for S4, just in a deeper context of the form[�{−}, R], and the external weakening and contraction rules are straightforwardly modelled byw ↓ and c ↓; all that remains is to capture the modalised splitting rule, which can be modelledby the 4 ↓, 5 ↓ and s rules. By these means, cut free proofs of system HS5 are mapped tocut-free proofs of KSg-S5, thus establishing the theorem described in section four.

Let us note, though, that the modalised splitting rule does not fix all of the problems arisingdue to lack of expressiveness of the cut-free sequent calculus. In particular, the modalisedsplitting rule gives the inferential strength of the 5 axiom only by also giving the inferentialstrength of the 4 axiom as well. To be more precise, hypersequent systems with the modalisedsplitting rule cannot be used to axiomatise systems B, KB, DB, K5 or D5, and it has notbe shown how the inferential strength of unrooted axioms can be achieved in a hypersequentcalculus other than by means of modalised splitting. Thus, while hypersequents achieve animportant result, one that we depend upon for the results of this paper, the means by which itachieves the goals appears to be fundamentally limited.

6 Review

The most important achievement of this paper bas been to provide a novel, modular approachto the proof theory of modal logic that begins to allow the kind of powerful proof analysisdescribed by Brunnler for several important modal logics (those for which we can prove up-fragment admissibility), and for which we have reason to hope can be extended to all the modallogic systems we have described. Furthermore we hope that our readers will find the system tobe elegant and provocative.

The most serious defect of the account given here from the point of view of formal aestheticsis that our proofs are external: our methods for showing cut-elimination are so far a ratherunderpowered semantical technique or by means of translations from a quite separate theory,so though we could pedantically claim to have provided a syntactic proof of cut-elimination,the proof is quite as external as with semantical proofs of cut-elimination. Furthermore, we aremostly left without resources to deal with the cases where we do not have appropriate cut-freeaxiomatisations, such as for system B. Nonetheless, it must be emphasised that these are defectsof our knowledge of how to prove results; to the best of this limited knowledge, the family ofproof systems are in excellent shape: no rival calculus appears to describe as many systems sosimply.

There are three principal avenues to explore in search of the missing proof techniques. Thefirst is to try to extend the semantic proofs; Avron’s paper on hypersequents succeeds in providing

13

369

Page 379: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

a semantical proof of cut elimination for HS5 and it is possible the technique can be adpatedand generalised. Second, Guglielmi [14] introduces a technique, called splitting, that may beregarded as the preferred way to give internal, syntactic proofs of cut-elimination in the calculusof structures, and in joint work with one of the authors (in preparation) has shown how thistechnique may be extended to deal with the case of SKS. Lastly, a very syntactically involvedtechnique of permutability of rules can be applied to prove cut elimination; two proofs of thisnature are due to Strassburger [22], as well as a proof by Brunnler [?]. Of these approaches, asplitting proof would be the most valuable.

A further issue relates to the role of proof analysis in the toolkit of the modal logician. Thesingle most important application of proof analysis in propositional logic has been its crucial rolein influencing the design of tableau methods. With modal logics, the ad hoc nature of sequentcharacterisations has been reflected by an equally ad hoc methodology for the design of modaltableau. Furthermore the designers of modal tableau have found it necessary to resort to uglytechniques equivalent to ‘analytic’ cut (a proof with an analytic cut is by virtue of this cut notanalytic; this is perhaps the most misleading piece of nomenclature in proof theory), so indeedthe degree of disorder among modal tableau is greater than among modal sequent calculi. Themost obvious and useful test of a claim to have provided a worthy modal proof theory is leveragethe theory to providing a principled approach to modal tableau. The application of the calculusof structures to the design of such systems is under intensive investigation, making use of insightsflowing from Dale Miller’s ‘proof search as computation’ slogan [7, 16].

References

[1] A. Avron The method of hypersequents in the proof theory of propositional non-classical logics. InW. Hodges et al. (eds.), Logic: From Foundations to Applications, pages 1–32. Oxford UniversityPress, 1996.

[2] N. Belnap. Display Logic. Journal of Philosophical Logic, 11:375–417, 1982.

[3] P. Blackburn, M. de Rijke, Y. Venema. Modal Logic. Cambridge University Press, 2001.

[4] T. Brauner. A cut-free Gentzen formulation of the modal logic S5. In the Logic Journal of theInterest Group in Pure and Applied Logics, volume 8(5), pages 629–643, 2000.

[5] K. Brunnler. Locality for classical logic. Submitted to Archive for Mathematical Logic, 2003.Preprint available http://www.wv.inf.tu-dresden.de/ kai/LocalityClassical.pdf.

[6] K. Brunnler. Deep Inference and Symmetry in Classical Logic. Logos Verlag, Berlin, 2004.

[7] P. Bruscoli. A Purely Logical Account of Sequentiality in Proof Search. In Proc. ICLP 2002,LNCS 2401:302–316. Springer-Verlag, 2002.

[8] R. A. Bull and K. Segerberg. Basic Modal Logic. In The Handbook of Philosophical Logic, volume2, pages 1–88. Kluwer, 1984.

[9] L. Cardelli and A. D. Gordon. Anytime, anywhere: Modal logics for mobile ambients. In Proc.Principles of Programming Languages 2000, pages 365–377. ACM Press, 2000.

[10] K. Dosen. Sequent-Systems for Modal Logic. In Journal of Symbolic Logic, volume 50(1), pages149–168, 1985.

[11] T. Forster. The modal aether. Manuscript.

[12] J. Garson. Modal Logic. In E. N. Zalta, editor, The Stan-ford Encyclopaedia of Philosophy, Winter 2001 edition. Availablehttp://plato.stanford.edu/archives/win2001/entries/logic-modal.

[13] A. Guglielmi and L. Straßburger. Non-commutativity and MELL in the Calculus of Structures.In Proc. Computer Science Logic 2001, LNCS 2142:54–68. Springer-Verlag, 2001.

[14] A. Guglielmi. A System of Interaction and Structure. Submitted to ACM Transac-tions on Computational Logic, 2002. Preprint available http://www.ki.inf.tu-dresden.de/~

guglielmi/Research/Gug/Gug.pdf.

14

370

Page 380: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[15] M. Kracht Power and weakness of the modal display calculus. In, H. Wansing (ed.), Proof Theoryof Modal Logic, pages 93–121. Kluwer, Dordrecht, 1996.

[16] R. Milner. Communicating and mobile systems: the pi-calculus. Cambridge University Press, 1999.

[17] M. Ohnishi and K. Matsumoto. Gentzen method in modal calculi, parts I and II. In OsakaMathematical Journal, volume 9, pages 113–130, 1957, and volume 11, pages 115–120, 1959.

[18] K. Schutte. Proof Theory. North-Holland, 1962.

[19] H. Schwichtenberg and A. Troelstra. Basic Proof Theory. Cambridge University Press, 1998.

[20] C. A. Stewart. Which 2-sided sequent systems are equivalent to 1-sided systems? Unpublishedmanuscript, 2004.

[21] P. Stouppa. The design of modal proof theories. MSc dissertation, Technische Universitat Dresden.In preparation.

[22] L. Straßburger. A local system for linear logic. In LPAR 02, 2002.

[23] S. Valenti. Cut-elimination in a modal sequent calculus for K. In Bolletino dell’Unione Mathe-matica Italiana, volume 1B, pages 119–130, 1982.

[24] S. Valenti. The sequent calculus for the modal logic D. In Bolletino dell’Unione MathematicaItaliana, volume 7A, pages 455–460, 1993.

[25] H. Wansing. Sequent Calculi for Normal Modal Propositional Logics. In Journal of Logic andComputation, volume 4(2), pages 125–142, 1994.

[26] H. Wansing. Displaying Modal Logic. Kluwer, Dordrecht, 1998.

15

371

Page 381: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Consistency proofs for systems of

multi-agent only knowing

Arild WaalerFinnmark College and University of Oslo, Norway

Email: [email protected]

Abstract

Various multi-modal systems are proposed for the representation ofbelief states in a multi-agent context. We introduce sequent calculi forthese systems and prove cut-elimination results. A corollary to theseresults is that a new and natural multi-modal extension of Levesque’ssystem of only knowing is consistent.

Keywords: Sequent calculus, cut-elimination, multi-modal logic ofbelief.

1 Introduction

Multi-agent belief logics can be viewed as systems designed for the represen-tation of representations (or languages) that agents use for reasoning aboutother agents’ cognitive states. A multi-agent only knowing system has lan-guage constructs for representing upper and lower bounds of beliefs, and inparticular the exact content of an agent’s belief state. The natural start-ing point for the design of such a system is to generalize the only knowingsystem of Levesque [8] to the multi-modal case.

The tricky part of this is hidden in an axiom in Levesque’s system (whichI shall refer to as the conv-axiom) to the effect that 3ϕ (ϕ is logicallypossible) is an axiom for each satisfiable, objective ϕ (“objective” becauseit does not contain any modal operators). The problem is how to generalizethe emphasized clause to multi-modal languages. One route to achieve thisis to generalize the semantical notion of satisfiability. This is what Halpernand Lakemeyer have attempted in a series of papers [7, 4, 5, 6], at the costof coding the satisfiability relation into the system.

Another possibility is to express the generalized conv-axiom syntacti-cally. It is then natural to take the syntactical counterpart of Levesque’s

1

372

Page 382: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

conv-axiom as the starting point: ` 3ϕ if ϕ 6` ⊥, where ϕ is any proposi-tional formula. This seemingly circular pattern is in the single agent casecompletely innocent since the axiom is restricted to ϕ’s outside the modalpart of the language. This constraint is clearly too restrictive in the multi-modal setting.

From the syntactical perspective, what seems to be appropriate is togeneralize the conv-axiom to the following: 3kϕ provided that ϕ 6` ⊥, wherek is an index individuating an agent, 3k is a conceivability operator foragent k and ϕ is any formula in which every k-modality which occurs init, occurs within the scope of a modality for another agent (Halpern andLakemayer call such a ϕ k-objective). This formulation arguably gives acleaner proof system than the systems studied by Halpern and Lakemeyer.Note that in the single agent case, the generalized conv-axiom is identicalto the single-agent formulation which we departed from. Also note thatthis generalization uses the whole language in the specification of the conv-axiom. This differs from Halpern and Lakemayer’s attempts, as they try toget around the complementary (co-belief) modalities in their analysis of theonly-knowing language.

However, the syntactically motivated conv-axiom has a circular pattern,and an argument is needed in order to show that the circularity is notvicious. Note, incidentally, that consistency cannot be obtained by standardsemantical methods. For to prove soundness of the conv-axiom one needs totransform the consistency condition to a corresponding semantical conditionin terms of satisfiability and this, in turn, requires a completeness argument.The standard way of proving completeness is via maximal consistent sets,which can only be used if consistency has already been established.

The scope of this paper is entirely syntactical. More precisely we intro-duce a sequent calculus for the multi-modal logic and prove cut-eliminationtheorems for it by constructive methods. Since the logic for the single agentcase is identical to Levesque’s system, the paper hence also presents a se-quent calculus for the propositional part of Levesque’s logic.

It follows from the subformula property of cut-free proofs in the sequentcalculus that the multi-modal extension of the syntactical form of the conv-axiom is indeed consistent. We shall also see that the cut-elimination resultsfor the core system can be generalized to a system in which a quasi-order isimposed on the index set individuating the modalities. The motivation forthe latter construct is to enable the representation of confidence levels foreach agent; this is necessary for the representation of certain multi-modaldefaults along the same lines as one can represent prioritized supernormaldefaults in an extension of Levesque’s system with confidence levels [9].

2

373

Page 383: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

2 The multi-agent logic LI

2.1 Axiomatic system

The object language contains a stock of propositional letters, the constants>and ⊥, the Boolean connectives ¬, ∨, ∧, ⊃, and ≡, and modal operators Bk

and Ck for each k in a non-empty index set I. The intended interpretation isthat the index set I represents the set of agents, Bk is a belief operator, andCk a complementary co-belief operator for agent k. We define the symbolOk by Okϕ = Bkϕ ∧Ck¬ϕ. Intuitively Okϕ means that ϕ is exactly whatagent k believes; the construction is inspired by the “all I know” operatorof Levesque [8]. Further abbreviations: bk is ¬Bk¬, ck is ¬Ck¬, 2kϕ isBkϕ ∧ Ckϕ, the dual 3kϕ is bkϕ ∨ ckϕ. The intended meaning of 2k islogical necessity, relativized to the conceivability space for agent k.

ϕ is a modal atom of modality k if it is of the form Bkψ or Ckψ for ak ∈ I. ϕ is a completely k-modalized formula if it is a Boolean combinationof modal atoms of modality k. ϕ is free of modality k if it is a Booleancombination of propositional letters and modal atoms not of modality k. ϕis a first-order formula if, for each k ∈ I and each subformula Bkψ or Ckψin ϕ, ψ is free of modality k.

The logic LI is defined by means of the axioms and inference rules below.We write ` ϕ if ϕ is theorem and ϕ1, . . . , ϕn ` ψ if (ϕ1 ∧ · · · ∧ ϕn) ⊃ ψ isa theorem; the provability symbol ‘`’ is used also for extensions of LI andshall always be understood relative to the system at hand. A formula ϕ isconsistent if ϕ 6` ⊥.

Let us say that a tautology is any substitution instance of a formula validin classical propositional logic such as 2kϕ ⊃ 2kϕ. LI is defined as the leastset that contains all tautologies, is closed under all instances of the rules

ϕ2kϕ

(RN)ϕ ϕ ⊃ ψ

ψ(MP)

and contains all instances of the following schemata for each k ∈ I:

KB : Bk(ϕ ⊃ ψ) ⊃ (Bkϕ ⊃ Bkψ)KC : Ck(ϕ ⊃ ψ) ⊃ (Ckϕ ⊃ Ckψ)B2: Bkϕ ⊃ 2kBkϕC2: Ckϕ ⊃ 2kCkϕ

B2: ¬Bkϕ ⊃ 2k¬Bkϕ

C2: ¬Ckϕ ⊃ 2k¬Ckϕ(conv): 3kϕ provided ϕ 6` ⊥, ϕ free of modality k

3

374

Page 384: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The (conv) axiom entails that 2kϕ is provable only if ϕ is provable.Hence, by RN, ` 2kϕ iff ` ϕ. The logic of 2k is hence S5. It is easy to showthat Bk and Ck are both K45 modalities. The axioms can be viewed asmodality reduction rules; the main properties are captured in the followingtwo lemmata, both of which are proved without the (conv)-axiom.

Lemma 1 Let κ be completely k-modalized and ϕ be any formula. Then` Bk(κ ∨ ϕ) ≡ (κ ∨ Bkϕ) and ` Ck(κ ∨ ϕ) ≡ (κ ∨Ckϕ). Hence ` Bkκ ≡(κ ∨Bk⊥) and ` Ckκ ≡ (κ ∨Ck⊥).

Proof. We first show, by induction on θ, that ` θ ⊃ 2kθ for each completelyk-modalized θ. The base case in which θ is Bkψ follows from axiom B2;likewise for the other base cases (using C2, B2, or C2). The inductionstep follows from the induction hypothesis and simple modal reasoning. Toestablish the first equivalence we use instances of the KB axiom, that ` κ ⊃Bkκ and that ` ¬κ ⊃ Bk¬κ. The second equivalence is proved similarly.The latter statements follow from the former since ` κ ≡ (κ ∨ ⊥). 2

Lemma 2 Any formula is provably equivalent to a first-order formula.

Proof. A modal atom Bkϕ is equivalent to a formula of the form Bk(ψ1 ∧· · · ∧ ψm) where each ψi is a disjunction of a formula of modality k and aformula free of modality k (either disjunct may be absent). If ϕ is not freeof modality k, we can distribute Bk over the conjunctions and use Lemma1 and the induction hypothesis on each Bkψj . 2

2.2 Sequent calculus

A sequent is an expression of the form Γ⇒∆, where the antecedent Γ andthe succedent ∆ are finite multisets of formulae, i.e. sets which may containseveral occurrences of the same formula. We use usual conventions likewriting Γ, ϕ for Γ ∪ {ϕ}. Intuitively a sequent Γ⇒∆ corresponds to theformula (∧Γ) ⊃ (∨∆), which we in the following shall denote Γ→ ∆; (∧Γ)is > when Γ = ∅ and (∨∆) is ⊥ when ∆ = ∅.

The rules of the sequent calculus for LI are given in Figure 1. In thedefinition of the rules the following notation is used: ΓBk is the set {Bkϕ |ϕ ∈ Γ}, ΓCk is defined correspondingly. When a set Γ is superscripted withk, like in Γk, it contains only modal atoms of modality k.

Proofs are trees regulated by the rules whose leaves are axioms. Notethat an axiom is a zero-premiss rule. Two inferences are successive if theconclusion of the first is a premiss of the next. In derivation figures a series

4

375

Page 385: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Structural rules

Γ⇒∆Γ, ϕ⇒∆ Lt

Γ⇒∆Γ⇒ ϕ,∆ Rt

Γ, ϕ, ϕ⇒∆Γ, ϕ⇒∆ Lc

Γ⇒ ϕ,ϕ,∆Γ⇒ ϕ,∆ Rc

Identity rules

ϕ⇒ ϕ Id

Γ1⇒ ϕ,∆1 Γ2, ϕ⇒∆2

Γ1,Γ2⇒∆1,∆2Cut

Logical rules for the connectives

Γ⇒ ϕ,∆Γ¬ϕ⇒∆ L¬

Γ, ϕ⇒∆Γ⇒¬ϕ,∆ R¬

Γ, ϕi⇒∆Γ, ϕ1 ∧ ϕ2⇒∆

L∧i (i = 1, 2)Γ1⇒ ϕ,∆1 Γ2⇒ ψ,∆2

Γ1,Γ2⇒ ϕ ∧ ψ,∆1,∆2R∧

Γ1, ϕ⇒∆1 Γ2, ψ⇒∆2

Γ1,Γ2, ϕ ∨ ψ⇒∆1,∆2L∨

Γ⇒ ϕi,∆Γ⇒ ϕ1 ∨ ϕ2,∆

R∨i (i = 1, 2)

Γ1⇒ ϕ,∆1 Γ2, ψ⇒∆2

Γ1,Γ2, ϕ ⊃ ψ⇒∆1,∆2L⊃

Γ, ϕ⇒ ψ,∆Γ⇒ ϕ ⊃ ψ,∆ R⊃

Γ1, ϕ, ψ⇒∆1 Γ2⇒ ϕ,ψ,∆2

Γ1,Γ2, ϕ ≡ ψ⇒∆1,∆2L≡

Γ1, ϕ⇒ ψ,∆1 Γ2, ψ⇒ ϕ,∆2

Γ1,Γ2⇒ ϕ ≡ ψ,∆1,∆2R≡

Logical rules for the modalities

Θk,Γ⇒∆k, ϕ

Θk,ΓBk ⇒∆k,BkϕLRBk

provided ∆k, ϕ is non-empty

Θk,Γ⇒∆k, ϕ

Θk,ΓCk ⇒∆k,CkϕLRCk

provided ∆k, ϕ is non-empty

Bkϕ,Ckψ⇒ Conv provided {¬ϕ,¬ψ} is LI -consistent

The ϕ may be absent in both LRBk and LRCk. In Conv ϕ and ψ are bothfree of modality k.

Figure 1: Sequent calculus rules for LI .

5

376

Page 386: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

of 0 or more successive structural inferences is indicated with a double line.The height h(π) of a proof π is defined as the number of successive inferences(including structural inferences) on the longest branch in π so if π is anaxiom, h(π) = 1.

Theorem 3 (Correctness) Γ⇒∆ is provable in the sequent calculus iff` Γ→ ∆.

Proof. “If” is proved by routine induction on the axiomatic derivation. Theaxioms are provable in the sequent calculus. RN is replaced by applicationsof LRBk and LRCk, while MP is replaced by Cut. The other direction isproved by induction on a proof π of Γ⇒∆. All the rules for the Booleanconnectives are standard. We consider the three modality rules.

Case 1: the last inference in π is

Θk,Γ⇒∆k, ϕ

Θk,ΓBk ⇒∆k,BkϕLRBk

The rule can be used with ϕ absent, provided that ∆ is non-empty. By induc-tion hypothesis, ` Θk,Γ → ∆k, ϕ. By RN and modal logic, ` Bk(Θk,Γ →∆k, ϕ). Let Θk = {θ1, . . . , θm} and ∆k = {δ1, . . . , δn}. Assume first thatϕ is present. It follows from Lemma 1 and standard modal logic that (i)` Bk(θ1 ∧ · · · ∧ θm) ≡ (θ1 ∧ · · · ∧ θm) ∨Bk⊥, (ii) ` Bk(δ1 ∨ · · · ∨ δn ∨ ϕ) ≡(δ1 ∨ · · · ∨ δn ∨ Bkϕ). By (i), (ii) and principles of normal modal logics itfollows that ` (θ1∧· · ·∧θm)∨Bk⊥,ΓBk → ∆,Bkϕ, from which we concludethat ` Θk,ΓBk → ∆k,Bkϕ. If ϕ is absent, we can reason as if ϕ is ⊥. Wethen get ` (θ1 ∧ · · · ∧ θm) ∨Bk⊥,ΓBk → ∆k,Bk⊥. Since ∆k is non-empty,` Bk⊥ → ∆k. Hence ` Θk,ΓBk → ∆k.

Case 2: the last inference in π is LRCk. Same as case 1.Case 3: the last inference in π is

Bkϕ,Ckψ⇒ Conv

We know from the side condition of this rule that {¬ϕ,¬ψ} is LI-consistentand that ϕ,ψ are free of modality k. Hence ¬ϕ ⊃ ψ is LI -consistent, i.e.3k(¬ϕ ⊃ ψ) is an axiom. The axiom is equivalent to Bk(¬ϕ ⊃ ψ) ⊃¬Ck(¬ϕ ⊃ ψ). By properties of normal modal logics Bkϕ ⊃ Bk(¬ϕ ⊃ ψ)and Ckψ ⊃ Ck(¬ϕ ⊃ ψ). Hence ` Bkϕ,Ckψ → ⊥. 2

2.3 Cut-elimination theorems

The proof of the Cut-elimination theorem is inspired by Girard [3] andShvarts [10]. As usual for such arguments, the proof breaks down into a

6

377

Page 387: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

list of cases, each of which is simple to handle. In our proof of the cut-elimination theorems we follow Gentzen’s construction in the original paper[2] and replace the Cut rule by the Mix:

Γ1⇒ (ϕ)m,∆1 Γ2, (ϕ)n⇒∆2

Γ1,Γ2⇒∆1,∆2Mix

where ϕ 6∈ ∆1, ϕ 6∈ Γ2, and (ϕ)m denotes m occurrences of ϕ. ϕ is calledthe mix formula of the inference. It is easy to show that a Cut can beaccomplished by a Mix and structural inferences, and vice versa.

Definition 4 The degree d(χ) of a formula χ is inductively defined by

(i) d(p) = 1, p a propositional letter.

(ii) d(¬ϕ) = d(Bkϕ) = d(Ckϕ) = d(ϕ) + 1.

(iii) d(ϕ ∧ ψ) = d(ϕ ∨ ψ) = d(ϕ ⊃ ψ) = d(ϕ ≡ ψ) = max{d(ϕ), d(ψ)} + 1.

The degree of a Mix inference is the degree of its mix formula. The degreed(π) of a proof π is the maximum d(ϕ) such that ϕ is the mix formula of aMix inference in π and such that the ϕ in the right premiss of the Mix isnot introduced by a Conv inference.

Lemma 5 (Principal lemma) Let Σ − ϕ be Σ with every occurrence ofϕ removed. Assume that π1 is a proof of Σ1⇒ Φ1 and π2 is a proof ofΣ2⇒ Φ2 such that d(π1) < d(ϕ) and d(π2) < d(ϕ). Then there is a proof πof Σ1,Σ2 − ϕ⇒ Φ1 − ϕ,Φ2 such that d(π) < d(ϕ).

Proof. Note first that if ϕ 6∈ Φ1, π can be constructed using thinnings:

π1...

Σ1⇒ Φ− ϕΣ1,Σ2 − ϕ⇒ Φ1 − ϕ,Φ2

In particular this is the case when π1 is an identity axiom ψ⇒ ψ, ψ 6= ϕ. Inthe same way we use thinnings when ϕ 6∈ Σ2. We assume therefore in thefollowing that ϕ ∈ Φ1 and ϕ ∈ Σ2.

Assume that the last rule r1 of π1 has premisses Σ′i⇒ Φ′i proved by π′iand that the last rule r2 of π2 has premisses Σ′′j ⇒ Φ′′j proved by π′′j . Thelemma is proved by induction on h(π1) + h(π2). There are several cases toconsider, but the only interesting one is really the case in which r1 and r2 are

7

378

Page 388: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

logical inferences such that ϕ is the principal formula of r1 in the succedentof the conclusion and ϕ is the principal formula of r2 in the antecedent ofthe conclusion. We consider the subcase where ϕ = Bkψ; the case when ϕis Ckψ is symmetrical, and all the other cases are standard. π1 must be asfollows:

π′1...

Θk1,Γ1⇒∆k

1 , (Bkψ)m, ψ

Θk1,Γ

Bk1 ⇒∆k

1, (Bkψ)m,Bkψr1

Case 1: r2 is LRBk. π2 is then

π′′1...

Θk2 ,Γ2, (ψ)j , (Bkψ)n⇒∆k

2 , χ

Θk2,Γ

Bk2 , (Bkψ)j , (Bkψ)n⇒∆k

2,Bkχr2

Apply the induction hypothesis to π′1 and π2; call the resulting proof π′.Then, apply the induction hypothesis to π′′1 and π1; this gives a proof π′′.The two proofs are then merged with a Mix of degree less than d(ϕ):

π′...

Θk1,Γ1,Θk

2 ,ΓBk2 ⇒∆k

1 ,∆k2,Bkχ,ψ

π′′...

Θk1,Γ

Bk1 ,Θk

2 ,Γ2, (ψ)j ⇒∆k1,∆

k2 , χ

Θk1,Θ

k1 ,Θ

k2 ,Θ

k2,Γ1,Γ2,Γ

Bk1 ,ΓBk

2 ⇒∆k1 ,∆

k1 ,∆

k2 ,∆

k2, χ

Mix

Θk1 ,Θ

k2,Γ1,Γ2,Γ

Bk1 ,ΓBk

2 ⇒∆k1,∆

k2 , χ

Θk1 ,Θ

k2,Γ

Bk1 ,ΓBk

1 ,ΓBk2 ,ΓBk

2 ⇒∆k1,∆

k2 ,Bkχ

LRBk

Θk1 ,Θ

k2 ,Γ

Bk1 ,ΓBk

2 ⇒∆k1 ,∆

k2 ,Bkχ

Case 2: π2 isBkψ,Ckχ⇒ Conv

The induction hypothesis gives a proof π′ of Θk1,Ckχ,Γ1⇒∆k

1, ψ. π is then

π′...

Θk1,Ckχ,Γ1⇒∆k

1, ψ

Θk1 ,Ckχ,Γ

Bk1 ⇒∆k

1,Bkψ Bkψ,Ckχ⇒ Conv

Θk1,Ckχ,Ckχ,Γ

Bk1 ⇒∆k

1

Cut

Θk1,Ckχ,Γ

Bk1 ⇒∆k

1

At this point a Cut is introduced in the proof. By definition of degree, thisCut inference does not increase the degree of the proof. 2

8

379

Page 389: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Theorem 6 (Basic Cut-elimination) Let Γ⇒∆ be LI-provable. Thenthere is a proof of Γ⇒∆ which contains a Cut inference only if the cutformula in the right premiss is introduced by Conv.

Proof. First we can prove the following result by induction on π (theinduction hypothesis uses the Principal lemma 5): Let π prove Γ⇒∆, andassume d(π) > 0. Then there is a proof π′ of Γ⇒∆ such that d(π′) < d(π).By repeated application of this result we can transform any proof into aproof with the same endsequent, and with degree 0. 2

Corollary 7 For each formula ϕ there is a first-order formula ψ such thatthe sequent ⇒ ϕ ≡ ψ has a cut-free proof.

Proof. We know from the proof of Lemma 2 that the equivalence canbe proved without reference to the (conv)-axiom. There is thus a sequentcalculus proof of it which does not use Conv. Hence, if the proof containsinstances of Mix, they can all be eliminated. 2

Lemma 8 Let L be a logic containing classical logic. Assume that {¬ϕ,¬ψ}is L-consistent and that θ1, . . . , θn⇒ ϕ is provable. Then there is a θi, 1 ≤i ≤ n, such that {¬θi,¬ψ} is L-consistent.

Proof. Assume the contrary, i.e. that ¬ψ⇒ θj is provable for each j ≤ n.Since we have a proof of θ1, . . . , θn⇒ ϕ, n applications of Cut give a proofof ¬ψ⇒ ϕ, contradicting the consistency of {¬ϕ,¬ψ}. 2

Theorem 9 (Cut-elimination for normal form input) Let Γ⇒∆ beLI-provable and assume that all formulae in Γ,∆ are first-order. Then thesequent has a cut-free proof.

Proof. By Theorem 6 there is a proof of Γ⇒∆ with degree 0. Thismeans that any right Cut formula is introduced by Conv. Consider thelast inference in the proof π.

π1...

Θ⇒Ψ,Bkϕ Bkϕ,Ckψ⇒Θ,Ckψ⇒Ψ Cut

We may without loss of generality assume that Θ and Ψ contain only first-order modal atoms of modality k. The idea is to replace π′ by another proofπ′′ and derive Θ,Ckψ⇒Ψ by thinnings from the endsequent of π′′. There

9

380

Page 390: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

are two cases. If the succedent occurrence of Bkϕ in the endsequent of π′

is not tracable to an axiom (i.e. it is never used to close a branch), we cansimply remove it from all sequents in π′ in which it occurs. This will give us aproof π′′ of Θ⇒Ψ, from which Θ,Ckψ⇒Ψ follows by thinning. Otherwise,we must use LRBk on Bkϕ, assume without loss of generality that this isthe lowermost logical inference in π′. The premiss of this inference mustbe of the form Γk, θ1, . . . , θn⇒∆k, ϕ, where Bkθ1, . . . ,Bkθn are in Θ. Sinceθ1, . . . , θn, ϕ are all first-order they will be lost in a further inverse applicationof a LR modality rule. Hence θ1, . . . , θn⇒ ϕ must be provable. Moreover, bydefinition of the Conv rule {¬ϕ,¬ψ} is LI -consistent and free of modalityk. By Lemma 8 there is a θj such that {¬θj,¬ψ} is LI-consistent. Thus thefollowing is an axiom:

Bkθj ,Ckψ⇒ Conv

The axiom now serves the function as π′′. Since Bkθj ∈ Θ, we see thatΘ,Ckψ⇒Ψ follows from Bkθj,Ckψ⇒ by thinnings. By repeating thisprocedure, we can remove every instance of Cut. 2

Let us now address why these results entail that LI is a consistent logic.Note that to show that LI is well-defined we must prove that for any instance3kϕ of (conv) the unprovability of ¬ϕ in the side condition of the axiom canbe established without any reference to this instance of (conv). The proof ofTheorem 3 shows that the structure of (conv) axioms in a proof is preservedin the corresponding sequent calculus proofs, and vice versa. We can henceaddress the problem by inspecting proofs in the sequent calculus.

So assume that Γ′⇒∆′ is a provable sequent. By selecting equivalencesaccording to Corollary 7, we see that there is a sequent Γ⇒∆ with onlyfirst-order formulae in it which is provable iff Γ′⇒∆′ is provable. More-over, any proof of Γ⇒∆ can be extended to a proof of Γ′⇒∆′ by a simplesequence of cuts. These cuts are analytic and unconnected to any instancesof Conv; they are therefore unproblematic from the point of view of consis-tency. By inspection of the cut-free proof of Γ⇒∆, it is clear that the onlypossible source of inconsistency is Conv. But Conv is well-defined sincethe consistency of {¬ϕ,¬ψ}, ϕ,ψ free of modality k, is unaffected by theaddition of the formula Bkϕ ∧Ckψ → ⊥, an alternative equivalent formu-lation of the (conv)-axiom. Hence, there is no circularity in proofs due tothe (conv) axiom. In particular it is clearly impossible to derive the emptysequent, which establishes the consistency of LI .

10

381

Page 391: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

3 Including a quasi-order on I

We will in this section address an interesting extension of the multi-modallogic by imposing further structure to the modalities, and show that themain results in this paper still hold.

Assume that there is a quasi-order ≤ (reflexive and transitive, not neces-sarily anti-symmetric) on I. Let iEj iff i ≤ j and j ≤ i. E is an equivalencerelation; if iEk, we say that i and k are E-equivalent. The equivalence classof k modulo E is denoted k. Extending the terminology used previously wesay that a modal atom of a modality E-equivalent to k is a modal k-atom;the notions of a formula free of k-modalities a completely k-modalized for-mula are defined analogously. By a first-order formula we now understanda formula in which the ψ in each subformula in the form Bkψ or Ckψ is freeof k-modalities, for any k ∈ I.

The idea is to use the equivalence classes of I modulo E to representagents. The indices in an equivalence class then denote different confidencelevels for the given agent.

The logic we address in this section extends LI in two ways. First, weintroduce a modality 2k for each k ∈ I, add the axiom

2kϕ ≡ 2iϕ if iEk

and change the occurrence of 2k to 2k in the definition of RN and theaxioms B2, CN , B2, and C2. Second, we change the (conv)-axiom andadd two persistence axioms to the system:

PB : Biϕ ⊃ Bkϕ, i ≤ kPC : Ckϕ ⊃ Ciϕ, i ≤ k(conv): 3kϕ provided ϕ 6` ⊥, ϕ free of modality k

The generalization of the two modality reduction lemmata 1 and 2 are easilyseen to hold.

The rules for the extended sequent calculus are as in the previous sec-tions, with the following two modifications. First, each part of the sequentin a modality rule in Figure 1 which is superscripted with a k is now super-scripted with k; for instance is Θk in LRBk now changed to Θk to indicatethat Θ contains only modal k-atoms. Second, the calculus contains in addi-tion the following rules provided i ≤ k:

Θk,Bkϕ⇒∆k

Θk,Biϕ⇒∆kPBi

Θk,Ciϕ⇒∆k

Θk,Ckϕ⇒∆kPCk

11

382

Page 392: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Correctness of the calculus is proved by extending the proof of Theorem 3and is straightforward.

To prove cut-elimination, we must change the definition of the degreeof a formula. According to Definition 4, d(Bkϕ) = d(Biϕ), but if i < k,we need to state that the degree of Bkϕ is less than the degree of Biϕ.Hence we change δ to d′, where d′(χ) = d(χ) if χ is not a modal atom,otherwise d′(Bkϕ) = d(Bkϕ) + ε(k), ε(k) < 1. ε is a function satisfyingε(Bkϕ) < ε(Biϕ) iff i < k, and can, since ≤ is a quasi-order, be defined.

The two new rules cause some new cases to be checked in the proof ofthe Principal lemma. We consider the new subcase to the case that we haveaddressed for the other system. Assume r1 and r2 are logical inferences suchthat ϕ is the principal formula of r1 in the succedent of the conclusion andϕ is the principal formula of r2 in the antecedent of the conclusion. Let ϕbe Biψ and π1 end with an instance r1 of LRBi:

π′1...

Θk1,Γ1⇒∆k

1 , (Biψ)m, ψ

Θk1 ,Γ

Bi1 ⇒∆k

1, (Biψ)m,Biψr1

We must check the case that π2 ends with a PBi:

π′′1...

Θk2,Bkψ, (Biψ)n⇒∆k

2

Θk2,Biψ, (Biψ)n⇒∆k

2

r2

Apply the induction hypothesis to π′1 and π2, giving π′, and to π1 andπ′′1 , giving π′′. In the proof π below the application of LRBi in π1 hasbeen changed to an instance of LRBk. The instance of PBi in π2 hasbeen changed to many applications of PBi (indicated by double lines in theproof):

π′...

Θk1,Θ

k2 ,Γ1⇒∆k

1,∆k2 , ψ

Θk1,Θ

k2 ,Γ

Bk1 ⇒∆k

1 ,∆k2 ,Bkψ

LRBkπ′′

...Θk

1 ,Θk2 ,Γ

Bi1 ,Bkψ⇒∆k

1 ,∆k2

Θk1,Θ

k1 ,Θ

k2 ,Θ

k2,Γ

Bk1 ,ΓBi

1 ⇒∆k1 ,∆

k1 ,∆

k2,∆

k2

Mix

Θk1 ,Θ

k2,Γ

Bk1 ,ΓBi

1 ⇒∆k1 ,∆

k2

Θk1,Θ

k2 ,Γ

Bi1 ,ΓBi

1 ⇒∆k1 ,∆

k2

PBi

Θk1,Θ

k2 ,Γ

Bi1 ⇒∆k

1,∆k2

12

383

Page 393: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

π contains a Mix on Bkψ. Note that d′(π) < d′(ϕ), even though d(π) =d(ϕ). This is why we must use d′ as a measure of degree instead of d inthe proof of the Principal lemma. The case in which Ck is principal in r1

is symmetrical. Consistency of the system follows from the cut-eliminationresults as for the LI logic.

Theorem 10 In the sequent calculus for the extended version of LI , a se-quent is provable iff it has a proof which contains a Cut only if the cutformula in the right premiss is introduced by Conv. If every formula in thesequent is first-order the sequent is provable iff it has a cut-free proof.

4 Summary and future work

We have introduced a sequent calculus for a syntactically based multi-modalextension of Levesque’s “only knowing” logic and proved cut-eliminationresults for it. The technique has been successfully extended to a generalizedversion of the logic which enables the representation of confidence levels ofbeliefs. The cut-elimination theorems establish consistency of the systems,a result which is not easily obtainable by standard semantical methods.

There is a natural possible-worlds semantics for this system [11], but adiscussion of this semantics is beyond the scope of this paper. The preciserelationship between the system proposed in this paper and the systemsof Halpern and Lakemayer is complex and has not yet been clarified. Thesemantical framework will facilitate a comparison.

The system presented in section 3 is capable of representing the interest-ing class of Gricean implicatures covered in [1]. The nature of implicaturesis multi-modal; they result from the audience’s reflections about what anutterer meant by an utterance put forth in a particular conversational con-text. The representation of implicatures must have a priority mechanism,since Gricean implicatures are governed by a simple priority mechanism toprevent conflicts.

To give an idea of the representation, let us address a simple example inthe logic LI . Let the index set be {A,U} for audience and utterer. Assumethat U has said “p or q” and that the context of utterance is represented bythe formula κ (typically what has been said in the conversation in additionto p ∨ q). The idea behind the implicatures addressed here (generated onthe basis of the Gricean Maxim of quantity) is that if someone says ϕ, ψproperly entails ϕ, and the context does not entail ψ, then the utteranceimplicates that ¬ψ. One type of implicature arises from the entailment(p ∧ q) ⊃ (p ∨ q); if U says p ∨ q and the context does not entail that U

13

384

Page 394: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

meant p ∧ q, then U implicates ¬(p ∧ q), i.e. that the disjunction should betaken exclusively. The mechanism behind this pragmatical inference (calleda scalar implicature) can be formalized by

¬BU¬(p ∧ q) ⊃ BA¬BU¬(p ∧ q).

Let us call this formula σ. The formula has the form of a supernormal defaultin a multi-modal language and expresses that if p ∧ q it is compatible withthe utterer’s beliefs, then the audience will know about this. Conversely, ifA does not explicitly know that U said p ∧ q, then A believes that U doesnot mean that p ∧ q. It is easy to prove that if κ,BU (p ∨ q) 6` ¬BU¬(p∧ q),then ` OA(κ∧BU (p∨ q)∧σ) ≡ OA(κ∧BU (p∨ q)∧BU¬(p∧ q)), otherwise` OA(κ ∧BU (p ∨ q) ∧ σ) ≡ OA(κ ∧BU (p ∨ q)).

Another type of implicature arises from implications like BUp ⊃ BU (p∨q). If U says p ∨ q and the context does not entail that U knows whichdisjunct is true, then A believes that U does not know this (this is called aclausal implicature). If we let ι(ϕ) be the formula BUϕ ⊃ BABUϕ, thereare four clausal implicatures in this example: ι(p) ∧ ι(¬p) ∧ ι(q) ∧ ι(¬q).Let us call this formula χ. We can now, e.g., prove that if κ,BU (p ∨q),¬BUp,¬BU¬p,¬BUq,¬BU¬q 6` ⊥, then ` OA(κ ∧ BU (p ∨ q) ∧ χ) ≡OA(κ ∧BU (p ∨ q) ∧ ¬BUp ∧ ¬BU¬p ∧ ¬BUq ∧ ¬BU¬q), as expected.

In some cases we can represent the clausal and scalar implicatures onthe same level, i.e. when the formula OA(κ ∧ BU (p ∨ q) ∧ χ ∧ σ) behavesas expected. This is, however, not always the case. In general there may bea conflict between clausal and scalar implicatures, in which case the clausalones should be given priority [1]. In the modal language we can capturethis by having two modalities for the audience, representing two confidencelevels: one for the beliefs after clausal implicatures have been accounted forand one for beliefs after the scalar implicatures have been processed. Thelanguage introduced in section 3 is strong enough to represent this prioritystructure [11]; this representation will be addressed in a separate study.

References

[1] Gazdard, G. Pragmatics: Implicatures, Presupposition and LogicalForm. Academic Press, 1979.

[2] Gentzen, G. Untersuchungen uber das logische Schliessen. Matema-tische Zeitschrift 39 (176-210, 405-431), 1935. Translation in Szabo,M.E., The collected papers of Gerhart Gentzen, Nort-Holland, 1969.

14

385

Page 395: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[3] Girard, J., Lafont, Y., Taylor, P. Proofs and Types. CambridgeTracts in Theoretical Computer Science 7, 1989.

[4] Halpern J.Y. Reasoning about Only Knowing with Many Agents. InProc. of the 11th National Conference on Artificial Intelligence (AAAI-93), 1993.

[5] Halpern J.Y. Theories of Knowledge and Ignorance for Many Agents.In Journal of Logic and Computation 7:1 (79-108), 1997.

[6] Halpern J.Y. and Lakemeyer, G. Multi-Agent Only Knowing. InJournal of Logic and Computation 11:1 (40-70), 2001.

[7] Lakemeyer, G. All They Know: A Study in Multi-Agent Autoepis-temic Reasoning. In Proc. of the 13th International Joint Conferenceon Artificial Intelligence (IJCAI-93) (376-381), 1993.

[8] Levesque, H. J. All I Know: Study in Autoepistemic Logic. ArtificialIntelligence 42 (263-309), 1990.

[9] Lian, E., Langholm, T. and Waaler, A. Only Knowing with Confi-dence Levels: Reductions and Complexity. To appear in the proceedingsfor JELIA 2004, LNAI no. 3229, Springer-Verlag, 2004.

[10] Shvarts , G. F. Gentzen Style Systems for K45 and K45D. In Meyer,A.R., Taitslin, M.A. (eds.), Logic at Botik’89, Lecture Notes in Com-puter Science 363, Springer-Verlag 1989.

[11] Waaler, A. Logical studies in Complementary Weak S5. Doctorial the-sis, Department of Philosophy, University of Oslo, 1994.

15

386

Page 396: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Connexive modal logic

Heinrich WansingDresden University of Technology

Institute of Philosophy 01062 Dresden, [email protected]

Abstract

In a first step, an axiomatic system of connexive propositional logic is pre-sented. This logic, C, is shown to be sound and complete with respect to a classof relational models. It seems that this semantics is the first known intuitivelyplausible interpretation of a system of connexive logic. The presentation ofC suggests that connexive logic is constructive. It is a variant of David Nel-son’s constructive logics with strong negation. In Nelson’s logic the verificationconditions of implications are dynamic, whereas all falsification conditions arestatic conditions of falsification on the spot. In C, both the verification and thefalsification conditions of implications are dynamic. This is enough to ensurethat C is connexive and can be given a comprehensible and clear interpretationin terms of information states.

In a second step, the language of the system C is extended by the modaloperators 2 and 3 to obtain a connexive analogue of the smallest normal modalpropositional logic K. Aiming at a connexive analogue of K that can be faithfullyembedded by a modal translation into QC, quantified C, we arrive at a systemthat will be called CK, connexive K. The system CK is a connexive version ofthe constructive modal logic FSKd characterized in [7]. CK is shown to be soundand complete with respect to relational models and to be decidable.

We shall also critically discuss the evaluation clauses for the modal opera-tors that are induced by the standard translation from modal propositional logicinto first-order logic. In the context of the connexive base logic, the falsificationclauses of formulas 2A induced by the standard translation appear to be intu-itively implausible. In any case, both syntactic duality axioms ∼ 2A↔ 3 ∼ Aand ∼ 3↔ 2 ∼ A fail to hold.

It seems that CK is the first system of connexive modal logic consideredin the modal logic literature. The present paper may therefore be seen as acontribution to establishing connexive modal logic as a respectable branch ofmodal logic.

Keywords: modal logic, connexive logic, Aristotle’s Thesis, Boethius’ Thesis, con-structive logic, strong negation, standard translation.

1

387

Page 397: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

1 Aristotle’s Theses and Boethius’ Theses

The following principle is well-known as “Aristotle’s Thesis”:

AT ∼ (∼ A→ A),

see, for example, [6]. AT is of interest, because although AT is not a theorem ofclassical logic, nevertheless in the history of logic, AT has sometimes been foundplausible, if it is viewed as a formalization of “it is not the case that not-A impliesA”. In classical logic, AT is, of course, logically equivalent with

AT ′ ∼ (A→ ∼ A).

Also against the background of logics in which AT and AT ′ fail to be logicallyequivalent, they would seem to constitute a pair of equally intuitive (or unintuitive)theses. There exists, thus, a tension between the supraclassicality and some intuitiveplausibility of AT and AT ′.

One may observe that, using intuitionistically acceptable means only, the pair oftheses AT and AT ′ is equivalent in deductive power with another pair of schemata,which in accordance with previously introduced terminology are here called (Strong)“Boethius’ Theses” BT and BT ′. Boethius’ theses are:

BT (A→ B) → ∼ (A→ ∼ B); BT ′ (A→∼ B) → ∼ (A→ B).

More precisely, BT and BT ′ each allow one to intuitionistically derive both ATand AT ′, and AT ′ allows one to intuitionistically derive both BT and BT ′, whereasAT neither classically implies BT nor classically implies BT ′. Moreover, the thesesBT and BT ′ can easily be shown to be intuitionistically equivalent. Whereas theconverses

BT c ∼ (A→ ∼ B) → (A→ B); BT ′c ∼ (A→ B) → (A→∼ B).

of BT and BT ′ are classically valid, only BT ′c is also intuitionistically valid.

2 Connexive logic

Let L be a language containing a unary connective ∼ (negation) and a binary connec-tive → (implication). A logical system in the language L is called a connexive logic,if AT , AT ′, BT , and BT ′ are valid schemata and, moreover, (A→ B)→ (B → A)fails to be a valid schema.1 The connective → in a system of connexive logic is saidto be a connexive implication.

1According to McCall [6, p. 416], a “system of connexive logic may range from one in which noproposition implies or is implied by its own negation to one in which Boethius’ thesis is asserted.”But, if “it is to be of any interest, it must exclude the characteristic thesis (p → q) → (q → p) ofequivalence” [6, p. 417, notation adjusted]

2

388

Page 398: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

In [6] McCall presents an axiomatization of a system of connexive logic suggestedby Angell [1]. The language of this propositional logic CC1 contains as primitive(notation adjusted) a unary connective ∼ (negation) and the binary connectives ∧(conjunction) and→ (implication). Disjunction ∨ and equivalence ↔ are defined inthe usual way. The schematic axioms and the rules of CC1 are as follows:

A1 (A→ B)→ ((B → C)→ (A→ C))A2 ((A→ A)→ B)→ B

A3 (A→ B)→ ((A ∧ C)→ (B ∧ C))A4 (A ∧A)→ (B → B)A5 (A ∧ (B ∧ C))→ (B ∧ (A ∧ C))A6 (A ∧A)→ ((A→ A)→ (A ∧A))A7 A→ (A ∧ (A ∧A))A8 ((A→ ∼ B) ∧B)→ ∼ AA9 (A ∧ ∼ (A ∧ ∼ B))→ B

A10 ∼ (A ∧ ∼ (A ∧A))A11 (∼ A ∨ ((A→ A)→ A)) ∨ (((A→ A) ∨ (A→ A))→ A)A12 (A→ A)→∼ (A→ ∼ A)R1 modus ponensR2 adjunction

Note that among these axiom schemata, only A12 is supraclassical. The systemCC1 is characterized by the following four-valued truth tables with designated values1 and 2:

∼1 42 33 24 1

∧ 1 2 3 41 1 2 3 42 1 2 4 33 3 3 3 44 4 3 4 3

→ 1 2 3 41 1 4 3 42 4 1 4 33 1 4 1 44 4 1 4 1

McCall emphasizes that the logic CC1 is only one among many possible systemssatisfying the theses of Aristotle and Boethius and that he “does not wish it to beregarded as the system of connexive logic” [6, p. 418]. Indeed, McCall does notsuggest any intuitive interpretation of the four truth values employed in the abovematrices and the fact hat the values 1 and 2 are designated, and hence the semanticsappears to be a purely formal method with little explanatory power.2 In contrast

2Note that the constant truth functions 1, 2, 3, and 4 can be defined as follows [6, p. 421]:1 := (p → p), 2 := ∼ (p↔ ∼ p), 3 := (p ↔ ∼ p), 4 := ∼ (p→ p), for some sentence letter p.Routley and Montgomery [15, p. 95] point out that CC1 “can be given a semantics by associatingthe matrix value 1 with logical necessity, value 4 with logical impossibility, value 2 with contingenttruth, and value 3 with contingent falsehood. However, many anomalies result; e.g. the conjunctionof two contingent truths yields a necessary truth”.

3

389

Page 399: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

to this, the system C of the present paper is meant to be a suitable candidate forthe title ‘the basic system of connexive logic’.

Moreover, McCall points out that CC1 has some properties that are difficultto justify, if the name connexive logic is meant to reflect the fact that in a validimplication A → B there exists a connection between the antecedent A and thesuccedent B. Axiom A4, for example, is bad in this respect. On the other hand,CC1 might be said to undergenerate, since (A∧A)→ A and A→ (A∧A) fail to betheorems of CC1. If the validity of Aristotle’s and Boethius’ theses is distinctive ofconnexive logics, it is not quite clear, however, how damaging the above criticism is.

More serious doubts have been cast on CC1 by Routley and Montgomery [15]and, as these authors have claimed, on connexive logics in general. The startingpoint of their criticism, however, is a certain subsystem Z1 of CC1 containing thecontraposition theorem (∼ A → B) → (∼ B → A). This troublesome schema failsto be a theorem of the connexive logic C, defined in Section 3.

3 A basic system of connexive logic

The key observation for obtaining a connexive logic admitting a transparent andilluminating semantical characterization is simple: in the presence of the doublenegation laws, it suffices to validate both BT ′ and BT ′c. In other words, an inter-pretation of the falsification conditions of implications is called for, which deviatesfrom the standard conditions. In Nelson’s systems of constructive logic, the doublenegation laws hold, but the falsification conditions of implications are the classicalones expressed by the schema ∼ (A→ B)↔ (A∧ ∼ B). As a result, provable equiv-alence fails to be a congruence relation. Otherwise, ∼∼ (A → B) ↔ ∼ (A ∧ ∼ B)and hence, due to the double negation and De Morgan laws, (A→ B) ↔ (∼ A∨B)would be provable. But this is not the case in a constructive logic, where theverification conditions of implications are those of intuitionistic implications. Toobtain a connexive implication, it is therefore enough to assume another inter-pretation of the falsification conditions of implications expressed by the schema∼ (A→ B)↔ (A→ ∼ B).

Consider the language L := {∧,→,∼} based on the denumerable set AtL ofpropositional variables. Disjunction ∨ and equivalence ↔ are defined as usual. Theschematic axioms and rules of the logic C are:

a1 the axioms of intuitionistic positive logica2 ∼∼A↔ Aa3 ∼ (A ∨B)↔ (∼ A ∧ ∼ B)a4 ∼ (A ∧B)↔ (∼ A ∨ ∼ B)a5 ∼ (A→ B)↔ (A→ ∼ B)}R1 modus ponens

4

390

Page 400: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Clearly, a5 is the only supraclassical axiom of C. Deducibility in C and the conse-quence relation `C are defined as usual.

A C-frame is a pair F = 〈W,≤〉, where ≤ is a reflexive and transitive binaryrelation on the non-empty set W . Let 〈W,≤〉+ be the set of all X ⊆ W such thatif u ∈ X and u ≤ w, then w ∈ X. A C-model is a structure M = 〈W,≤, v+, v−〉,where 〈W,≤〉 is a C-frame and v+ and v− are valuation functions from AtL into〈W,≤〉+. Intuitively, W is a set of information states. The function v+ sends anatom p to the states in W that support the truth of p, whereas v− sends p to thestates that support the falsity of p. M = 〈W,≤, v+, v−〉 is said to be the modelbased on the frame 〈W,≤〉. The relations M, t |=+ A (M supports the truth of Aat t) and M, t |=− A (M supports the falsity of A at t) are inductively defined asfollows:

M, t |=+ p iff t ∈ v+(p)M, t |=− p iff t ∈ v−(p)

M, t |=+ (A ∧B) iff M, t |=+ A and M, t |=+ B tM, t |=− (A ∧B) iff M, t |=− A or M, t |=− B t

M, t |=+ (A→ B) iff ∀v ≥ t (M, v |=+ A implies M, v |=+ B)M, t |=− (A→ B) iff ∀v ≥ t (M, v |=+ A implies M, v |=− B)

M, t |=+∼ A iff M, t |=− AM, t |=−∼ A iff M, t |=+ A

Validity of a formula A in a C-model (M |= A) and validity of A on a frame(F |= A) are defined in the usual way. This means that if M = 〈W,≤, v+, v−〉is a C-model, then M |= A iff for every t ∈ W , M, t |=+ A. F |= A holds iffM |= A for every model M based on F . A formula is C-valid iff it is valid on everyframe. Support of truth and support of falsity for arbitrary formulas are persistentwith respect to the relation ≤ of possible expansion of information states. Thatis, for any C-model M = 〈W,≤, v+, v−〉, s, t ∈ W , and formula A, if s ≤ t, thenM, s |=+ A implies M, t |=+ A and M, s |=− A implies M, t |=− A. It can easilybe shown that a negation normal form theorem holds. Using familiar methods (seealso Section 4), the following can be shown:

Proposition 1 For any L-formula A, C ` A iff A is C-valid.

Proposition 2 The logic C satisfies the disjunction and the constructible falsityproperties. If C ` A ∨B, then C ` A or C ` B. If C ` ∼ (A ∧B), then C ` ∼ A orC ` ∼ B.

Proposition 3 The connexive logic C is decidable.

It is obvious form the above presentation that C differs from Nelson’s four-valuedconstructive logic N4 only with respect to the falsification (or support of falsity)

5

391

Page 401: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

conditions of implications. This modification is significant, as it not only turns →into a connexive implication but also leads to a constructive logic with intuition-istic implication in which the De Morgan laws hold and provable equivalence is acongruence relation.

Proposition 4 The set {A | C ` A} is closed under the rule A↔ B/C(A)↔ C(B).

4 First-order connexive logic

In this section we define a first-order extension QC of C. The system differs from thefirst-order extension QN4 of Nelson’s four-valued constructive logic characterized in[7, Section 4.1] only with respect to the treatment of negated implications. In orderto keep this paper self-contained, we shall here briefly present the axiomatizationand semantic characterization of QC, following the presentation in [7].

We extend the propositional language L to a first-order language by addingdenumerably many constant and predicate symbols. Let TL (CTL) denote the setof L-terms (closed L-terms), and let AtL be the set of atomic formulas and ForLbe the set of all L-formulas. The schematic axioms and rules of QC are those of Ctogether with:

∼ ∃xA↔ ∀x ∼ A; ∼ ∀xA↔ ∃x ∼ A

A(t)→ ∃xA(x) (t is free for x in A)

∀xA(x)→ A(t) (t is free for x in A)

A→ B(x)A→ ∀xB(x) (x not free in A); A(x)→ B

∃xA(x)→ B(x not free in B)

Deducibility in QC and the consequence relation `QC are defined in the usual way.A QC-model is a structure 〈W,≤,∆, D, v+, v−〉, where 〈W,≤〉 is a C-frame; ∆ is

a set such that CTL ⊆ ∆ ⊆ TL, and D is a function from W to subsets of ∆ suchthat (i) CTL ⊆ Du for all u ∈W and (ii) Du ⊆ Dt whenever u ≤ t. Finally, v+ andv− are functions from AtL to 〈W,≤〉+ such that if u ∈ v+(−)(P (a1, . . . , an)), thena1, . . . , an ∈ Du. The function D assigns to every state u ∈ W its domain Du. IfM = 〈W,≤,∆, D, v+, v−〉 is a model and t ∈ W , the notions M, t |=+ A (state tsupports the truth of A in M) and M, t |=− A (t supports the falsity of A in M)are inductively defined as for C, except that in addition we have:

M, t |=+ ∀xA(x) iff (∀u ∈W ) if t ≤ u, then (∀c ∈ Du) M, u |=+ A(c)M, t |=− ∀xA(x) iff (∃c ∈ Dt) M, t |=+ A(c)M, t |=+ ∃xA(x) iff (∃c ∈ Dt) M, t |=+ A(c)M, t |=− ∃xA(x) iff (∀u ∈W ) if t ≤ u, then (∀c ∈ Du) M, u |=− A(c)

6

392

Page 402: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Again, support of truth and support of falsity for arbitrary formulas are per-sistent with respect to ≤. A formula A ∈ ForL is QC-valid if for any modelM = 〈W,≤,∆, D, v+, v−〉 and t ∈ W , M, t |=+ A. Soundness can be shown byinduction on the length of proofs.

Proposition 5 For any A ∈ ForL, if QC ` A, then A is QC-valid.

As in the case of QN4, completeness can be shown by a faithful embedding intoQInt+, positive first-order intuitionistic logic, see [2]. L+, the language of QInt+,is the result of deleting ∼ from L. A QInt+-model is a structure 〈W,≤,∆, D, v〉defined like a QC-model, except that it has only one valuation function v from AtLto 〈W,≤〉+ such that if u ∈ v(P (a1, . . . , an)), then a1, . . . , an ∈ Du.

LetM = 〈W,≤,∆, D, v〉 be a QInt+-model, t ∈W and A ∈ ForL+ . The relationM, t |= A (A is true at t in M) is defined exactly as the relation M, t |=+ A wasdefined for positive connectives and quantifies (with v+ replaced by v in case ofatomic formulas). A formula A ∈ ForL+ is QInt+-valid if for any QInt+-modelM = 〈W,≤,∆, D, v〉 and t ∈W , A is true at t in M.

Proposition 6 For any A ∈ ForL+, QInt ` A iff A is QInt+-valid.

The proof is essentially the same as the completeness proof for full first-order intu-itionistic logic [2], omitting the part concerning negation.

We say that a formula A ∈ ForL is in negation normal form (nnf), if it containsnegations only in front of atomic formulas, in other words, if ∼ B is a subformulaof A, then B is atomic.

Definition 1 We define the transformation (·) on ForL as follows:

1. B := B, ∼ B :=∼ B for B ∈ AtL.

2. ∼∼ A := A for a formula A.

3. A �B := A �B, where A and B are formulas and � ∈ {∨,∧,→}.

4. For any formula A and Q ∈ {∀,∃}, QxA := QxA.

5. For any formulas A and B, ∼ (A ∨B) := ∼ A∧∼ B, ∼ (A ∧B) := ∼ A∨∼ B,and ∼ (A→ B) := A→ ∼ B.

6. For any formula A, ∼ ∀xA := ∃x∼ A and ∼ ∃xA := ∀x∼ A.

Proposition 7 For any formula A, A is in negation normal form and QC ` A↔ A.

We now define the positive intuitionistic first-order language L′+ by deleting ∼ fromL and adding for each predicate symbol P of L a new predicate symbol P∼ of thesame arity.

7

393

Page 403: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Definition 2 The transformation (·)∗ of formulas from ForL into ForL′+ is definedas follows:

1. For an atomic P (a), (P (a))∗ := P (a) and (∼ P (a))∗ := P∼(a).

2. (A �B)∗ := A∗ �B∗, where A and B are formulas in nnf and � ∈ {∨,∧,→}.

3. (QxA)∗ := QxA∗, where A is a formula in nnf and Q ∈ {∀,∃}.

4. (A)∗ := (A)∗ for any formula A not in nnf.

Proposition 8 For any A ∈ ForL, QC ` A iff QInt+ ` A∗.

Proof. If QC ` A, one can easily check by induction on the length of proofs that A∗ isprovable in QInt+. If QInt ` A∗ and B0, . . . Bn = A∗ is a proof, then B′0, . . . B

′n = A,

where B′i is the result of replacing any subformula P∼(a) by ∼ P (a), is a proof ofthe nnf of A in QC. By Proposition 7, we then have QC ` A.

Proposition 9 For any A ∈ ForL, QC ` A iff A is QC-valid.

Proof. Let A0 be a QC-valid formula. Assume that QC 6` A0. Then QInt+ 6` (A0)∗

by Proposition 8. This means that there is a QInt+-model M = 〈W,≤,∆, D, v〉 ofthe language L′+ and t0 ∈ W such that M, t0 6|= (A0)∗. We define a QC-modelM′ = 〈W,≤,∆, D, v+, v−〉 with the same W , ≤, and ∆. The assignment functionsv+ and v− are defined as follows:

v+(P (a)) := v(P (a)) and v−(P (a)) := v(P∼(a)).

Using induction on the structure of a formula, we can show that

M, t |= A∗ if and only if M′, t |=+ A

for any t ∈ W and formula A in nnf. Thus, M′, t0 6|=+ A0 and M′, t0 6|=+ A0 byProposition 7, which conflicts with the QC-validity of A0.

5 The connexive modal logic CK

Let now L2,3 := {∧,→,∼,2,3}. To obtain a connexive analogue of the smallestnormal modal logic K, we consider the translation Tx from ForL2,3 into the first-order language L of QC:

8

394

Page 404: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

pTx7−→ P (x)

∼ p Tx7−→ ∼ P (x)

A ∗B Tx7−→ Tx(A) ∗ Tx(B), ∗ ∈ {∧,→}∼ (A ∗B) Tx7−→ ∼ (Tx(A) ∗ Tx(B)), ∗ ∈ {∧,→}

2ATx7−→ ∀y(R(x, y)→ Ty(A))

3ATx7−→ ∃y(R(x, y) ∧ Ty(A))

∼2ATx7−→ ∃y(R(x, y) ∧ Ty(∼A))

∼3ATx7−→ ∀y(R(x, y)→ Ty(∼A))

Consider the following axioms and rules.

a6 2A ∧2B → 2(A ∧B)a7 2(A→ A)a8 3(A ∨B)→ 3A ∨3Ba9 3(A→ B)→ (2A→ 3B)a10 (3A→ 2B)→ 2(A→ B)a11 ∼2A↔ 3 ∼Aa12 ∼3A↔ 2 ∼AR2 A→ B /2A→ 2BR3 A→ B /3A→ 3B

The connexive modal logic CK is defined as the deductive closure of axioms a1–a12 under the rules R1, R2, and R3. If ∆ is a set of L2,3-formulas and A anL2,3-formula, the relation ∆ ` A holds iff A belongs to the closure of CK ∪ ∆under R1. Axioms a9 and a10 ensure that 2 and 3 can be interpreted semanticallywith respect to a single accessibility relation R (instead of a pair of independentaccessibility relations R2 and R3) (see [4], [8], [17]) and that CK can be faithfullyembedded into QC.

A CK-frame is a triple F = 〈W,≤, R〉, where 〈W,≤〉 is a C-frame and R is abinary relation on W such that:

1. ≤ is reflexive and transitive;

2. ≤−1 ◦R ⊆ R◦ ≤−1;

3. R◦ ≤ ⊆ ≤ ◦R.

A CK-model is a structure M = 〈W,≤, R, v+, v−〉, where 〈W,≤, R〉 is a CK-frameand v+ and v− are valuation functions from AtL into 〈W,≤〉+. The relationsM, t |=+ A and M, t |=− A are inductively defined as for C, except that in ad-dition we have:

9

395

Page 405: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

M, t |=+ 2A iff ∀u ≥ t ∀v(uRv implies M, v |=+ A)M, t |=− 2A iff ∃u(tRu and M, u |=− A)

M, t |=+ 3A iff ∃u(tRu and M, u |=+ A)M, t |=− 3A iff ∀u ≥ t ∀v(uRv implies M, v |=− A)

A discussion of the support of truth conditions for the modal operators can be foundin [16]. Support of truth and support of falsity are persistent with respect to therelation ≤ of possible expansion of information states. A formula A is valid in amodel M = 〈W,≤, R, v+, v−〉 (M |= A) if M, t |=+ A for all t ∈W ; A is said to bevalid on a frame F (F |= A) if A is valid in any model based on F .

Proposition 10 For any L-formula A, CK ` A iff A is valid on every CK-frame.

The proof is basically the completeness proof for the constructive modal logic FSKd,see [8]. A canonical model is defined and shown to satisfy the above conditions 1.–3.on its underlying frame.

Definition 3 We define the transformation (·) on ForL2,3 as in Definition 1 exceptthat the clause for quantified formulas is replaced by the following:

6. For any L2,3-formula A, ∼ 2A := 3∼ A and ∼ 3A := 2∼ A.

Proposition 11 For any L2,3-formula A, A is in negation normal form and CK `A↔ A.

This negation normal form theorem allows CK to be rewritten as FS+, the positivefragment of Fischer Servi’s intuitionistic modal logic FS, see [3], [4], [5], [16].

Proposition 12 The translation Tx is a faithful embedding of CK into QC.

Proof. Observe that the translation of L2,3-formulas in negation normal form intothe language L2,3

′+ that maps every propositional variable p to itself, sends everynegated propositional variable ∼p to a fresh propositional variable p′, and commuteswith the positive connectives is a faithful embedding of CK into FS+. This is enoughto establish the claim, since FS+ is faithfully embedded by the standard translation,coinciding on negation-free formulas with Tx, into QInt+ = CK+, positive CK.

Since FS is decidable, we obtain the following:

Corollary 1 The connexive modal logic CK is decidable.

Proposition 13 The logic CK satisfies the disjunction and the constructible falsityproperties.

Proposition 14 The set {A | CK ` A} is closed under the rule A ↔ B/C(A) ↔C(B).

10

396

Page 406: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

Note hat Tx differs from the standard translation STx from ForL2,3 into ForL:

pSTx

7−→ P (x)

∼ ASTx

7−→ ∼ STx(A)

A ∗BSTx

7−→ STx(A) ∗ STx(B), ∗ ∈ {∧,→}2A

STx

7−→ ∀y(R(x, y)→ STy(A))

3ASTx

7−→ ∃y(R(x, y) ∧ STy(A))

Whereas Tx guarantees the syntactic duality between 2 and 3, the standard trans-lation of ∼ 2A is equivalent with ∃y(R(x, y)→ ∼ STy(A)), which is not equivalentwith the standard translation of 3 ∼ A. Also ∼ 3A and 2 ∼ A are not equiva-lent under STx. The standard translation of ∼ 3A yields a formula equivalent with∀y(∼ R(x, y)∨∼ STy(A)). The support of falsity clauses for modal formulas inducedby STx are:

M, t |=− 3A iff ∀s ≥ t ∀u(sR∼u orM, u |=− A)M, t |=− 2A iff ∃s∀u ≥ s(tRs implies M, u |=− A)

where R∼ is an impossibility relation. The support of falsity clause for formulas 3Ais plausible: if for every expansion s of information state t, every state u is such thateither u is impossible relative to s or u supports the falsity of A, then t supports thefalsity of 3A. The support of falsity clause for 2A, however, appears to be muchmore problematic. To support the falsity of 2A, it is enough for a state t that thereexists one inaccessible state, which is not very convincing.

6 Another system of connexive logic

Axiom a5 calls for some explanation. Whereas the direction from right to left canbe justified by rejecting the view that if A implies B and A is inconsistent, A impliesany formula, in particular B, the direction from left to right seems rather strong. Ifthe verification conditions of implications are dynamic, then a5 indicates that thefalsification conditions of implications are dynamic as well. The falsity of (A→ B)thus implies that if A is true, B is false. Yet, one might wonder why it is not requiredthat the falsity of (A → B) implies that if if A is true, B is not true. This cannotbe expressed in a language with just one negation ∼ expressing falsity instead ofabsence of truth. If one adds to C the further axiom ∼ A → (A → B) to obtaina connexive variant of Nelson’s three-valued logic N3, intuitionistic negation ¬ isdefinable by setting: ¬A := A→∼ A. Then a5 could be replaced by

a5′ ∼ (A→ B)↔ (A→ ¬B).

11

397

Page 407: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

The resulting system is another system of connexive logic with a transparent seman-tics and an intuitive interpretation in terms of information states. It satisfies AT ,AT ′, BT , and BT ′, because A→ ¬ ∼ A and ∼ A→ ¬A are theorems.

7 Some further topics

Aristotle’s and Boethius’ Theses express, as it seems, some pre-theoretical intuitionsabout meaning relations between negation and implication. But is not clear that alanguage must contain only one negation operation and only one implication con-nective. The language of systems of consequential implication comprises two impli-cation connectives and one negation, see [9], [10], [11], [12]. In [13], the notion of anormal system of consequential logic is defined. The smallest normal consequentiallogic that satisfies AT is called CI. Alternatively, CI can be characterized as thesmallest normal system that satisfies Weak Boethius’ Thesis:

(A→ B) ⊃∼ (A→ ∼ B),

where → is consequential and ⊃ is material implication. In the full version of thepaper we shall make comments on Aristotle’s and Boethius’ Theses in the presenceof more than one implication and more than one negation (cf. Section 6). Moreover,we shall briefly discuss Popper’s philosophy of science in the light of connexive logic.

References

[1] R. Angell, A Propositional Logic with Subjunctive Conditionals, Journal of SymbolicLogic 27 (1962), 327–343.

[2] D. van Dalen, Intuitionistic Logic, in: D. Gabbay and F. Guenthner (eds.), Handbookof Philosophical Logic Vol. III, Reidel, Dordrecht, 1986, 225–339.

[3] G. Fischer Servi. Axiomatizations for some Intuitionistic Modal Logics, Red. Sem. Mat.Universi. Politec. Torino 42, 179–194, 1984.

[4] D. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-Dimensional ModalLogics: Theory and Applications , Elsevier, Amsterdam, 2003.

[5] C. Grefe. Fischer Servi’s Intuitionistic Modal Logic has the Finite Model Property, in:M. Kracht et al. (eds.), Advances in Modal logic. Vol. 1., CSLI Lecture Notes 87, 1998,85–98.

[6] S. McCall, Connexive Implication, Journal of Symbolic Logic 31 (1966), 415–433.

[7] S.P. Odintsov and H. Wansing, Inconsistency-tolerant Description Logic: Motivationand Basic Systems, in: V.F. Hendricks and J. Malinowski (eds.), Trends in Logic: 50Years of Studia Logica, Kluwer Academic Publishers, Dordrecht, 2003, 287–321.

[8] S.P. Odintsov and H. Wansing, Constructive Predicate Logic and Constructive ModalLogic. Formal Duality versus Semantical Duality, to appear in: V. Hendricks etal. (eds.), Proceedings FOL75, Logos Verlag, Berlin, 2004.

12

398

Page 408: pdfs. · PDF filePreface Advances in Modal Logic is the main international forum at which research on all aspects of modal logic is presented. The Advances in Modal Logic Initiative

[9] C. Pizzi, Boethius’ Thesis and Conditional Logic, Journal of Philosophical Logic 6(1977), 283–302.

[10] C. Pizzi, Decision Procedures for Logics of Consequential Implication, Notre DameJournal of Formal Logic 32 (1991), 618–636.

[11] C. Pizzi, Consequential Implication: A Correction Notre Dame Journal of Formal Logic34 (1993), 621–624.

[12] C. Pizzi, Weak vs. Strong Boethius’ Thesis: A Problem in the Analysis of ConsequentialImplication, in: A. Ursini and P. Aglinano (eds.), Logic and Algebra, Marcel Dekker,New York, 1996, 647–654.

[13] C. Pizzi and T. Williamson, Strong Boethius’ Thesis and Consequential Implication,Journal of Philosophical Logic 26 (1997), 569–588.

[14] R. Routley, Semantical Analyses of Propositional Systems of Fitch and Nelson, StudiaLogica, 33 (1974), 283–298.

[15] R. Routley and H. Montgomery, On Systems Containing Aritotle’s Thesis, Journal ofSymbolic Logic 33 (1968), 82–96.

[16] A.K.Simpson. The Proof Theory and Semantics of Intuitionistic Modal Logics, PhDThesis, University of Edinburg, 1994.

[17] V. Sotirov. Modal Theories With Intuitionistic Logic, in: Mathematical Logic. Proceed-ings of the Conference on Mathematical Logic, Dedicated to the Memory of A.A. Markov(1903-1979), Sofia, September 22-23, 1980, Sofia, 1984, 139–171.

13

399