Region-based theories of space - univ-toulouse.fr...Handbook of Incidence Geometry. Elsevier (1995)...

Region-based theories of space

Philippe Balbiani

Logic, Interaction, Language and ComputationInstitut de recherche en informatique de Toulouse

CNRS — Universite de Toulouse

Contents

1. Historical excursion2. Algebras of regions, models and representation theory3. First-order mereotopology4. Qualitative spatial reasoning using constraint calculi5. Modal logic and mereotopology

Historical excursionPoints

Point-based theories of spaceI Classical Euclidean geometryI Topology

PointI Simplest spatial entityI Without dimension and internal structure

Historical excursionTopology

T = (X ,O)

I X : nonempty set of “points”I O: set of subsets of X such that

I ∅ ∈ OI X ∈ OI if u, v ∈ O then u ∩ v ∈ OI if ui ∈ O for each i ∈ I then

⋃{ui : x ∈ I} ∈ O


Let T = (X ,O) be a topology

OpenI subset u of X such that u ∈ O

ClosedI subset u of X such that X \ u ∈ O



Interior of u ⊆ X : In(u)

I greatest open set contained in uClosure of u ⊆ X : Cl(u)

I least closed set containing u

Historical excursionBodies

Idea: develop an alternative theory of spaceI Basic notion

I Solid bodiesI Basic relations

I “One solid is part of another solid”I “Two solids overlap”I “One solid touches another solid”

Abstract philosophical disciplinesI Ontology

I Theory of “existent”I Mereology

I Theory of “part-whole” relations

Historical excursionMereology

Founders of mereologyI Lesnewski (1927–1931)I Tarski (1927)

Pointless approachI De Laguna (1922)

I “Point, line, and surface, as sets of solids”I Whitehead (1929)

I “Process and Reality”I Pointless geometry

Historical excursionDe Laguna (1922) and Whitehead (1929)

De Laguna (1922)I SolidsI Ternary relation between solids

I “x connects y with z”

Whitehead (1929)I RegionsI Binary relation between regions

I “x is connected with y ”

Historical excursionWhitehead (1929)

Whitehead (1929)I RegionsI Binary relation between regions

xCy : “x is in contact with y ”I Mereological relations

6: “Part-of”O: “Overlap”

Cext : “External contact”6◦: “Tangential inclusion”�: “Nontangential inclusion”


Some formal properties of the contact relation(W1): ∀x xCx(W2): ∀x ∀y (xCy → yCx)

(W3): ∀x ∀y (x = y ↔ ∀z (xCz ↔ yCz))

Some other Whitehead’s spatial relations between regions“Part-of”: x 6 y ::= ∀z (xCz → yCz)

“Overlap”: xOy ::= ∃z (z 6 x ∧ z 6 y)

“External contact”: xCexty ::= xCy ∧ xOy“Tangential inclusion”: x 6◦ y ::= x 6 y ∧ ∃z (zCextx ∧ zCexty)

“Nontangential inclusion”: x � y ::= x 6 y ∧ x6◦y


Abstractive setI Set α of regions such that

I α is totally ordered by the nontangential inclusionI There is no region included in every element of α

Covering relationI An abstractive set α covers an abstractive set β if every

region in α contains a region in β

Equivalence relationI α ≡ β provided α covers β and β covers α


Geometrical elementI Class of equivalence modulo ≡

Incidence relationI Order relation induced by the covering relation in the set of

all geometrical elements

PointI Minimal (with respect to the incidence relation) geometrical

element

Historical excursionTarski (1927) and Grzegorczyk (1960)

Tarski (1927)I SpheresI Binary relation between spheres

I “x is part of y ”

Grzegorczyk (1960)I BodiesI Binary relations between bodies

I Part-ofI Separation

Historical excursionGrzegorczyk (1960)

Grzegorczyk’s pointless geometry (R,6,C)

(G0): (R,6) is a mereological field, i.e. a completeBoolean algebra with deleted zero element

(G1): C is a reflexive relation in R(G2): C is a symmetric relation in R(G3): C is monotone with respect to 6 in the sense that

we have: x 6 y → ∀z ∈ R (xCz → yCz)

Relation� of nontangential inclusionI Defined in the same way as by Whitehead


Representative of a point in (R,6,C)I Set p of regions such that

1. p has no minimum and is totally ordered by the relation�2. Given two regions x and y , if we have zOx and zOy for

every z ∈ p, then xCy

Point in (R,6,C)

I Filter P in R generated by a representative of a pointI P belongs to a region x if x is a member of P

NotationsΠ: Set of all points of (R,6,C)

π(x): Set of all points of a region x


Grzegorczyk’s pointless geometry (R,6,C)

(G0): (R,6) is a mereological field, i.e. a completeBoolean algebra with deleted zero element

(G1): C is a reflexive relation in R(G2): C is a symmetric relation in R(G3): C is monotone with respect to 6 in the sense that

we have: x 6 y → ∀z ∈ R (xCz → yCz)

Two additional axioms are introduced(G4): Every region has at least one point(G5): If xCy , then x and y overlap with every member of

P for some point P



Regular openI subset u of X such that In(Cl(u)) = u

Regular closedI subset u of X such that Cl(In(u)) = u



The set RO(T ) of all regular open subsets of T constitutes aBoolean algebra

I 0T = ∅, 1T = XI −T u = In(X \ u)

I u ⊕T v = In(Cl(u ∪ v)), u ⊗T v = u ∩ vThe set RC(T ) of all regular closed subsets of T constitutes aBoolean algebra

I 0T = ∅, 1T = XI −T u = Cl(X \ u)

I u ⊕T v = u ∪ v , u ⊗T v = Cl(In(u ∩ v))


Theorem (Grzegorczyk)I Let T = (X ,O) be a Hausdorff topological space and

R = RO(T ) be the set of all its nonempty regular-opensets. For any x , y ∈ R, we set xCy if and only ifCl(x) ∩ Cl(y) 6= ∅. Then (R,⊆,C) satisfies (G0)–(G3). Ifevery point of X is the intersection of a decreasing (withrespect to�) family of open sets, then axioms (G4) and(G5) are also satisfied.

Theorem (Grzegorczyk)I Suppose that (R,6,C) satisfies (G0)–(G5). Let O be the

topology in Π generated by the set {π(x): x ∈ R}. Then{π(x): x ∈ R} coincides with the set of all nonemptyregular-open sets of (Π,O) and π is an isomorphism.


The notions of a point used by Whitehead and Grzegorczyk areequivalent when

(G6): If x � y , then x � z and z � y for some z ∈ R(G6′): If xCy , then xCz and z?Cy for some z ∈ R

Unpleasant feature of Grzegorczyk’s systemI It is not a first-order system

Historical excursionClarke (1981, 1985)

Clarke’s system (R,C)

(C1): ∀x xCx(C2): ∀x ∀y (xCy → yCx)

(C3): ∀x ∀y (x = y ↔ ∀z (xCz ↔ yCz))

(C4): If A is a nonempty subset of R, thenC(x) =

⋃{C(y): y ∈ A} for some x ∈ R

PointsI Certains subsets P of R satisfying some closure conditions

Additional axiom(C5): If xCy then x , y ∈ P for some point P

Historical excursionClarke (1981, 1985)

The relations of contact and overlap used by Clarke areequivalent

I xCy if and only if xOy

Unpleasant feature of Clarke’s systemI It is not a first-order system

Historical excursionQualitative spatial reasoning

Interval logic for reasoning about spaceI Pointless approach to the theory of spaceI Ontological primitives

I Physical objectsI Regions

I Primitive relationsI Part-ofI Contact

Important systemsI Boolean contact algebrasI Polygonal plane mereotopologyI Region connection calculus

Historical excursionBibliography

Bennett, B.: A categorical axiomatisation of region-basedgeometry. Fundamenta Informaticæ46 (2001) 145–158.Biacino, L., Gerla, G.: Connection structures. Notre DameJournal of Formal Logic 32 (1991) 242–247.Biacino, L., Gerla, G.: Connection structures: Grzegorczyk’sand Whitehead’s definitions of point. Notre Dame Journal ofFormal Logic 37 (1996) 431–440.Clarke, B.: Individuals and points. Notre Dame Journal ofFormal Logic 26 (1985) 61–75.Clarke, B.: A calculus of individuals based on ‘connection’.Notre Dame Journal of Formal Logic 22 (1981) 204–218.


Cohn, A., Hazarika, S.: Qualitative spatial representation andreasoning: an overview. Fundamenta Informaticæ46 (2001)1–29.Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: a proximity approach — I. FundamentaInformaticæ74 (2006) 209–249.Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: proximity approach — II. FundamentaInformaticæ74 (2006) 251–282.Duntsch, I., Wang, H., McCloskey, S.: A relation-algebraicapproach to the region connection calculus. TheoreticalComputer Science 255 (2001) 63–83.Duntsch, I., Winter, M.: A representation theorem for Booleancontact algebras. Theoretical Computer Science 347 (2005)498–512.


Gerla, G.: Pointless geometries. In Buekenhout, F. (editor):Handbook of Incidence Geometry. Elsevier (1995) 1015–1031.Grzegorczyk, A.: The systems of Lesniewski in relation tocontemporary logical research. Studia Logica 3 (1955) 77–95.Grzegorczyk, A.: Axiomatizability of geometry without points.Synthese 12 (1960) 228–235.De Laguna, T.: Point, line, and surface, as sets of solids.Journal of Philosophy 19 (1922) 449–461.Mormann, T.: Continuous lattices and Whiteheadian theory ofspace. Logic and Logical Philosophy 6 (1998) 35–54.


Naimpally, S., Warrack, B.: Proximity Spaces. CambridgeUniversity Press (1970).Pratt-Hartmann, I.: First-order mereotopology. In Aiello, M.,Pratt-Hartmann, I., van Benthem, J. (editors): Handbook ofSpatial Logics. Springer (2007) 13–97.Randell, D., Cui, Z., Cohn, A.: A spatial logic based onregions and connection. In Nebel, B., Rich, C., Swartout, W.(editors): Principles of Knowledge Representation andReasoning. Morgan Kaufmann Publishers (1992) 165–176.Roeper, P.: Region-based topology. Journal of PhilosophicalLogic 26 (1997) 251–309.Stell, J.: Boolean connection algebras: A new approach to theRegion-Connection Calculus. Artificial Intelligence 122 (2000111–136.


Tarski, A.: Foundations of the geometry of solids. In: Logic,Semantics, Metamathematics. Clarendon Press (1956) 24–29.Vakarelov, D., Dimov, G., Duntsch, I., Bennett, B.: Aproximity approach to some region-based theories of space.Journal of Applied Non-Classical Logics 12 (2002) 527–559.Vakarelov, D., Duntsch, I., Bennett, B.: A note on proximityspaces and connection based mereology. In Guarino N., Smith,B., Welty, C. (editors): Fois’01. 2nd International Conferenceon Formal Ontology in Information Systems. ACM (2001)139–150.De Vries, H.: Compact Spaces and Compactifications. VonGorcum (1962).Whitehead, A.: Process and Reality. MacMillan (1929).

Contents


Algebras of regions, models and representation theoryContact algebras: Dimov and Vakarelov (2006)

Contact algebra B = (B,0,1, ·,+, ?,C)

(DiVa0): (B,0,1, ·,+, ?) is a Boolean algebra with ? as theBoolean complement

(DiVa1): xCy implies x 6= 0 and y 6= 0(DiVa2): (x + y)Cz is equivalent to xCz or yCz(DiVa3): xC(y + z) is equivalent to xCy or xCz(DiVa4): x 6= 0 implies xCx(DiVa5): xCy implies yCx

Relation� of nontangential inclusionI x � y ::= xCy?


Contact algebra of regular-closed setsI Let P be a point-based topology. LetRCP = (RCP ,0P ,1P , ·P ,+P , ?P ,CP) be the structuredefined as follows:

I (RCP ,0P ,1P , ·P ,+P , ?P) is the Boolean algebra of theregular-closed sets of P,

I xCPy iff x ∩ y 6= ∅.Then RCP meets constraints (DiVa1)–(DiVa5). Remarkthat

I x �P y iff x ⊆ InP(y).


Contact algebra of regular-open setsI Let P be a point-based topology. LetROP = (ROP ,0P ,1P , ·P ,+P , ?P ,CP) be the structuredefined as follows:

I (ROP ,0P ,1P , ·P ,+P , ?P) is the Boolean algebra of theregular-open sets of P,

I xCPy iff ClP(x) ∩ ClP(y) 6= ∅.Then ROP meets constraints (DiVa1)–(DiVa5). Remarkthat

I x �P y iff ClP(x) ⊆ y .


Let B = (B,0,1, ·,+, ?,C) be a contact algebraI Mereological relations 6 and O in B

6: x 6 y ::= x · y = xO: xOy ::= x · y 6= 0

I Mereotopological relation Con in BCon: Con(x) ::=

∀y ∀z (y 6= 0 ∧ z 6= 0 ∧ x = y + z → yCz)


Let B = (B,0,1, ·,+, ?,C) be a contact algebraI RCC − 8 basic mereotopological relations in B between

two nonzero regionsDC: DC(x , y) ::= xCyEC: EC(x , y) ::= xCy ∧ xOyPO: PO(x , y) ::= xOy ∧ x6y ∧ y6x

TPP: TPP(x , y) ::= x 6 y ∧ x�y ∧ y6xTPPI: TPPI(x , y) ::= y 6 x ∧ y�x ∧ x6y

NTPP: NTPP(x , y) ::= x � y ∧ y6xNTPPI: NTPPI(x , y) ::= y � x ∧ x6y

Id : Id(x , y) ::= x 6 y ∧ y 6 x

Algebras of regions, models and representation theoryExtensions of contact algebras by adding new axioms

Let B = (B,0,1, ·,+, ?,C) be a contact algebraI B is connected:

(Con): x � x implies x = 0 or x = 1I B is extensional:

(Ext): x 6= 1 implies x � y for some y ∈ B such thaty 6= 1

I B is normal:(Nor): x � y implies x � z and z � y for some

z ∈ B


Let B = (B,0,1, ·,+, ?,C) be a contact algebraI (Ext) is equivalent to each of the following axioms:

(Ext ′): x 6 y iff xCz implies yCz for every z ∈ B(Ext ′′): x = y iff xCz is equivalent to yCz for every

z ∈ B(Ext ′′′): x 6= 0 implies y � x for some y ∈ B such that

y 6= 0


A point-based topology P issemiregular iff it has a base of regular-closed sets

normal iff every pair of closed disjoint sets can beseparated by a pair of open sets

χ-normal iff every pair of regular-closed disjoint sets can beseparated by a pair of open sets

extensional iff RCP satisfies axiom (Ext)weakly regular iff it is semiregular and for every nonempty

open set x there exists a nonempty open set ysuch that ClP(x) ⊆ y


A point-based topology P isconnected iff it cannot be represented as the sum of two

disjoint nonempty open setsT0 iff for every two different points there exists an

open set that contains one of them and does notcontain the other

T1 iff every one-point set is a closed setHausdorff iff every two different points can be separated by a

pair of disjoint open setscompact iff every cover of P composed of open sets

contains a finite subset which also covers P


PropositionI Let P be a semiregular point-based topology

1. P is weakly regular iff RCP satisfies axiom (Ext).2. P is χ-normal iff RCP satisfies axiom (Nor).3. P is connected iff RCP satisfies axiom (Con).4. If P is Hausdorff and compact then RCP satisfies axioms

(Ext) and (Nor).5. If P is normal and Hausdorff then RCP satisfies axiom

(Nor).

Algebras of regions, models and representation theoryPoints in contact algebras and topological representation theorems

Let P be a point-based topology and u ∈ P be a pointI The set Γu = {x ∈ RCP : u ∈ x} satisfies the following

conditions:I P ∈ Γu,I x ∪ y ∈ Γu iff x ∈ Γu or y ∈ Γu,I x ∈ Γu and y ∈ Γu imply xCPy .


Let B = (B,0,1, ·,+, ?,C) be a contact algebraI A set Γ ⊆ B of regions is called a clan iff it satisfies the

following conditions:I 1 ∈ Γ,I x + y ∈ Γ iff x ∈ Γ or y ∈ Γ,I x ∈ Γ and y ∈ Γ imply xCy .

I A clan is said to be maximal if it is maximal with respect toinclusion

NotationsCLANSB: Set of all clans of B

MaxCLANSB: Set of all maximal clans of B


Let B = (B,0,1, ·,+, ?,C) be a contact algebraI Every ultrafilter of B is a clanI If (Γi)i is a collection of ultrafilters of B such that xCy for

every x , y ∈⋃

i Γi then⋃

i Γi is a clan and every clan canbe obtained by such a construction


Lemma (Dimov and Vakarelov)I Let B = (B,0,1, ·,+, ?,C) be a contact algebra and

x 7→ h(x) = {Γ ∈ CLANSB: x ∈ Γ} be a function of B in2CLANSB . Then

I h(x + y) = h(x) ∪ h(y),I h(0) = ∅,I h(1) = CLANSB,I x 6 y iff h(x) ⊆ h(y),I x = 1 iff h(x) = CLANSB,I xCy iff h(x) ∩ h(y) 6= ∅.


Theorem (Representation of contact algebras)I Let B = (B,0,1, ·,+, ?,C) be a contact algebra. Then there

exists a semiregular T0 compact space P and a denseC-separable embedding h of B in the contact algebra ofregular closed sets RCP . Moreover,

I B satisfies (Con) iff P is connected,I B satisfies (Ext) iff P is weakly regular,I B satisfies (Nor) iff P is χ-normal.

Algebras of regions, models and representation theoryAnother topological representation of contact algebras

Theorem (Representation of extensional contact algebras)I Let B = (B,0,1, ·,+, ?,C) be a contact algebra satisfying

axiom (Ext). Then there exists a weakly regular T1compact space P and a dense C-separable embedding hof B in the contact algebra of regular closed sets RCP .Moreover,

I B satisfies (Con) iff P is connected,I B satisfies (Nor) iff P is χ-normal.

Algebras of regions, models and representation theoryAnother topological representation of contact algebras

Theorem (Representation of extensional normal contactalgebras)

I Let B = (B,0,1, ·,+, ?,C) be a contact algebra satisfyingaxioms (Ext) and (Nor). Then there exists a Hausdorffcompact space P and a dense C-separable embedding hof B in the contact algebra of regular closed sets RCP .Moreover,

I B satisfies (Con) iff P is connected.

Algebras of regions, models and representation theoryBibliography

Dimov, G., Vakarelov, D.: Topological representation ofprecontact algebras. In MacCaull, W., Winter, M., Duntsch, I.(editors): Relational Methods in Computer Science. SpringerLNCS 3929 (2006) 1–16.Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: a proximity approach — I. FundamentaInformaticæ74 (2006) 209–249.Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: proximity approach — II. FundamentaInformaticæ74 (2006) 251–282.Duntsch, I., Vakarelov, D.: Region-based theory of discretespaces: a proximity approach. Annals of Mathematics andArtificial Intelligence 49 (2007) 5–14.Duntsch, I., Winter, M.: A representation theorem for Booleancontact algebras. Theoretical Computer Science 347 (2005)498–512.


Duntsch, I., Winter, M.: Weak contact structures. In MacCaull,W., Winter, M., Duntsch, I. (editors): Relational Methods inComputer Science. Springer LNCS 3929 (2006) 73–82.Gabelaia, D., Konchakov, R., Kurucz, A., Wolter, F.,Zakharyaschev, M.: Combining spatial and temporal logics:expressiveness versus complexity. Journal of ArtificialIntelligence Research 23 (2005) 167–243.Galton, A.: Qualitative Spatial Change. Oxford UniversityPress (2001).Randell, D., Cui, Z., Cohn, A.: A spatial logic based onregions and connection. In Nebel, B., Rich, C., Swartout, W.(editors): Principles of Knowledge Representation andReasoning. Morgan Kaufmann Publishers (1992) 165–176.Roeper, P.: Region-based topology. Journal of PhilosophicalLogic 26 (1997) 251–309.


Stell, J.: Boolean connection algebras: A new approach to theRegion-Connection Calculus. Artificial Intelligence 122 (2000)111–136.Vakarelov, D., Dimov, G., Duntsch, I., Bennett, B.: Aproximity approach to some region-based theories of space.Journal of Applied Non-Classical Logics 12 (2002) 527–559.Vakarelov, D., Duntsch, I., Bennett, B.: A note on proximityspaces and connection based mereology. In Guarino N., Smith,B., Welty, C. (editors): Fois’01. 2nd International Conference onFormal Ontology in Information Systems. ACM (2001)139–150.De Vries, H.: Compact Spaces and Compactifications. VonGorcum (1962).

Contents


MereotopologiesMereotopology of open sets / Mereotopology of closed sets


Mereotopology of open sets over TI Boolean sub-algebra (D,0T ,1T ,−T ,⊕T ,⊗T ) of

(RO(T ),0T ,1T ,−T ,⊕T ,⊗T ) such thatI if U ∈ X , u is open in T and U ∈ u then there is v ∈ D such

that U ∈ v and v ⊆ u

Mereotopology of closed sets over TI Boolean sub-algebra (D,0T ,1T ,−T ,⊕T ,⊗T ) of

(RC(T ),0T ,1T ,−T ,⊕T ,⊗T ) such thatI if U ∈ X , u is closed in T and U ∈ u then there is v ∈ D

such that U ∈ v and v ⊆ u

MereotopologiesExamples of mereotopologies

Let n ≥ 1

Semi-algebraic subset of IRn

I Boolean combination of polynomial inequationsSemi-linear subset (polytope) of IRn

I Boolean combination of linear inequationsAlgebraic polytope of IRn

I Boolean combination of polynomial inequations withalgebraic coefficients

Rational polytope of IRn

I Boolean combination of polynomial inequations withrational coefficients

MereotopologiesExamples of mereotopologies

Let n ≥ 1

RO(IRn) / RC(IRn)

I regular open / closed subsets of IRn

ROS(IRn) / RCS(IRn)

I regular open / closed semi-alg. subsets of IRn

ROP(IRn) / RCP(IRn)

I regular open / closed polytopes of IRn

ROPalg(IRn) / RCPalg(IRn)

I regular open / closed alg. polytopes of IRn

ROPrat (IRn) / RCPrat (IRn)

I regular open / closed rat. polytopes of IRn

Euclidean logicsSyntax

TermsI s ::= x | 0 | 1 | −s | (s ⊕ t) | (s ⊗ t) where x ∈ VAR

FormulasI φ ::= s ≤ t | s ≥ t | C(s, t) | c(s) | ⊥ | ¬φ | (φ ∨ ψ) | ∀xφ

AbbreviationI s = t ::= s ≤ t ∧ s ≥ t

Euclidean logicsSyntax

ReadingI s ≤ t : “s is contained in t”I s ≥ t : “s contains t”I C(s, t): “s is in contact with t”I c(s): “s is connected”

ExamplesI C(x , y)→ ∃z(C(x , z) ∧ c(z) ∧ z ≤ y)

I c(x) ∧ c(y) ∧ c(z) ∧ c(x ⊕ y ⊕ z)→ c(x ⊕ y) ∨ c(x ⊕ z)

I c(x) ∧ c(y)→ c(x ⊗ y) ∨ C(−x ,−y)

Euclidean logicsSemantics

Let T = (X ,O) be a topology and M = (D,0T ,1T ,−T ,⊕T ,⊗T )be a mereotopology over T

InterpretationI function θ: x 7→ θ(x) ∈ D

s 7→ θ(s) ∈ DSatisfaction

I M, θ |= s ≤ t iff θ(s) is contained θ(t)I M, θ |= s ≥ t iff θ(s) contains θ(t)I M, θ |= C(s, t) iff θ(s) is in contact with θ(t)I M, θ |= c(s) iff θ(s) is connected

ValidityI M |= φ iff for all interpretations θ, M, θ |= φ

Euclidean logicsExamples

If φ = C(x , y)→ ∃z(C(x , z) ∧ c(z) ∧ z ≤ y) thenI RO(IR2) 6|= φ

I for all finitary mereotopologies M, M |= φ

If φ = c(x) ∧ c(y) ∧ c(z) ∧ c(x ⊕ y ⊕ z)→ c(x ⊕ y) ∨ c(x ⊕ z)then

I RO(IR2) 6|= φ

I ROS(IR2) |= φ

If φ = c(x) ∧ c(y)→ c(x ⊗ y) ∨ C(−x ,−y) thenI RO(IR2) |= φ

I RO(torus) 6|= φ

ExpressivityDefinition of ≤ and ≥ by means of C

Let M = (D,0T ,1T ,−T ,⊕T ,⊗T ) be a mereotopology over IRn

ThenI M |= x ≤ y ↔ ∀z(C(x , z)→ C(y , z))

I M |= x ≥ y ↔ ∀z(C(y , z)→ C(x , z))

ExpressivityDefinition of c by means of C


If M respects components thenI M |= c(x)↔∀x1∀x2(x1 6= 0 ∧ x2 6= 0 ∧ x1 ⊕ x2 = x ∧ x1 ⊗ x2 = 0→∃y1∃y2(x1 ≥ y1 ∧ x2 ≥ y2 ∧ C(y1, y2) ∧ ¬C(y1 ⊕ y2,−x)))

ExpressivityDefinition of RCC8 by means of C

I DC(x , y) ::= ¬C(x , y)

I EC(x , y) ::= C(x , y) ∧ ∀z(x ≥ z ∧ y ≥ z → z = 0)

I PO(x , y) ::= ∃z(x ≥ z ∧ y ≥ z ∧ z 6= 0) ∧ x 6≤ y ∧ x 6≥ yI TPP(x , y) ::= x ≤ y ∧ ∃z(EC(x , z) ∧ EC(y , z))

I NTPP(x , y) ::= x ≤ y ∧ ∀z(¬EC(x , z) ∨ ¬EC(y , z))

I TPPI(x , y) ::= x ≥ y ∧ ∃z(EC(x , z) ∧ EC(y , z))

I NTPPI(x , y) ::= x ≥ y ∧ ∀z(¬EC(x , z) ∨ ¬EC(y , z))

I EQ(x , y) ::= x ≤ y ∧ x ≥ y

ExpressivityDefinition of halfspace by means of ≤, ≥ and convex


ThenI M |= halfspace(x)↔

x 6= 0 ∧ −x 6= 0 ∧ convex(x) ∧ convex(−x)

ExpressivityDefinition of parallel by means of ≤, ≥ and convex


ThenI M |= parallel(x , y)↔

halfspace(x) ∧ halfspace(y) ∧ x 6= y ∧ x 6= −y∧(x ⊗ y = 0 ∨ x ⊗−y = 0 ∨ −x ⊗ y = 0 ∨ −x ⊗−y = 0)

ExpressivityDefinition of C by means of ≤, ≥ and convex


If M is finitary thenI M |= C(x , y)↔∃x ′∃y ′(x ≥ x ′ ∧ convex(x ′) ∧ y ≥ y ′ ∧ convex(y ′)∧∀x ′′∀y ′′(x ′ ≤ x ′′ ∧ y ′ ≤ y ′′ ∧ parallel(x ′′, y ′′)→x ′′ ⊗ y ′′ 6= 0))

Decidability/complexityThe SAT problem

Let Σ be a signature and C be a class of mereotopologies

Let SAT (Σ, C) be the following decision problemI input: a formula φ over ΣI output: determine whether there exists

I a mereotopology M = (D,0,1,−,⊕,⊗) in CI an interpretation θ: x 7→ θ(x) ∈ D

such that M, θ |= φ


With quantification

In the table below, n ≥ 2 and every signature contains 0, 1, −,⊕ and ⊗

all RC(IR) RC(IRn)

≤,≥ NEXPTIME-h NEXPTIME-h NEXPTIME-hin EXPSPACE in EXPSPACE in EXPSPACE

≤,≥, c ? non-elem. undec.≤,≥,C ? non-elem. undec.≤,≥,C, c ? non-elem. undec.


Without quantification

In the table below, n ≥ 3 and every signature contains 0, 1, −,⊕ and ⊗

all RC(IR) RC(IR2) RC(IRn)

≤,≥ NPTIME-c NPTIME-c NPTIME-c NPTIME-c≤,≥, c EXPTIME-c NPTIME-c undec. EXPTIME-h≤,≥,C NPTIME-c PSPACE-c PSPACE-c PSPACE-c≤,≥,C, c EXPTIME-c PSPACE-c undec. EXPTIME-h

Bibliography

Bennett, B., Duntsch, I.: Axioms, algebras and topology. InAiello, M., Pratt-Hartmann, I., van Benthem, J. (editors):Handbook of Spatial Logics. Springer (2007) 99–159.Cohn, A., Hazarika, S.: Qualitative spatial representation andreasoning: an overview. Fundamenta Informaticæ46 (2001)1–29.Cohn, A., Renz, J.: Qualitative spatial representation andreasoning. In van Hermelen, F., Lifschitz, V., Porter, B. (editors):Handbook of Knowledge Representation. Elsevier (2008)551–596.

Bibliography

Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: a proximity approach — I. FundamentaInformaticæ74 (2006) 209–249.Dimov, G., Vakarelov, D.: Contact algebras and region-basedtheory of space: proximity approach — II. FundamentaInformaticæ74 (2006) 251–282.Kontchakov, R., Pratt-Hartmann, I., Wolter, F.,Zakharyaschev, M.: Spatial logics with connectednesspredicates. Logical Methods in Computer Science 6 (2010)1–43.

Bibliography

Pratt-Hartmann, I.: First-order mereotopology. In Aiello, M.,Pratt-Hartmann, I., van Benthem, J. (editors): Handbook ofSpatial Logics. Springer (2007) 13–97.Renz, J., Nebel, B.: Qualitative spatial reasoning usingconstraint calculi. In Aiello, M., Pratt-Hartmann, I., vanBenthem, J. (editors): Handbook of Spatial Logics. Springer(2007) 161–215.Vakarelov, D., Dimov, G., Duntsch, I., Bennett, B.: Aproximity approach to some region-based theories of space.Journal of Applied Non-Classical Logics 12 (2002) 527–559.

Contents


Qualitative spatial reasoning using constraint calculiConstraint-based methods

ExamplesI “The room is 5 metres in length and 6 in breadth”I “The desk should be placed in front of the window”I “The table should be placed between the sofa and the

armchair”


A partial method for determining inconsistency of a CSP is thepath-consistency method

I A CSP is path-consistent iff for any partial instantiation ofany two variables satisfying the constraints between thetwo variables, it is possible for any third variable to extendthe partial instantiation to this third variable satisfying theconstraints between the three variables

A straightforward way to enforce path-consistency on a binaryCSP is to strengthen relations by successively applying thefollowing operation until a fixed point is reached

I Rij := Rij ∩ (Rik ◦ Rkj)

The resulting CSP is equivalent to the original CSP


A straightforward way to enforce path-consistency on a binaryCSP is to strengthen relations by successively applying thefollowing operation until a fixed point is reached

I Rij := Rij ∩ (Rik ◦ Rkj)

The resulting CSP is equivalent to the original CSP

The algorithm sketched has a runningI time of O(n5)

Advanced algorithms can enforce path-consistency inI time O(n3)


Relation algebras based on JEPD relationsI Jointly exhaustive and pairwise disjoint (JEPD) relations

are sometimes called atomic, basic or base relations

ExamplesI Interval algebra (Allen, 1983)I Region connection calculus (Randell, Cui and Cohn, 1992)


B being a finite set of JEPD binary relations, the consistencyproblem CSPSAT (S) for S ⊆ 2B is defined as follows

input: a finite set V of variables over a domain D and afinite set Θ of binary constraints R(xi , xj) whereR ∈ S and xi , xj ∈ V

output: is there an instantiation of all variables in V withvalues from D such that all constraints in Θ aresatisfied?

Qualitative spatial reasoning using constraint calculiTemporal constraint calculi

Point algebra (Vilain and Kautz)I temporal beings: points in dimension 1I basic relations: <, Id , >I temporal relations: ∅, <, Id , >, 6, 6=, >, ?I composition table

< Id >

< < < ?

Id < Id >

> ? > >


Point algebra (Vilain and Kautz)I composition in the point algebra

∅ < Id > 6 6= > ?

∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅< ∅ < < ? < ? ? ?

Id ∅ < Id > 6 6= > ?

> ∅ ? > > ? ? > ?

6 ∅ < 6 ? 6 ? ? ?

6= ∅ ? 6= ? ? ? ? ?

> ∅ ? > > ? ? > ?

? ∅ ? ? ? ? ? ? ?


Point network: (n,C)

I n is a positive integerI C: (i , j) 7→ {∅, <, Id , >,6, 6=,>, ?}

Given a point network (n,C), by successively applying thefollowing operation until a fixed point is reached

I Cij := Cij ∩ (Cik ◦ Ckj)

one can compute an equivalent path-consistent point network(n,C′) in polynomial time


Proposition (Vilain and Kautz)I Let (n,C) be a point network and (n,C′) be an equivalent

path-consistent point network. The following conditions areequivalent:

1. (n,C) is consistent.2. (n,C′) does not contain the ∅ relation.

PropositionI The consistency problem of point networks is in P.


Interval algebra (Allen)I temporal beings: intervals in dimension 1I basic relations: p, m, o, s, d , fi , Id , f , di , si , oi , mi , piI temporal relations: subsets of{p,m,o, s,d , fi , Id , f ,di , si ,oi ,mi ,pi}

I composition table

p m o · · ·p p p p · · ·m p p p · · ·o p p p,m,o · · ·...

......

.... . .


Interval network: (n,C)

I n is a positive integerI C: (i , j) 7→ subsets of {p,m,o, s,d , fi , Id , f ,di , si ,oi ,mi ,pi}

Given an interval network (n,C), by successively applying thefollowing operation until a fixed point is reached


one can compute an equivalent path-consistent intervalnetwork (n,C′) in polynomial time


There exists interval networks (n,C) such thatI (n,C) is inconsistentI (n,C) is path-consistent

Proposition (Vilain and Kautz)I The consistency problem of interval network is NP-hard.

Proof: Reduction of 3SAT.


Many interval relations can be encoded in the point algebraI x {p,m,o} y can be encoded as x− < x+, y− < y+,

x− < y−, x+ < y+

I x {d} y can be encoded as x− < x+, y− < y+, x− > y−,x+ < y+

PropositionI The consistency problem of pointizable interval networks is

in P.


Convex interval relationsI DefinitionI Properties

PropositionI Let (n,C) be a convex interval network and (n,C′) be an

equivalent path-consistent interval network. The followingconditions are equivalent:


PropositionI The consistency problem of convex interval networks is in

P.


Preconvex interval relations (Ligozat)I DefinitionI Properties

Proposition (Ligozat)I Let (n,C) be a preconvex interval network and (n,C′) be

an equivalent path-consistent interval network. Thefollowing conditions are equivalent:


Proposition (Ligozat)I The consistency problem of preconvex interval networks is

in P.


ORD-Horn interval relations (Nebel and Burckert)I DefinitionI Properties

Proposition (Nebel and Burckert)I Let (n,C) be a ORD-Horn interval network and (n,C′) be

an equivalent path-consistent interval network. Thefollowing conditions are equivalent:


Proposition (Nebel and Burckert)I The consistency problem of ORD-Horn interval networks is

in P.


PropositionI A relation in Allen’s algebra is preconvex iff it is ORD-Horn.

Proposition (Ligozat)I The preconvex subclass is the unique greatest tractable

subclass that contains all basic relations.

Proposition (Nebel and Burckert)I The ORD-Horn subclass is the unique greatest tractable

subclass that contains all basic relations.

Qualitative spatial reasoning using constraint calculiSpatial constraint calculi

Let us consider the following set of JEPD binary relationsbetween nonempty regular closed regions in a topologicalspace

DC: DC(x , y) ::= x and y are disconnectedEC: EC(x , y) ::= x and y are externally connectedPO: PO(x , y) ::= x and y partially overlaps

TPP: TPP(x , y) ::= x is a tangential proper part of yTPPI: TPPI(x , y) ::= y is a tangential proper part of x

NTPP: NTPP(x , y) ::= x is a nontangential proper part ofy

NTPPI: NTPPI(x , y) ::= y is a nontangential proper part ofx

Id : Id(x , y) ::= x and y are equal


RCC8 (Randell, Cui and Cohn)I spatial beings: nonempty regular closed regions in a

topological spaceI basic relations: DC, EC, PO, TPP, TPPI, NTPP, NTPPI,

IdI spatial relations: subsets of{DC,EC,PO,TPP,TPPI,NTPP,NTPPI, Id}


Composition table

DC EC · · ·DC ? {DC,EC,PO,TPP,NTPP} · · ·EC {DC,EC,PO,TPPI,NTPPI} ? · · ·

......

.... . .


RCC8 network: (n,C)

I n is a positive integerI C: (i , j) 7→ subsets of{DC,EC,PO,TPP,TPPI,NTPP,NTPPI, Id}

Given an RCC8 network (n,C), by successively applying thefollowing operation until a fixed point is reached


one can compute an equivalent path-consistent RCC8 network(n,C′) in polynomial time


There exists RCC8 networks (n,C) such thatI (n,C) is inconsistentI (n,C) is path-consistent

Proposition (Renz and Nebel)I The consistency problem of RCC8 networks is NP-hard.

Proof: Reduction of NOT-ALL-EQUAL-3SAT.

Qualitative spatial reasoning using constraint calculiComputational complexity

Proposition (Bennett, Nutt)I The consistency problem of RCC8 networks is in PSPACE .

Proof: Arbitrarily topological set constraints can be translatedinto multimodal formulas that have an S4-operator I and anS5-operator �

I π(DC(x , y)) ::= �¬(x ∧ y)

I π(EC(x , y)) ::= �¬(Ix ∧ Iy) ∧ ♦(x ∧ y)

I π(PO(x , y)) ::= ♦(Ix ∧ Iy) ∧ ♦(x ∧ ¬y) ∧ ♦(¬x ∧ y)

I π(TPP(x , y)) ::= �(¬x ∨ y) ∧ ♦(x ∧ ¬Iy) ∧ ♦(¬x ∧ y)

I π(NTPP(x , y)) ::= �(¬x ∨ Iy) ∧ ♦(¬x ∧ y)

I π(Id(x , y)) ::= �(¬x ∨ y) ∧�(x ∨ ¬y)

Qualitative spatial reasoning using constraint calculiComputational complexity

Proposition (Renz and Nebel)I The consistency problem of RCC8 networks is in NP.

Proof: The RCC8 network (n,C) is consistent iff its translation

π(Cij(xi , xj)) : 1 ≤ i < j ≤ n

�(xi ↔ ¬I¬Ixi) : 1 ≤ i ≤ n

♦xi : 1 ≤ i ≤ n

into multimodal formulas that have an S4-operator I and anS5-operator � is satisfied in a polynomial model of depth 1.

Qualitative spatial reasoning using constraint calculiIdentifying tractable subsets of spatial CSPs

ORD-Horn RCC8 relations (Renz and Nebel)I DefinitionI Properties

Proposition (Renz and Nebel)I Let (n,C) be a ORD-Horn RCC8 network and (n,C′) be an

equivalent path-consistent RCC8 network. The followingconditions are equivalent:


Proposition (Renz and Nebel)I The consistency problem of ORD-Horn RCC8 networks is

in P.

Qualitative spatial reasoning using constraint calculiVariants: combining topological and size information

Combining topological and size information (Gerevini andRenz)

I There are 8 JEPD binary relations between nonemptyregular closed regions in a topological space: DC, EC,PO, TPP, TPPI, NTPP, NTPPI, Idtopo

I Assuming that our regions are measurable subsets in IRn,there are 3 JEPD binary relations between their size: <,Idsize, >


Combining topological and size informationI Example of a constraint network combining topological (Θ)

and size (Σ) informationΘ: x0{PO,TPPI, Idtopo}x1, x0{TPP, Idtopo}x2,

x1{TPP, Idtopo}x2, x3{TPPI, Idtopo}x4Σ: size(x0) < size(x2), size(x1) > size(x3),

size(x2) 6 size(x4)


Combining topological and size informationI The topological relations and the relative size relations are

not independentDC 7−→ <, Idsize, >EC 7−→ <, Idsize, >PO 7−→ <, Idsize, >

TPP 7−→ <TPPI 7−→ >

NTPP 7−→ <NTPPI 7−→ >

Idtopo 7−→ Idsize


Combining topological and size informationI The topological relations and the relative size relations are

not independent< 7−→ DC, EC, PO, TPP, NTPP

Idsize 7−→ DC, EC, PO, Idtopo> 7−→ DC, EC, PO, TPPI, NTPPI


Combining topological and size informationI Example of a constraint network combining topological (Θ)

and size (Σ) informationΘ: x0{PO,TPPI, Idtopo}x1, x0{TPP, Idtopo}x2,

x1{TPP, Idtopo}x2, x3{TPPI, Idtopo}x4Σ: size(x0) < size(x2), size(x1) > size(x3),

size(x2) 6 size(x4)

Remark that Θ and Σ are independently consistent, buttheir union is not consistent


Combining topological and size informationI The following propagation algorithm does not always

detect inconsistencies of a constraint network combining aORD-Horn topological network (n,T ) and a size network(n,S)

I successively apply the following operations until a fixedpoint is reached

1. enforce path-consistency to (n, T );2. enforce path-consistency to (n, S);3. extend (n, T ) with the topological constraints entailed by the

size constraints in (n, S);4. extend (n, S) with the size constraints entailed by the

topological constraints in (n, T );


Combining topological and size informationI The following propagation algorithm always detects

inconsistencies of a constraint network combining aORD-Horn topological network (n,T ) and a size network(n,S)

I successively apply the following operations until a fixedpoint is reached

1. Tij := (Tij∩reltopo(Sij))∩((Tik∩reltopo(Sik ))◦(Tkj∩reltopo(Skj)))2. Sij := (Sij ∩ relsize(Tij))∩((Sik ∩ relsize(Tik ))◦(Skj ∩ relsize(Tkj)))


Proposition (Gerevini and Renz)I Let (n,T ) be a ORD-Horn topological network and (n,S)

be a size network. Let (n,T ′) and (n,S′) be the networksobtained by means of the previous propagation algorithm.The following conditions are equivalent:

1. the constraint network combining (n,T ) and (n,S) isconsistent.

2. (n,T ′) does not contain the ∅topo relation and (n,S′) doesnot contain the ∅size relation.

Proposition (Gerevini and Renz)I The consistency problem of constraint networks combining

ORD-Horn topological networks and size networks is in P.

Qualitative spatial reasoning using constraint calculiVariants: rectangle algebra

Rectangle algebra (Condotta)I spatial beings: isothetic rectangles in dimension 2

Since isothetic rectangles in dimension 2I can be seen as pairs of intervals

then basic relations between themI can be seen as pairs of basic relations between intervals

and spatial relations between themI can be seen as sets of pairs of basic relations between

intervals


Rectangle algebraI spatial beings: isothetic rectangles in dimension 2I basic relations: pairs of the form (A,B) where

A,B ∈ {p,m,o, s,d , fi , Id , f ,di , si ,oi ,mi ,pi}I spatial relations: sets of pairs of the form (A,B) where

A,B ∈ {p,m,o, s,d , fi , Id , f ,di , si ,oi ,mi ,pi}I Composition table

I (A1,B1) ◦rec (A2,B2) = (A1 ◦int A2)× (B1 ◦int B2)I Example: (o,p) ◦rec (s,mi) = {o} × {p,m,o, s,d}


Rectangle network: (n,C)

I n is a positive integerI C: (i , j) 7→ sets of pairs of the form (A,B) where

A,B ∈ {p,m,o, s,d , fi , Id , f ,di , si ,oi ,mi ,pi}

Given a rectangle network (n,C), by successively applying thefollowing operation until a fixed point is reached


one can compute an equivalent path-consistent rectanglenetwork (n,C′) in polynomial time


There exists rectangle networks (n,C) such thatI (n,C) is inconsistentI (n,C) is path-consistent

PropositionI The consistency problem of rectangle networks is

NP-hard.

PropositionI The consistency problem of rectangle networks is in NP.


Strongly preconvex relationsI DefinitionI Properties

Proposition (Condotta)I Let (n,C) be a strongly preconvex rectangle network and

(n,C′) be an equivalent path-consistent rectangle network.The following conditions are equivalent:


Proposition (Condotta)I The consistency problem of strongly preconvex rectangle

networks is in P.

Qualitative spatial reasoning using constraint calculiVariants: RCC5

Let us consider the following set of JEPD binary relationsbetween nonempty subsets of a nonempty set

DR: DR(x , y) ::= x and y are separatedPO: PO(x , y) ::= x and y partially overlapsPP: PP(x , y) ::= x is a proper part of yPPI: PPI(x , y) ::= y is a proper part of x

Id : Id(x , y) ::= x and y are equal

Qualitative spatial reasoning using constraint calculiVariants: RCC5

RCC5I spatial beings: nonempty subsets of a nonempty setI basic relations: DR, PO, PP, PPI, IdI spatial relations: subsets of {DR,PO,PP,PPI, Id}

Composition table

DR PO · · ·DR ? {DR,PO,PP} · · ·PO {DR,PO,PPI} ? · · ·

......

.... . .

Qualitative spatial reasoning using constraint calculiVariants: egg yolk representation

Egg yolk representation (Cohn and Gotts)I How many JEPD relations are there between non crisp

regions?

If x− and x+ are regions such thatI x− is a nontangential proper part of x+

then one can see (x−, x+) as a vague region such thatI x− denotes the set of all points that necessarily belong to

the vague regionI x+ denotes the set of all points that possibly belong to the

region

Qualitative spatial reasoning using constraint calculiVariants: egg yolk representation

Egg yolk representationI spatial beings: pairs (x−, x+) of regions such that x− is a

nontangential proper part of x+

I basic relations: quadruples(

A−− A−+

A+− A++

)where

A−−,A−+,A+−,A++ ∈{DC,EC,PO,TPP,TPPI,NTPP,NTPPI, Id}

I spatial relations: sets of quadruples(

A−− A−+

A+− A++

)where

A−−,A−+,A+−,A++ ∈{DC,EC,PO,TPP,TPPI,NTPP,NTPPI, Id}

Qualitative spatial reasoning using constraint calculiBibliography

Allen, J.: Maintaining knowledge about temporal intervals.Communications of the ACM 26 (1983) 832–843.Bennett, B.: Modal logics for qualitative spatial reasoning.Journal of the IGPL 4 (1996) 23–45.Cohn, A., Bennett, B., Gooday, J., Gotts, N.: Qualitativespatial representation and reasoning with the region connectioncalculus. GeoInformatica 1 (1997) 275–316.Gerevini, A., Renz, J.: Combining topological and sizeinformation for spatial reasoning. Artificial Intelligence 137(2002) 1–42.Ligozat, G.: “Corner” relations in Allen’s algebra. Constraints 3(1998) 165–177.

Qualitative spatial reasoning using constraint calculiBibliography

Nebel, B., Burckert, H.-J.: Reasoning about temporalrelations: a maximal tractable subclass of Allen’s intervalalgebra. Journal of the ACM 42 (1995) 43–66.Nutt, W.: On the translation of qualitative spatial reasoningproblems into modal logics. In Burgard, W., Christaller, T.,Cremers, A. (editors): KI-99. Springer LNAI 1701 (1999)113–124.Randell, D., Cui, Z., Cohn, A.: A spatial logic based onregions and connection. In Nebel, B., Rich, C., Swartout, W.(editors): Principles of Knowledge Representation andReasoning. Morgan Kaufmann Publishers (1992) 165–176.Renz, J., Nebel, B.: On the complexity of qualitative spatialreasoning: a maximal tractable fragment of the regionconnection calculus. Artificial Intelligence 108 (1999) 69–123.Vilain, M., Kautz, H.: Constraint propagation algorithms fortemporal reasoning. In: AAAI-86 Proceedings. AAAI (1986)377–382.

Contents


Modal logic and mereotopologyModal logics for mereotopological relations

Mereotopological interpretation of modal logic: syntaxI φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | [r ]φ

I where r ∈ {dc,ec,po, tpp, tppi ,ntpp,ntppi}I Abbreviations

I Standard definitions for the remaining Boolean operationsI 〈r〉φ ::= ¬[r ]¬φ where r ∈ {dc,ec,po, tpp, tppi ,ntpp,ntppi}

I Intuitive meaningI The interpretation of [r ]φ is φ holds in every region r -related

to the current oneI The interpretation of 〈r〉φ is φ holds in some region

r -related to the current one


Mereotopological interpretation of modal logic: framesI Concrete frames are structures of the form F = (W ,R)

whereI W is a nonempty set of nonempty regular closed regions in

a topological space (X ,O)I R: r ∈ {dc,ec,po, tpp, tppi ,ntpp,ntppi} 7−→ R(r) ⊆W ×W

is a function such thatI x R(dc) y iff x ∩ y = ∅I x R(ec) y iff x ∩ y 6= ∅ and In(x) ∩ In(y) = ∅I x R(po) y iff In(x) ∩ In(y) 6= ∅, x 6⊆ y and x 6⊇ yI x R(tpp) y iff x ⊆ y , x 6⊆ In(y) and x 6⊇ yI x R(tppi) y iff x ⊇ y , In(x) 6⊇ y and x 6⊆ yI x R(ntpp) y iff x ⊆ In(y) and x 6⊇ yI x R(ntppi) y iff In(x) ⊇ y and x 6⊆ y


Mereotopological interpretation of modal logic: framesI Examples of concrete frames

1. the set Preg of all nonempty regular closed regions in apoint-based topological space P

2. the set IRnreg of all nonempty regular closed regions in IRn

3. the set IRncon of all nonempty convex regular closed regions

in IRn

4. the set IRnrec of all nonempty rectangle regular closed

regions in IRn, i.e. regions of the form Π{Ci : 1 ≤ i ≤ n}where C1, . . . ,Cn are non-singleton closed intervals in IR


Mereotopological interpretation of modal logic: framesI Abstract frames are structures of the form F = (W ,R)

whereI W is a nonempty set of regionsI R: r ∈ {dc,ec,po, tpp, tppi ,ntpp,ntppi} 7−→ R(r) ⊆W ×W

is a function such thatI R(dc), R(ec), R(po), R(tpp), R(tppi), R(ntpp) and R(ntppi)

are JEPD binary relations on WI R(dc), R(ec) and R(po) are symmetricalI R(tppi) is the converse of R(tpp) and R(ntppi) is the

converse of R(ntpp)I for any entry q1, . . . , qk in row r1 and column r2 of the

composition table for RCC8,R(r1) ◦ R(r2) ⊆ R(q1) ∪ . . . ∪ R(qk )


Proposition (Lutz and Wolter)1. Every concrete frame is an abstract frame.2. Every abstract frame is isomorphic to a concrete frame.


Mereotopological interpretation of modal logic: simpleobservationsDifference modality: [d ]φ ::= [dc]φ ∧ . . . ∧ [ntppi]φUniversal modality: [u]φ ::= φ ∧ [dc]φ ∧ . . . ∧ [ntppi]φ

Nominal: nom(φ) ::= 〈u〉(φ ∧ [d ]¬φ)

Proper part: I [pp]φ ::= [tpp]φ ∧ [ntpp]φI [ppi]φ ::= [tppi]φ ∧ [ntppi]φ


Mereotopological interpretation of modal logic: examplesI [u](harborcity ↔ (city ∧ 〈ppi〉harbor))

I [u](harbor → (〈ec〉river ∨ 〈ec〉sea))

I [u](Dresden→ harborcity)

I [u](Elbe→ river)

I [u](Dresden→ ¬sea ∧ [ec]¬sea ∧ [po]¬sea ∧ [tpp]¬sea ∧[tppi]¬sea ∧ [ntpp]¬sea ∧ [ntppi]¬sea)

I [u](Dresden→ 〈po〉Elbe ∧ (river → Elbe) ∧ [ec](river →Elbe) ∧ [po](river → Elbe) ∧ [tpp](river →Elbe) ∧ [tppi](river → Elbe) ∧ [ntpp](river →Elbe) ∧ [ntppi](river → Elbe))


Some modal logics for mereotopological relationsI LRCC8(TOP): the logic of the class of all concrete frames of

the form Preg

I LRCC8(IRnreg): the logic of the concrete frame consisting in

all nonempty regular closed regions in IRn

I LRCC8(IRncon): the logic of the concrete frame consisting in

all nonempty convex regular closed regions in IRn

I LRCC8(IRnrec): the logic of the concrete frame consisting in

all nonempty rectangle regular closed regions in IRn

I LRCC8(RS): the logic of the class of all abstract frames


Relationship between some modal logics for mereotopologicalrelations

I LRCC8(RS) LRCC8(TOP), LRCC8(IRn

reg), LRCC8(IRncon), LRCC8(IRn

rec)

Proof: nom(p) ∧ nom(q) ∧ 〈u〉(p ∧ 〈dc〉q)→〈u〉(〈ppi〉p ∧ 〈ppi〉q) is valid in TOP, IRn

reg ,IRn

con and IRnrec but not in RS

I LRCC8(TOP) LRCC8(IRnreg)

Proof: 〈ppi〉> is valid in IRnreg but not in TOP


Proposition (Lutz and Wolter)I LRCC8(TOP), LRCC8(IRn

reg), LRCC8(IRncon), LRCC8(IRn

rec) andLRCC8(RS) are undecidable.

Proposition (Lutz and Wolter)1. LRCC8(TOP), LRCC8(IRn

reg) and LRCC8(IRncon) are not

recursively enumerable.2. LRCC8(RS) is recursively enumerable.


One of the main open problems formulated by Lutz and Wolterwas to find

I decidable modal logics based on a reasonable set ofmereotopological relations

Nenov and Vakarelov present such a logic based on thefollowing mereotopological relations

I overlap O and underlap OI part-of 6 and its converse >I contact C and dual contact CI interior part-of� and its converse�

Modal logic and mereotopologyThe first-order logic of mereotopological structures

Contact algebra B = (B,0,1, ·,+, ?,C)

I (B,0,1, ·,+, ?) is a Boolean algebra with ? as the Booleancomplement

I xCy implies x 6= 0 and y 6= 0I (x + y)Cz is equivalent to xCz or yCzI xC(y + z) is equivalent to xCy or xCzI x 6= 0 implies xCxI xCy implies yCx

The complement of C is denoted by C


Relations O of overlap and O of underlapI x O y ::= x · y 6= 0 and x O y ::= x + y 6= 1

Relations 6 of part-of and its converse >I x 6 y ::= x · y? = 0 and x > y ::= x? · y = 0

Relation C of dual contactI x C y ::= x? C y?

Relations� of interior part-of and its converse�I x � y ::= x C y? and x � y ::= x? C y


Example of a contact algebraI If P is a point-based topology andRCP = (RCP ,0P ,1P , ·P ,+P , ?P ,CP) is the structuredefined as follows:

I (RCP ,0P ,1P , ·P ,+P , ?P) is the Boolean algebra of theregular-closed sets of P

I x CP y iff x ∩ y 6= ∅then RCP is a contact algebra such that

I x OP y iff InP(x ∩ y) 6= ∅ and x OP y iff x ∪ y 6= PI x 6P y iff x ⊆ y and x >P y iff x ⊇ yI x CP y iff InP(x) ∪ InP(y) 6= PI x �P y iff x ⊆ InP(y) and x �P y iff InP(x) ⊇ y


Example of a contact algebraI If G = (N,E) is a (reflexive and symmetric) graph then the

structure S(G) = (2N , ∅,N,∩,∪, \,CG) defined as follows:I (2N , ∅,N,∩,∪, \) is the Boolean algebra of the subsets of NI x CG y iff E(x) ∩ E(y) 6= ∅

is a contact algebra such thatI x OG y iff x ∩ y 6= ∅ and x OG y iff x ∪ y 6= NI x 6N y iff x ⊆ y and x >N y iff x ⊇ yI x CN y iff E(N \ x) ∩ E(N \ y) 6= ∅I x �G y iff E(x) ∩ E(N \ y) = ∅ and x �P y iff

E(N \ x) ∩ E(y) = ∅


LemmaI The relations O, O, 6, >, C, C,� and� satisfy the

following first-order conditions:1. x 6 x ,2. if x 6 y and y 6 x then x = y ,3. if x 6 y and y 6 z then x 6 z,4. if x O y then y O x ,5. if x O y then x O x ,6. if x O x then x 6 y ,7. if x O y and y 6 z then x O z,8. x O x or x O x ,9. if x O y and x ¯O z then y 6 z,

10. if x C y then y C x ,11. if x O y then x C y ,12. if x C y then x O x ,13. . . .


LemmaI The relations O, O, 6, >, C, C,� and� satisfy the

following first-order conditions:1. x > x ,2. if x > y and y > x then x = y ,3. if x > y and y > z then x > z,4. if x O y then y O x ,5. if x O y then x O x ,6. if x ¯O x then x > y ,7. if x O y and y > z then x O z,8. x O x or x O x ,9. if x ¯O y and x O z then y > z,

10. if x C y then y C x ,11. if x O y then x C y ,12. if x C y then x O x ,13. . . .


A relational structure W = (W ,O, O,6,>,C, C,�,�) isI a mereotopological structure iff it satisfies the first-order

conditions of the two above lemmasI a standard (full standard) mereotopological structure iff

there exists a contact algebra B = (B,0,1, ·,+, ?,C) suchthat

I W ⊆ B (W = B)I for all x , y ∈ B, x O y iff x · y 6= 0, x O y iff x + y 6= 1, x 6 y

iff x · y? = 0, x > y iff x? · y = 0, x C y iff x? C y?, x � y iffx C y? and x � y iff x? C y


A relational structure W = (W ,O, O,6,>,C, C,�,�) isI a mereotopological structure iff it satisfies the first-order

conditions of the two above lemmasI a completely standard (full completely standard)

mereotopological structure iff there exists a contactalgebra B = (B,0,1, ·,+, ?,C) of regular closed subsets ofsome topological space such that

I W ⊆ B (W = B)I for all x , y ∈ B, x O y iff x · y 6= 0, x O y iff x + y 6= 1, x 6 y

iff x · y? = 0, x > y iff x? · y = 0, x C y iff x? C y?, x � y iffx C y? and x � y iff x? C y


Proposition (Dimov and Vakarelov)I Every contact algebra B = (B,0,1, ·,+, ?,C) can be

isomorphically embedded into the contact algebraRCP = (RCP ,0P ,1P , ·P ,+P , ?P ,CP) over somepoint-based topology P.

CorollaryI A mereotopological structure is standard iff it is completely

standard.

Proposition (Nenov and Vakarelov)I Every mereotopological structure is completely standard.

Modal logic and mereotopologyA modal logic for mereotopological structures

A polymodal logic based on mereotopological structures:MTML

I φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | [r ]φ | [U]φ

I where r ∈ {O, O,6,>,C, C,�,�}I Abbreviations

I Standard definitions for the remaining Boolean operationsI 〈r〉φ ::= ¬[r ]¬φ where r ∈ {O, O,6,>,C, C,�,�}I 〈U〉φ ::= ¬[U]¬φ

I Intuitive meaningI The interpretation of [r ]φ is φ holds in every region r -related

to the current oneI The interpretation of 〈r〉φ is φ holds in some region

r -related to the current oneI The interpretation of [U]φ is φ holds in every regionI The interpretation of 〈U〉φ is φ holds in some region


In the two above lemmas, the following first-order conditionsare not modally definable

I if x 6 y and y 6 x then x = yI if x > y and y > x then x = y

A generalized mereotopological structure is a generalization ofthe notion of a mereotopological structure by dropping theseconditions and by adding the following conditions:

I if x 6 y and y O y then x = y

I if x > y and y ¯O y then x = y

I if x 6 y , y O z and x ¯O z then x = y

I if x > y , y ¯O z and x O z then x = y


PropositionI Every mereotopological structure is a generalized

mereotopological structure.

Proposition (Nenov and Vakarelov)I Every generalized mereotopological structure is a

p-morphic image of a mereotopological structure.


Axiomatization: Let MTML be the least normal logic in orlanguage based on

I Sahlqvist axiom schemes corresponding to the first-orderconditions defining generalized mereotopologicalstructures

I the following axiom schemes:I [U]φ→ φI φ→ [U]〈U〉φI [U]φ→ [U][U]φI [U]φ→ [O]φ∧ [O]φ∧ [6]φ∧ [>]φ∧ [C]φ∧ [C]φ∧ [�]φ∧ [�]φ


Proposition (Nenov and Vakarelov): The following conditionsare equivalent for any modal formula φ:

1. φ is a theorem of MTML.2. φ is valid in the class of all generalized mereotopological

structures.3. φ is valid in the class of all mereotopological structures.4. φ is valid in the class of all standard and completely

standard mereotopological structures.


Proposition (Nenov and Vakarelov)I MTML does not possess the finite model property with

respect to its standard semantics.

Proof: The Grzegorczyk formula [6]([6](p → [6]p)→ p)→ pI is true in all finite mereotopological structuresI is falsified in the generalized mereotopological structures

W = (W ,O, O,6,>,C, C,�,�) defined byI W = {a,b}I O, O, 6, >, C, C,� and� coincides with W ×W


Proposition (Nenov and Vakarelov)I MTML admits filtration with respect to its nonstandard

semantics and hence is decidable.

Modal logic and mereotopologyBibliography

Balbiani, P., Tinchev, T., Vakarelov, D.: Modal logics forregion-based theory of space. Fundamenta Informaticæ81(2007) 29–82.Ivanov, N., Vakarelov, D.: Relational syllogistics. Inpreparation.Lutz, C., Wolter, F.: Modal logics of topological relations.Logical Methods in Computer Science 2 (2006) 1–41.Nenchev, V.: Logics for stable and unstable mereotopologicalrelations. In preparation.

Modal logic and mereotopologyBibliography

Nenov, Y., Vakarelov, D.: Modal logics for mereotopologicalrelations. In Areces, C., Goldblatt, R. (editors): Advances inModal Logic. College Publications (2008) 249–272.Tinchev, T., Vakarelov, D.: Logics of space withconnectedness predicates: complete axiomatizations. InBeklemishev, L., Goranko, V., Shehtman, V. (editors): Advancesin Modal Logic. College Publications (2010) 434–453.Vakarelov, D.: Region-based theory of space: algebras ofregions, representation theory and logics. In Gabbay, D.,Goncharov, S., Zakharyaschev, M. (editors): MathematicalProblems from Applied Logic II. Logics for the XXIst Century.Springer (2007) 267–348.Vakarelov, D.: Dynamic mereotopology: a point-free theory ofchanging regions. I. Stable and unstable mereotopologicalrelations. Fundamenta Informaticæ100 (2010) 159–180.

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