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Modal logic programming revisitedLinh Anh Nguyen aa University of Warsaw, Institute of Informatic , ul. Banacha 2, 02-097 , Warsaw , PolandPublished online: 13 Apr 2012.

To cite this article: Linh Anh Nguyen (2009) Modal logic programming revisited, Journal of Applied Non-Classical Logics,19:2, 167-181, DOI: 10.3166/jancl.19.167-181

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Modal logic programming revisited

Linh Anh Nguyen

University of WarsawInstitute of Informaticsul. Banacha 202-097 Warsaw (Poland)

nguyen@mimuw.edu.pl

ABSTRACT. We present optimizations for the modal logic programming system MProlog, in-cluding the standard form for resolution cycles, optimizedsets of rules used as meta-clauses,optimizations for the version of MProlog without existential modal operators, as well as iter-ative deepening search and tabulation. Our SLD-resolutioncalculi for MProlog in a numberof modal logics are still strongly complete when resolutioncycles are in the standard form andoptimized sets of rules are used. We also show that the labelling technique used in our direct ap-proach is relatively better than the Skolemization technique used in the translation approachesfor modal logic programming.

KEYWORDS:modal logic, logic programming, MProlog.

DOI:10.3166/JANCL.19.167–181c© 2009 Lavoisier, Paris

1. Introduction

Modal logic programming is the field that extends classical logic programming todeal with modalities. As modal logics can be used, among others, to reason aboutknowledge and belief, developing a good formalism for modallogic programs and anefficient computational procedure for it is desirable.

The first work on extending logic programming with modal logic is (Fariñas delCerro, 1986) on the implemented system Molog (Fariñas del Cerro et al., 1986). WithMolog, the user can fix a modal logic and define or choose the rules to deal withmodal operators. Molog can be viewed as a framework which canbe instantiated withparticular modal logics. As an extension of Molog, the Toulouse Inference Machine(Balbiani et al., 1991) together with an abstract machine model called TARSKI forimplementation (Alliotet al., 1992) makes it possible for a user to select clauses whichcannot exactly unify with the current goal, but just resemble it in some way.

Journal of Applied Non-Classical Logics. Volume 19 – No. 2/2009, pages 167 to 181

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Apart from the mentioned works, modal logic programming hasbeen studied byseveral authors in (Balbianiet al., 1988; Akama, 1989; Debartet al., 1992; Nonnen-gart, 1994; Baldoniet al., 1996) and by us in (Nguyen, 2003; Nguyen, 2004; Nguyen,2006). The works (Balbianiet al., 1988; Baldoniet al., 1996) and our mentionedworks use the direct approach, the works (Akama, 1989; Debart et al., 1992) usethe functional translation approach, and the work (Nonnengart, 1994) uses the semi-functional approach. See (Orgunet al., 1994; Nguyen, 2006) for further information.

In (Nguyen, 2006), we gave a framework for developing the least model semantics,fixpoint semantics, and SLD-resolution calculi for positive modal programs (calledMProlog programs) in modal logics whose frame restrictionsconsist of the serial-ity conditions (i.e.∀x∃y Ri(x, y) for every modal indexi) and some classical Hornclauses. We have applied the framework for basic serial monomodal logics (Nguyen,2003), multimodal logics of belief (Nguyen, 2006), the classBSMM of basic serialmultimodal logics (Nguyen, 2006), and the classsCFG of serial context-free gram-mar logics (Nguyen, 2007).

The special feature of our framework is that it uses the direct approach and doesnot assume any special restriction on occurrences of modal operators1, while the work(Balbianiet al., 1988) assumes that universal modal operators do not occur in bodies ofprogram clauses and goals, and the work (Baldoniet al., 1996) assumes that existentialmodal operators do not occur in program clauses and goals.

Using our framework we have designed and implemented the modal logic pro-gramming system MProlog (Nguyen, 2004; Nguyen, 2008b). Thetheoretical founda-tion of the MProlog system is very different than that of Molog (Fariñas del Cerroetal., 1986). In MProlog, the labelling technique is used for existential modal opera-tors instead of Skolemization. Our system uses new technicalities like normal formsof modalities and pre-orders between modal operators. It also eliminates some draw-backs of Molog, e.g., MProlog gives substitutions as computed answers, while Mologcan only answer “yes” or “no” (where “yes” means there existsa correct answer).

In this paper, we present new results and optimization techniques for modal logicprogramming. One of the theoretical results is the theorem about strongcomplete-ness of our SLD-resolution calculi for MProlog. We give various optimizations forMProlog that are both interesting from the theoretical point of view and useful forthe implementation, including the standard form for resolution cycles, optimized setsof rules used as meta-clauses, optimizations for the version of MProlog without ex-istential modal operators, as well as iterative deepening search and tabulation. Theother theoretical results are that our SLD-resolution calculi for MProlog in a numberof modal logics are still strongly complete when resolutioncycles are in the standardform and optimized sets of rules are used. We also show that the labelling techniqueused in our direct approach is relatively better than the Skolemization technique usedin the translation approaches for modal logic programming.

1. Programs and goals in our framework are of a normal form butthe language is as expressiveas the general modal Horn fragment.

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Modal logic programming revisited 169

This paper can be treated as a supplement to (Nguyen, 2006). We assume that thereader is familiar with modal logic and logic programming. Despite that we have madethis paper self-contained to a certain extent, for a more comprehensive explanationwe recommend the reader to read some parts of (Nguyen, 2006),e.g. the illustratingexample given in (Nguyen, 2006, Section 1). Due to the lack ofspace, the proofs ofthe theorems given in this paper are presented only in the technical report (Nguyen,2008a).

2. Preliminaries

2.1. Considered modal logics

Modal logics considered in our framework of modal logic programming arequan-tified modal logics with fixed domain and rigid terms. Their language is an exten-sion of the language of classical first-order logic with modal operators�i and♦i, for1 ≤ i ≤ m (wherem is a fixed number). Ifm = 1 then we ignore the subscripti andwrite � and♦. The operators�i are calleduniversalmodal operators, while♦i arecalledexistentialmodal operators.

We restrict ourselves to modal logics that extend the quantified modal logicK(m)

with axioms(D) : �iϕ → ♦iϕ (for 1 ≤ i ≤ m) and some axioms that correspondto frame restrictions of the form of a Horn clause. The axiom(D) for modal indexicorresponds to the seriality condition∀x∃y Ri(x, y). When�iϕ is read as “agentibelieves thatϕ holds” (and♦iϕ is treated as¬�i¬ϕ), the axiom says that beliefs ofagenti are consistent.

In some examples we use the modal logicsKDI4s andKDI4s5. In these logics,�iϕ means “ϕ is believed with degree at leasti”. These logics are axiomatised as:

KDI4s = K(m) + (D) + (I) + (4s)

KDI4s5 = K(m) + (D) + (I) + (4s) + (5)

where the schemata of the additional axioms are:2

(I) : �iϕ→ �jϕ if i > j,(4s) : �iϕ→ �j�iϕ (strong positive introspection),(5) : ¬�iϕ→ �i¬�iϕ (negative introspection).

2.2. The logical formalism MProlog

A universal modalityis a (possibly empty) sequence of universal modal operators.We use� to denote a universal modality. Similarly as in classical logic programming,

2. Axiom (5s) : ¬�iϕ→ �j¬�iϕ (strong negative introspection) is derivable inKDI4s5.

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we write�(ϕ← ψ1, . . . , ψn) to denote the formula∀(�(ϕ ∨ ¬ψ1 . . . ∨ ¬ψn)).3 WeuseE to denote a classical atom.

A program clauseis a formula of the form�(A ← B1, . . . , Bn), wheren ≥ 0andA, B1, . . . ,Bn are formulas of the formE, �iE, or ♦iE. � is called themodalcontext, A thehead, and(B1, . . . , Bn) thebodyof the program clause. AnMPrologprogramis a finite set of program clauses.

An MProlog goal atomis a formula of the form�E or �♦iE, where� is calledthe modal contextof the goal atom. AnMProlog goal is a formula written in theclausal form← α1, . . . , αk, where eachαi is an MProlog goal atom.

LetP be an MProlog program andG an MProlog goal of the form← α1, . . . , αk.A substitutionθ is acorrect answerin a modal logicL for P ∪ {G} if the domain ofθ consists of variables occurring inG andP |=L ∀((α1 ∧ . . . ∧ αk)θ).

It is shown in (Nguyen, 2008a) that MProlog has the same expressive power asthe general Horn fragment in normal modal logics that are characterised by a classof Kripke structures. For a specific modal logicL, we may adopt some restrictionson modal contexts of MProlog program clauses and MProlog goal atoms and call theobtained languageL-MProlog. (If no restriction is adopted thenL-MProlog is the sameas MProlog.) Such restrictions either follow from equivalencies inL or are acceptablefrom the practical point of view, and furthermore, they do not reduce expressivenessof the language.

For example, inKDI4s5 we have the equivalence∇∇′ϕ ≡ ∇′ϕ, where∇ and∇′ are arbitrary modal operators. Hence we can assume that the modal context of aKDI4s5-MProlog program clausehas length 1 or 0, and aKDI4s5-MProlog goalatomis a formula of the formE, �iE or ♦iE, withE being a classical atom.

2.3. A framework of SLD-resolution for MProlog

In (Nguyen, 2006), we gave a framework for developing fixpoint semantics, theleast model semantics, and SLD-resolution calculi forL-MProlog, whereL is a se-rial modal logic whose frame restrictions except serialityare Horn clauses (of classi-cal first-order logic). We outline here the fragment involving SLD-resolution of thatframework.

From now on, by amodal operatorwe mean�i, ♦j , or 〈S〉k, where〈S〉k is♦k labelled byS, which is either a classical atom or a variable for classicalatoms(called anatom variable). For further information on labelled modal operators, see(Nguyen, 2006). We use∇ and∇′ to denote modal operators.

A modalityis a (possibly empty) sequence of modal operators. We use△ to denotea modality. Recall that we useE to denote a classical atom. Amodal atomis a formula

3. By ∀(ϕ) we denote theuniversal closureof ϕ, which is the formula obtained by adding auniversal quantifier for every variable having a free occurrence inϕ.

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of the form△E. A simple modal atomis a formula of the formE or∇E. We useA,B to denote simple modal atoms, andα, β to denote modal atoms.

There may exist a compact form for modalities inL. For each specific modal logicL, we defineL-normal form of modalities. For example, a modality is inKDI4s5-normal form if its length is 0 or 1. It is possible that no restriction is adopted forL-normal form of modalities. A modality is inL-normal labelled formif it is in L-normal form and does not contain unlabelled existential modal operators♦i. A modalatom△E is in L-normal (labelled) form if△ is in L-normal (labelled) form. Given aground modal atom, theNFL operatorconverts it toL-normal form.

Given a modal atomα, one can derive other modal atoms fromα using axiomsof L. The corresponding operator is called theSatL operator(whereSat stands for“saturation”). The direct consequence operatorTL,P is defined usingSatL andNFL.An SLD-resolution calculus can be viewed as a reversed analogue of a direct conse-quence operator. Hence, to define an SLD-resolution calculus forL-MProlog we needreversed analogues of the operatorsSatL andNFL. These operators are called therSatL operatorand therNFL operator, respectively. See (Nguyen, 2006) for formaldefinitions of the operatorsSatL,NFL, rSatL, andrNFL.

TherSatL andrNFL operators are each specified by a finite set of rules of theform α ← β, whereα andβ are (schemata of) modal atoms. The rules are used asmeta-clauses in SLD-derivations. Such rules can be accompanied by conditions whichspecify when the rule are applicable.4

For example, therNFL andrSatL operators forL = KDI4s5 are specified bythe following rules, whereX is a fresh atom variable:5

rNFL: (a) ∇E ← 〈X〉i∇E;

rSatL: (a) ∇∇′E ← ∇′E,(b) ♦iE ← ♦jE if i > j,(c) ♦iE ← 〈X〉iE.

A goal is a clause of the form← α1, . . . , αk, where eachαi is a modal atom.Resolventsof a goal← α1, . . . , αk and anrSatL/rNFL rule α ← β are definedin the usual way (Nguyen, 2006). For example, resolving← �1♦2p(x) with therule∇∇′E ← ∇′E results in← ♦2p(x), since∇ is instantiated to�1, and∇′ isinstantiated to♦2.

For each specific modal logicL, we define a pre-order�L to compare modal op-erators. For example, forL = KDI4s5, the pre-order�L is the least reflexive andtransitive binary relation between modal operators such that: ♦i �L 〈S〉i �L �i,and if i < j then�i �L �j and♦j �L ♦i. If ∇ �L ∇

′ then we say that∇ is anL-instanceof∇′. We say that an atom△E is anL-instanceof△′E′ if △ and△′ have

4. In general, a rule is of the form(α← ϕ, β, ψ), whereα andβ stand for modal atoms,ϕ is apre-condition, andψ is apost-computation.5. This means thatstandardizing variables apartis also needed for atom variables.

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the same lengthk and there exists a substitutionθ such thatE = E′θ and the modaloperator at positioni of△ is anL-instance of the modal operator at positioni of△′θfor every1 ≤ i ≤ k.

If � and�′ are universal modalities, and furthermore,� is the modal context of

an L-MProlog program clause, then we say that�′ is anL-context instanceof � if

�ϕ → �′ϕ is a theorem inL for an arbitraryϕ. For example,�1 is aKDI4s5-

context instance of�2.

Theforward labelled formof an atomα is the atomα′ such that ifα is of the form△♦iE thenα′ = △〈E〉iE, elseα′ = α. For example, the forward labelled form of♦1s(a) is 〈s(a)〉1s(a).

Let G = ← α1, . . . , αi, . . . , αk be a goal andϕ = �(A ← B1, . . . , Bn) a pro-gram clause. ThenG′ is derivedfromG andϕ in L using a most general unifier (mgu)θ, and called anL-resolventof G andϕ, if the following conditions hold:

– αi = △′A′, with△′ in L-normal labelled form, is called theselected atom.– △′ is anL-instance of�′ which in turn is anL-context instance of�.– θ is an mgu such that:A′θ has the same classical atom asAθ, andA′θ is an

L-instance of the forward labelled form ofAθ.– G′ is the goal← (α1, . . . , αi−1,△

′B1, . . . ,△′Bn, αi+1, . . . , αk)θ.

For example, the onlyKDI4s5-resolvent of← �1p(x) and�2(p(x)← ♦2q(x))is ← �1♦2q(x) (here,� = �2 and△′ = �

′ = �1). As another example, theonlyKDI4s5-resolvent of← 〈Y 〉1〈X〉1r(x), 〈X〉1s(x) and�1(�1r(x) ← s(x)) is← 〈Y 〉1s(x), 〈X〉1s(x) (here,� = �

′ = �1 and△′ = 〈Y 〉1).

SLD-derivation from anL-MProlog program and anL-MProlog goal inL is definedusing two kinds of steps: a) deriving anL-resolvent of a goal and a variant of a programclause, b) deriving a resolvent of a goal and a variant of anrSatL/rNFL rule. Thenotions of SLD-refutation and computed answer forL-MProlog are defined in theusual way. See Figure 1 for an example of SLD-refutation.

A selection ruleis a function that maps an SLD-derivation with the last goal← α1, . . . , αk to some atomαi (1 ≤ i ≤ k). Let P be anL-MProlog program,GanL-MProlog goal, andR a selection rule. AnSLD-refutationof P ∪ {G} in L viaRis an SLD-refutation ofP ∪ {G} in L that usesR to select atoms.

Using the given framework, in (Nguyen, 2003; Nguyen, 2006; Nguyen, 2007) wehave specified SLD-resolution calculi forL-MProlog in basic serial monomodal log-ics, multimodal logics of belief, the classBSMM of basic serial multimodal log-ics, and the classsCFG of serial context-free grammar logics. See (Nguyen, 2003;Nguyen, 2006; Nguyen, 2008a) for the proofs of the followingtheorem.

THEOREM 1. — Let L be one of the serial modal logics studied in (Nguyen, 2003;Nguyen, 2006; Nguyen, 2007), i.e. KD, T, KDB, B, KD4, S4, KD5,KD45, S5,KDI4s,KDI4, KDI4s5, KDI45, KD4s5s, KD45(m), KD4Ig5a, BSMM , or sCFG.Then the SLD-resolution calculus given in (Nguyen, 2003; Nguyen, 2006; Nguyen,2007) for L-MProlog is sound and strongly complete:

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Modal logic programming revisited 173

Consider the goalG =← �1p(x) and the following programP :

ϕ1 = �2(p(x)← ♦2q(x))ϕ2 = �1(q(x)← r(x), s(x))ϕ3 = �1(�1r(x)← s(x))ϕ4 = ♦1s(a)←

Here is an SLD-refutation ofP ∪ {G} in L = KDI4s5:

Goals Input clauses/rules MGUs← �1p(x)← �1♦2q(x) ϕ1 {x1/x}← ♦2q(x) rSatL(a)← ♦1q(x) rSatL(b)← 〈X〉1q(x) rSatL(c)← 〈X〉1r(x), 〈X〉1s(x) ϕ2 {x5/x}← 〈Y 〉1〈X〉1r(x), 〈X〉1s(x) rNFL(a)← 〈Y 〉1s(x), 〈X〉1s(x) ϕ3 {x7/x}← 〈X〉1s(a) ϕ4 {x/a, Y/s(a)}the empty clause ϕ4 {X/s(a)}

(Variables of the input clause of stepi are subscripted byi.)

Figure 1. An example of SLD-resolution for MProlog

Soundness: LetP be an L-MProlog program andG an L-MProlog goal. Then everycomputed answer in L forP ∪ {G} is a correct answer in L forP ∪ {G}.

Strong completeness: LetP be an L-MProlog program,G an L-MProlog goal, andR a selection rule. For every correct answerθ in L of P ∪ {G}, there existsan SLD-refutation ofP ∪ {G} in L via R with computed answerγ such thatGθ = Gγδ for some substitutionδ.

3. Optimizations

3.1. The standard form for resolution cycles

Roughly speaking, an SLD-derivation from a programP and a goalG in L is anapplication of a sequence of program clauses ofP and rSatL/rNFL rules to thegoalG. (By an application of a program clause or a rule we mean an application of itsvariant.) Aresolution cycleis defined to be a fragment of an SLD-derivation that startseither immediately after an application of a program clauseor from the beginning ofthe derivation, and ends with an application of a program clause. Note that an SLD-refutation can be divided into a sequence of resolution cycles.

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A selection rule is standardif in every resolution cycle only atoms at the sameposition are selected. A resolution cycle in an SLD-derivation via a standard selectionrule is thus an application of a sequence ofrSatL/rNFL rules with a program clauseat the end to the selected atom.

A resolution cycle is in thestandard formif it is a resolution cycle via a standardselection rule with the property that,rSatL rules are applied beforerNFL rules,which in turn are applied before the used program clause.6

THEOREM 2. — Let L be one of the modal logics mentioned in Theorem 1. Thenthe SLD-resolution calculus given in (Nguyen, 2003; Nguyen, 2006; Nguyen, 2007)for L-MProlog is still strongly complete when adopting the restriction that resolutioncycles are in the standard form.

See (Nguyen, 2008a, Section 5.1) for the proof of this theorem.

3.2. Optimizing the set of rSatL rules

Consider, for example, the SLD-resolution calculus given in (Nguyen, 2006) forKDI4s-MProlog. This calculus uses the followingrSatKDI4s

rules:

△♦iE ←△〈X〉iE△∇iα←△�jα if i ≤ j△♦iE ←△♦jE if i > j△∇�iα←△�iα△♦iE ←△〈X〉j♦iE

Recall thatE stands for a classical atom, whileα stands for a modal atom (of theform△E). As shown in (Nguyen, 2008a), without loss of (strong) completeness ofthe SLD-resolution calculus forKDI4s-MProlog, the occurrences ofα in the aboverules can be replaced byE (a more restricted form). The change results in a moreefficient SLD-resolution calculus forKDI4s-MProlog.

The intuition behind such an optimization is based on properties of theSatL oper-ator of the fixpoint semantics. When constructing a leastL-model forP and realizing aformula of the form(A← B1, . . . , Bn) at a possible worldw, to checkB1, . . . ,Bn atw, or equivalently, to derive modal atoms△B1, . . . ,△Bn, where△ = w, onlySatLrules that manipulatesuffixesof modal atoms are needed. This kind of optimizationfor the set of rules specifyingSatL makes a corresponding optimization for the set ofrules specifyingrSatL.

The mentioned optimization is applicable for SLD-resolution calculi ofL-MPrologin a number of other modal logics. See (Nguyen, 2008a, Section 5.2) for a proof ofthe following theorem.

6. Note that a resolution cycle may use norSatL/rNFL rules. The operatorrNFL is usuallydeterministic and one application of anrNFL rule is usually enough for one resolution cycle.

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THEOREM 3. — Let L be one of the serial monomodal logics KD, T, KDB, B,KD4, S4, KD5, KD45, S5, or one of the multimodal logics of belief studied in(Nguyen, 2006), i.e.KDI4s, KDI4, KDI4s5, KDI45, KD4s5s, KD45(m), orKD4Ig5a. By changing everyrSatL rule of the form△△′α ← △△′′α of theSLD-resolution calculus given in (Nguyen, 2003; Nguyen, 2006) for L-MProlog tothe more restricted form△△′E ← △△′′E, the obtained SLD-calculus is still soundand strongly complete for L-MProlog (independently from whether resolution cyclesare required to be in the standard form or not).

The mentioned optimization, however, is not applicable forthe SLD-resolution cal-culi given in (Nguyen, 2006; Nguyen, 2007) for MProlog in thelarge classesBSMMandsCFG of multimodal logics. The reason is that, a chain of applications ofrSatLrules forL ∈ {BSMM, sCFG}may be too complex, and we may need to apply thefirst rules of that chain on the middle of the considered modality.

As another optimization, we can avoid explicit use of the (general)�-lifting rule“△∇α ← △�iα if ∇ �L �i ” by embedding it into the otherrSatL rules. Forexample, the optimized set ofrSatKDI4s

rules consists of the following:

△♦iE ←△〈X〉iE△♦iE ←△♦jE if i > j△∇∇′

iE ←△�iE△♦iE ←△〈X〉j♦iE

where∇′

i denotes an arbitrary modal operator with indexi. See (Nguyen, 2008a) foroptimized sets ofrSatL rules forL-MProlog in other modal logics.

3.3. The case of MProlog-�

An L-MProlog-� program(resp.goal) is anL-MProlog program (resp. goal) with-out existential modal operators. In this subsection, letL denote one of the modallogics considered in (Nguyen, 2003; Nguyen, 2006; Nguyen, 2007) exceptBSMM .Consider the original or optimized SLD-resolution calculus given in (Nguyen, 2003;Nguyen, 2006; Nguyen, 2007; Nguyen, 2008a) forL-MProlog. We show that the cal-culus can be significantly simplified forL-MProlog-�.

Let P be anL-MProlog-� program andG anL-MProlog-� goal. Observe that:

1) For every rule specifyingrSatL or rNFL, if the r.h.s. atom contains an unla-belled existential modal operator then the l.h.s. atom alsocontains an unlabelled ex-istential modal operator. Hence all therSatL rules containing unlabelled existentialmodal operators can be deleted without affecting SLD-refutations ofP ∪ {G} in L.

2) If we replace the operators〈X〉i, 〈X〉j , and〈X〉 in the rules specifyingrNFL

respectively by�i, �j, and�, then all SLD-refutations ofP ∪ {G} change accord-ingly and remain correct. So, we can assume this replacementfor L-MProlog-�.Similarly, we can replace therSatL rule△α ← △〈X〉�α for L ∈ {KDB,B} by△α←△��α.

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3) With the modifications mentioned in the two preceding items, every SLD-derivation fromP ∪ {G} in L does not contain any (labelled or unlabelled) existentialmodal operator. Consequently, modal operators of the form∇i in the (remaining)rules specifyingrSatL or rNFL can be replaced by�i. In general, the rules can bereformulated so that they contain no operators of the form∇.

The mentioned modifications create a more efficient, sound and strongly com-plete SLD-resolution calculus forL-MProlog-�. For example, we need only onerSatKDI4s

rule△�j�iE ← △�iE (and norNFKDI4srules) forKDI4s-MPro-

log-�. As another example, forKDI4s5-MProlog-� we need only the followingrules:

rNFKDI4s5 : �iE ← �j�iErSatKDI4s5 : �j�iE ← �iE

3.4. Restrictions and iterative deepening search

In this and the next subsection, we present additional optimizations that have beenimplemented for the modal logic programming MProlog version 2.0 (Nguyen, 2008b).

It is not easy for users to imagine and control the behaviour of MProlog programs.One of the reasons is that MProlog usesrSatL/rNFL rules as meta clauses and theusers may not be fully aware of all possible effects of the rules. Even when they under-stand the rules well, they may not have enough control on the rules without modifyingthe interpreter of the used MProlog system. Another reason is that a complete SLD-resolution forL-MProlog may need a rule like△�iE ← △�i�iE, which can beapplied repeatedly forever. This difficulty suggests that programming in MProlog isbetter treated as a mixture of “programming” and theorem proving for the modal Hornfragment.

To restrict the search space for MProlog, one can apply some restrictions like:

1) a limit on the lengths of modalities that can occur in derivations,2) a limit on the number of applications ofrSatL (resp.rNFL) rules in a resolu-

tion cycle,3) a limit on the nesting depth of function symbols occurringin modal atoms,4) a limit on the length of derivation,5) a limit on the number of applications ofrSatL/rNFL rules in a derivation.

These restrictions may affect completeness of the used calculus. For real applica-tions, however, it is reasonable to set the limits mentionedin the first three items ofthe above list to some low values. The remaining limits can bedealt with by iterativedeepening search. For example, iterative deepening searchw.r.t. the number of ap-plications ofrSatL/rNFL rules in a derivation can be done by ignoring the limit onthe length of derivation (or setting it to a high enough value), and at each deepeningiteration, increasing the limit on the number of applications ofrSatL/rNFL rules bya certain value (e.g. by a constant specified by a parameter).

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3.5. Tabulation

Setting the limits mentioned in the previous subsection to some concrete values,using the depth first search strategy an execution of an MProlog program may stillloop forever as in the case of Prolog. A solution for this is touse some tabulation(tabling) mechanism.

There are advanced tabulation methods for Prolog like OLDT-resolution (Tamakiet al., 1986), linear tabulated resolution (Shenet al., 2001; Zhouet al., 2003). Thesemethods use sophisticated techniques (e.g. the suspension-resumption mechanism andthe stack-wise representation of OLDT) that are better implemented by the underlyingProlog abstract machine. Besides, these methods try to reach “answer completion” forsubgoals as early as possible.

For the MProlog system (version 2.0), which has been writtenin Prolog as a mod-ule, we adopted a tabulation mechanism with the essential feature that, looping foranswer completion is done only at the most outer level for thetop goal. The main loopcontinues when more answers have been tabulated (for some subgoals) during the lastiteration. At each iteration of the loop, when the system encounters a subgoal that hasa variant called earlier during the iteration, the subgoal is resolved only with the tab-ulated answers of its variant. As applications of differentsequences ofrSatL/rNFL

rules may give the same effect, our tabulation method that delays answer completionfor subgoals has an advantage in quickly finding the first answer for the top goal.

EXAMPLE 4. — Consider the goalG =← ♦s(x) and the following programP inthe modal logicKD:

ϕ1 = ♦p(x)← q(x)ϕ2 = ♦p(x)← r(x)ϕ3 = q(a)←ϕ4 = r(a)←ϕ5 = �(s(x)← p(x), t(x), u(x))ϕ6 = �(t(x)← p(x))

Here is an SLD-derivation fromP ∪ {G} in KD:

Goals Input clauses/rulesG =← ♦s(x)G1 =← 〈Y 〉s(x) rSatKD

G2 =← 〈Y 〉p(x), 〈Y 〉t(x), 〈Y 〉u(x) ϕ5

G3 =← q(x), 〈p(x)〉t(x), 〈p(x)〉u(x) ϕ1

G4 =← 〈p(a)〉t(a), 〈p(a)〉u(a) ϕ3

G5 =← 〈p(a)〉p(a), 〈p(a)〉u(a) ϕ6

G6 =← q(a), 〈p(a)〉u(a) ϕ1

G7 =← 〈p(a)〉u(a) ϕ3

failure

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At the failure withG7 the system backtracks toG2 and resolves the goal atom〈Y 〉p(x) with the next program clauseϕ2, resulting in:

Goals Input clauses/rulesG⋆

3 =← r(x), 〈p(x)〉t(x), 〈p(x)〉u(x) ϕ2

G⋆4 =← 〈p(a)〉t(a), 〈p(a)〉u(a) ϕ4

As the goal atom〈p(a)〉t(a) has been called earlier and succeeded (the informationremains through the mentioned backtracking), the next goalis G⋆

5 =← 〈p(a)〉u(a),which fails. At this point, the systems backtracks to the topgoalG. Since somenew answers for subgoals (namely,q(a), 〈p(a)〉p(a), 〈p(a)〉t(a)) have been tabulated,the system makes another round for solvingG after deleting information about whatsubgoals have been called. This round does not tabulate any new answer for subgoals,so when the system backtracks again to the top goalG, it stops with a failure result.�

4. On the usefulness of the direct approach

The relationship between the direct approach and the translation approaches(Debartet al., 1992; Nonnengart, 1994) for modal logic programming has been dis-cussed in our papers (Nguyen, 2003; Nguyen, 2006). Further information can be foundin (Nguyen, 2008a). In this section, we give some additionalremarks on the usefulnessof the direct approach.

As shown in our works (Nguyen, 2003; Nguyen, 2006), the direct approach man-ages very well with modal logics with variants of axiom (5). As discussed in (Ohlbach,1988), functional translation cannot deals with axiom (5) in a pure way. Hence it is notstraightforward to extend the functional translation approach by Debartet al. for pro-gramming in modal logics with axiom (5). The semi-functional translation approachby Nonnengart does not share this problem, but note that it produces atoms of the ac-cessibility relations (for translating universal modal operators), which are loosely re-lated with atoms of the other predicates. Hence, it is harderto use the semi-functionaltranslation approach to develop an efficient search procedure for modal logic program-ming.

In the rest of this section, we argue that the labelling technique used in our directapproach is relatively better than the Skolemization technique used in the translationapproaches for modal logic programming.

Consider the following logic programP in the modal logicL = KD:

♦p← q q ←♦p← r r ←

The direct consequence operatorTL,P of our fixpoint semantics has the least fixpoint{q, r, 〈p〉p}. The functional translation ofP results in the following programP ′:

p(ε!a)← q(ε) q(ε)←p(ε!b)← r(ε) r(ε)←

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Modal logic programming revisited 179

The semi-functional translation ofP results inP ′′ = P ′ ∪ {R(x, x!y)←}.7 The di-rect consequence operator ofP ′ has the least fixpoint{q(ε), r(ε), p(ε!a), p(ε!b)}.Thus, the bottom-up computation forP ′ or P ′′ gives two atomsp(ε!a) andp(ε!b) in-stead of one atom〈p〉p that is created by our bottom-up computation forP . As shownbelow, this problem corresponds to another problem in top-down computation.

Reconsider the programP and the goalG given in Example 4. Functional transla-tion ofG andP givesG′ =← s(ε!z, x) and the following programP ′:

ψ1 = p(ε!f(x), x)← q(ε, x)ψ2 = p(ε!g(x), x)← r(ε, x)ψ3 = q(ε, a)←ψ4 = r(ε, a)←ψ5 = s(ε!y, x)← p(ε!y, x), t(ε!y, x), u(ε!y, x)ψ6 = t(ε!y, x)← p(ε!y, x)

Consider SLD-resolution with tabulation forP ′ ∪ {G′}. The first derivation is:

Goals Input clausesG′ =← s(ε!z, x)G′

1 =← p(ε!z, x), t(ε!z, x), u(ε!z, x) ψ5

G′

2 =← q(ε, x), t(ε!f(x), x), u(ε!f(x), x) ψ1

G′

3 =← t(ε!f(a), a), u(ε!f(a), a) ψ3

G′

4 =← p(ε!f(a), a), u(ε!f(a), a) ψ6

G′

5 =← q(ε, a), u(ε!f(a), a) ψ1

G′

6 =← u(ε!f(a), a) ψ3

failure

At the failure withG′

6 the system backtracks toG′

1 and resolves the goal atomp(ε!z, x) with the next program clauseψ2, resulting in:

Goals Input clausesG′′

2 =← r(ε, x), t(ε!g(x), x), u(ε!g(x), x) ψ2

G′′

3 =← t(ε!g(a), a), u(ε!g(a), a) ψ4

G′′

4 =← p(ε!g(a), a), u(ε!g(a), a) ψ6

G′′

5 =← q(ε, a), u(ε!g(a), a) ψ1

G′′

6 =← u(ε!g(a), a) ψ3

failure

The problem is that tabulation does not work as it did in Example 4 for the di-rect approach, becauset(ε!g(a), a) is not the same ast(ε!f(a), a). That is, using thefunctional translation approach, more computation is needed for this example.

7. (Debartet al., 1992) uses ‘!’ to constructpath expressions, while (Nonnengart, 1994) uses ‘:’.

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The same problem occurs for the semi-functional translation approach. We givebelow the translation for the considered programP and goalG and leave detailedanalysis for the reader. The results of the translation (using the modal logicKD) arethe goalG′′ =← s(z, x), R(ε, z) and the following programP ′′:

p(ε!f(x), x)← q(ε, x)p(ε!g(x), x)← r(ε, x)q(ε, a)←r(ε, a)←s(y, x)← p(y, x), t(y, x), u(y, x), R(ε, y)t(y, x)← p(y, x), R(ε, y)R(x, x!y)←

5. Concluding remarks

We have presented significant optimizations for the theory of MProlog. They arevery useful for increasing efficiency of the implemented system MProlog. We havealso shown that the labelling technique used in our direct approach is relatively betterthan the Skolemization technique used in the translation approaches.

There are some problems deserving investigation: i) developing semantics formodal logic programs with negation, ii) developing a good methodology for practi-cal programming in modal logics (i.e. a methodology with conceivable behaviour ofmodal logic programs), iii) applying modal logic programming for practical problems.

Acknowledgements

This work has been supported by grant N N206 3982 33 from the Polish Ministryof Science and Higher Education. I would like to thank also Professor Andrzej Szałasand an anonymous reviewer for their useful comments on this paper.

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