Logics for Topological Reasoning

ESSLLI Summer School

August 2000

University of Birmingham, UK

Brandon BennettSchool of Computer Studies

University of LeedsLeeds LS2 9JT, UK

brandon@scs.leeds.ac.uk

Contents

1 Introduction 2

2 Basic Classical Topology 22.1 Interior and Closure Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Region Connection Calculus 43.1 Interpretation in Point-Set Topology . . . . . . . . . . . . . . . . . . . . . . . 53.2 RCC Relations Representable in Interior Algebra . . . . . . . . . . . . . . . . 5

4 The Modal Logic S4 74.1 Kripke Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Modal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Algebraic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Power-Set Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.5 Mapping Between Algebraic and Logical Expressions . . . . . . . . . . . . . . 104.6 Entailment among Modal Algebraic Equations . . . . . . . . . . . . . . . . . 114.7 Relating S4 Modal-Algebraic Entailment to Deducibility . . . . . . . . . . . . 11

5 Encoding Topological Relations in S4 145.1 RCC Relations Representable in S4 . . . . . . . . . . . . . . . . . . . . . . . 14

6 Negative Equations and their S4 Representation 15

7 The Extended Modal Logic, S4+ 167.1 Determining Entailments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Representing RCC Relations in S4+ 17

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9 Eliminating Entailment Constraints by Using S4u 199.1 An Example Entailment Encoded in S4u . . . . . . . . . . . . . . . . . . . . . 20

10 Intuitionistic Encoding 21

11 A Non-Modal Encoding 2211.0.1 Boundary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2311.0.2 Adequacy of the Representation . . . . . . . . . . . . . . . . . . . . . 2411.0.3 A Model-Building Procedure . . . . . . . . . . . . . . . . . . . . . . . 25

12 Complexity 25

13 Beyond Topology 2613.1 Region-Based Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.2 Spatio-Temporal Reasoning in PSTL . . . . . . . . . . . . . . . . . . . . . . 30

1 Introduction

Spatial concepts and reasoning are pervasive in our everyday experience and are becomingincreasingly important in computer applications (e.g. GIS, CAD/CAM, robotics). Tradition-ally spatial data has been represented by numerical coordinates, which describe models interms of the points of a Cartesian field. However, the spatial facts we have available, and thegoals we wish to achieve based on this information, in many cases involve high-level conceptsrather than precise geometrical objects (Davis 1990). From the need to handle such abstractspatial concepts has arisen the sub-field of AI known as Qualitative Spatial Reasoning (Cohn1997).

Of all our spatial concepts, it can be argued that topological concepts are the most basic.Moreover there is strong evidence of their psychological significance (Knauff, Rauh and Renz1997). However, perhaps because they are so basic, their importance is often overlooked; and,although spatial reasoning is becoming increasingly central to a wide range of software appli-cations, very few computer systems make use of an explicit model of topological information.Nevertheless, in the last few years considerable effort has been put into developing suitablerepresentations (Randell, Cui and Cohn 1992, Asher and Vieu 1995, Cohn, Bennett, Goodayand Gotts 1997) and a coherent theoretical picture has begun to emerge (Pratt and Lemon1997), together with a number of successful computationally oriented techniques (Bennett1994, Nebel 1995, Bennett 1997, Renz and Nebel 1997a, Renz 1998, Renz and Nebel 1999).In this seminar series we shall examine the logical foundations of this work.

2 Basic Classical Topology

A topological space can be formally defined in a number of ways. Perhaps the simplest is asa set of sets, which is closed under arbitrary unions and finite intersections. This is the set ofopen subsets of the space. The largest open set (which is the same as the union of all opensets) is called the universe of the topology. The empty set (which is equal to a union of anempty set of open sets) must also be included in the open sets. A topology T is often referredto as a tuple T = 〈U , O〉, where U is the universe and O is the set of open sets.

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In a topology T = 〈U , O〉, given an arbitrary subset X of U , the interior of X is the largestmember of O that is a subset of X. The interior of X is often denoted X◦. The mappingfrom subsets of U to their interiors may be identified with a unary function, i. Because ofthe closure conditions on the set of open sets, i must satisfy the following axioms (sometimescalled Kuratowski’s axioms (Kuratowski 1972)):

T1) i(X) ⊆ XT2) i(i(X)) = i(X)T3) i(U) = UT4) i(X ∩ Y ) = i(X) ∩ i(Y )

where X and Y are any subsets of U . Moreover, given a set U , any function i that mapssubsets of U to subsets of U and obeys the above axioms determines a unique topology 〈U , O〉:the elements of O are simply those subsets X of U such that i(X) = X. Hence, any topology〈U , O〉 can be alternatively characterised by a structure 〈U, i〉, where i is an interior function.

The complement of X is written −X or sometimes X. This is the set of elements of Uthat are not elements of X. The function c(X) =def −i− (X) is called the closure operator.

2.1 Interior and Closure Algebras

The theory of topological spaces is traditionally stated in the language of set theory. But,if we are concerned only with the structure of a topological space with respect to Booleancombinations of regions and the interior and closure operations on these regions, we can dowithout the full language of set theory and give a purely algebraic account of the space,which does not involve any use of the elementhood relation, ‘∈’. This abstraction results ina Boolean algebra with an additional operator obeying appropriate conditions for either aninterior or a closure function.

The first comprehensive treatment of these algebras (McKinsey and Tarski 1944) theclosure operator was taken as an extra primitive added to a Boolean algebra and the resultingalgebra called a closure algebra. I shall more often refer to the dual structure of an interioralgebra, which is a structure 〈S,∪,−, i〉, where 〈S,∪,−〉 is a Boolean Algebra and i satisfiesthe equations characterising an interior operator.

Interior algebraic equations provide a simple constraint language for describing topolog-ical relationships between arbitrary sets of points in a topological space. Some of the moresignificant constraints which can be expressed are given in table 1

Constraint MeaningX = i(X) X is open

X = −i− (X) X is closedX = i− i− (X) X is regular openX ∪ Y = Y X is part of Y

X ∪ i(Y ) = i(Y ) X is part of the interior of YX ∩ Y = ∅ X and Y are disjoint

i(X) ∩ i(Y ) = ∅ The interiors of X and Y are disjointX = Y ∪ Z X is the union of Y and Z

Table 1: Some constraints expressible as interior algebra equations.

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Figure 1: Basic relations in the RCC theory

3 Region Connection Calculus

The Region Connection Calculus (RCC) is an axiomatisation of certain spatial concepts andrelations in classical 1st-order predicate calculus. The basic theory assumes just one primitivedyadic relation: C(x, y) read as ‘x connects with y’. Individuals can be interpreted as denotingspatial regions. The relation C(x, y) is reflexive and symmetric, which is ensured by thefollowing two axioms:

RCC1) ∀xC(x, x)RCC2) ∀xy[C(x, y)→ C(y, x)]

We also require C to be extensional:1

RCC3) ∀xy[∀z[C(z, x) ↔ C(z, y)] → x = y]

Using C(x, y) a very large class of intuitively significant relations can be defined. Someof the most useful of these are illustrated in Figure 1 and their definitions are given inTable 2. The non-symmetrical relations P, PP, TPP and NTPP have inverses which we writeas Ri, where R ∈ {P,PP,TPP,NTPP}. These relations are defined by definitions of the formRi(x, y) ≡def R(y, x). Of the defined relations, DC,EC,PO,EQ,TPP,NTPP, TPPi and NTPPihave been proven to form a jointly exhaustive and pairwise disjoint set, which is known asRCC-8.

RCC also incorporates a constant denoting the universal region, a sum function andpartial functions giving the product of any two overlapping regions and the complement ofevery region except the universe. These are defined as follows:

RCCD1) x = U ≡def ∀y[C(x, y)]RCCD2) x = y + z ≡def ∀w[C(w, x) ↔ [C(w, y) ∨ C(w, z)]]RCCD3) Prod(x, y, z) ≡def ∀u[C(u, z) ↔ ∃v[P(v, x) ∧ P(v, y) ∧ C(u, v)]]RCCD4) Compl(x, y) ≡def ∀z[(C(z, y) ↔ ¬NTPP(z, x)) ∧ (O(z, y)↔ ¬P(z, x))]]

It should be noted that within the RCC theory there is no such thing as a null (or empty)region. Thus there is no product of discrete regions or complement of the universal region.

1(Randell et al. 1992) does not contain an explicit extensionality axiom; but it can be shown to follow fromthe definition of the sum operator given in that paper.

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Relation Interpretation DefinitionDC(x, y) x is DisConnected from y ¬C(x, y)P(x, y) x is a Part of y ∀z[C(z, x)→ C(z, y)]PP(x, y) x is a Proper Part of y P(x, y) ∧ ¬P(y, x)EQ(x, y) x is EQual to y P(x, y) ∧ P(y, x)O(x, y) x Overlaps y ∃z[P(z, x) ∧ P(z, y)]DR(x, y) x is DiscRete from y ¬O(x, y)PO(x, y) x Partially Overlaps y O(x, y) ∧ ¬P(x, y) ∧ ¬P(y, x)EC(x, y) x is Externally Connected to y C(x, y) ∧ ¬O(x, y)TPP(x, y) x is a Tangential Proper Part of y PP(x, y) ∧

∃z[EC(z, x) ∧ EC(z, y)]NTPP(x, y) x is a Non-Tang’l Proper Part of y PP(x, y) ∧

¬∃z[EC(z, x) ∧ EC(z, y)]

Table 2: Some significant relations definable within the RCC theory.

This means we do not have a full Boolean algebra of regions; but, in order that appropriateregions exist to fulfil the requirements of the quasi-Boolean structure suggested by the abovedefinitions, the basic RCC theory is supplemented with the following existential axioms:2

RCC4) ∃x∀y[C(x, y)]RCC5) ∀xy∃z∀w[C(z, w) ↔ [C(w, x) ∨ C(w, y)]]RCC6) ∀xy[O(x, y) → ∃z[Prod(x, y, z)]RCC7) ∀x[¬(x = U) ↔ ∃y[Compl(x, y)]

3.1 Interpretation in Point-Set Topology

As one would expect, the regions and relations of the RCC theory can be interpreted in termsof classical point-set topology. In fact there are two dual interpretation that are equallyreasonable.

Closed Interpretation:

• A region is identified with a regular closed set of points.• Regions are connected if they share at least one point.• Regions overlap if their interiors share at least one point.

Open Interpretation:

• A region is identified with a regular open set of points.• Regions are connected if their closures share at least one point.• Regions overlap if they share at least one point.

3.2 RCC Relations Representable in Interior Algebra

I now consider how RCC relations can be represented in interior algebra. If we assume theOpen Interpretation of RCC (given above) we see that the the relations O and C be formally

2Arguably, the axioms for U and + are redundant because (in standard 1st-order logic) denotations ofconstants and functional terms always exist.

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Interior Algebra EquationRCC Relation Open Interpretation Closed Interpretation

DC(x, y) i(X ) ∪ i(Y ) = U X ∩ Y = UDR(x, y) X ∩ Y = U i(X) ∩ i(Y ) = UP(x, y) X ∪ Y = U X ∪ Y = UPi(x, y) X ∪ Y = U X ∪ Y = UNTP(x, y) i(X ) ∪ Y = U X ∪ i(Y ) = UNTPi(x, y) X ∪ i(Y ) = U i(X) ∪ Y = UEQ(x, y) (X ∪ Y ) ∩ (X ∪ Y ) = U (X ∪ Y ) ∩ (X ∪ Y ) = U

Table 3: Seven relations defined by interior algebra equations

defined in terms of point sets by:

O(x, y) ≡def ∃π[π ∈ X ∧ π ∈ Y ]

C(x, y) ≡def ∃π[π ∈ c(X) ∧ π ∈ c(Y )]

However, these definitions make use of a highly expressive set-theoretic language, includingboth quantification and the element relation. Given that the relations are intuitively verysimple, one may wonder whether it is possible to give an alternative characterisation of C andO in the much less expressive language of interior algebraic equations.

As it happens the negations of each of these relations can be quite easily defined as follows:

DC(x, y) ≡def i(X ) ∪ i(Y ) = U

DR(x, y) ≡def X ∩ Y = U

But C and O cannot themselves be defined as interior algebraic equations. This followsfrom the general observation that purely equational constraints are always consistent withany purely equational theory (there must always be at least a trivial one-element model, inwhich all constants denote the same individual). Thus if the negation of some constraintcan be expressed as an equation, then the constraint itself cannot be equationally expressible(otherwise that constraint would be consistent with its own negation).

To define C and O we would need both interior algebraic equations and the negations ofsuch equations, however many useful relations can be specified with equations alone. Table 3gives definitions in interior algebra of seven binary relations: DC, DR, P, Pi, NTP, NTPiand EQ. Equations corresponding to both the open and closed interpretations of RCC areshown. This set, which will be called RCC-7, is of particular significance because, as willbe shown below, each of the RCC-8 relations can be expressed as a conjunction of positiveand negative RCC-7 relations. Note that RCC-7 is neither jointly exhaustive nor pairwisedisjoint: if two regions partially overlap, they stand in none of the seven relations; and DR(being the disjunction of DC and EC) can hold of two regions which are also DC. A numberof other binary RCC relations are expressible by means of interior/closure algebra equations.For example, EQ(sum(x, y), u) can be expressed by X ∪ Y = U .

It turns out that all the RCC-8 relations can be specified by a conjunction of interioralgebraic equalities and disequalities as given in Table 4.

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Table 4: The RCC-8 relations represented as interior algebra constraints

RCC Rel. Equivalent Algebraic Constraint(s)

DC(x, y) — (i(x) ∪ i(y) = U)EC(x, y) DR(x, y) ∧ ¬DC(x, y) (x ∩ y = U) ∧ (i(x) ∪ i(y) 6= U)PO(x, y) ¬DR(x, y) ∧ ¬P(x, y) ∧ ¬Pi(x, y) (x ∩ y 6= U) ∧ (x ∪ y 6= U) ∧ (x ∪ y 6= U)TPP(x, y) P(x, y) ∧ ¬EQ(x, y) ∧ ¬NTPP(x, y) (x ∪ y = U) ∧ (x 6= y) ∧ (i(x) ∪ y 6= U)TPPi(x, y) Pi(x, y) ∧ ¬EQ(x, y) ∧ ¬NTPPi(x, y) (x ∪ y = U) ∧ (x 6= y) ∧ (x ∪ i(y) 6= U)NTPP(x, y) — (i(x) ∪ y = U)NTPPi(x, y) — (x ∪ i(y) = U)EQ(x, y) — (x = y)

To accord fully with the topological interpretations of RCC, the equational representationmust also constrain the regions to be regular open or closed depending on which of the twointerpretations it used. In the first case one should add an equation X = i− i− (X) for eachregion variable X; and in the second case one should add X = −i− i(X).

4 The Modal Logic S4

S4 is one of the simpler and better known modal logics. It may also be called KT4 since itis obtained from classical propositional logic by adding the the rule of necessitation and thefollowing axiom schemas:

K. �(φ → ψ) → (�φ → �ψ)T. �φ → φ

4. �φ → ��φ

A modal logic which satisfies the schema K, as well as obeying the rule of necessitation,is known as normal.

4.1 Kripke Semantics

Currently the best known interpretations of modal logics are those in terms of Kripke se-mantics. In a Kripke semantics a model consists of a set of possible worlds together withan accessibility relation — a binary relation between worlds — associated with each modaloperator. Propositions denote sets of possible worlds (the set of worlds in which they aretrue). A Kripke model, M, is thus a structure 〈W,R, P, d〉, where W is a set of worlds, R isthe accessibility relation, P is a set of constants, {pi}, and d is a function mapping elementsof P to subsets of W .

Such a model determines the truth of each modal formula at each possible world. Classicalformulae are interpreted as follows:

• Atomic formulae, pi are true in exactly the worlds in the set d(pi).

• Conjunctions, φ ∧ ψ, are true in worlds where both φ and ψ are true.

• Disjunctions, φ ∨ ψ, are true in worlds where either φ or ψ (or both) is true.

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• Negations, ¬φ, are true in worlds where φ is not true.

We write |=Mα φ to mean that formula φ is true at world α in model M. A modal operator,�, is then interpreted as follows: in a model M = 〈W,R, P, d〉

|=Mα �φ iff |=Mβ φ for all β ∈W s.t. R(α, β)

A frame is a set of all Kripke models satisfying some specification of the properties of theaccessibility relation, R. For example, the set of all Kripke models in which R is reflexive andsymmetric constitutes a frame. Finally we say that a formula is valid in some frame, F , if itis true at every world in every model in F .

The logic S4 is characterised by the frame, FS4, consisting of all Kripke models whoseaccessibility relations are reflexive and transitive (R is a quasi-ordering on W ). Every theoremprovable according to the proof system for S4 specified above is valid in FS4; and converselyevery formula valid in FS4 is provable in the proof system.

A vast spectrum of different modal operators can be specified by placing more or lessgeneral restrictions on the corresponding accessibility relation. Furthermore, Kripke semanticsallows one to specify operators whose logic seems to correspond well with intuitive propertiesof modal concepts employed in natural language. Indeed, a number of logics proposed fornatural language modalities, which were originally specified proof theoretically (by axiomschemata intended to capture intuitive properties of modal concepts) can be captured veryeasily within the Kripke paradigm by quite simple restrictions on the accessibility relation.

Whilst the Kripke approach certainly provides a very flexible approach to modal semantics,its generality is often overstated. Consequently, many researchers in both AI and philosoph-ical logic tend to think of possible worlds semantics as essentially based upon accessibilityrelations. However, although Kripke models may be appropriate for certain types of modaloperator, in other cases it may be more natural to suppose a quite different structuring ofpossible worlds or even a semantics that is not based on possible worlds at all.

4.2 Modal Algebras

A modal algebra is a mathematical structure that provides a semantics for modal logics whichis more general than a Kripke model. Just as the formulae of classical propositional logic canbe interpreted as referring to elements of a Boolean algebra, modal formulae can be interpretedas elements of a Boolean algebra supplemented with an additional unary operation obeyingcertain constraints. This is a modal algebra. Boolean algebras with additional operatorswere first studied in detail by Jonsson and Tarski (1951). Their connection to modal logicswas investigated by Lemmon (1966a, 1966b). A clear account of the essential properties ofmodal algebras and their relation to Kripke semantics is given by Hughes and Cresswell (1968,Chapter 17) and a much more detailed examination can be found in (Goldblatt 1976).

A modal algebra can be represented by a structure M = 〈S,+,−, �〉, where 〈S,+,−〉 isa Boolean algebra and, for all elements x and y of the algebra, the operator ‘�’ satisfies theequation

�(x+ y) = �x+ � y (add)

Operators obeying this equation are known as additive.3 The maximal and minimal elementsof the algebra will respectively be denoted 1 and 0.

3It is additive operators which are the primary focus of the investigations of Jonsson and Tarski (1951).

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For our purposes, it will be more convenient to deal with the dual modal algebraic struc-tures 〈S, ·,−,�〉, where · is the Boolean product. � corresponds, of course, to the modal �operator and satisfies the additivity condition in the form:

�(x · y) = �x+ � y (add)

4.3 Algebraic Models

We can now define an algebraic model for a modal language as a structure 〈S, ·,−,�, P, δ〉,where 〈S, ·,−,�〉 is a modal algebra, P is the set of constants of the language and δ is afunction mapping modal formulae to elements of S. For each constant p ∈ P , δ[p] may beany element of S. This assignment to the constants determines the value δ[φ] of all complexformulae according to the following recursive specification:4

• δ[α ∧ β] = δ[α] · δ[β]

• δ[¬α] = −δ[α]

• δ[�α] = �(δ[α])

The algebraic equation characterising additivity corresponds to the modal schema

�(φ ∧ ψ) ↔ (�φ ∧ �ψ) ,

which is true in every normal modal logic.We say that a formula, φ, is universal in a model 〈S, ·,−,�, P, δ〉 iff δ[φ] = 1 — i.e. if the

model assigns to the formula the unit (universal) element of the modal algebra 〈S, ·,−,�〉.An algebraic frame, FE , is a set of all algebraic models whose algebras satisfy some set ofequations, E, constraining the ‘�’ operator. Finally we say that a formula is valid with respectto some algebraic frame, FE , if it is universal in every model in FE .

In order that algebraic models provide a semantics for some modal logic, L, we must finda set of characteristic equations, EL such that a formula φ is valid in the frame FEL if andonly if it is a theorem of L. For brevity I shall denote the frame associated with the logic Lby FL, rather than FEL . For instance, the frame FS4 is the set of all models satisfying theequations:

�x ≤ x (i.e. x · �x = �x) (epis)5

� 1 = 1 (norm)

�(�(x)) = �(x) (idem)

It is known that a formula is valid with respect to FS4 iff it is a theorem of the logic S4(Hughes and Cresswell 1968, Chapter 17).

Note that, if φ ↔ ψ is a theorem of some logic L, then φ and ψ must have the samedenotation in every algebra in FL. Thus, since � is interpreted as an extensional algebraicfunction, �φ ↔ �ψ must be a theorem of L. Hence, any modal logic which can be givenan algebraic semantics will be closed under the rule of equivalence: if ` φ ↔ ψ then `�φ ↔ �ψ, which I shall refer to as RE.

4Specifications for the connectives ∨ , → , ↔ and ♦ can easily be derived from their definitions in termsof ¬, ∧ and �.

5So called because it is required for an epistemological interpretation of the modality, since if ‘I know φ’then φ must be true.

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4.4 Power-Set Algebras

According to Stone’s representation theorem (Stone 1936)6 every Boolean algebra is isomor-phic to a Boolean algebra whose elements are sets and whose operators are identified with theusual union, intersection and complementation operations of elementary set theory. More-over, such an algebra can always be embedded in a Boolean algebra whose elements are allthe subsets of some (universal) set W .

Jonsson and Tarski (1951) showed that a similar theorem holds for Boolean algebras withadditional additive operators. This means that every modal algebra can be isomorphicallyembedded in a modal algebra whose elements are all members of the power set, 2W, of someset, W . One may think of the elements of W as possible worlds; and since each proposition,p, of the modal language is interpreted as an element, α, in the modal algebra, α may beregarded as the set of worlds in which p is true.

Where an algebraic model is based on a power-set algebra, I shall represent it by a structure〈U,∩,−,�, P, δ〉, where the product operator is ‘∩’ to indicate that the Boolean operatorscorrespond to the operators of elementary set theory. The elements of the algebra are nowthe members of the power set of U and the maximal element, 1, of a power-set algebra isequal to U itself. A modal operator, �, in a power-set algebra, maps every subset, X, of Uto another subset �(X).

The power-set algebras are representative of the whole class of modal algebras in thesense that an equation which is true in all power-set algebras is true in every modal algebra(because every modal algebra can be embedded in a power-set algebra). This means that incharacterising validity in terms of algebraic frames we can restrict the frames to contain onlymodels based on power-set algebras. In the sequel I shall assume that we always consideronly models based on power-sets and I shall refer to the resulting semantics as algebraic setsemantics.

4.5 Mapping Between Algebraic and Logical Expressions

As with the classical set-semantics it will be useful to introduce meta-level notation for refer-ring to the mapping between modal formulae and modal algebraic terms. I assume that theseterms are interpreted as sets in a power-set algebra. Thus MAT[φ] is the modal algebraicterm obtained from the formulae φ by replacing the connectives ¬, ∨ , ∧ and � by theoperators −, ∪, ∩ and � and the 0-order constants, pi, by set constants, Pi. Since the ♦operator is equivalent to ¬�¬ this is replaced by the algebraic operator −�−. The functionMF is the inverse of MAT so that MF[τ ] is the formulae φ such that MAT[φ] = τ . I shallwrite φ MF

MAT τ to refer to the mapping in the form of a relation.I also define the transform MFe[ξ], such that MFe[τ = 1] = MF[τ ] for universal equations

and MFe[τ1 = τ2] = MF[τ1] ↔ MF[τ2], for non-universal equations. The expression MFe[ξ]refers to a modal formula which (because of the correspondence theorem, Mcorr, which willbe given in section 4.7) may be regarded as representative of the modal algebraic equationξ. However, because of the form of the entailment correspondence theorem, S4ECT, alsoproved in section 4.7, one might say that an equation ξ constraining an S4 modal algebra isbetter represented by �MFe[ξ] rather than MFe[ξ].

Equations characterising a class of algebraic structures (a frame) will in general containfree variables which are taken as implicitly universally quantified — the equations hold for

6A comprehensive study of this theorem can be found in (Johnstone 1982).

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all elements of the algebra. Thus, an equation with free variables will correspond to a classof modal formulae, which can be represented as a formula schema. Because of this, it isconvenient to generalise MF so as to operate on terms with free variables. In such a casethe resulting expression will be a modal schema rather than a formula and schematic logicalvariables will take the place of the free variables in the algebraic term. Accordingly, MFecan also be allowed to operate on equations containing free variables — again the result willbe a schema rather than a formula.

By means of MFe, a set of algebraic equations defining a frame FL can be translateddirectly into a set of modal schemas which specify the proof system of the corresponding logicL. To ensure the proof system is complete it will also be necessary to add the inference ruleRE which is intrinsic to algebraic semantics (as explained at the end of section 4.3).

4.6 Entailment among Modal Algebraic Equations

If some entities of interest (in our case these will be spatial regions) are identified with elementsin an algebra, then equations between algebraic terms can be used to specify relationshipsbetween these entities. One can then reason about these relations in terms of entailmentsamong algebraic equations. Since set algebras are representative of the class of modal algebrasthe notion of entailment among modal algebraic equations can be defined in terms of possibleset assignments to a language of modal algebraic terms.

A set assignment to a language of algebraic terms is a structure Σ = 〈S,U, σ,m〉, where:

• S is a set of constants,• U is the universal set,• σ : S → 2U assigns a subset of U to each constant in S (the logical constants 0

and 1 are assigned ∅ and U respectively),• and m : 2U → 2U specifies the modal operator � as a set function.

If τ is a term built from the constants in S by means of Boolean and modal operators, thenΣ[τ ] is the set assigned to τ by Σ. This is determined by σ, m and the usual interpretationof Boolean operations on sets. If Σ[τ1] = Σ[τ2] we say that Σ satisfies the equation τ1 = τ2.

Entailment relations among modal-algebraic equations can now be specified as follows:

τ1 = υi, . . . , τn = υn |=MALτ0 = υ0

means that, for every assignment Σ = 〈S,U, σ,m〉 (where S includes all the constants oc-curring in the terms τi and υi) satisfying the equations associated with the frame FL, if Σsatisfies the equations τ1 = υ1, . . . , τn = υn it also satisfies the equation τ0 = υ0.

4.7 Relating S4 Modal-Algebraic Entailment to Deducibility

If a modal logic L is characterised by a modal algebraic frame FL there is a correspondencebetween deduction in the logic and entailment between algebraic equations in the algebras inFL. Because of this we can use modal logics to reason about algebraic equations.

From the definition of an algebraic frame for the logic L we have the following correspon-dence between universal set equations and logical theorems:

|=MALτ = 1 iff `L φ, where φ MF

MAT τ (Mcorr)

More generally, the following correspondence between the entailment relation among universalset equations constraining algebras in FS4 and the deducibility relation of S4 can be proved:

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S4 Entailment Correspondence Theorem:

τ1 = 1, . . . , τn = 1 |=MAS4τ0 = 1 iff �φ1, . . . ,�φn `S4 φ0 (S4ECT)

where φi MFMAT τi

This is the main theorem justifying the use of S4 for topological reasoning (Bennett 1997).

Proof of S4ECT:

Since S4 is an extension of classical logic it obeys the deduction theorem: φ1, . . . , φn `S4

φ0 iff `S4 (φ1 ∧ . . . ∧ φn) → φ0. By combining this with Mcorr we get the more generalcorrespondence

φ1, . . . , φn `S4 φ0 iff |=MAS4(τ1 ∩ . . . ∩ τn) ∪ τ0 = 1 .

Hence�φ1, . . . ,�φn `S4 φ0 iff |=MAS4

(� τ1 ∩ . . . ∩ � τn) ∪ τ0 = 1 .

Because of the additivity of � the equation on the r.h.s. is equivalent to �(τ1 ∩ . . .∩ τn) ⊆ τ0,so we can establish S4ECT by showing that

|=MAS4�(τ1 ∩ . . . ∩ τn) ⊆ τ0 iff τ1 = U , . . . , τn = 1 |=MAS4

τ0 = 1 .

The r.h.s. can then be re-written to give

|=MAS4

�(τ1 ∩ . . . ∩ τn) ⊆ τ0 iff (τ1 ∩ . . . ∩ τn) = 1 |=MAS4

τ0 = 1

and this equivalence can be more succinctly expressed as

|=MAS4�(υ) ⊆ τ0 iff υ = 1 |=MAS4

τ0 = 1 (†) .

Clearly the left to right direction of (†) must hold for any modal algebra, which satisfies� 1 = 1 (i.e. any normal modal algebra) and hence any algebra in FS4.

The right to left direction of (†) is harder to show. I prove it by proving the contrapositive— i.e.:

if 6|=MAS4�(υ) ⊆ τ0 then υ = 1 6|=MAS4

τ0 = 1 (††)

Let S be the set of all constants occurring in the terms υ and τ0. If the antecedent of (††) istrue, there must be some assignment Σ = 〈S,U, σ,m〉 satisfying the equational constraints ofthe frame FS4 and such that Σ[�(υ)] 6⊆ Σ[τ0]. From Σ we can construct an assignment, Σ′,which verifies the consequent of (††) — i.e. Σ′[υ] = 1 but Σ′[τ0] 6= 1:

Let U ′ = Σ[�(υ)]. Note that U ′ is an open subset of U . We now define Σ′ =〈S,U ′, σ′,m′〉 by stipulating that:

• σ′[κ] = σ[κ] ∩ U ′, for all constants, κ ∈ S,• m′(X) = m(X) for all sets X ⊆ U ′.

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It can then be shown that for any term, τ (made up of constants in S), Σ′[τ ] = Σ[τ ]∩U ′.We know this identity holds for atomic terms because of the definition of σ′, so to show itinductively for all terms we need to show that, if it holds for α and β, it must hold for α,α ∪ β, α ∩ β and �α. For the Boolean operators the required identities are demonstrated bythe following sequences of equations:

Σ′[α] = U ′ − Σ′[α] = U ′ − (Σ[α] ∩ U ′) = U ′ − Σ[α] = (U − Σ[α]) ∩ U ′ = Σ[α] ∩ U ′

Σ′[α ∪ β] = Σ′[α] ∪ Σ′[β] = (Σ[α] ∩ U ′) ∪ (Σ[β] ∩ U ′) = (Σ[α] ∪ Σ[β]) ∩ U ′ = Σ[α ∪ β] ∩ U ′

Σ′[α ∩ β] = Σ′[α] ∩ Σ′[β] = (Σ[α] ∩ U ′) ∩ (Σ[β] ∩ U ′) = (Σ[α] ∩ Σ[β]) ∩ U ′ = Σ[α ∩ β] ∩ U ′

(In the first of these the identity U ′−Σ[α] = (U−Σ[α])∩U ′ depends on the fact that U ′ ⊆ U .)For the case of �α we have:

Σ′[�α] = m′(Σ′[α]) = m(Σ′[α]) = m(Σ[α] ∩ U ′) = m(Σ[α]) ∩m(U ′) = Σ[�α]) ∩ U ′

We must verify that the algebra specified by Σ′ is a member of FS4. I have establishedthat for every term (built from constants in S) Σ′[τ ] = Σ[τ ] ∩ U ′. This means that everyequation, τ1 = τ2, satisfied by Σ will also be satisfied by Σ′. Since, by hypothesis, Σ mustsatisfy all the frame equations of FS4, Σ′ must also satisfy these frame equations.

To complete the proof I must show that Σ′ verifies the r.h.s. of (††). Since the algebragenerated by Σ′ is in FS4, it must satisfy epis, which means that for any term, τ , Σ′[� τ ] ⊆Σ′[τ ]. We know that Σ′[� υ] = U ′, so Σ′[υ] ⊇ U ′; but Σ′[υ] = Σ[υ] ∩U ′, so Σ′[υ] = U ′. Recallthat Σ was chosen to verify the antecedent of (††) because Σ[� υ] 6⊆ Σ[τ0]. Thus, U ′ 6⊆ Σ[τ0];and from this it follows that Σ[τ0] ∩ U ′ $ U ′. Hence we have Σ′[τ0] $ U ′. �

By means of the MFe meta-function, an arbitrary modal set equation can be directlytransformed into universal form and the formula MFe[ξ] can be regarded as representingthe equational constraint ξ. The modal logic S4 can thus be used to reason about arbitraryequations constraining algebras in the frame FS4 according to the following generalisation ofS4ECT:

ξ1, . . . , ξn |=MAS4ξ0 iff �MFe[ξ1], . . . ,�MFe[ξ1] `S4 MFe[ξ0] .

The form of S4ECT is a bit awkward in that in the S4 deduction corresponding to anentailment between equations, we need to add an extra � operator to the formulae on theleft of `S4 but not to the formula on the right.7

This means that the question “What is the S4 representation of the equation ξ?” does nothave a simple answer. However, it is easily shown that a sequent �φ1, . . . ,�φn `S4 φ0 is infact valid if and only if �φ1, . . . ,�φn `S4 �φ0. Thus, for the purpose of testing entailments,it can be said that the representation of an equation ξ is �MFe[ξ].

7Note that if we do not add �s as required the correspondence fails. For example x = 1 |=MAS4�x = 1

but X 6`S4 �X.

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5 Encoding Topological Relations in S4

It was established by Tarski and McKinsey (1948) that the S4 box operator can be modelledalgebraically by an interior operator. We have seen that the algebraic semantics for S4 satisfiesthe same equations as an interior algebra, so these algebras are equivalent. By making use ofthe meta-level notation relating modal algebraic equations and corresponding modal formulaewe can also examine the relationship between closure/modal algebraic equations and modalformulae. The representation of a closure/modal algebraic equation ξ in modal logic is theformula MFe[ξ]. Because the equations specifying properties of the closure operation containfree variables they will be mapped to modal schemata rather than formulae. The characteristicequations of a closure algebra and corresponding modal schemata are given in Table 5.

Interior Axioms Modal Schemata

i(X) ∪X = X (�φ ∨ φ) ↔ φ (T’)i(i(X)) = i(X) ��φ ↔ �φ (4+)i(U) = U �> (N)

i(X ∩ Y ) = i(X) ∩ i(Y ) �(φ ∧ ψ) ↔ (�φ ∧ �ψ) (R)

Table 5: Interior Axioms and Corresponding Modal Schemata

Clearly T’ is equivalent to the schema T, �φ → φ and, given that T holds, 4+ can beweakened to �φ → ��φ, which is the schema 4. Furthermore it is well known that theschemata N and R in conjunction with the rule RE are equivalent to the combination ofschema K and the rule of necessitation, RN. Thus specifying that N, R and RE hold isan alternative way of specifying that a modal logic is normal (see (Chellas 1980, chapter 4)).Recall that RE holds in any algebraic semantics for a modal operator. Hence, the modal logicderived from an interior or closure algebra by transforming equational algebraic constraintsinto modal schemata is exactly the logic S4. Consequently, in virtue of the correspondencetheorem S4ECT, deduction in S4 can be used to reason about closure algebraic equations.

5.1 RCC Relations Representable in S4

Since the S4 modality can be interpreted as an interior function over a topological space, wecan use this interpretation to encode topological relations as S4 formulae. The basis of thisrepresentation is exactly the same as for the C representation but by use of the additionalmodal operator it is possible to make a distinction between connection and overlapping whichcannot be expressed in C. Table 6 shows the S4 formula corresponding to each of the RCC-7relations according to the open set interpretation of RCC regions. The middle column showsthe algebraic set-equation associated with the relation. We see that, if the interior operatori is identified with the corresponding modal algebra operator �, then the interior algebraicequation ξ, is represented by the S4 formula �MFe[ξ].

I now illustrate how the correspondence theorem S4ECT, enables deduction in S4 tobe used to reason about entailment among certain RCC relations. Consider the followingargument:

NTP(a, b) ∧ DR(b, c) |= DC(a, c)

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RCC Relation Interior Algebra Equation (ξ) S4 formula (�MFe[ξ])

DC(x, y) i(X ) ∪ i(Y ) = U �(�¬x ∨ �¬y)DR(x, y) X ∩ Y = U �¬(x ∧ y)P(x, y) X ∪ Y = U �(¬x ∨ y)Pi(x, y) X ∪ Y = U �(x ∨ ¬y)NTP(x, y) i(X ) ∪ Y = U �(�¬x ∨ y)NTPi(x, y) X ∪ i(Y ) = U �(x ∨ �¬y)EQ(x, y) (X ∪ Y ) ∩ (X ∪X) = U �((¬x ∨ y) ∧ (x ∨ ¬y))

Table 6: Seven relations defined by interior algebra equations and corresponding S4 formulae

This corresponds to the following entailment between interior algebraic equations:

i(A ) ∪B= U , B ∩ C = U , A = i(A), B = i(B), C = i(C) |= i(A ) ∪ i(C ) = U .

Here the equations of the form α = i(α) constrain the regions to correspond to open sets.8

By appealing to S4ECT this can be shown to be valid because we have

�(�¬a ∨ b), �¬(b ∧ c), �(a ↔ � a), �(b ↔ � b), �(c ↔ � c) `S4 �(�¬a ∨ �¬c) .

The S4 representation is quite expressive but does have serious limitations. For instance,although both disconnection, DC(x, y), and discreteness, DR(x, y), can be represented it isstill not possible to specify the relation of external connection, EC(x, y). We have also seenthat (although their negations can be represented) the fundamental relations C and O cannotbe represented. In order to overcome these deficiencies we need a language in which one canexpress closure-algebraic inequalities as well as equalities.

6 Negative Equations and their S4 Representation

In order to represent the complete set of RCC-8 relations we need to be able to represent notonly equational constraints in interior algebra but the negations of such constraints. However,the following theorem allows us to reduce the problem of testing consistency of sets of bothpositive and negative equational constraints to the problem of testing entailment for positiveequations:

Consistency of Equational Literals (ELcons)

µ1 = ν1, . . . , µm = νm, ¬(σ1 = τ1), . . . ,¬(σn = τn) |=iff

µ1 = ν1, . . . , µm = νm |= σi = τi for some i ∈ {1, . . . n}

ELcons can be established by considering possible proofs of inconsistency in some proofsystem for 1st-order logic with equality, which is known to be refutation complete. One suchsystem, is that where the only proof rules are binary resolution, paramodulation and factoring

8In general, to be faithful to RCC, one should ensure that regions are regular open by adding the strongerconstraint α = i− i− (α); but the inference in this example is valid for any open regions.

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(Duffy 1991). Since we are dealing with sets of literals (i.e. only unit clauses), factoring isnot required and a simplified version of paramodulation can be employed. The details of therules that are used do not matter, since ELcons can be demonstrated from quite generalobservations. The proof is as follows:

Proof of ELcons: Suppose we refute a set of equational literals by means of binaryresolution and paramodulation. Once an application of binary resolution can be made, in-consistency is proved immediately; so any successful refutation must consist of a series ofparamodulations followed by a single binary resolution. Note also that each paramodulationeither involves two positive literals and generates a new positive literal or it involves a positiveand a negative literal and generates a new negative literal. These observations enable us toshow that any refutation makes essential use of exactly one negative literal. The key pointsare that the derivation of a positive literal cannot involve any negative literals and that norule operates on more than one negative literal.

Consider the final step in the refutation; this is a resolution between a positive and anegative literal. The positive literal is either in the original set of literals or has been derivedby a sequence of paramodulations involving only positive literals. The negative literal is eitherin the original set or has been generated from a positive and a negative literal. In the lattercase, the positive literal must have been derived from only positive literals and the negativeliteral is either in the original set or is in turn derived from a positive and negative literal.However long this sequence continues, it is clear that exactly one negative literal from theoriginal set is involved in the proof. �

7 The Extended Modal Logic, S4+

In order to increase the expressive power of S4, so that we can represent both positive andnegative algebraic constraints we can define an augmented representation language, S4+,whose expressions are pairs 〈M, E〉, where M and E are formulae of S4. These formula setsare called respectively model and entailment constraints. The reason for this terminologyis that the model constraints can be regarded a constraining possible topological modelswhereas entailment constraints forbid certain entailments from the model constraints. Hence,we stipulate that:

An S4+ expression 〈M, E〉 is consistent if and only ifno formula in E is entailed by the set M.

In virtue of ELcons this accords with an interpretation under which the formulae in Mare identified with corresponding algebraic equations, where as the formulae in E are identifiedwith the negations of such equations.

7.1 Determining Entailments

Computing inconsistency of S4+ expressions is a special case of determining entailmentsbetween situation descriptions characterisable in S4. To refer to such an entailment, I shalluse the notation 〈M, E〉 |=S4+ 〈M′, E ′〉. We can express the meaning of this as an entailmentbetween set-equations as follows:

m1 = U ∧ . . . ∧ mh = U ∧ e1 6= U ∧ . . . ∧ ei 6= U |=m′1 = U ∧ . . . ∧ m′j = U ∧ e′1 6= U ∧ . . . ∧ e′k 6= U

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If we then bring the r.h.s. over to the left and move the resulting negation inwards we get:

m1 = U ∧ . . . ∧ mh = U ∧ e1 6= U ∧ . . . ∧ ei 6= U ∧(m′1 6= U ∨. . .∨ m′j 6= U ∨ e′1 = U ∨. . .∨ e′k = U ) |= .

To show the validity of this we must show that whichever of the equations in the disjunctionis chosen the resulting equation set is inconsistent. This is equivalent to showing that:

for all p ∈M′ we have 〈M, E ∪ {p}〉 |=S4+ and for all q ∈ E ′ we have 〈M ∪ {q}, E〉 |=S4+

Another equivalent way of expressing these which is more convenient from the point ofview of actually calculating the entailments is the following:

S4+ Entailment Theorem (S4+ET)

〈M, E〉 |=S4+ 〈M′, E ′〉 iffeither 〈M, E〉 |=S4+ or ( for all φ ∈M′ : 〈M, {φ}〉 |=S4+

and for all ψ ∈ E ′ : 〈M ∪ {ψ}, E〉 |=S4+ )

Informally, this means that a sequent is valid iff: either 〈M, E〉 is itself inconsistent; or, eachof the model constraints in M′ is entailed by the model constraints M and also each of theentailment constraints in E ′ in conjunction with the model constraints M entails one of theentailment constraints in E . Determining the validity of a S4+ entailment has thus beenreduced to determining the inconsistency of certain S4+ expressions and we already knowthat such an expression is inconsistent iff one of its entailment constraints is entailed by itsmodel constraints.

8 Representing RCC Relations in S4+

Since S4+ can represent both equations and inequalities between terms made up of Booleanoperations and an interior operator, it can express a very large class of spatial relationships.In particular, it can represent all those RCC relations which can be expressed as a conjunctionof positive and negative RCC-7 relations.

The representations of the RCC-8 relations under the open set interpretation are givenin table 7. The alternative closed set representation is given below in table 8. The way theyare obtained can be summarised as follows: express the RCC-8 relations in terms of RCC-7relations and interpret these as equational constraints on interior algebras as given in table 4.Then translate these constraints into S4 according to table 6. The formulae correspondingto positive RCC-7 relations become model constraints in the S4+ representation and thosecorresponding to negated RCC-7 relations become entailment constraints. Note that the S4+

correspondence theorem requires that model constraints have an extra initial � added tothe result of applying MFe to the modal algebraic equation but this is not required in theentailment constraints. This asymmetry stems from S4ECT.

Let us now consider how the S4+ representation can be used to test the consistency of asimple set of spatial relations. Take for example the following conjunction of RCC-8 relations:

TPP(a, b) ∧ DC(b, c) ∧ PO(a, c) .

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Relation Model Constraint Entailment ConstraintsDC(x, y) �(�¬x ∨ �¬y) ¬x, ¬yEC(x, y) �¬(x ∧ y) �¬x ∨ �¬y, ¬x, ¬yPO(x, y) — ¬(x ∧ y), x → y, y → x, ¬x, ¬yTPP(x, y) �(x → y) �¬x ∨ y, y → x, ¬x, ¬yTPPi(x, y) �(y → x) �¬y ∨ x, x → y, ¬x, ¬yNTPP(x, y) �(�¬x ∨ y) y → x, ¬x, ¬yNTPPi(x, y) �(�¬y ∨ x) x → y, ¬x, ¬yEQ(x, y) �(x ↔ y) ¬x, ¬yC(x, y) — �¬x ∨ �¬y ¬x, ¬yEQ(x, sum(y, z)) �(x ↔ �♦(y ∨ z)) ¬x, ¬yRegOpen(x) x ↔ ♦�x —

Table 7: The S4+ encoding of some RCC relations (open set interpretation)

Translating into S4+ according to table 7 we get the following representation:

〈{�(a → b), �(�¬b ∨ �¬c)}, {�¬a ∨ b, b → a, ¬(a ∧ c), a → c, c → a, ¬a, ¬b, ¬c}〉

This is an ordered pair consisting of two sets of S4 formulae, the first set being model con-straints and the second entailment constraints. Appealing to part 3 of S4+CT we determinethat the relations are inconsistent because

�(a → b), �(�¬b ∨ �¬c) `S4 ¬(a ∧ c)

i.e. one of the entailment constraints is entailed by the model constraints.

Relation Model Constraint Entailment ConstraintsDC(x, y) �¬(x ∧ y) ¬x, ¬yEC(x, y) �¬(�x ∧ � y) ¬(x ∧ y), ¬x, ¬yPO(x, y) — ¬(�x ∧ � y), x → y, y → x, ¬x, ¬yTPP(x, y) �(x → y) x → � y, y → x, ¬x, ¬yTPPi(x, y) �(y → x) y → �x, x → y, ¬x, ¬yNTPP(x, y) �(x → � y) y → x, ¬x, ¬yNTPPi(x, y) �(y → �x) x → y, ¬x, ¬yEQ(x, y) �(x ↔ y) ¬x, ¬yC(x, y) — ¬(x ∧ y), ¬x, ¬yEQ(x, sum(y, z)) �(x ↔ (y ∨ z)) ¬x, ¬yRegClosed(x) x ↔ ♦�x —

Table 8: S4+ encoding based on the closed set interpretation of RCC

Notice that the entailment constraints for each RCC relation R(x, y) contain the formulae¬x and ¬y. These correspond the the negative equational constraints X 6= U and X 6= U ,ensuring that X and Y must be non-empty regions of space as required by the RCC theory.

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If our modal encoding is to agree fully with the intended meaning of RCC regions we alsoneed to enforce the condition that the regions are regular. Thus in the open set interpretationwe should add to M additional formulae x ↔ �♦x for each variable X occurring in the setof RCC relations we are encoding. Under the closed set interpretation we would similarlyadd formulae of the form x ↔ ♦�x It must be noted that these conditions are not generalschemata such that every instance must be true. They only apply to formulae which can bedirectly associated with RCC regions.

The regularity condition is also relevant to the encoding of Boolean functions of regions.For instance in the open set interpretation we see in table 7 that the sum of y and z isrepresented by the formula �♦(y ∨ z).

9 Eliminating Entailment Constraints by Using S4u

In reasoning with the extended 0-order language S4+ the meanings of the two types of con-straint are handled at the meta-level: determining entailments in these languages involveschecking a number of different object-level entailments in the logic S4. A set of algebraicconstraints encoded in an S4+ expression 〈M, E〉 is consistent if and only if none of its entail-ment constraints in E is entailed by the set of all model constraints inM. A natural questionregarding this representation is whether it might be possible to extend the S4 language itselfso that the semantics of the two types of constraint was built directly into the object language.This would mean that computation of entailments could be carried out entirely at the objectlevel.

In terms of algebraic semantics it is quite easy to introduce a new modal operator ∀ bymeans of which the model/entailment constraint distinction can be made at the object level.If δ(φ) is the algebraic denotation of a formula φ, we define ∀ by:

• δ(∀φ) = U iff δ(φ) = U .• δ(∀φ) = ∅ iff δ(φ) 6= U .

A dual operator ∃φ is defined by ∃φ ≡def ¬∀¬φ. Considered on its own, ∀ can be regardedas an S5 modal operator. This means it satisfies all the schemata obeyed by an S4 modalityand in addition the schema

∃∀φ → ∀φ .

Our intended interpretation of the new ∀ operator is that ∀φ is true at every point/worldin the model. However, the S5 axioms permit Kripke models where accessibility is an equiv-alence relation rather than the universal relation. This means that ∀φ could be satisfied in amodel where φ is not true at every world but just at all the worlds within some equivalenceclass. Normally this is of little importance, since the truth of a formula can only dependon worlds accessible from the actual world. Consequently the other worlds can be ignored,and all consistent formulae have models where accessibility is actually the universal relation.However, where we have other modalities, such as the topologically interpreted S4 operator,the accessibility relation of such an operator might join worlds in different equivalence classes.In this case certain formulae may be satisfiable only in models where the S5 accessibility rela-tion separates the set of worlds into more than one equivalence class. Consequently we cannotassume that ∀φ can be assumed to mean that φ it true at all possible worlds.

To avoid this possibility and ensure the correct logical behaviour of the new ∀ operator

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we add to our bi-modal logic the schema

♦φ → ∃φ . (CONNECT)

This means that the S4 accessibility relation cannot take us outside the S5 equivalence classoccupied by the actual world. It can then be shown that a formulae of this S4/S5 hybrid isconsistent with the axiom schemata if and only if it is satisfiable in a model where the S5accessibility is the universal relation. The resulting logic is thus called S4u. It is known tobe decidable (Goranko and Passy 1992).

Given the definition of ∀, we have ¬∀φ= U iff φ 6= U . Thus, negations of universal setequations (and hence all equations) can be converted into positive equations. This obviatesthe need for entailment constraints, since a model constraint ¬∀φ has the same meaning asφ taken as an entailment constraint. More specifically, the translation of an S4+ expression

〈{φ1, . . . , φj}, {ψ1 . . . ψk}〉

into S4u is the formula

∀φ1 ∧ . . . ∧ ∀φj ∧ ¬∀ψ1 ∧ . . . ∧ ¬∀ψk .

Consequently any expression of S4+ can be represented by a simple object level formula inthe multi-modal language S4u.

In the S4u encoding a set of key RCC relations are represented as as follows:

Relation Open set encoding Closed set encodingC(x, y) ∃(♦x ∧ ♦ y) ∃(x ∧ y)O(x, y) ∃(x ∧ y) ∃(�x ∧ � y)P(x, y) ∀(x → y) ∀(x → y)TP(x, y) ∀(x → y) ∧ ∃(♦x ∧ ¬y) ∀(x → y) ∧ ∃(x ∧ ♦¬y)NTP(x, y) ∀(♦x → y) ∀(x → � y)NE(x) ∃x ∃xRegular(x) ∀(x ↔ �♦x) ∀(x ↔ ♦�x)

Table 9: S4u encodings of RCC relations

All the RCC-8 relations can be expressed by conjunctions of these relations and theirnegations. Note that we can now explicitly represent relations such as C and O as well astheir negations DC and DR. The non-emptiness condition (NE(x)) is also now explicitlyexpressed by the simple formula ∃x.

9.1 An Example Entailment Encoded in S4u

Let us look at a simple example of spatial reasoning carried out in S4+. We shall considerthe transitivity of the proper part relation, PP:

PP(a, b) ∧ PP(b, c) |= PP(a, c) .

PP(x, y) is equivalent to P(x, y) ∧ ¬P(y, x) and we also require that x and y are non-null.Thus the modal representation of PP(a, b) is:

∀(a → b) ∧ ¬∀(b → a) ∧ ∃a ∧ ∃b

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Hence the transitivity of PP corresponds to the entailment:

∀(a → b) ∧ ¬∀(b → a), ∀(b → c) ∧ ¬∀(c → b), ∃a, ∃b, ∃c|= ∀(a → c) ∧ ¬∀(c → a) ∧ ∃a ∧ ∃c

In testing the validity of this entailment it is natural to proceed as follows. Since the r.h.s.is a conjunction, the sequent is valid iff each of the four sequents with the same l.h.s. but justone conjunct on the r.h.s. is valid. Of these four sequents, the two with ∃a and ∃c on ther.h.s. are trivially valid because these formulae also occur on the l.h.s.. To prove the validityof the other two, it is convenient to move all conjuncts on the l.h.s. which have an initial ∃operator over to the right and then to replace ¬∃ prefixes by ∀¬. We shall then have thefollowing two sequents:

∀(a → b) ∧ ∀(b → c) |= ∀(a → c) ∨ ∀(b → a) ∨ ∀(c → b) ∨ ∀¬a ∨ ∀¬b ∨ ∀¬c

∀(a → b) ∧ ∀(b → c) ∧ ∀(c → a) |= ∀(b → a) ∨ ∀(c → b) ∨ ∀¬a ∨ ∀¬b ∨ ∀¬c

We can verify these proof-theoretically by the application of just one modal rule (togetherwith ordinary classical reasoning). This is the rule RK which holds in any normal modallogic:

(φ1 ∧ . . . ∧ φn) → φ[RK]

(∀φ1 ∧ . . . ∧ ∀φn) → ∀φ

This rule together with the deduction theorem means that

if φ1, . . . , φn |= φ then ∀φ1, . . . ,∀φn |= ∀φ

Application of this principle validates both of our sequents, since

a → b, b → c |= a → c and a → b, b → c, c → a |= b → a.

10 Intuitionistic Encoding

Prior to the topological interpretation of S4, Tarski (1938) had already showed that theintuitionistic propositional calculus9 (henceforth I) can be given an interpretation in whicheach propositional letter, p, corresponds to an open set, P , within a topological space andthe connectives are associated with Boolean and topological operations as given in Table 10.Thus, each formula φ of I corresponds to an interior algebraic term, τ(φ). Tarski showedthat φ is a theorem of I just in case it corresponds to a term which denotes the universalset, U , in every topological model (i.e. for every assignment of open sets in a topologicalspace to the constants occurring in τ(φ), we find that τ(φ) = U). More generally φ0 can beintuitionistically deduced from a set of formulae {φi} iff every topological model satisfyingthe equations τ(φi) = U also satisfies τ(φ0) = U (see (Bennett 1994)).

9For an introduction to intuitionistic logic see e.g. (Nerode 1990).

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p ∧ q P ∩Qp ∨ q P ∪Q∼ p i(P )p⇒ q i(P ∪Q )

Table 10: Topological interpretation of I

Table 11: Representation of the RCC-7 relations in I

RCC Algebraic Constraint I formula

DC(x, y) i(x) ∪ i(y) = U ∼x ∨ ∼ yDR(x, y) x ∩ y = U ∼(x ∧ y)P(x, y) x ∪ y = U x⇒ yPi(x, y) x ∪ y = U y⇒ xNTPP(x, y) i(x) ∪ y = U ∼x ∨ yNTPPi(x, y) x ∪ i(y) = U x ∨ ∼ yEQ(x, y) x = y x⇔ y

Bennett (1994) used this correspondence to construct a topological reasoning algorithmbased upon an intuitionistic theorem prover. Table 11 shows how each of the RCC-7 relationscan be encoded in I.10 The regularity of a region x can also be encoded by ∼∼x ⇒ x.Negative topological equations can also be handled using exactly the same technique as wasemployed in the construction of S4+.

Consider, for example, the configuration P(a, b) ∧ DC(b, c) ∧ PO(a, c). P and DC corre-spond directly to I formulae as given in Table 11. PO must first be analysed as ¬DR ∧¬P ∧ ¬Pi as shown in Table 4. The (positive) model constraints correspond to the formulaset {a⇒ b, ∼ b ∨ ∼ c} and the (negative) entailment constraints (including the non-null con-straints) to {a ⇒ c, c ⇒ a, ∼(a ∧ c), ∼ a, ∼ b, ∼ c}. These constraints are inconsistentbecause a⇒ b, ∼ b ∨ ∼ c `I ∼(a ∧ c) — i.e. one of the entailment constraints in entailed bythe model constraints.

11 A Non-Modal Encoding

A spatial interpretation of a Kripke models for S4u is obtained by identifying possible worldswith spatial points. Thus a formula of the form ∀φ ensures that the topological conditionencoded by φ holds at every point in space. Similarly, ∃φ means that there is some point psatisfying the condition represented by φ. p can be thought of as a sample point, which bearswitness to some topological constraint. For instance, where two regions x and y overlap, thecorresponding modal formula ∃(�(x) ∧ �(y)) ensures the existence of a point which is in theinterior of both x and y.

The accessibility relation for the S4 operator � may be thought of as the relation of twopoints being ‘arbitrarily close’. Thus, the formula �x is true at a point p if region x occupies

10Note that in I the formula ∼ a ∨ ∼ b is strictly stronger than the (classically equivalent) formula ∼(a ∧ b).Similarly, ∼ a ∨ b is strictly stronger than a⇒ b.

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point p and also all points arbitrarily close to p. This means that p must be an interior pointof x. Consider, for example, a boundary point p of a regular (closed) region x. This fails tosatisfy the left hand disjunct, �(¬x), of the regularity condition, because there are arbitrarilyclose points (e.g. p itself) which are occupied by x. Consequently, p must satisfy ♦(�(x)).This says that there is a point say q which is arbitrarily close to p and such that every pointarbitrarily close to q is occupied by x.

The complexity results of Renz and Nebel (1999) show that to reason with only theRCC-8 relations, the full power of the S4 encoding is not required. Indeed, a much simplerrepresentation can be used. We note that, when we are dealing with regular regions, everypoint is either an interior, exterior or boundary point of each region. We describe these statesformally using the following relations between a point and a region: I(p, r), E(p, r) and B(p, r).These three conditions are jointly exhaustive and pairwise disjoint.

We represent topological information in terms of the existence of witness points and byconstraints imposed on the witness points. The following conditions introduce witness points:

NE(x) ∃p [I(p, x)]O(x, y) ∃p [I(p, x) ∧ I(p, y)]¬P(x, y) ∃p [I(p, x) ∧ E(p, y)]C(x, y) ∃p [(I(p, x) ∧ I(p, y)) ∨ (B(p, x) ∧ B(p, y))]

We also have the following global constraints on the witness points:11

DC(x, y) ∀p [E(p, x) ∨ E(p, y)]NTP(x, y) ∀p [E(p, x) ∨ I(p, y)]DR(x, y) ∀p [E(p, x) ∨ E(p, y) ∨ (B(p, x) ∧ B(p, y))]P(x, y) ∀p [E(p, x) ∨ I(p, y) ∨ (B(p, x) ∧ B(p, y))]

(The usual regularity constraint on regions will be taken account of implicitly by the treatmentof boundary points given in the next section.)

11.0.1 Boundary Points

In constructing a model for a spatial configuration, boundary points have a special significancebecause each boundary point of a regular region is arbitrarily close to both its interior and itsexterior. This means that if a region r has an interior point which coincides with a boundarypoint of s then r must also have interior points which coincide with both interior and exteriorpoints of s. Similarly, if r has an exterior point coinciding with a boundary point of s then rmust have exterior points coinciding with both interior and exterior points of s.

Figure 2 illustrates how the different types of point are distributed at and near a boundarypoint. The middle diagram shows the case where regions r and s are externally tangent (wewrite ET(r, s)) and the right-hand diagram shows the case where r is internally tangent to s(we write IT(r, s)).

When one of the conditions ET(x, y) and IT(x, y) holds, it must be witnessed by a specialconfiguration of three points, which we call a boundary cluster. This consists of a boundary

11There might be cases where we would want to use further constraints (such as ∃p [E(p, x) ∧ E(p, y)], whichholds iff x and y do not jointly fill the entire space).

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Figure 2: Boundary Points

point pb and two ‘arbitrarily close’ neighbouring points p1 and p2 (one on each side of theboundary). The existence of and topological constraints on these points are specified asfollows:

IT(x, y) ∃pbp1p2 [BC(p1, pb, b2) ∧ B(pb, x) ∧ B(pb, y)∧ I(p1, x) ∧ I(p1, y) ∧ E(p2, x) ∧ E(p2, y)]

ET(x, y) ∃pbp1p2 [BC(p1, pb, b2) ∧ B(pb, x) ∧ B(pb, y)∧ I(p1, x) ∧ E(p1, y) ∧ E(p2, x) ∧ I(p2, y)]

Because a boundary point is arbitrarily close to the other points in the cluster, wheneverit is in the interior of some region r the whole cluster is interior to r; and, if the boundarypoint is exterior to r, so is the whole cluster. Formally:

∀r, x, y, z [(BC(x, y, z) ∧ I(y, r)) → (I(x, r) ∧ I(z, r))]

∀r, x, y, z [(BC(x, y, z) ∧ E(y, r)) → (E(x, r) ∧ E(z, r))]

11.0.2 Adequacy of the Representation

It is easy to see that a topological model of a set of RCC-8 relations (according to theirstandard interpretation over regular closed point-sets)will satisfy the corresponding witnesspoint constraints given in the last section, under the natural interpretation of I,B,E and BC.To demonstrate the adequacy of the representation we show equivalence of satisfiability inthe opposite direction: from the witness point representation corresponding to a set of RCCrelations and satisfying the stipulated conditions on BC, we can construct a topological spacetogether with an assignment of a subset of this space to each region name occurring in theRCC relations; this topological space satisfies the RCC relations under the standard regularclosed point-set interpretation.

From the witness point model we construct a topological space 〈U,O〉. First we augmentthe constraining relations so that for each point p and each region r we have either E(p, r),B(p, r) or I(p, r) — any such extension will be consistent but, for definiteness, whenever pand r are unrelated we can add E(p, r). Let U be the set of witness points. We now definethe set O of open sets as the smallest set which is closed under union and intersection and issuch that: U ∈ O; for each region r, {p | I(p, r)} ∈ O and {p | E(p, r)} ∈ O. Within the space〈U,O〉, each region r is identified with the (closed) set of witness points {p | I(p, r) ∨ B(p, r)}.

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We need to ensure that regions and the closure operation have the correct interpretation.This means that for each region we must have {p | E(p, r)} is the largest open set disjoint from{p | I(p, r)}. This condition is violated just in case there is some region whose interior pointsinclude boundary points of r but no exterior points of r. But the constraints on boundaryclusters ensure that if a boundary point of r is in the interior of s, then there are also bothinterior and exterior points of r which are in the interior of s. Similarly, if a boundary pointof r is in the exterior of s, there are both interior and exterior points of r in the exteriorof s. It is now easy to verify that: a) 〈U,O〉 is a topological space; b) for each region rnamed in the situation description, the topological closure operation on {p | I(p, r)} is just{p | I(p, r) ∨ B(p, r)}.

11.0.3 A Model-Building Procedure

In virtue of the representational adequacy just proved, we can use the witness point repre-sentation as the basis for a model-building procedure to test the consistency of sets of RCCrelations. First we create data-structures representing the witness points and boundary clus-ters as required by the existential constraints on the witness point model. For each point andboundary cluster, we then attempt to satisfy the global constraints on witness point togetherwith the special conditions on boundary clusters. This can be done using a straightforwardbacktracking search. It is worth noting that the satisfaction problems for each witness pointand boundary cluster are independent and so could be carried out in parallel.

12 Complexity

One of the principal motivations of developing the modal representations of topological con-cepts was to provide a formalism that is amenable to computational manipulation. Unfortu-nately I do not have time to cover complexity results in any detail so I shall merely summarisethe most important results:

• The intuitionistic encoding of (Bennett 1994) showed that reasoning with a significantsubset of RCC relations is decidable. (As we have seen, the encodable relations includeall the RCC-8 relations and Boolean combinations of these relations as well as the(quasi-)Boolean relationships between regions.)

• (Nebel 1995) showed that consistency checking of sets of basic RCC-8 relations istractable and in the complexity class NC. A naive algorithm takes order n3 time, wheren is the number of RCC-8 relations; but by the use of parallelism runtime can be reducedto O(n log n).

• (Nebel 1995) also shows that a set S of RCC-8 relations is consistent iff it is pathconsistent — i.e. computing all compositions two relations in the set does not produceany new relation that is stronger than those already in the set; or in other words S isa fixed point relative to compositional inferences.

• Tractable sublanguages of disjunctive constraints over the RCC-8 relations have beenidentified in (Renz and Nebel 1997b, Renz and Nebel 1999).

• (Renz 1998) investigates canonical models for satisfiable RCC-8 constraint networks. Itshows that the set of modal formulae representing RCC-8 relations is satisfiable iff it

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has a model in which the S4 accessibility relation has no chains of length greater thanone.

• Dornheim (1998) proves that it is undecidable whether a given formula of 1st-orderRCC language is true for regions in the plane. (A similar result, without the planarityrestriction on the interpretation was sketched by (Gotts 1996) and is based on muchearlier results of (Grzegorczyk 1951)).

13 Beyond Topology

To conclude these notes I shall mention two recently developed logical systems which takeus beyond the confines of purely topological information. The first of these is Region-BasedGeometry which is a fully expressive geometrical theory in which regions, rather than points,are taken as the basic entities. Secondly I briefly mention the modal language PSTL whichis a two-dimensional combination of the topological modal language S4u and the well-knowntemporal logic PTL.

13.1 Region-Based Geometry

For many applications, confining spatial vocabulary to topological notions is too restrictive.We would like to employ a much wider range of concepts for describing shapes, relative sizesand positions of spatial objects. The results of Davis, Gotts and Cohn (1999) and Pratt (1999)tell us that even adding a predicate as apparently simple as convexity immediately gives usan extremely expressive system which is essentially as expressive as the 1st-order languageof polynomial constraints over reals numbers. However, from the point of view constructinga general purpose spatial ontology (within which one might hope to embed more restrictedcomputationally-oriented representations) the possibility of constructing a fully expressivegeometrical language based on regions is very attractive. Hence I sketch here the theory ofRegion Based Geometry (RGB) as presented in (Bennett, Cohn, Torrini and Hazarika 2000a)and (Bennett, Cohn, Torrini and Hazarika 2000b).

We begin with a formal theory of the parthood relation, P(x, y). As a basis for theaxiomatisation we take the classical Mereology of Lesneiwski (Lesniewski 1927-1931) (seealso (Tarski 1929, Woodger 1937, Simons 1987)):

D1) PP(x, y) ≡def (P(x, y) ∧ ¬(x = y))D2) DR(x, y) ≡def ¬∃z[P(z, x) ∧ P(z, y)]D3) SUM(α, x) ≡def ∀y[y ∈ α → P(y, x)] ∧ ¬∃z[P(z, x) ∧ ∀y[y ∈ α → DR(y, z)]]

In D3, α is a 2nd-order variable, which can denote any subset of the domain of regions.x ∈ α is of course true just in case the denotation of x is a member of the set denoted by α;but our object language does not include any other set-theoretic apparatus.12

In addition to the usual principles of classical logic and the theory of sets, the system isrequired to satisfy the following specifically mereological postulates:

A1) ∀x∀y∀z[P(x, y) ∧ P(y, z) → P(x, z)]A2) ∀α[∃x[x ∈ α] → ∃!x[SUM(α, x)]]

12In fact the form x ∈ α could be written as α(x), in the style of a 2nd-order language without set theory.

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These ensure firstly that the part relation is transitive and secondly (and slightly contro-versially) that for any non-empty set of individuals there is a unique individual which is thesum of that set.

We now develop a theory which we call Region-Based Geometry, inspired by Tarski’sGeometry of Solids (Tarski 1929). Whereas Tarski’s presentation is not formalised (andin some respects somewhat unclear) we shall give a fully formal axiomatisation. FollowingTarski, we build on Lesneiwski’s mereology by introducing a new primitive sphere predicate,which we write S(x). In terms of P and S a series of geometrical relationships and conceptsare defined and a set of postulates is given.

We shall often want to quantify over just the spherical regions in the domain. For conve-nience we introduce the notations

• ∀◦x[φ] ≡def ∀x[S(x) → φ]• ∃◦x[φ] ≡def ∃x[S(x) ∧ φ]

As in (Tarski 1929) we define the relations of external tangency (ET), internal tangency(IT), external diametricity (ED), internal diametricity (ID) and concentricity (x}y). SeeFig. 3 for 2D illustrations.

D4) ET(a, b) ≡def (S(a) ∧ S(b) ∧ DR(a, b)∧ ∀◦xy[(P(a, x) ∧ P(a, y) ∧ DR(b, x) ∧ DR(b, y)) → (P(x, y) ∨ P(y, x))])

D5) IT(a, b) ≡def (S(a) ∧ S(b) ∧ PP(a, b) ∧∀◦xy[(P(a, x) ∧ P(a, y) ∧ P(x, b) ∧ P(y, b)) → (P(x, y) ∨ P(y, x))])

D6) ED(a, b, c) ≡def (S(a) ∧ S(b) ∧ S(c) ∧ ET(a, c) ∧ ET(b, c) ∧∀◦xy[(DR(x, c) ∧ DR(y, c) ∧ P(a, x) ∧ P(b, y)) → DR(x, y)])

D7) ID(a, b, c) ≡def (S(a) ∧ S(b) ∧ S(c) ∧ IT(a, c) ∧ IT(b, c) ∧∀◦xy[(DR(x, c) ∧ DR(y, c) ∧ ET(a, x) ∧ ET(b, y)) → DR(x, y)])

D8) a}b ≡def S(a) ∧ S(b) ∧ [ (a = b)∨ (PP(a, b) ∧ ∀◦xy[(ED(x, y, a) ∧ IT(x, b) ∧ IT(y, b)) → ID(x, y, b)])∨ (PP(b, a) ∧ ∀◦xy[(ED(x, y, b) ∧ IT(x, a) ∧ IT(y, a)) → ID(x, y, a)]) ]

Figure 3: Relations among spheres defined by Tarski

We now define some fundamental relations involving spheres:

D9) B(x, y, z) ≡def x = y ∨ y = z ∨∃vw[ED(x, y, v) ∧ ED(v, w, y) ∧ ED(y, z, w)]

D10) COB(s, r) ≡def S(s) ∧ ∀s′[s′}s → (O(s′, r) ∧ ¬P(s′, r))]D11) EQD(x, y, z) ≡def ∃◦z′[z′}z ∧ COB(y, z′) ∧ COB(x, z′)]D12) Mid(x, y, z) ≡def B(x, y, z) ∧ ∃◦y′[y′}y ∧ COB(x, y′) ∧ COB(z, y′)]D13) EQD(w, x, y, z) ≡def ∃◦uv[Mid(w, u, y) ∧ Mid(x, u, v) ∧ EQD(v, z, y)]D14) Nearer(w, x, y, z) ≡def ∃◦x′[B(w, x, x′) ∧ ¬(x}x′) ∧ EQD(w, x′, y, z)]

B(x, y, z) holds when the centre of y is between the centres of x and z (or coincides with

27

one of these). COB(s, r) means that sphere s is Centred On the Boundary of r. EQD(x, y, z)says that the centres of x and y are equidistant from the centre of z. Mid(x, y, z) says thatthe centre of y lies mid-way between the centres of x and z; and EQD(w, x, y, z) holds whenthe distance between the centres of w and x is the same as the distance between the centresof y and z. Nearer(w, x, y, z) means that the centres of w and x are closer than the centres ofy and z.

Since the concepts B and EQD are definable, we can write the axioms of n-dimensionalElementary Geometry (Tarski 1959) (these axioms are reproduced in Appendix ??), withinour language (the value of n is fixed by appropriate choice of upper and lower dimensionaxioms). (Tarski 1929) takes this approach to prove that his geometry of solids is categoricaland is modelled by n-dimensional Euclidean space in which spheres are interpreted as openballs and ‘solids’ are regular open sets. We take a similar approach; however, whereas Tarskiintroduced points as sets of spheres, our relations concern spheres but they hold just in case thecentre points of the spheres satisfy the corresponding point relations. Thus the quantifiers ofthe point-based geometry axioms can be replaced by quantifiers over spheres and the equalityrelation replaced by the}relation. Hence, in addition to A1 and A2 of Mereology, our theorycontains:

A3) A complete axiom set for n-dimensional geometry (e.g. (Tarski 1959)) encodedin terms of the B, EQD and } relations.

A4) ∀◦xyz[(x}y ∧ y}z) → x}z]A5) ∀◦xx′yzw[(EQD(x, y, z, w) ∧ x′}x) → EQD(x′, y, z, w)]

Axioms A4 and A5 ensure that}behaves like equality relative to the geometrical axioms.13

To get a categorical axiomatisation we have to ensure that the class of regular open sets ofcentre points of spheres coincides with the class of regions and that the P relation correspondsto the inclusion relation among the centre points. Rather than stating these as a meta-levelconstraints (as Tarski does) we enforce them directly by axioms. First, we define relationsthat hold when the centre point of a sphere is within the interior of a region:

D15) InI(s, r) ≡def ∃s′[s′}s ∧ P(s′, r)]

We can now specify the axioms:

A6) ∀◦xy[¬(x}y) → ∃◦s[s}x ∧ ∀◦z[InI(z, s) ↔ Nearer(x, z, x, y)]]A7) ∀◦x∃◦y[¬(x}y) ∧ ∀◦z[InI(z, x) ↔ Nearer(x, z, x, y)]]A8) ∀xy[P(x, y) ↔ ∀◦s[InI(s, x) → InI(s, y)]]A9) ∀r∃◦s[P(s, r)]

A6 ensures that for every pair of distinct points x and y there is a sphere centred at oneand bounded by the other. A7 says that all spheres can be constructed in this way. A8means that P(x, y) holds just in case every interior point of x is an interior point of y (thisactually makes A1 redundant). A9 states that every region has a spherical part (from thisit can be proofed that every region is equal to the sum of its spherical parts). The theoryspecified by the axioms A1–9 we call RBGn (Region-Based Geometry).

Let an n-dimensional classical interpretation for RBGn be a function =n which assignsa non-empty regular open subset of Rn to each 1st-order variable of RBGn and a set ofnon-empty regular open subsets of Rn to each of its 2nd-order variables. Under a classical

13Reflexivity and symmetry are implicit in the definition of}.

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interpretation P(x, y) holds just in case =n(x) ⊆ =n(y); S(x) holds just in case =n(x) is anopen n-ball.

Theorem 1. Axioms RBGn provide a categorical axiom system for n-dimensional region-based geometry, such that every model is isomorphic to a classical interpretation =n.

Proof: (Tarski 1959) proves that all models of the axioms specified by A3-A5 have thestructure of n-dimensional Cartesian spaces over R. This guarantees that there are spherescentred at all and only the points of Rn. A6 and A7 ensure that there is a sphere correspond-ing to every open n-ball in the space, such that InI(s, s′) holds just in case the centre point ofs is in the ball corresponding to s′. A8 fixes the interpretation of P(x, y) to coincide with thecondition {s|InI(s, x)} ⊆ {s|InI(s, y)}. The interpretation of S is now completely fixed relativeto Rn and InI and P also have there intended interpretations over the domain of spheres. Wenow fix the interpretation of the regions.

From A9 and D3 it is easy to show that every region is the sum of its spherical parts:

∀x[SUM({y | P(y, x) ∧ S(y)}, x)]

This, with A2, ensures that the set of regions coincides with the SUMs of arbitrary sets ofspheres. Let the set of ‘interior-points’ of a region r be the set of centre points of all spheress such that InI(s, r). Clearly, determining this set for all regions completely determines the Prelation. We now show that for any set of spheres α such that SUM(α, s), the set S of interiorpoints of s coincides with the smallest regular open set R containing all interior points of allspheres in α.

First note that from A1, A2, D8 and D15 one can prove:

∀◦s∀r[InI(s, r) ↔ ∃◦s′[P(s′, r) ∧ InI(s, s′)]] ,

which means that the interior points of a region are just those interior to its spherical parts.Consequently the interior points of any region form an open set. Since S must be open and Ris regular open, then if S were larger than R it would contain some sphere disjoint from R andhence disjoint from all spheres in α and hence disjoint from s. Thus S ⊆ R. Since R is openwe can exactly cover all its points by the interior points of a set of spheres ρ. By A2 theremust be a region r such that SUM(ρ, r). The regularity of R then means that interior pointsof r are exactly those in R. Now suppose S $ R, then using A8 and the mereological axiomsone can show that ∃x[P(x, r) ∧ DJ(x, s)] and thence (using 9) ∃◦x[P(x, r) ∧ ¬DJ(x, s)]. Butif r includes a sphere which is disjoint from s then, contrary to our supposition R cannot bethe smallest regular open set including all spheres in α. Therefore we must have S = R. �

Because of the 2nd-order nature of the theory we cannot get a truly complete axiomsystem. However, the categoricity result means the theory would be complete if we had anoracle for 2nd-order logic, so the meaning of all non-logical vocabulary is completely fixed.

Theorem 2. RBGn is undecidable for n ≥ 2.

Proof: For n = 2 this follows from the results of (Grzegorczyk 1951) (see also (Gotts1996, Dornheim 1998)). Undecidability for higher dimensions can be demonstrated by defininga ‘slice’ of n-space that is infinitely extended in two dimensions but of a fixed finite thicknessin the others. Two dimensional regions can be simulated by parts of the slice such that theirboundaries within the slice are orthogonal to the faces of the slice.14 �

14RBG1 may also be undecidable; so far we have no result on this.

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13.2 Spatio-Temporal Reasoning in PSTL

Recent work of Wolter and Zakharyaschev (2000) has shown how modal encodings can beused to construct decidable spatio-temporal languages encompassing the S4u topological rep-resentation. By using multi-dimensional modal logic, topological representations in S4u canbe combined with the temporal logic PTL to yield the very expressive spatio-temporal logicPSTL.15

The models for this logic correspond to the structure illustrated in figure 4, where eachhorizontal slice is an S4u model and the vertical structure along the time dimension providesa PTL model. Thus each ‘possible world’ in the model has both a spatial and a temporalindex.

Figure 4: The spatio-temporal structure of PSTL models

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