Logics for Topological Reasoning ESSLLI Summer School August 2000 University of Birmingham, UK Brandon Bennett School of Computer Studies University of Leeds Leeds LS2 9JT, UK
[email protected]
Contents
1 Introduction 2 Basic Classical Topology
2 2
3 Region Connection Calculus
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2.1 Interior and Closure Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Interpretation in Point-Set Topology . . . . . . . . . . . . . . . . . . . . . . . 3.2 RCC Relations Representable in Interior Algebra . . . . . . . . . . . . . . . .
4 The Modal Logic S 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
Kripke Semantics . . . . . . . . . . . . . . . . . . . . . . Modal Algebras . . . . . . . . . . . . . . . . . . . . . . . Algebraic Models . . . . . . . . . . . . . . . . . . . . . . Power-Set Algebras . . . . . . . . . . . . . . . . . . . . . Mapping Between Algebraic and Logical Expressions . . Entailment among Modal Algebraic Equations . . . . . Relating S 4 Modal-Algebraic Entailment to Deducibility
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5 Encoding Topological Relations in S4
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6 Negative Equations and their S 4 Representation 7 The Extended Modal Logic, S 4+
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8 Representing RCC Relations in S 4+
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5.1 RCC Relations Representable in S 4 . . . . . . . . . . . . . . . . . . . . . . .
7.1 Determining Entailments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
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9 Eliminating Entailment Constraints by Using S4u
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10 Intuitionistic Encoding 11 A Non-Modal Encoding
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12 Complexity 13 Beyond Topology
25 26
9.1 An Example Entailment Encoded in S 4u . . . . . . . . . . . . . . . . . . . . .
11.0.1 Boundary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.0.2 Adequacy of the Representation . . . . . . . . . . . . . . . . . . . . . 11.0.3 A Model-Building Procedure . . . . . . . . . . . . . . . . . . . . . . .
13.1 Region-Based Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Spatio-Temporal Reasoning in PSTL . . . . . . . . . . . . . . . . . . . . . .
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23 24 25
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1 Introduction Spatial concepts and reasoning are pervasive in our everyday experience and are becoming increasingly important in computer applications (e.g. GIS, CAD/CAM, robotics). Traditionally spatial data has been represented by numerical coordinates, which describe models in terms of the points of a Cartesian eld. However, the spatial facts we have available, and the goals we wish to achieve based on this information, in many cases involve high-level concepts rather than precise geometrical objects (Davis 1990). From the need to handle such abstract spatial concepts has arisen the sub- eld of AI known as Qualitative Spatial Reasoning (Cohn 1997). Of all our spatial concepts, it can be argued that topological concepts are the most basic. Moreover there is strong evidence of their psychological signi cance (Knau , Rauh and Renz 1997). 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 applications, very few computer systems make use of an explicit model of topological information. Nevertheless, in the last few years considerable e ort has been put into developing suitable representations (Randell, Cui and Cohn 1992, Asher and Vieu 1995, Cohn, Bennett, Gooday and Gotts 1997) and a coherent theoretical picture has begun to emerge (Pratt and Lemon 1997), together with a number of successful computationally oriented techniques (Bennett 1994, 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 de ned in a number of ways. Perhaps the simplest is as a set of sets, which is closed under arbitrary unions and nite intersections. This is the set of open subsets of the space. The largest open set (which is the same as the union of all open sets) is called the universe of the topology. The empty set (which is equal to a union of an empty set of open sets) must also be included in the open sets. A topology T is often referred to as a tuple T = hU ; Oi, where U is the universe and O is the set of open sets. 2
In a topology T = hU ; Oi, given an arbitrary subset X of U , the interior of X is the largest member of O that is a subset of X . The interior of X is often denoted X . The mapping from subsets of U to their interiors may be identi ed with a unary function, i. Because of the closure conditions on the set of open sets, i must satisfy the following axioms (sometimes called Kuratowski's axioms (Kuratowski 1972)): T1) i(X ) X T2) i(i(X )) = i(X ) T3) i(U ) = U T4) 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 maps subsets of U to subsets of U and obeys the above axioms determines a unique topology hU ; Oi: the elements of O are simply those subsets X of U such that i(X ) = X . Hence, any topology hU ; Oi can be alternatively characterised by a structure hU; ii, where i is an interior function. The complement of X is written ?X or sometimes X . This is the set of elements of U that 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 Boolean combinations of regions and the interior and closure operations on these regions, we can do without 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, `2'. This abstraction results in a Boolean algebra with an additional operator obeying appropriate conditions for either an interior or a closure function. The rst comprehensive treatment of these algebras (McKinsey and Tarski 1944) the closure operator was taken as an extra primitive added to a Boolean algebra and the resulting algebra called a closure algebra. I shall more often refer to the dual structure of an interior algebra, which is a structure hS; [; ?; ii, where hS; [; ?i is a Boolean Algebra and i satis es the equations characterising an interior operator. Interior algebraic equations provide a simple constraint language for describing topological relationships between arbitrary sets of points in a topological space. Some of the more signi cant constraints which can be expressed are given in table 1 Constraint Meaning X = i(X ) X is open X = ?i ? (X ) X is closed X = i ? i ? (X ) X is regular open X [Y = Y X is part of Y X [ i(Y ) = i(Y ) X is part of the interior of Y X \Y = ; X and Y are disjoint i(X ) \ i(Y ) = ; The interiors of X and Y are disjoint X = Y [Z X is the union of Y and Z Table 1: Some constraints expressible as interior algebra equations. 3
a
a
b
b
DC(a,b)
EC(a,b)
a
b
TPP(a,b)
a
b
TPPi(a,b)
a b PO(a,b)
a b
a b EQ(a,b)
NTPP(a,b)
a
b
NTPPi(a,b)
Figure 1: Basic relations in the RCC theory
3 Region Connection Calculus The Region Connection Calculus (RCC) is an axiomatisation of certain spatial concepts and relations in classical 1st-order predicate calculus. The basic theory assumes just one primitive dyadic relation: C(x; y ) read as `x connects with y '. Individuals can be interpreted as denoting spatial regions. The relation C(x; y ) is re exive and symmetric, which is ensured by the following two axioms: RCC1) 8xC(x; x) RCC2) 8xy[C(x; y) ! C(y; x)] We also require C to be extensional:1 RCC3) 8xy[8z[C(z; x) $ C(z; y)] ! x = y] Using C(x; y ) a very large class of intuitively signi cant relations can be de ned. Some of the most useful of these are illustrated in Figure 1 and their de nitions are given in Table 2. The non-symmetrical relations P, PP, TPP and NTPP have inverses which we write as Ri, where R 2 fP; PP; TPP; NTPPg. These relations are de ned by de nitions of the form Ri(x; y) def R(y; x). Of the de ned relations, DC,EC,PO,EQ,TPP,NTPP, TPPi and NTPPi have been proven to form a jointly exhaustive and pairwise disjoint set, which is known as RCC-8. RCC also incorporates a constant denoting the universal region, a sum function and partial functions giving the product of any two overlapping regions and the complement of every region except the universe. These are de ned as follows: RCCD1) x = U def 8y[C(x; y)] RCCD2) x = y + z def 8w[C(w; x) $ [C(w; y) _ C(w; z)]] RCCD3) Prod(x; y; z) def 8u[C(u; z) $ 9v[P(v; x) ^ P(v; y) ^ C(u; v)]] RCCD4) Compl(x; y) def 8z[(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 from
the de nition of the sum operator given in that paper.
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Relation DC(x; y ) P(x; y ) PP(x; y ) EQ(x; y ) O(x; y ) DR(x; y ) PO(x; y ) EC(x; y ) TPP(x; y )
Interpretation x is DisConnected from y x is a Part of y x is a Proper Part of y x is EQual to y x Overlaps y x is DiscRete from y x Partially Overlaps y x is Externally Connected to y x is a Tangential Proper Part of y
De nition :C(x; y) 8z[C(z; x) ! C(z; y)] P(x; y ) ^ :P(y; x) P(x; y ) ^ P(y; x) 9z[P(z; x) ^ P(z; y)] :O(x; y) O(x; y ) ^ :P(x; y ) ^ :P(y; x) C(x; y ) ^ :O(x; y ) PP(x; y ) ^ 9z[EC(z; x) ^ EC(z; y)] NTPP(x; y ) x is a Non-Tang'l Proper Part of y PP(x; y ) ^ :9z[EC(z; x) ^ EC(z; y)]
Table 2: Some signi cant relations de nable within the RCC theory. This means we do not have a full Boolean algebra of regions; but, in order that appropriate regions exist to ful l the requirements of the quasi-Boolean structure suggested by the above de nitions, the basic RCC theory is supplemented with the following existential axioms:2 RCC4) 9x8y[C(x; y)] RCC5) 8xy9z8w[C(z; w) $ [C(w; x) _ C(w; y)]] RCC6) 8xy[O(x; y) ! 9z[Prod(x; y; z)] RCC7) 8x[:(x = U ) $ 9y[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 terms of classical point-set topology. In fact there are two dual interpretation that are equally reasonable. Closed Interpretation: A region is identi ed 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 identi ed 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 the Open Interpretation of RCC (given above) we see that the the relations O and C be formally 2 Arguably, the axioms for U and + are redundant because (in standard 1st-order logic) denotations of
constants and functional terms always exist.
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Interior Algebra Equation RCC Relation Open Interpretation Closed Interpretation DC(x; y ) i( X ) [ i( Y ) = U X \Y = U DR(x; y ) X \Y = U i(X ) \ i(Y ) = U P(x; y ) X[Y =U X [Y = U Pi(x; y ) X[Y =U X [Y = U NTP(x; y ) i( X ) [ Y = U X [ i(Y ) = U NTPi(x; y ) X [ i( Y ) = U i(X ) [ Y = U EQ(x; y ) (X [ Y ) \ (X [ Y ) = U (X [ Y ) \ (X [ Y ) = U
Table 3: Seven relations de ned by interior algebra equations de ned in terms of point sets by: O(x; y ) def 9 [ 2 X ^ 2 Y ] C(x; y ) def 9 [ 2 c(X ) ^ 2 c(Y )] However, these de nitions make use of a highly expressive set-theoretic language, including both quanti cation and the element relation. Given that the relations are intuitively very simple, one may wonder whether it is possible to give an alternative characterisation of C and O in the much less expressive language of interior algebraic equations. As it happens the negations of each of these relations can be quite easily de ned 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 de ned as interior algebraic equations. This follows from the general observation that purely equational constraints are always consistent with any purely equational theory (there must always be at least a trivial one-element model, in which all constants denote the same individual). Thus if the negation of some constraint can be expressed as an equation, then the constraint itself cannot be equationally expressible (otherwise that constraint would be consistent with its own negation). To de ne C and O we would need both interior algebraic equations and the negations of such equations, however many useful relations can be speci ed with equations alone. Table 3 gives de nitions in interior algebra of seven binary relations: DC, DR, P, Pi, NTP, NTPi and EQ. Equations corresponding to both the open and closed interpretations of RCC are shown. This set, which will be called RCC-7, is of particular signi cance because, as will be shown below, each of the RCC-8 relations can be expressed as a conjunction of positive and negative RCC-7 relations. Note that RCC-7 is neither jointly exhaustive nor pairwise disjoint: 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 number of 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 speci ed by a conjunction of interior algebraic 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. DC( ) EC( ) PO( ) TPP( ) TPPi( ) NTPP( ) NTPPi( ) EQ( ) x; y
x; y
x; y
x; y
x; y
x; y
x; y
x; y
Equivalent Algebraic Constraint(s) | ( ( ) [ ( ) = U) DR( ) ^ :DC( ) ( \ = U ) ^ ( ( ) [ ( ) 6= U ) :DR( ) ^ :P( ) ^ :Pi( ) ( \ 6= U ) ^ ( [ 6= U ) ^ ( [ 6= U ) P( ) ^ :EQ( ) ^ :NTPP( ) ( [ = U ) ^ ( 6= ) ^ ( ( ) [ 6= U ) Pi( ) ^ :EQ( ) ^ :NTPPi( ) ( [ = U ) ^ ( 6= ) ^ ( [ ( ) 6= U ) | ( ( ) [ = U) | ( [ ( ) = U) | ( = ) x; y
x; y
x; y
x; y
x; y
x; y
x; y
x; y
x
x
x; y
x; y
x; y
y
y
x
i x
i y
i x
i y
y
x
x
y
x
y
i x
x
y
x
y
x
y
y
i y
i x
x
y
i y
x
y
To accord fully with the topological interpretations of RCC, the equational representation must also constrain the regions to be regular open or closed depending on which of the two interpretations it used. In the rst case one should add an equation X = i ? i ? (X ) for each region variable X ; and in the second case one should add X = ?i ? i(X ).
4 The Modal Logic S 4 S 4 is one of the simpler and better known modal logics. It may also be called KT 4 since it
is obtained from classical propositional logic by adding the the rule of necessitation and the following axiom schemas: K. ( ! ) ! ( ! )
T. ! 4. !
A modal logic which satis es 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 semantics. In a Kripke semantics a model consists of a set of possible worlds together with an accessibility relation | a binary relation between worlds | associated with each modal operator. Propositions denote sets of possible worlds (the set of worlds in which they are true). A Kripke model, M, is thus a structure hW; R; P; di, where W is a set of worlds, R is the accessibility relation, P is a set of constants, fpig, and d is a function mapping elements of P to subsets of W . Such a model determines the truth of each modal formula at each possible world. Classical formulae are interpreted as follows:
Atomic formulae, pi are true in exactly the worlds in the set d(pi). Conjunctions, ^ , are true in worlds where both and are true. Disjunctions, _ , are true in worlds where either or (or both) is true. 7
Negations, : , are true in worlds where is not true. We write j=M to mean that formula is true at world in model M. A modal operator, , is then interpreted as follows: in a model M = hW; R; P; di
j=M
i
j=M for all 2 W s.t. R( ; )
A frame is a set of all Kripke models satisfying some speci cation of the properties of the accessibility relation, R. For example, the set of all Kripke models in which R is re exive and symmetric constitutes a frame. Finally we say that a formula is valid in some frame, F , if it is true at every world in every model in F . The logic S 4 is characterised by the frame, FS 4 , consisting of all Kripke models whose accessibility relations are re exive and transitive (R is a quasi-ordering on W ). Every theorem provable according to the proof system for S 4 speci ed above is valid in FS 4 ; and conversely every formula valid in FS 4 is provable in the proof system. A vast spectrum of di erent modal operators can be speci ed by placing more or less general restrictions on the corresponding accessibility relation. Furthermore, Kripke semantics allows one to specify operators whose logic seems to correspond well with intuitive properties of modal concepts employed in natural language. Indeed, a number of logics proposed for natural language modalities, which were originally speci ed proof theoretically (by axiom schemata intended to capture intuitive properties of modal concepts) can be captured very easily within the Kripke paradigm by quite simple restrictions on the accessibility relation. Whilst the Kripke approach certainly provides a very exible approach to modal semantics, its generality is often overstated. Consequently, many researchers in both AI and philosophical logic tend to think of possible worlds semantics as essentially based upon accessibility relations. However, although Kripke models may be appropriate for certain types of modal operator, in other cases it may be more natural to suppose a quite di erent structuring of possible 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 which is more general than a Kripke model. Just as the formulae of classical propositional logic can be interpreted as referring to elements of a Boolean algebra, modal formulae can be interpreted as elements of a Boolean algebra supplemented with an additional unary operation obeying certain constraints. This is a modal algebra. Boolean algebras with additional operators were rst studied in detail by Jonsson and Tarski (1951). Their connection to modal logics was investigated by Lemmon (1966a, 1966b). A clear account of the essential properties of modal 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 = hS; +; ?; i, where hS; +; ?i is a Boolean algebra and, for all elements x and y of the algebra, the operator `' satis es the equation (x + y ) = x + y (add) Operators obeying this equation are known as additive.3 The maximal and minimal elements of the algebra will respectively be denoted 1 and 0. 3 It 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 structures hS; ; ?; i, where is the Boolean product. corresponds, of course, to the modal operator and satis es the additivity condition in the form: (x y ) = x + y (add)
4.3 Algebraic Models
We can now de ne an algebraic model for a modal language as a structure hS; ; ?; ; P; i, where hS; ; ?; i is a modal algebra, P is the set of constants of the language and is a function mapping modal formulae to elements of S . For each constant p 2 P , [p] may be any element of S . This assignment to the constants determines the value [ ] of all complex formulae according to the following recursive speci cation:4 [ ^ ] = [ ] [ ] [: ] = ?[ ] [ ] = ([ ]) The algebraic equation characterising additivity corresponds to the modal schema ( ^ ) $ ( ^ ) ; which is true in every normal modal logic. We say that a formula, , is universal in a model hS; ; ?; ; P; i i [ ] = 1 | i.e. if the model assigns to the formula the unit (universal) element of the modal algebra hS; ; ?; i. An algebraic frame, FE , is a set of all algebraic models whose algebras satisfy some set of equations, E , constraining the `' operator. Finally we say that a formula is valid with respect to 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 nd a set of characteristic equations, EL such that a formula is valid in the frame FEL if and only if it is a theorem of L. For brevity I shall denote the frame associated with the logic L by FL , rather than FEL . For instance, the frame FS 4 is the set of all models satisfying the equations: x x (i.e. x x = x) (epis)5 (norm) (idem) It is known that a formula is valid with respect to FS 4 i it is a theorem of the logic S 4 (Hughes and Cresswell 1968, Chapter 17). Note that, if $ is a theorem of some logic L, then and must have the same denotation in every algebra in FL . Thus, since is interpreted as an extensional algebraic function, $ must be a theorem of L. Hence, any modal logic which can be given an algebraic semantics will be closed under the rule of equivalence: if ` $ then ` $ , which I shall refer to as RE. 1 = 1 ((x)) = (x)
4 Speci cations for the connectives _ , ! , $ and can easily be derived from their de nitions in terms of :, ^ and . 5 So called because it is required for an epistemological interpretation of the modality, since if `I know ' then 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 isomorphic to a Boolean algebra whose elements are sets and whose operators are identi ed with the usual union, intersection and complementation operations of elementary set theory. Moreover, such an algebra can always be embedded in a Boolean algebra whose elements are all the subsets of some (universal) set W . Jonsson and Tarski (1951) showed that a similar theorem holds for Boolean algebras with additional additive operators. This means that every modal algebra can be isomorphically embedded in a modal algebra whose elements are all members of the power set, 2W, of some set, 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, , in the modal algebra, may be regarded 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 hU; \; ?; ; P; i, where the product operator is `\' to indicate that the Boolean operators correspond to the operators of elementary set theory. The elements of the algebra are now the members of the power set of U and the maximal element, 1, of a power-set algebra is equal to U itself. A modal operator, , in a power-set algebra, maps every subset, X , of U to another subset (X ). The power-set algebras are representative of the whole class of modal algebras in the sense 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 in characterising validity in terms of algebraic frames we can restrict the frames to contain only models based on power-set algebras. In the sequel I shall assume that we always consider only models based on power-sets and I shall refer to the resulting semantics as algebraic set semantics.
4.5 Mapping Between Algebraic and Logical Expressions
As with the classical set-semantics it will be useful to introduce meta-level notation for referring to the mapping between modal formulae and modal algebraic terms. I assume that these terms are interpreted as sets in a power-set algebra. Thus MAT[ ] is the modal algebraic term obtained from the formulae by replacing the connectives :, _ , ^ and by the operators ?, [, \ 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 function MF is the inverse of MAT so that MF[ ] is the formulae such that MAT[ ] = . I shall write MF MAT to refer to the mapping in the form of a relation. I also de ne the transform MFe[ ], such that MFe[ = 1] = MF[ ] for universal equations and MFe[ 1 = 2 ] = MF[ 1] $ MF[ 2], for non-universal equations. The expression MFe[ ] refers to a modal formula which (because of the correspondence theorem, Mcorr, which will be given in section 4.7) may be regarded as representative of the modal algebraic equation . However, because of the form of the entailment correspondence theorem, S4ECT, also proved in section 4.7, one might say that an equation constraining an S 4 modal algebra is better represented by MFe[ ] rather than MFe[ ]. Equations characterising a class of algebraic structures (a frame) will in general contain free variables which are taken as implicitly universally quanti ed | the equations hold for 6 A 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 class of modal formulae, which can be represented as a formula schema. Because of this, it is convenient to generalise MF so as to operate on terms with free variables. In such a case the resulting expression will be a modal schema rather than a formula and schematic logical variables will take the place of the free variables in the algebraic term. Accordingly, MFe can also be allowed to operate on equations containing free variables | again the result will be a schema rather than a formula. By means of MFe, a set of algebraic equations de ning a frame FL can be translated directly into a set of modal schemas which specify the proof system of the corresponding logic L. To ensure the proof system is complete it will also be necessary to add the inference rule RE 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 identi ed with elements in an algebra, then equations between algebraic terms can be used to specify relationships between these entities. One can then reason about these relations in terms of entailments among algebraic equations. Since set algebras are representative of the class of modal algebras the notion of entailment among modal algebraic equations can be de ned in terms of possible set assignments to a language of modal algebraic terms. A set assignment to a language of algebraic terms is a structure = hS; U; ; mi, where: S is a set of constants, U is the universal set, : 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 speci es the modal operator as a set function. If is a term built from the constants in S by means of Boolean and modal operators, then [ ] is the set assigned to by . This is determined by , m and the usual interpretation of Boolean operations on sets. If [ 1] = [ 2] we say that satis es the equation 1 = 2 . Entailment relations among modal-algebraic equations can now be speci ed as follows: 1 = i; : : : ; n = n j=MAL 0 = 0 means that, for every assignment = hS; U; ; mi (where S includes all the constants occurring in the terms i and i ) satisfying the equations associated with the frame FL , if satis es the equations 1 = 1; : : : ; n = n it also satis es the equation 0 = 0.
4.7 Relating S 4 Modal-Algebraic Entailment to Deducibility
If a modal logic L is characterised by a modal algebraic frame FL there is a correspondence between deduction in the logic and entailment between algebraic equations in the algebras in FL. Because of this we can use modal logics to reason about algebraic equations. From the de nition of an algebraic frame for the logic L we have the following correspondence between universal set equations and logical theorems: j=MAL = 1 i `L ; where MF MAT (Mcorr) More generally, the following correspondence between the entailment relation among universal set equations constraining algebras in FS 4 and the deducibility relation of S 4 can be proved: 11
S4
Entailment Correspondence Theorem:
(S4ECT) 1; : : : ; n `S4 0 where i MF MAT i This is the main theorem justifying the use of S 4 for topological reasoning (Bennett 1997).
1 = 1; : : : ; n = 1 j=MAS4 0 = 1
i
Proof of S4ECT:
Since S 4 is an extension of classical logic it obeys the deduction theorem: 1 ; : : : ; n `S 4 0 i `S4 ( 1 ^ : : : ^ n ) ! 0 . By combining this with Mcorr we get the more general correspondence
1; : : : ; n `S4 0 Hence
j=MAS4 ( 1 \ : : : \ n) [ 0 = 1 :
i
1; : : : ; n `S4 0
i j=MAS4 ( 1 \ : : : \ n) [ 0 = 1 : Because of the additivity of the equation on the r.h.s. is equivalent to ( 1 \ : : : \ n ) 0, so we can establish S4ECT by showing that
j=MAS4 ( 1 \ : : : \ n ) 0
1 = U ; : : : ; n = 1 j=MAS4 0 = 1 :
i
The r.h.s. can then be re-written to give
j=MAS4 ( 1 \ : : : \ n) 0
i
( 1 \ : : : \ n ) = 1 j=MAS4 0 = 1 and this equivalence can be more succinctly expressed as
j=MAS4 ( ) 0
i
= 1 j=MAS4 0 = 1
(y) :
Clearly the left to right direction of (y) must hold for any modal algebra, which satis es
1 = 1 (i.e. any normal modal algebra) and hence any algebra in FS 4.
The right to left direction of (y) is harder to show. I prove it by proving the contrapositive | i.e.: (yy) if 6j=MAS4 ( ) 0 then = 1 6j=MAS4 0 = 1 Let S be the set of all constants occurring in the terms and 0 . If the antecedent of (yy) is true, there must be some assignment = hS; U; ; mi satisfying the equational constraints of the frame FS 4 and such that [( )] 6 [ 0]. From we can construct an assignment, 0, which veri es the consequent of (yy) | i.e. 0[ ] = 1 but 0[ 0] 6= 1: Let U 0 = [( )]. Note that U 0 is an open subset of U . We now de ne 0 = hS; U 0; 0; m0i by stipulating that: 0[] = [] \ U 0, for all constants, 2 S , m0(X ) = m(X ) for all sets X U 0.
12
It can then be shown that for any term, (made up of constants in S ), 0[ ] = [ ] \ U 0. We know this identity holds for atomic terms because of the de nition of 0, so to show it inductively for all terms we need to show that, if it holds for and , it must hold for , [ , \ and . For the Boolean operators the required identities are demonstrated by the following sequences of equations: 0[ ] = U 0 ? 0[ ] = U 0 ? ([ ] \ U 0 ) = U 0 ? [ ] = (U ? [ ]) \ U 0 = [ ] \ U 0 0 [ [ ] = 0 [ ] [ 0[ ] = ([ ] \ U 0 ) [ ([ ] \ U 0 ) = ([ ] [ [ ]) \ U 0 = [ [ ] \ U 0 0 [ \ ] = 0 [ ] \ 0[ ] = ([ ] \ U 0 ) \ ([ ] \ U 0 ) = ([ ] \ [ ]) \ U 0 = [ \ ] \ U 0 (In the rst of these the identity U 0 ? [ ] = (U ? [ ]) \ U 0 depends on the fact that U 0 U .) For the case of we have: 0 [ ] = m0(0 [ ]) = m(0[ ]) = m([ ] \ U 0) = m([ ]) \ m(U 0) = [ ]) \ U 0 We must verify that the algebra speci ed by 0 is a member of FS 4. I have established that for every term (built from constants in S ) 0 [ ] = [ ] \ U 0. This means that every equation, 1 = 2, satis ed by will also be satis ed by 0. Since, by hypothesis, must satisfy all the frame equations of FS 4, 0 must also satisfy these frame equations. To complete the proof I must show that 0 veri es the r.h.s. of (yy). Since the algebra generated by 0 is in FS 4 , it must satisfy epis, which means that for any term, , 0 [ ] 0 [ ]. We know that 0 [ ] = U 0, so 0 [ ] U 0 ; but 0[ ] = [ ] \ U 0, so 0 [ ] = U 0. Recall that was chosen to verify the antecedent of (yy) because [ ] 6 [ 0]. Thus, U 0 6 [ 0]; and from this it follows that [ 0] \ U 0 $ U 0. Hence we have 0[ 0 ] $ U 0. By means of the MFe meta-function, an arbitrary modal set equation can be directly transformed into universal form and the formula MFe[ ] can be regarded as representing the equational constraint . The modal logic S 4 can thus be used to reason about arbitrary equations constraining algebras in the frame FS 4 according to the following generalisation of S4ECT:
MFe[1]; : : : ; MFe[1] `S4 MFe[0] : The form of S4ECT is a bit awkward in that in the S 4 deduction corresponding to an entailment between equations, we need to add an extra operator to the formulae on the left of `S 4 but not to the formula on the right.7 This means that the question \What is the S 4 representation of the equation ?" does not have a simple answer. However, it is easily shown that a sequent 1 ; : : : ; n `S 4 0 is in fact valid if and only if 1 ; : : : ; n `S 4 0. Thus, for the purpose of testing entailments, it can be said that the representation of an equation is MFe[ ]. 1; : : : ; n j=MAS4 0
i
7 Note that if we do not add s as required the correspondence fails. For example x = 1 j= MAS4 x = 1 but X 6`S4 X .
13
5 Encoding Topological Relations in S4 It was established by Tarski and McKinsey (1948) that the S 4 box operator can be modelled algebraically by an interior operator. We have seen that the algebraic semantics for S 4 satis es the same equations as an interior algebra, so these algebras are equivalent. By making use of the meta-level notation relating modal algebraic equations and corresponding modal formulae we can also examine the relationship between closure/modal algebraic equations and modal formulae. The representation of a closure/modal algebraic equation in modal logic is the formula MFe[ ]. Because the equations specifying properties of the closure operation contain free variables they will be mapped to modal schemata rather than formulae. The characteristic equations of a closure algebra and corresponding modal schemata are given in Table 5. Interior Axioms
Modal Schemata
i(X ) [ X = X ( _ ) $ i(i(X )) = i(X ) $ i(U ) = U > i(X \ Y ) = i(X ) \ i(Y ) ( ^ ) $ ( ^ )
(T') (4+) (N) (R)
Table 5: Interior Axioms and Corresponding Modal Schemata Clearly T' is equivalent to the schema T, ! and, given that T holds, 4+ can be weakened to ! , which is the schema 4. Furthermore it is well known that the schemata N and R in conjunction with the rule RE are equivalent to the combination of schema K and the rule of necessitation, RN. Thus specifying that N, R and RE hold is an 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 logic derived from an interior or closure algebra by transforming equational algebraic constraints into modal schemata is exactly the logic S 4. Consequently, in virtue of the correspondence theorem S4ECT, deduction in S 4 can be used to reason about closure algebraic equations.
5.1 RCC Relations Representable in S 4
Since the S 4 modality can be interpreted as an interior function over a topological space, we can use this interpretation to encode topological relations as S 4 formulae. The basis of this representation is exactly the same as for the C representation but by use of the additional modal operator it is possible to make a distinction between connection and overlapping which cannot be expressed in C . Table 6 shows the S 4 formula corresponding to each of the RCC-7 relations according to the open set interpretation of RCC regions. The middle column shows the algebraic set-equation associated with the relation. We see that, if the interior operator i is identi ed with the corresponding modal algebra operator , then the interior algebraic equation , is represented by the S 4 formula MFe[ ]. I now illustrate how the correspondence theorem S4ECT, enables deduction in S 4 to be used to reason about entailment among certain RCC relations. Consider the following argument: NTP(a; b) ^ DR(b; c) j= DC(a; c) 14
RCC Relation Interior Algebra Equation ( ) S 4 formula ( MFe[ ]) 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 de ned by interior algebra equations and corresponding S 4 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 ) j= i( A ) [ i( C ) = U : Here the equations of the form = i( ) 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 S 4 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 is still not possible to specify the relation of external connection, EC(x; y ). We have also seen
that (although their negations can be represented) the fundamental relations C and O cannot be represented. In order to overcome these de ciencies we need a language in which one can express closure-algebraic inequalities as well as equalities.
6 Negative Equations and their S 4 Representation In order to represent the complete set of RCC-8 relations we need to be able to represent not only 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 both positive and negative equational constraints to the problem of testing entailment for positive equations:
Consistency of Equational Literals (ELcons)
1 = 1 ; : : : ; m = m ; :(1 = 1); : : : ; :(n = n) j= i
1 = 1 ; : : : ; m = m j= i = i for some i 2 f1; : : :ng
ELcons can be established by considering possible proofs of inconsistency in some proof system for 1st-order logic with equality, which is known to be refutation complete. One such system, is that where the only proof rules are binary resolution, paramodulation and factoring 8 In general, to be faithful to RCC, one should ensure that regions are regular open by adding the stronger
constraint = i ? i ? ( ); but the inference in this example is valid for any open regions.
15
(Du y 1991). Since we are dealing with sets of literals (i.e. only unit clauses), factoring is not required and a simpli ed version of paramodulation can be employed. The details of the rules that are used do not matter, since ELcons can be demonstrated from quite general observations. The proof is as follows: Proof of ELcons: Suppose we refute a set of equational literals by means of binary resolution and paramodulation. Once an application of binary resolution can be made, inconsistency is proved immediately; so any successful refutation must consist of a series of paramodulations followed by a single binary resolution. Note also that each paramodulation either involves two positive literals and generates a new positive literal or it involves a positive and a negative literal and generates a new negative literal. These observations enable us to show that any refutation makes essential use of exactly one negative literal. The key points are that the derivation of a positive literal cannot involve any negative literals and that no rule operates on more than one negative literal. Consider the nal step in the refutation; this is a resolution between a positive and a negative literal. The positive literal is either in the original set of literals or has been derived by a sequence of paramodulations involving only positive literals. The negative literal is either in the original set or has been generated from a positive and a negative literal. In the latter case, the positive literal must have been derived from only positive literals and the negative literal 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 the original set is involved in the proof.
7 The Extended Modal Logic, S 4+ In order to increase the expressive power of S 4, so that we can represent both positive and negative algebraic constraints we can de ne an augmented representation language, S 4+, whose expressions are pairs hM; Ei, where M and E are formulae of S 4. These formula sets are called respectively model and entailment constraints. The reason for this terminology is that the model constraints can be regarded a constraining possible topological models whereas entailment constraints forbid certain entailments from the model constraints. Hence, we stipulate that: An S 4+ expression hM; Ei is consistent if and only if no formula in E is entailed by the set M. In virtue of ELcons this accords with an interpretation under which the formulae in M are identi ed with corresponding algebraic equations, where as the formulae in E are identi ed with the negations of such equations.
7.1 Determining Entailments
Computing inconsistency of S 4+ expressions is a special case of determining entailments between situation descriptions characterisable in S 4. To refer to such an entailment, I shall use the notation hM; Ei j=S 4+ hM0; E 0i. We can express the meaning of this as an entailment between set-equations as follows:
m1 = U ^ : : : ^ mh = U ^ e1 6= U ^ : : : ^ ei 6= U j= m01 = U ^ : : : ^ m0j = U ^ e01 6= U ^ : : : ^ e0k 6= U 16
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 ^ (m01 = 6 U _: : : _ m0j 6= U _ e01 = U _: : : _ e0k = U ) j= : To show the validity of this we must show that whichever of the equations in the disjunction is chosen the resulting equation set is inconsistent. This is equivalent to showing that: for all p 2 M0 we have hM; E [ fpgi j=S 4+
and for all q 2 E 0 we have hM [ fq g; Ei j=S 4+
Another equivalent way of expressing these which is more convenient from the point of view of actually calculating the entailments is the following: S 4+
Entailment Theorem (S4+ET)
hM; Ei j=S4+ hM0; E 0i i either hM; Ei j=S 4+ or ( for all 2 M0 : hM; f gi j=S 4+ and for all 2 E 0 : hM [ f g; Ei j=S 4+ ) Informally, this means that a sequent is valid i : either hM; Ei is itself inconsistent; or, each of the model constraints in M0 is entailed by the model constraints M and also each of the entailment constraints in E 0 in conjunction with the model constraints M entails one of the entailment constraints in E . Determining the validity of a S 4+ entailment has thus been reduced to determining the inconsistency of certain S 4+ expressions and we already know that such an expression is inconsistent i one of its entailment constraints is entailed by its model constraints.
8 Representing RCC Relations in S 4+ Since S 4+ can represent both equations and inequalities between terms made up of Boolean operations 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 conjunction of positive and negative RCC-7 relations. The representations of the RCC-8 relations under the open set interpretation are given in table 7. The alternative closed set representation is given below in table 8. The way they are obtained can be summarised as follows: express the RCC-8 relations in terms of RCC-7 relations and interpret these as equational constraints on interior algebras as given in table 4. Then translate these constraints into S 4 according to table 6. The formulae corresponding to positive RCC-7 relations become model constraints in the S 4+ representation and those corresponding to negated RCC-7 relations become entailment constraints. Note that the S 4+ correspondence theorem requires that model constraints have an extra initial added to the result of applying MFe to the modal algebraic equation but this is not required in the entailment constraints. This asymmetry stems from S4ECT. Let us now consider how the S 4+ representation can be used to test the consistency of a simple set of spatial relations. Take for example the following conjunction of RCC-8 relations: TPP(a; b) ^ DC(b; c) ^ PO(a; c) :
17
Relation Model Constraint Entailment Constraints DC(x; y ) :x; :y ( :x _ :y) EC(x; y ) :(x ^ y) :x _ :y; :x; :y PO(x; y ) | :(x ^ y); x ! y; y ! x; :x; :y TPP(x; y ) (x ! y) :x _ y; y ! x; :x; :y TPPi(x; y ) (y ! x) :y _ x; x ! y; :x; :y NTPP(x; y ) y ! x; :x; :y ( :x _ y) NTPPi(x; y ) x ! y; :x; :y ( :y _ x) EQ(x; y ) :x; :y ( x $ y ) C(x; y ) | :x _ :y :x; :y EQ(x; sum(y; z )) (x $ (y _ z )) :x; :y RegOpen (x) x $ x |
Table 7: The S 4+ encoding of some RCC relations (open set interpretation) Translating into S 4+ according to table 7 we get the following representation: hf(a ! b); ( :b _ :c)g; f :a _ b; b ! a; :(a ^ c); a ! c; c ! a; :a; :b; :cgi This is an ordered pair consisting of two sets of S 4 formulae, the rst set being model constraints and the second entailment constraints. Appealing to part 3 of S4+CT we determine that 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 Constraints DC(x; y ) :x; :y :(x ^ y) EC(x; y ) : ( x ^ y); :x; :y : ( x ^ y ) PO(x; y ) | :( x ^ y); x ! y; y ! x; :x; :y TPP(x; y ) x ! y; y ! x; :x; :y (x ! y) TPPi(x; y ) y ! x; x ! y; :x; :y (y ! x) NTPP(x; y ) y ! x; :x; :y (x ! y) ( y ! x ) NTPPi(x; y ) x ! y; :x; :y EQ(x; y ) :x; :y (x $ y) C(x; y ) | :(x ^ y); :x; :y EQ(x; sum(y; z )) (x $ (y _ z )) :x; :y RegClosed(x) x $ x |
Table 8: S 4+ 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. 18
If our modal encoding is to agree fully with the intended meaning of RCC regions we also need to enforce the condition that the regions are regular. Thus in the open set interpretation we should add to M additional formulae x $ x for each variable X occurring in the set of RCC relations we are encoding. Under the closed set interpretation we would similarly add formulae of the form x $ x It must be noted that these conditions are not general schemata such that every instance must be true. They only apply to formulae which can be directly 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 is represented by the formula (y _ z ).
9 Eliminating Entailment Constraints by Using S4u In reasoning with the extended 0-order language S 4+ the meanings of the two types of constraint are handled at the meta-level: determining entailments in these languages involves checking a number of di erent object-level entailments in the logic S 4. A set of algebraic constraints encoded in an S 4+ expression hM; Ei is consistent if and only if none of its entailment constraints in E is entailed by the set of all model constraints in M. A natural question regarding this representation is whether it might be possible to extend the S 4 language itself so 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 object level. In terms of algebraic semantics it is quite easy to introduce a new modal operator 8 by means of which the model/entailment constraint distinction can be made at the object level. If ( ) is the algebraic denotation of a formula , we de ne 8 by: (8 ) = U i ( ) = U . (8 ) = ; i ( ) 6= U . A dual operator 9 is de ned by 9 def :8: . Considered on its own, 8 can be regarded as an S 5 modal operator. This means it satis es all the schemata obeyed by an S 4 modality and in addition the schema
98 ! 8 : Our intended interpretation of the new 8 operator is that 8 is true at every point/world in the model. However, the S 5 axioms permit Kripke models where accessibility is an equivalence relation rather than the universal relation. This means that 8 could be satis ed in a model where is not true at every world but just at all the worlds within some equivalence
class. Normally this is of little importance, since the truth of a formula can only depend on 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 S 4 operator, the accessibility relation of such an operator might join worlds in di erent equivalence classes. In this case certain formulae may be satis able only in models where the S 5 accessibility relation separates the set of worlds into more than one equivalence class. Consequently we cannot assume that 8 can be assumed to mean that it true at all possible worlds. To avoid this possibility and ensure the correct logical behaviour of the new 8 operator 19
we add to our bi-modal logic the schema
(CONNECT) This means that the S 4 accessibility relation cannot take us outside the S 5 equivalence class occupied by the actual world. It can then be shown that a formulae of this S 4=S 5 hybrid is consistent with the axiom schemata if and only if it is satis able in a model where the S 5 accessibility is the universal relation. The resulting logic is thus called S 4u . It is known to be decidable (Goranko and Passy 1992). Given the de nition of 8, we have :8 = U i 6= U . Thus, negations of universal set equations (and hence all equations) can be converted into positive equations. This obviates the need for entailment constraints, since a model constraint :8 has the same meaning as taken as an entailment constraint. More speci cally, the translation of an S 4+ expression
! 9 :
hf 1; : : : ; j g; f 1 : : : k gi into S 4u is the formula
8 1 ^ : : : ^ 8 j ^ :8 1 ^ : : : ^ :8 k : Consequently any expression of S 4+ can be represented by a simple object level formula in the multi-modal language S 4u. In the S 4u encoding a set of key RCC relations are represented as as follows: Relation Open set encoding Closed set encoding C(x; y ) 9( x ^ y) 9(x ^ y) O(x; y ) 9(x ^ y) 9( x ^ y) P(x; y ) 8(x ! y) 8(x ! y) TP(x; y ) 8(x ! y ) ^ 9( x ^ :y ) 8(x ! y ) ^ 9(x ^ :y ) NTP(x; y ) 8 ( x ! y ) 8(x ! y) NE(x) 9x 9x Regular(x) 8(x $ x) 8(x $ x)
Table 9: S 4u encodings of RCC relations All the RCC-8 relations can be expressed by conjunctions of these relations and their negations. Note that we can now explicitly represent relations such as C and O as well as their negations DC and DR. The non-emptiness condition (NE(x)) is also now explicitly expressed by the simple formula 9x.
9.1 An Example Entailment Encoded in S 4u
Let us look at a simple example of spatial reasoning carried out in S 4+. We shall consider the transitivity of the proper part relation, PP: PP(a; b) ^ PP(b; c) j= 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: 8(a ! b) ^ :8(b ! a) ^ 9a ^ 9b 20
Hence the transitivity of PP corresponds to the entailment:
8(a ! b) ^ :8(b ! a); 8(b ! c) ^ :8(c ! b); 9a; 9b; 9c j= 8(a ! c) ^ :8(c ! a) ^ 9a ^ 9c 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 i each of the four sequents with the same l.h.s. but just one conjunct on the r.h.s. is valid. Of these four sequents, the two with 9a and 9c on the r.h.s. are trivially valid because these formulae also occur on the l.h.s.. To prove the validity of the other two, it is convenient to move all conjuncts on the l.h.s. which have an initial 9 operator over to the right and then to replace :9 pre xes by 8:. We shall then have the following two sequents:
8(a ! b) ^ 8(b ! c) j= 8(a ! c) _ 8(b ! a) _ 8(c ! b) _ 8:a _ 8:b _ 8:c 8(a ! b) ^ 8(b ! c) ^ 8(c ! a) j= 8(b ! a) _ 8(c ! b) _ 8:a _ 8:b _ 8:c We can verify these proof-theoretically by the application of just one modal rule (together with ordinary classical reasoning). This is the rule RK which holds in any normal modal logic: ( 1 ^ : : : ^ n ) ! [RK] (8 1 ^ : : : ^ 8 n ) ! 8 This rule together with the deduction theorem means that if
1 ; : : : ; n j=
then
8 1; : : : ; 8 n j= 8
Application of this principle validates both of our sequents, since
a ! b; b ! c j= a ! c
a ! b; b ! c; c ! a j= b ! a:
and
10 Intuitionistic Encoding Prior to the topological interpretation of S 4, Tarski (1938) had already showed that the intuitionistic propositional calculus9 (henceforth I ) can be given an interpretation in which each propositional letter, p, corresponds to an open set, P , within a topological space and the connectives are associated with Boolean and topological operations as given in Table 10. Thus, each formula of I corresponds to an interior algebraic term, ( ). Tarski showed that is a theorem of I just in case it corresponds to a term which denotes the universal set, U , in every topological model (i.e. for every assignment of open sets in a topological space to the constants occurring in ( ), we nd that ( ) = U ). More generally 0 can be intuitionistically deduced from a set of formulae f i g i every topological model satisfying the equations ( i ) = U also satis es ( 0 ) = U (see (Bennett 1994)). 9 For an introduction to intuitionistic logic see e.g. (Nerode 1990).
21
p^q p_q p p)q
P \Q P [Q i( P ) i( P [ Q )
Table 10: Topological interpretation of I Table 11: Representation of the RCC-7 relations in I RCC DC(x; y ) DR(x; y ) P(x; y ) Pi(x; y ) NTPP(x; y ) NTPPi(x; y ) EQ(x; y )
Algebraic Constraint i(x) [ i(y) = U x\y = U x[y = U x[y = U i(x) [ y = U x [ i(y) = U x=y
I formula x _ y (x ^ y) x)y y)x x _ y x _ y x,y
Bennett (1994) used this correspondence to construct a topological reasoning algorithm based upon an intuitionistic theorem prover. Table 11 shows how each of the RCC-7 relations can 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 was employed in the construction of S 4+ . Consider, for example, the con guration P(a; b) ^ DC(b; c) ^ PO(a; c). P and DC correspond directly to I formulae as given in Table 11. PO must rst be analysed as :DR ^ :P ^ :Pi as shown in Table 4. The (positive) model constraints correspond to the formula set fa ) b; b _ cg and the (negative) entailment constraints (including the non-null constraints) to fa ) c; c ) a; (a ^ c); a; b; cg. These constraints are inconsistent because a ) b; b _ c `I (a ^ c) | i.e. one of the entailment constraints in entailed by the model constraints.
11 A Non-Modal Encoding
A spatial interpretation of a Kripke models for S4u is obtained by identifying possible worlds with spatial points. Thus a formula of the form 8 ensures that the topological condition encoded by holds at every point in space. Similarly, 9 means that there is some point p satisfying the condition represented by . p can be thought of as a sample point, which bears witness to some topological constraint. For instance, where two regions x and y overlap, the corresponding modal formula 9((x) ^ (y )) ensures the existence of a point which is in the interior of both x and y . The accessibility relation for the S 4 operator may be thought of as the relation of two points being `arbitrarily close'. Thus, the formula x is true at a point p if region x occupies 10 Note 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.
22
point p and also all points arbitrarily close to p. This means that p must be an interior point of x. Consider, for example, a boundary point p of a regular (closed) region x. This fails to satisfy the left hand disjunct, (:x), of the regularity condition, because there are arbitrarily close 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 point arbitrarily close to q is occupied by x. The complexity results of Renz and Nebel (1999) show that to reason with only the RCC-8 relations, the full power of the S 4 encoding is not required. Indeed, a much simpler representation can be used. We note that, when we are dealing with regular regions, every point is either an interior, exterior or boundary point of each region. We describe these states formally 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 by constraints imposed on the witness points. The following conditions introduce witness points: NE(x) O(x; y ) :P(x; y) C(x; y )
9p [I(p; x)] 9p [I(p; x) ^ I(p; y)] 9p [I(p; x) ^ E(p; y)] 9p [(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 ) NTP(x; y ) DR(x; y ) P(x; y )
8p [E(p; x) _ E(p; y)] 8p [E(p; x) _ I(p; y)] 8p [E(p; x) _ E(p; y) _ (B(p; x) ^ B(p; y))] 8p [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 treatment of boundary points given in the next section.)
11.0.1 Boundary Points In constructing a model for a spatial con guration, boundary points have a special signi cance because each boundary point of a regular region is arbitrarily close to both its interior and its exterior. This means that if a region r has an interior point which coincides with a boundary point of s then r must also have interior points which coincide with both interior and exterior points of s. Similarly, if r has an exterior point coinciding with a boundary point of s then r must have exterior points coinciding with both interior and exterior points of s. Figure 2 illustrates how the di erent types of point are distributed at and near a boundary point. The middle diagram shows the case where regions r and s are externally tangent (we write 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 special con guration of three points, which we call a boundary cluster. This consists of a boundary
11 There might be cases where we would want to use further constraints (such as 9p [E(p; x) ^ E(p; y)], which holds i x and y do not jointly ll the entire space).
23
E(r)
B(r) I(r)
E(r) I(s)
A boundary point of r
B(r) I(r) B(s) E(s)
ET(r,s)
E(r) E(s)
B(r) I(r) B(s) I(s)
IT(r,s)
Figure 2: Boundary Points point pb and two `arbitrarily close' neighbouring points p1 and p2 (one on each side of the boundary). The existence of and topological constraints on these points are speci ed as follows: IT(x; y )
9pbp1p2 [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 ) 9pb p1 p2 [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, whenever it is in the interior of some region r the whole cluster is interior to r; and, if the boundary point is exterior to r, so is the whole cluster. Formally:
8r; x; y; z [(BC(x; y; z) ^ I(y; r)) ! (I(x; r) ^ I(z; r))] 8r; 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 their standard interpretation over regular closed point-sets)will satisfy the corresponding witness point 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 satis ability in the opposite direction: from the witness point representation corresponding to a set of RCC relations and satisfying the stipulated conditions on BC, we can construct a topological space together with an assignment of a subset of this space to each region name occurring in the RCC relations; this topological space satis es the RCC relations under the standard regular closed point-set interpretation. From the witness point model we construct a topological space hU; Oi. First we augment the 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 de niteness, whenever p and r are unrelated we can add E(p; r). Let U be the set of witness points. We now de ne the set O of open sets as the smallest set which is closed under union and intersection and is such that: U 2 O; for each region r, fp j I(p; r)g 2 O and fp j E(p; r)g 2 O. Within the space hU; Oi, each region r is identi ed with the (closed) set of witness points fp j I(p; r) _ B(p; r)g. 24
We need to ensure that regions and the closure operation have the correct interpretation. This means that for each region we must have fp j E(p; r)g is the largest open set disjoint from fp j I(p; r)g. This condition is violated just in case there is some region whose interior points include boundary points of r but no exterior points of r. But the constraints on boundary clusters ensure that if a boundary point of r is in the interior of s, then there are also both interior and exterior points of r which are in the interior of s. Similarly, if a boundary point of r is in the exterior of s, there are both interior and exterior points of r in the exterior of s. It is now easy to verify that: a) hU; Oi is a topological space; b) for each region r named in the situation description, the topological closure operation on fp j I(p; r)g is just fp j I(p; r) _ B(p; r)g.
11.0.3 A Model-Building Procedure In virtue of the representational adequacy just proved, we can use the witness point representation as the basis for a model-building procedure to test the consistency of sets of RCC relations. First we create data-structures representing the witness points and boundary clusters as required by the existential constraints on the witness point model. For each point and boundary cluster, we then attempt to satisfy the global constraints on witness point together with the special conditions on boundary clusters. This can be done using a straightforward backtracking search. It is worth noting that the satisfaction problems for each witness point and 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 concepts was to provide a formalism that is amenable to computational manipulation. Unfortunately I do not have time to cover complexity results in any detail so I shall merely summarise the most important results: The intuitionistic encoding of (Bennett 1994) showed that reasoning with a signi cant subset of RCC relations is decidable. (As we have seen, the encodable relations include all 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 is tractable and in the complexity class NC. A naive algorithm takes order n3 time, where n is the number of RCC-8 relations; but by the use of parallelism runtime can be reduced to O(n log n). (Nebel 1995) also shows that a set S of RCC-8 relations is consistent i it is path consistent | i.e. computing all compositions two relations in the set does not produce any new relation that is stronger than those already in the set; or in other words S is a xed point relative to compositional inferences. Tractable sublanguages of disjunctive constraints over the RCC-8 relations have been identi ed in (Renz and Nebel 1997b, Renz and Nebel 1999). (Renz 1998) investigates canonical models for satis able RCC-8 constraint networks. It shows that the set of modal formulae representing RCC-8 relations is satis able i it 25
has a model in which the S 4 accessibility relation has no chains of length greater than one. Dornheim (1998) proves that it is undecidable whether a given formula of 1st-order RCC language is true for regions in the plane. (A similar result, without the planarity restriction on the interpretation was sketched by (Gotts 1996) and is based on much earlier results of (Grzegorczyk 1951)).
13 Beyond Topology To conclude these notes I shall mention two recently developed logical systems which take us beyond the con nes of purely topological information. The rst of these is Region-Based Geometry which is a fully expressive geometrical theory in which regions, rather than points, are taken as the basic entities. Secondly I brie y mention the modal language PSTL which is a two-dimensional combination of the topological modal language S4u and the well-known temporal logic PTL.
13.1 Region-Based Geometry
For many applications, con ning spatial vocabulary to topological notions is too restrictive. We would like to employ a much wider range of concepts for describing shapes, relative sizes and 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 us an extremely expressive system which is essentially as expressive as the 1st-order language of polynomial constraints over reals numbers. However, from the point of view constructing a general purpose spatial ontology (within which one might hope to embed more restricted computationally-oriented representations) the possibility of constructing a fully expressive geometrical language based on regions is very attractive. Hence I sketch here the theory of Region 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 the axiomatisation we take the classical Mereology of Lesneiwski (Lesniewski 1927-1931) (see also (Tarski 1929, Woodger 1937, Simons 1987)): D1) PP(x; y) def (P(x; y) ^ :(x = y)) D2) DR(x; y) def :9z[P(z; x) ^ P(z; y)] D3) SUM( ; x) def 8y[y 2 ! P(y; x)] ^ :9z[P(z; x) ^ 8y[y 2 ! DR(y; z)]] In D3, is a 2nd-order variable, which can denote any subset of the domain of regions. x 2 is of course true just in case the denotation of x is a member of the set denoted by ; 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 is required to satisfy the following speci cally mereological postulates: A1) 8x8y8z[P(x; y) ^ P(y; z) ! P(x; z)] A2) 8 [9x[x 2 ] ! 9!x[SUM( ; x)]] 12 In fact the form x 2 could be written as (x), in the style of a 2nd-order language without set theory.
26
These ensure rstly that the part relation is transitive and secondly (and slightly controversially) that for any non-empty set of individuals there is a unique individual which is the sum of that set. We now develop a theory which we call Region-Based Geometry, inspired by Tarski's Geometry of Solids (Tarski 1929). Whereas Tarski's presentation is not formalised (and in some respects somewhat unclear) we shall give a fully formal axiomatisation. Following Tarski, 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 concepts are de ned and a set of postulates is given. We shall often want to quantify over just the spherical regions in the domain. For convenience we introduce the notations 8x[ ] def 8x[S(x) ! ] 9x[ ] def 9x[S(x) ^ ] As in (Tarski 1929) we de ne the relations of external tangency (ET), internal tangency (IT), external diametricity (ED), internal diametricity (ID) and concentricity (x } y ). See Fig. 3 for 2D illustrations. D4) ET(a; b) def (S(a) ^ S(b) ^ DR(a; b) ^ 8xy[(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) ^ 8xy[(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) ^ 8xy[(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) ^ 8xy[(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) ^ 8xy[(ED(x; y; a) ^ IT(x; b) ^ IT(y; b)) ! ID(x; y; b)]) _ (PP(b; a) ^ 8xy[(ED(x; y; b) ^ IT(x; a) ^ IT(y; a)) ! ID(x; y; a)]) ] a
b
b
a
a
c
b
a
b
a b
c ET(a,b)
IT(a,b)
ED(a,b,c)
ID(a,b,c)
CONC(a,b)
Figure 3: Relations among spheres de ned by Tarski We now de ne some fundamental relations involving spheres: D9) B(x; y; z) def x = y _ y = z _ 9vw[ED(x; y; v) ^ ED(v; w; y) ^ ED(y; z; w)] 0 0 D10) COB(s; r) def S(s) ^ 8s [s } s ! (O(s0; r) ^ :P(s0; r))] D11) EQD(x; y; z) def 9z0[z0 } z ^ COB(y; z0) ^ COB(x; z0)] D12) Mid(x; y; z) def B(x; y; z) ^ 9y0[y0 } y ^ COB(x; y0) ^ COB(z; y0)] D13) EQD(w; x; y; z) def 9uv[Mid(w; u; y) ^ Mid(x; u; v) ^ EQD(v; z; y)] D14) Nearer(w; x; y; z) def 9x0[B(w; x; x0) ^ :(x } x0) ^ EQD(w; x0; 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 that the centre of y lies mid-way between the centres of x and z ; and EQD(w; x; y; z ) holds when the distance between the centres of w and x is the same as the distance between the centres of y and z . Nearer(w; x; y; z ) means that the centres of w and x are closer than the centres of y and z. Since the concepts B and EQD are de nable, we can write the axioms of n-dimensional Elementary Geometry (Tarski 1959) (these axioms are reproduced in Appendix ??), within our language (the value of n is xed by appropriate choice of upper and lower dimension axioms). (Tarski 1929) takes this approach to prove that his geometry of solids is categorical and is modelled by n-dimensional Euclidean space in which spheres are interpreted as open balls and `solids' are regular open sets. We take a similar approach; however, whereas Tarski introduced points as sets of spheres, our relations concern spheres but they hold just in case the centre points of the spheres satisfy the corresponding point relations. Thus the quanti ers of the point-based geometry axioms can be replaced by quanti ers over spheres and the equality relation replaced by the } relation. Hence, in addition to A1 and A2 of Mereology, our theory contains: A3) A complete axiom set for n-dimensional geometry (e.g. (Tarski 1959)) encoded in terms of the B, EQD and } relations. A4) 8xyz[(x } y ^ y } z) ! x } z] A5) 8xx0yzw[(EQD(x; y; z; w) ^ x0 } x) ! EQD(x0; 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 of centre points of spheres coincides with the class of regions and that the P relation corresponds to the inclusion relation among the centre points. Rather than stating these as a meta-level constraints (as Tarski does) we enforce them directly by axioms. First, we de ne relations that hold when the centre point of a sphere is within the interior of a region: D15) InI(s; r) def 9s0[s0 } s ^ P(s0; r)] We can now specify the axioms: A6) 8xy[:(x } y) ! 9s[s } x ^ 8z[InI(z; s) $ Nearer(x; z; x; y)]] A7) 8x9y[:(x } y) ^ 8z[InI(z; x) $ Nearer(x; z; x; y)]] A8) 8xy[P(x; y) $ 8s[InI(s; x) ! InI(s; y)]] A9) 8r9s[P(s; r)] A6 ensures that for every pair of distinct points x and y there is a sphere centred at one and bounded by the other. A7 says that all spheres can be constructed in this way. A8 means that P(x; y ) holds just in case every interior point of x is an interior point of y (this actually makes A1 redundant). A9 states that every region has a spherical part (from this it can be proofed that every region is equal to the sum of its spherical parts). The theory speci ed by the axioms A1{9 we call RBGn (Region-Based Geometry). Let an n-dimensional classical interpretation for RBGn be a function =n which assigns a non-empty regular open subset of Rn to each 1st-order variable of RBGn and a set of non-empty regular open subsets of Rn to each of its 2nd-order variables. Under a classical 13 Re exivity and symmetry are implicit in the de nition of }.
28
interpretation P(x; y ) holds just in case =n (x) =n (y ); S(x) holds just in case =n (x) is an open n-ball. Theorem 1. Axioms RBGn provide a categorical axiom system for n-dimensional regionbased geometry, such that every model is isomorphic to a classical interpretation =n . Proof: (Tarski 1959) proves that all models of the axioms speci ed by A3-A5 have the structure of n-dimensional Cartesian spaces over R. This guarantees that there are spheres centred at all and only the points of Rn. A6 and A7 ensure that there is a sphere corresponding to every open n-ball in the space, such that InI(s; s0) holds just in case the centre point of s is in the ball corresponding to s0 . A8 xes the interpretation of P(x; y) to coincide with the condition fsjInI(s; x)g fsjInI(s; y )g. The interpretation of S is now completely xed relative to Rn and InI and P also have there intended interpretations over the domain of spheres. We now x the interpretation of the regions. From A9 and D3 it is easy to show that every region is the sum of its spherical parts: 8x[SUM(fy j P(y; x) ^ S(y)g; x)] This, with A2, ensures that the set of regions coincides with the SUMs of arbitrary sets of spheres. Let the set of `interior-points' of a region r be the set of centre points of all spheres s such that InI(s; r). Clearly, determining this set for all regions completely determines the P relation. We now show that for any set of spheres such that SUM( ; s), the set S of interior points of s coincides with the smallest regular open set R containing all interior points of all spheres in . First note that from A1, A2, D8 and D15 one can prove: 8s8r[InI(s; r) $ 9s0[P(s0; r) ^ InI(s; s0)]] ; 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 R is regular open, then if S were larger than R it would contain some sphere disjoint from R and hence disjoint from all spheres in and hence disjoint from s. Thus S R. Since R is open we can exactly cover all its points by the interior points of a set of spheres . By A2 there must be a region r such that SUM(; r). The regularity of R then means that interior points of r are exactly those in R. Now suppose S $ R, then using A8 and the mereological axioms one can show that 9x[P(x; r) ^ DJ(x; s)] and thence (using 9) 9 x[P(x; r) ^ :DJ(x; s)]. But if r includes a sphere which is disjoint from s then, contrary to our supposition R cannot be the smallest regular open set including all spheres in . Therefore we must have S = R. Because of the 2nd-order nature of the theory we cannot get a truly complete axiom system. However, the categoricity result means the theory would be complete if we had an oracle for 2nd-order logic, so the meaning of all non-logical vocabulary is completely xed. Theorem 2. RBGn is undecidable for n 2. Proof: For n = 2 this follows from the results of (Grzegorczyk 1951) (see also (Gotts 1996, Dornheim 1998)). Undecidability for higher dimensions can be demonstrated by de ning a `slice' of n-space that is in nitely extended in two dimensions but of a xed nite thickness in the others. Two dimensional regions can be simulated by parts of the slice such that their boundaries within the slice are orthogonal to the faces of the slice.14 14 RBG1 may also be undecidable; so far we have no result on this. 29
13.2 Spatio-Temporal Reasoning in PSTL
Recent work of Wolter and Zakharyaschev (2000) has shown how modal encodings can be used to construct decidable spatio-temporal languages encompassing the S 4u topological representation. By using multi-dimensional modal logic, topological representations in S 4u can be combined with the temporal logic PTL to yield the very expressive spatio-temporal logic PSTL.15 The models for this logic correspond to the structure illustrated in gure 4, where each horizontal slice is an S 4u model and the vertical structure along the time dimension provides a PTL model. Thus each `possible world' in the model has both a spatial and a temporal index. Space t+1 Space t Time
Figure 4: The spatio-temporal structure of PSTL models
References Asher, N. and Vieu, L.: 1995, Toward a geometry of common sense: A semantics and a complete axiomatization of mereotopology, Proceedings of the International Joint Conference on Arti cial Intelligence (IJCAI-95), Montreal. Bennett, B.: 1994, Spatial reasoning with propositional logics, in J. Doyle, E. Sandewall and P. Torasso (eds), Principles of Knowledge Representation and Reasoning: Proceedings of the 4th International Conference (KR94), Morgan Kaufmann, San Francisco, CA. Bennett, B.: 1997, Logical Representations for Automated Reasoning about Spatial Relationships, PhD thesis, School of Computer Studies, The University of Leeds. abstract and postscript at http://www.scs.leeds.ac.uk/brandon/thesis.html. Bennett, B., Cohn, A. G., Torrini, P. and Hazarika, S. M.: 2000a, Describing rigid body motions in a qualitative theory of spatial regions, in H. A. Kautz and B. Porter (eds), Proceedings of AAAI-2000. Bennett, B., Cohn, A. G., Torrini, P. and Hazarika, S. M.: 2000b, A foundation for regionbased qualitative geometry, in W. Horn (ed.), Proceedings of ECAI-2000. Chellas, B. F.: 1980, Modal Logic: An Introduction, Cambridge University Press. 15 Note that this name for the logic has been coined after the paper (Wolter and Zakharyaschev 2000), where
it was just called ML.
30
Cohn, A.: 1997, Qualitative spatial representation and reasoning techniques, in G. Brewka, C. Habel and B. Nebel (eds), Proceedings of KI-97, Vol. 1303 of LNAI, Springer Verlag, pp. 1{30. Cohn, A. G., Bennett, B., Gooday, J. and Gotts, N.: 1997, RCC: a calculus for region based qualitative spatial reasoning, GeoInformatica 1, 275{316. Davis, E.: 1990, Representations of Commonsense Knowledge, Morgan Kaufmann, San Mateo. Davis, E., Gotts, N. and Cohn, A. G.: 1999, Constraint networks of topological relations and convexity, Constraints 4(3), 241{280. Dornheim, C.: 1998, Undecidability of plane polygonal mereotopology, in A. G. Cohn, L. Schubert and S. Shapiro (eds), Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), Morgan Kaufman, pp. 342{353. Du y, D.: 1991, Priniples of Automated Theorem Proving, Wiley. Goldblatt, R. I.: 1976, Metamathematics of modal logic, parts I and II, Reports on Mathematical Logic. 6 41{78 and 7 21{52. Also in Goldblatt, R.I. Metamathematics of Modality, CLSI Publications (1993). Goranko, V. and Passy, S.: 1992, Using the universal modality: Gains and questions, Journal of Logic and Computation 2, 5{30. Gotts, N. M.: 1996, Using the RCC formalism to describe the topology of spherical regions, Technical report, Report 96.24, School of Computer Studies, University of Leeds. Grzegorczyk, A.: 1951, Undecidability of some topological theories, Fundamenta Mathematicae 38, 137{152. Hughes, G. E. and Cresswell, M. J.: 1968, An Introduction to Modal Logic, Methuen, London. Johnstone, P. T.: 1982, Stone Spaces, Cambridge University Press. Jonsson, B. and Tarski, A.: 1951, Boolean algebras with operators, part I, American Journal of Matehematics 73, 891{939. Knau , M., Rauh, R. and Renz, J.: 1997, A cognitive assessment of topological spatial relations: Results from an empirical investigation,, Proceedings of the 3rd International Conference on Spatial Information Theory (COSIT'97),, Springer-Verlag. Kuratowski, K.: 1972, Introduction to Set Theory and Topology, 2nd edn, Pergamon Press. Lemmon, E. J.: 1966a, Algebraic semantics for modal logics I, Journal of Symbolic Logic 31, 46{65. Lemmon, E. J.: 1966b, Algebraic semantics for modal logics II, Journal of Symbolic Logic 31, 191{218. Lesniewski, S.: 1927-1931, O podstawach matematyki, Przeglad Filozo czny. 31
McKinsey, J. and Tarski, A.: 1944, The algebra of topology, Annals of Methematics 45, 141{ 191. Nebel, B.: 1995, Computational properties of qualitative spatial reasoning: First results, in I. Wachsmuth, C.-R. Rollinger and W. Brauer (eds), Procedings of the 19th German AI Conference (KI-95, Vol. 981 of LNCS, Springer-Verlag, pp. 233{244. Nerode, A.: 1990, Some lectures on intuitionistic logic, in S. Homer, A. Nerode, R. Platek, , G. Sacks and A. Scedrov (eds), Logic and Computer Science, Vol. 1429 of Lecture Notes in Mathematics, Springer-Verlag, pp. 12{59. Pratt, I.: 1999, First-order qualitative spatial representation languages with convexity, Spatial Cognition and Computation 1, 181{204. Pratt, I. and Lemon, O.: 1997, Ontologies for Plane, Polygonal Mereotopology, Notre Dame Journal of Formal Logic 38(2), 225{245. Randell, D., Cui, Z. and Cohn, A.: 1992, A spatial logic based on regions and connection, Proc. 3rd Int. Conf. on Knowledge Representation and Reasoning, Morgan Kaufmann, San Mateo, pp. 165{176. Renz, J.: 1998, A canonical model of the Region Connection Calculus, in A. G. Cohn, L. Schubert and S. Shapiro (eds), Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98), Morgan Kaufman, pp. 330{341. Renz, J. and Nebel, B.: 1997a, On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus, Proceedings of IJCAI-97. Renz, J. and Nebel, B.: 1997b, On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus, Proceedings of IJCAI-97. Renz, J. and Nebel, B.: 1999, On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus, Arti cial Intelligence 108(1{2), 69{ 123. Simons, P.: 1987, Parts: A Study In Ontology, Clarendon Press, Oxford. Stone, M.: 1936, The theory of representations for Boolean algebras, Transactions of the American Mathematical Society 40, 37{111. Tarski, A.: 1929, Les fondaments de la geometrie des corps, Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego pp. 29{33. A suplement to Annales de la Societe Polonaise de Mathematique. English translation, `Foundations of the Geometry of Solids', in A. Tarski, Logic, Semantics, Metamathematics, Oxford Clarendon Press, 1956. Tarski, A.: 1938, Der Aussagenkalkul und die Topologie [sentential calculus and topology], Fundamenta Mathematicae 31, 103{134. English translation in A. Tarski, Logic, Semantics, Metamathematics, Oxford Clarendon Press, 1956. Tarski, A.: 1959, What is elementary geometry?, in L. Henkin, P. Suppes and A. Tarski (eds), The Axiomatic Method (with special reference to geometry and physics), North-Holland, Amsterdam, pp. 16{29. 32
Tarski, A. and McKinsey, J.: 1948, Some theorems about the sentential calculi of Lewis and Heyting, Journal of Symbolic Logic 13, 1{15. Wolter, F. and Zakharyaschev, M.: 2000, Spatio-temporal representation and reasoning based on RCC-8, Proceedings of the seventh Conference on Principles of Knowledge Representation and Reasoning, Morgan Kaufman, Montreal, Canada, pp. 3{14. Woodger, J.: 1937, The Axiomatic Method in Biology, Cambridge University Press.
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