Applied Intelligence 17, 239–251, 2002c© 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Multi-Dimensional Modal Logic as a Frameworkfor Spatio-Temporal Reasoning

BRANDON BENNETT∗ AND ANTHONY G. COHNSchool of Computing, University of Leeds, LS2 9JT, UK

brandon@comp.leeds.ac.uk

agc@comp.leeds.ac.uk

FRANK WOLTERInstitut fur Informatik, Universitat Leipzig, Augustus-Platz 10-11, 04109 Leipzig, Germany

wolter@informatik.uni-leipzig.de

MICHAEL ZAKHARYASCHEVDepartment of Computer Science, King’s College, Strand, London WC2R 2LS, UK

mz@dcs.kcl.ac.uk

Abstract. In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledgerepresentation and, in particular, for representing spatio-temporal information. We construct a two-dimensionallogic capable of describing topological relationships that change over time. This logic, called PSTL (PropositionalSpatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u,which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whetherthe full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions(both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidabilityand complexity results that are relevant to computation with multi-dimensional formalisms and discuss possibledirections for further research.

Keywords: spatio-temporal reasoning, modal logic, multi-dimensional logic

1. Introduction

It is widely accepted that many kinds of AI appli-cations require high-level reasoning involving spa-tial and temporal concepts (see e.g. [1–3]). Severalapproaches have been applied to representing theseconcepts: some researchers have developed spe-cialised computation-oriented representations such assolid geometry [4], constraints techniques [5] andspatio-temporal database architectures [6], others have

∗Supported by the EPSRC under grants GR/K65041 andGR/M56807.

employed techniques of formal logics. Typically, log-ical representations have made use of the expres-sive power of first-order languages [7–12]; but, sincefirst-order logic is undecidable, such languages donot provide an effective reasoning algorithm un-less supplemented by some special purpose inferencemechanism.

An alternative to first-order logic is the frameworkof propositional modal logic, which extends classicalpropositional calculus with one or more ‘modal’ oper-ators, interpreted on some relational or algebraic struc-tures. Appropriate structures may be chosen to corre-spond to an aspect of reality, such as time or space,

240 Bennett et al.

which one wishes to describe within a logical repre-sentation.

For example, one may think of time as a sequenceof points represented by the natural numbers 〈N, <〉 onwhich temporal operators—‘some time in the future’,‘since’ etc.—can be defined. For instance ♦+ϕ is trueat a time point t ∈ N (in symbols t |= ♦+ϕ) if and onlyif t ′ |= ϕ for some t ′ > t . The resulting propositionaltemporal logic, PTL, is decidable in PSPACE [13] andby Kamp’s theorem [14] it is as expressive (on 〈N, <〉)as the first-order language with unary predicates Pi (x),< and =. Temporal logics of this sort are widely usedin computer science [15, 16].

Modal representations of space are relatively newto computer science. However, the possibility of giv-ing topological interpretations of intuitionistic proposi-tional logic has been known since the thirties [17, 18].Tang [19] and later McKinsey and Tarski [20] inter-preted the necessity operator of the Lewis modal sys-tem S4 as the interior operator of topological spacesand proved that S4 is sound and complete with respectto this interpretation (for details see e.g. Chagrov andZakharyaschev [21]). Since S4 is decidable in PSPACE,this suggests that it might provide a computationallyviable language for describing topological informa-tion. That S4 really can be used for qualitative spatialrepresentation and reasoning was observed by Bennett[22, 23], who embedded the spatial language RCC-8into S4u—i.e. S4 extended with the universal modality∀ (for all points in the space). This encoding, whichwill be explained in Section 2, provides a decision pro-cedure for a spatial language that can express a largeclass of significant topological relations including allthose in RCC-8.

As we have seen, propositional modal and temporallogics may be decidable and hence amenable to au-tomated reasoning [24–29]. This of course means thatthey are less expressive than first-order logic. Moreoverit is clear that modalities of single type (say, only tem-poral or only spatial) are not comprehensive enoughfor many real applications, which require reasoningwith a range of different concepts. What we actuallyneed is systems combining several types of modal op-erator. Such multi-modal logics have been the subjectof much recent research—e.g. [30–35]. Central prob-lems in this field concern how modalities can be com-bined, whether interplay between operators preservesdecidability and other meta-logical properties, andwhat is the computational complexity of the resultingsystems.

t+1

t

Time

Space

Space

Figure 1. Motion modelled in terms of a Cartesian product of spaceand time.

Whereas modalities can be successfully combined aslogically independent operators [30, 36], a key featureof most of the more interesting systems is the interac-tion between different modalities. This applies in par-ticular to modal operators relevant to describing spatialand temporal concepts. Consider, for instance, a mov-ing region (the black disk) depicted in Fig. 1. If we canmodel space as an S4u-frame (i.e., a quasi-order) F

and if the flow of time is represented by a linear orderG, then the whole spatio-temporal ‘universe’ can beviewed as the Cartesian product F × G, in which theS4u-modalities act ‘horizontally’ to talk about spatialregions, while the temporal operators act ‘vertically’taking care of their movements in time.

Cartesian products of Kripke frames are typical ex-amples of multi-dimensional structures that serve asmodels of multi-dimensional modal logics. The ideaof a multi-dimensional modal logic was introduced bySegerberg [37] who presented an axiomatisation for a2-dimensional extension of S5. Products of modal log-ics were investigated in [38–40]. In computer scienceand AI, multi-dimensional modal logics are used e.g.for constructing temporal epistemic logics for multi-agent and distributed systems [41], temporal logics ofparallel processes [42], temporal, epistemic, and dy-namic description logics [43–46]. Further details anddevelopment of multi-dimensional modal logic can befound in [35].

Although the idea of constructing multi-dimensionalmodal logics may appear natural and simple, theresulting ‘hybrids’ often turn out to be very com-plex or undecidable, even if the one-dimensionalcomponents are ‘almost tractable’. As with otherframeworks for knowledge representation there is adelicate balance between expressive power and com-putational complexity. Let us consider, for instance, the

Multi-Dimensional Modal Logic 241

following naıve example of a ‘compass’ spatial logic[47].

In our everyday practice, we often connect spatialstructures with this or that system of coordinates. Forinstance, in geographical maps we have four com-pass directions: North, South, West, and East. Usingthese we can say e.g. that Moscow is to the North-East of London. Spatial relations of this sort can beexpressed in a modal language with four operators♦N (‘somewhere to the North’), . . . ,♦S (‘somewhereto the South’) interpreted in the Cartesian product oftwo linear orders, say, 〈R, <〉 × 〈R, <〉 in the standardKripke-style manner:

(x, y) |= ♦N ϕ iff ∃y′ > y (x, y′) |= ϕ,

(x, y) |= ♦W ϕ iff ∃x ′ < x (x ′, y) |= ϕ,

etc.

That ‘Moscow is to the North-East of London’ can berepresented in this language by the formula London →♦N ♦E Moscow. It is well known from modal logic thatthe satisfiability problem for a language with a single♦ operator is NP-complete for the frames 〈R, <〉 or〈N, <〉, [48]. However, the satisfiability problem for abi-modal language (with ♦N and ♦E ) in 〈R,<〉×〈R,<〉or 〈N, <〉 × 〈N, <〉 turns out to be undecidable, even�1

1-complete [31, 49, 50]. The interval temporal logicof Halpern and Shoham [51] can be interpreted in the{(x, y) ∈ N × N : x ≤ y} part of 〈N, <〉 × 〈N, <〉 (see[47]), and this logic is also not recursively enumerable.There are many other examples of undecidable multi-dimensional modal logics, e.g. S5 × S5 × S5 [52] oreven any logic between K × K × K and S5 × S5 × S5[53].

Thus, we see that multi-dimensional modal logicsare not easy to deal with, and we need to be careful inconstructing effective and expressive spatio-temporalformalisms. For example, the straightforward attack onthe problem by means of using the Cartesian productsof frames for S4 and the flow of time 〈N, <〉 (or anyother infinite linear order) has not brought any resultyet: whether the logic of such 2-dimensional frames isdecidable remains one of the challenging open prob-lems in the field.

In the rest of this paper, we discuss possible usesof multi-dimensional modal logics for spatio-temporalrepresentation and reasoning, giving initial results al-ready obtained in this direction. Thereby we would likeboth to attract the attention of the knowledge represen-tation community to this novel and promising approachand also to give incentive to logicians to investigate

modal languages that may turn out to have practicalutility.

2. Region Connection Calculusand a Modal Interpretation

The Region Connection Calculus (RCC) is a first-ordertheory proposed by Randell et al. [54] for qualita-tive spatial representation and reasoning.1 The basiclanguage of RCC contains only one primitive pred-icate C(X, Y ), read as ‘region X is connected withregion Y ’.2

Many other spatial relations can be defined in termsof the C primitive. Of particular significance are theeight relations depicted in Fig. 2. These form a pair-wise disjoint and exhaustive set of relations known asRCC-8. Essentially the same set (but applied to themore restricted class of connected regions) has beenindependently identified as useful in the context of Ge-ographical Information Systems [55]. In English, therelations can be described as: Dis-Connection, ExternalConnection, Partial Overlap, Tangential Proper Part,Non-Tangential Proper Part and Equality. Formal no-tations for these relations are given under the diagrams.The part relations are asymmetric, so each has an in-verse, suffixed by ‘i’.

The RCC formalism was originally presented as anaıve theory in the spirit of Hayes [1], so no spe-cific model was assumed. However, it has been foundthat the theory can be interpreted in classical point-settopology [23, 56, 57]. Thus, we can take as modelsof RCC, topological spaces, T = 〈U, I〉, where U isa non-empty set, the universe of the space, and I aninterior operator on U .3 Individual variables of RCCrange over non-empty regular closed sets of T, i.e.,an assignment in T is a map a associating with every

a

abb

a

b

DC(a,b)

b

a

a b

TPP(a,b)EC(a,b) TPPi(a,b)

a

ba

NTPP(a,b)PO(a,b)

b

EQ(a,b) NTPPi(a,b)

a b

Figure 2. Basic relations in the RCC theory.

242 Bennett et al.

variable X a set a(X ) ⊆ U such that a(X ) = CIa(X ),where C(X ) =def −I(−X ) is the closure operator onU dual to I. The connection relation C(X, Y ) is theninterpreted as meaning that the point sets denoted byX and Y share at least one point:

T |=a C(X, Y ) iff a(X ) ∩ a(Y ) �= ∅.

The full first-order theory of RCC is too expres-sive to be computationally useful and is in fact un-decidable (this follows from [58]; results applyingmore specifically to RCC can be found in [59] and[60]). Fortunately, there are various decidable (andeven tractable) fragments of RCC.

As was mentioned above, for certain applicationsone can limit the relations employed to the set RCC-8. Moreover, according to the experiments reported byKnauff et al. [61] the eight predicates turn out to be‘cognitively adequate’ in the sense that people indeeddistinguish between those relations. Formally, the lan-guage of RCC-8 consists of a set of individual vari-ables X0, X1, . . . , called region variables, the eightbinary predicates DC, EC, PO, EQ, TPP, TPPi, NTPP,NTPPi, and the Boolean connectives (¬, ∧, ∨, →), outof which we can construct spatial formulas.

RCC-8 is interpreted in topological spaces T =〈U, I〉: as before, the region variables range over non-empty regular closed sets in T, and the eight predicatesare defined in the following way:

DC(X1, X2) ¬∃x[x ∈ X1 ∩ X2]

EC(X1, X2) ∃x[x ∈ (X1 ∩ X2)]

∧ ¬ ∃x[x ∈ (IX1 ∩ IX2)]

PO(X1, X2) ∃x[x ∈ (IX1 ∩ IX2)]

∧ ∃x[x ∈ (I(X1) ∩ −X2)]

∧ ∃x[x ∈ (−X1 ∩ IX2)]

EQ(X1, X2) ∀x[x ∈ X1 ↔ x ∈ X2]

TPP(X1, X2) ∀x[x ∈ (−X1 ∪ X2)]

∧ ∃x[x ∈ (X1 ∩ −IX2)]

∧ ∃x[x ∈ (−X1 ∩ X2)]

NTPP(X1, X2) ∀x[x ∈ (−X1 ∪ IX2)]

∧ ∃x[x ∈ (−X1 ∩ X2)].

For example, according to this definition, EC(X, Y )means that X and Y share at least one point but do notshare any interior point (i.e. they only share boundarypoints).

The main reasoning task for RCC-8 can beformulated as follows: given a finite set � of spa-

tial formulas, decide whether � is satisfiable in atopological space, i.e., whether there exists a topo-logical space T and an assignment a in it such thatT |=a �.

That this satisfiability problem is decidable was ob-served by Bennett [62], who exploited the results ofTarski [18] to encode RCC-8 into intuitionistic propo-sitional logic. Subsequently, Bennett [22] gave anotherembedding of RCC-8 into the more expressive logiccontaining Lewis’s S4 modality and an additional uni-versal modality, whose meaning will be explained be-low. In the current paper we shall refer to this bi-modallogic as S4u. The embedding is based on the result ofMcKinsey and Tarski [20] according to which S4 ischaracterised by the class of topological spaces in thefollowing sense:

Given a topological space T = 〈U, I〉 and a modalformula ϕ, we interpret the propositional variables inϕ as subsets of U , the Boolean connectives as the cor-responding set-theoretic operations, the necessity op-erator I as I, and the possibility operator C as C (wedenote the usual box � and diamond ♦ of S4 by I andC to emphasize their topological meaning). Now, if thevalue of ϕ is the whole space U , no matter what setsare assigned to its variables and what topological spaceis taken, then ϕ is provable in S4, and vice versa.

According to [63], this completeness theorem stillholds if we extend S4 with the universal modalities∀ and ∃ interpreted in T as ‘for all points in T’ and‘there is a point in T’, respectively. That is the valueof ∀ϕ (under a certain interpretation) is U if the valueof ϕ (under this interpretation) is U , otherwise it is ∅;∃ϕ ≡def ¬∀¬ϕ.

It is easy to see that the language of the resultinglogic S4u is expressive enough to encode the topologi-cal meaning of the RCC-8 formulas as defined above.Given such a formula ϕ, we replace in it occurrencesof RCC-8 predicates with the corresponding modal for-mulas, e.g.

DC(Xi , X j ) with ¬∃(pi ∧ p j ),

NTPP(Xi , X j ) with ∀(¬pi ∨ I p j ) ∧ ∃(¬pi ∧ p j ),

etc.

(pi , p j are propositional variables) and add to the resultthe conjunct

∃pi ∧ ∀(pi ↔ CI pi )

for every region variable Xi in ϕ to ensure that thepropositional variables are interpreted by non-empty

Multi-Dimensional Modal Logic 243

regular closed sets of topological spaces. The resultingmodal formula is denoted by ϕ†.

Now, if we recall that some (in particular, all finite)topological spaces are determined by Kripke frames〈W, R〉 for S4—every such frame gives rise to the topo-logical space 〈W, I〉, where

IX = {x ∈ X : ∀y ∈ W (x Ry → y ∈ X )}for any X ⊆ W —and that S4u has the finite modelproperty [63], then we immediately obtain

Theorem 1. For every RCC-8 formula ϕ, the follow-ing conditions are equivalent:

(i) ϕ is satisfiable in a topological space,(ii) ϕ† is satisfiable in a topological space,

(iii) ϕ† is satisfiable in some finite Kripke frame for S4.

Thus we reduce the satisfiability problem for RCC-8formulas to the satisfiability problem for propositionalbimodal formulas in Kripke frames for S4u, which isdecidable [63].

Moreover, one can show that every satisfiable for-mula of the form ϕ† can be satisfied in a partially or-dered Kripke frame each point in which has at most two(incomparable) successors, and the number of worldsin this frame is linear in the number of symbols inϕ† (and consequently also linear in the length of theRCC-8 formula ϕ). This result was obtained by Renz[64]; a somewhat more general theorem is proved in[65]. It follows in particular that the satisfiability prob-lem for RCC-8 formulas is NP-complete; see [66, 67]where maximal tractable fragments of RCC-8 are alsodescribed. It also follows that all satisfiable RCC-8 for-mulas can be satisfied in Rn for any n ≥ 1. (Note how-ever that S4u is not complete with respect to Rn .)

3. Point-Based Temporal RCC-8

One approach to constructing spatio-temporal logics isto combine RCC-8 with point-based temporal logics,for instance, the well-known propositional temporallogic PTL interpreted on the flow of time 〈N, <〉 andhaving the temporal operators S (Since) and U (Until).Other standard operators can be defined:

• ◦ϕ ≡def ϕ U ϕ (‘at the next moment ϕ’),• ♦+ϕ ≡def (p ∨ ¬p)Uϕ (‘at some time in the

future ϕ’)• �+ϕ ≡def ¬♦+¬ϕ (‘always in the future ϕ’),

and similarly for their past counterparts◦−, �− and ♦−.

The spatial regions occupied by the objects underconsideration may change with time passing by, butthe topological space in which they are moving alwaysremains the same. This naıve picture is formalised bythe following concept of topological temporal model.

Definition 1. A topological temporal model (ortt-model, for short) based on a topological space T =〈U, I〉 is a triple of the form M = 〈T, N, a〉, where a,an assignment in T, associates with every region vari-able X and every moment of time n ∈ N a non-emptyregular closed subset of U . For each n, we take an tobe the function defined by an(X ) = a(X, n).

There are several different ways of introducing atemporal dimension into the syntax of RCC-8. The mostobvious is to allow applications of the operators S andU (along with the Booleans) to spatial formulas. Wecall the resulting spatio-temporal language ST0.

Definition 2. For a tt-model M = 〈T, N, a〉, anST0-formula ϕ, and n ∈ N, define the truth-relation(M, n) |= ϕ, meaning ‘ϕ holds in M at moment n,’by induction on the construction of ϕ:

• if ϕ contains no temporal operators, then (M, n) |=ϕ iff T |=an ϕ;

• (M, n) |= ϕUψ iff there is k > n such that (M, k) |=ψ and (M, l) |= ϕ for every l such that n < l < k;

• (M, n) |= ϕSψ iff there is k < n such that (M, k) |=ψ and (M, l) |= ϕ for every l such that k < l < n.

ST0 allows us to say many things about changesof spatial relationships over time. For example, ¬ +P(Kosovo, Yugoslavia) means that ‘Kosovo will not al-ways be part of Yugoslavia.’ It is also expressive enoughto constrain movements to be continuous, in so far asone can describe possible continuous transitions amongthe RCC-8 relations that can hold between two regions(such transitions were identified by Randell et al. [54]and the importance has been emphasised more recentlyby Muller [68]):

�+(DC(X, Y ) →◦(DC(X, Y ) ∨ EC(X, Y ))),

�+(EC(X, Y ) →◦(DC(X, Y ) ∨ EC(X, Y )

∨ PO(X, Y ))),

�+(PO(X, Y ) →◦(EC(X, Y ) ∨ PO(X, Y )

∨ TPP(X, Y ) ∨ EQ(X, Y )

∨ TPPi(X, Y ))),

etc.

244 Bennett et al.

However, the expressive power of ST0 is limited in thatone can only employ the RCC-8 relations to compareregion variables at the same point in time; so it doesnot allow one to describe spatial relations between theextensions of a region variable at different times. Forexample, there is no way to say something like ‘Theextension of the EU is a part of what its extension willbe next year (i.e. at the next time point).’

To overcome this limitation we introduce the logicST1, which extends ST0 by allowing applications of thenext-time operator ◦ not only to formulas but also toregion variables. Thus, arguments of the RCC-8 pred-icates are now region terms, which consist of a regionvariable that may be prefixed by an arbitrarily long se-quence of ◦ operators. For instance ◦◦EU could de-note the region occupied by the EU in two years timeand one can write formulas such as P(EU,◦◦EU).The semantics of ST1 is just that of ST0 extended bythe following clause:

• a(◦X, n) = a(X, n + 1).

Using ST1 we can now talk about the changing ex-tensions of individual region variables. For instance�+P(EU,◦EU) says that ‘the EU will never shrink’.The new construct may also be used to refine the con-tinuity assumption by requiring that �+(EQ(X,◦X ) ∨O(X,◦X )), i.e., ‘regions X and ◦X either coincideor overlap.’ We can also express the condition thatthe extension of a region variable X is fixed for thefuture: �+EQ(X,◦X ); or that it has at most twodistinct states, one on even days, another on oddones: �+EQ(X,◦◦X ). Note, that the ST1-formula�+NTPP(X,◦X ) is satisfiable only in models basedon infinite topological spaces, whereas for formulas ofST0 (and of course RCC-8) finite topological spacesalways suffice.

It may appear that ST1 can compare regions thatare separated in time only by limited sequences of timepoints. However, using an auxiliary variable, whose ex-tension is constrained to be constant over time, we canwrite, for instance,

�+EQ(X,◦X ) ∧ ♦+EQ(X, EU ) ∧ P(Russia, X ),

which is satisfiable iff ‘someday in the future thepresent territory of Russia will be part of the EU.’ Thiscontrasts with the formula ♦+P(Russia, EU) mean-ing that there will be a day when Russia (its terri-tory on that day—perhaps without Chechnya but withByellorussia) becomes part of the EU.

Imagine now that we want to say that every lo-cation in Europe will pass through the Euro-zone,but only the land currently occupied by Germanywill use the Euro forever. Unfortunately, we do notknow which countries will form within Europe inthe future, so we can’t simply write down all for-mulas of the form ♦+P(X, ♦Euro-zone). What weneed is the possibility of constructing regions ♦+ Xcontaining all the points that will belong to regionX at some time in the future and �+ X contain-ing just those points included in all future states ofX . Then we can write: P(Europe, ♦+Euro-zone) andEQ(Germany, �+Euro-zone). Similarly, the formulaP(Russia, ♦+EU) says that all points of the present ter-ritory of Russia will belong to the EU at some time inthe future (but perhaps at different moments of time).

To permit fully general application of temporal op-erators to region variables we define the language ST2

in which region variables may be prefixed by arbitrarystrings of the operators ◦, �+, ♦+. The semantics ofthe new operators are given by:

• a(�+t, n) = CI (⋂

k>n a(t, k)),• a(♦+t, n) = CI (

⋃k>n a(t, k)).

Finally, we can make our languages STi , for i =0, 1, 2, even more expressive by allowing applicationsof the Boolean operations to region terms. Their se-mantical meaning is defined as follows:

• a(t ∨ t ′, n) = CI (a(t, n) ∪ a(t ′, n)),• a(t ∧ t ′, n) = C I(a(t, n) ∩ a(t ′, n)),• a(¬t, n) = CI (U − a(t, n)).

Let ST+i be the resulting family of languages. In

these languages we can write formulas such asEQ(UK, Great Britain ∨ Northern Ireland), meaningthat the extension of the UK is the sum of the extensionsof Great Britain and Northern Ireland.

4. Modal Encoding of ST+2

The decidability and complexity results for the STi lan-guages were proved in [69] by embedding them into thetwo-dimensional propositional modal logic S4u×PTL,the Cartesian product of the logics S4u and PTL. Wecall this logic Propositional Spatio-Temporal Logic orPSTL. Its connectives are: the Booleans, the necessityand possibility operators I and C of S4, the univer-sal necessity and possibility operators ∀ and ∃, and the

Multi-Dimensional Modal Logic 245

temporal operators S and U . It is not known whetherthe full PSTL (or even S4 × PTL) is decidable. How-ever, its fragments corresponding to the STi languagesare quite manageable.

The modal translation from ST+2 into PSTL is de-

fined by extending the † transformation specified inSection 2 to handle region terms containing Booleanand temporal operators. The region terms are alreadyostensibly formulas of PSTL; however, we must bearin mind that these terms are intended to denote onlyregular closed subsets of the space. The easiest way totake this into account is to prefix C I to every subtermoccurring in an ST+

2 -formula.What is the intended semantics of PSTL? As we

have seen, when encoding pure RCC-8 into S4u wecan specify the semantics with either topological mod-els or Kripke models and these are equivalent in termsof validity. However, the addition of the temporal com-ponent makes the situation more complicated in thatthe topological and Kripke semantics do not perfectlyagree on the class of valid formulas. Let us define thesetwo types of model structures:

Definition 3. A Kripke PSTL-model is a triple K =〈F, N, V〉 where F = 〈W, R〉 is a quasi-order (a framefor S4) and V, a valuation, is a map associating withevery propositional variable p and every n ∈ N a subsetVV (p, n) ⊆ W . For each spatial point u ∈ W andeach time point n ∈ N, the truth-relation (u, n) |=K ϕ

is defined by:

• (u, n) |=K p iff u ∈ V(p, n),• (u, n) |=K ∀ψ iff (v, n) |=K ψ for all v ∈ W ,• (u, n) |=K Iψ iff (v, n) |=K ψ for all v ∈ W such

that u Rv,• (u, n) |=K ψUχ iff there is k > n such that

(u, k) |=K χ and (u, m) |=K ψ for all m, n < m < k,

plus the standard clauses for S and the Booleans. APSTL-formula ϕ is satisfied in K if (u, n) |=K ϕ forsome u ∈ W and n ∈ N.

Definition 4. A topological PSTL-model is a structureN = 〈T, N, U〉 in which T = 〈U, I〉 is a topologicalspace and U is a map associating with every proposi-tional variable p and every n ∈ N a set U(p, n) ⊆ U .U is then extended to arbitrary PSTL-formulas in thefollowing way:

• U(ψ ∧ χ, n) = U(ψ, n) ∩ U(χ, n),• U(¬ψ, n) = U − U(ψ, n),

• U(∀ψ, n) = U if U(ψ, n) = U , and U(∀ψ, n) = ∅otherwise,

• U(Iψ, n) = IU(ψ, n),• x ∈ U(ψUχ, n) iff there is a k > n such that x ∈

U(χ, k) and x ∈ U(ψ, m) for all m, n < m < k.

Consequently the defined temporal operators are inter-preted by

• U(◦ψ, n) = U(◦ψ, n + 1),• U(♦+ψ, n) = ⋃

k>n U(ψ, k),• U(�+ψ, n) = ⋂

k>n U(ψ, k).

A PSTL-formula ϕ is satisfied in N if U(ϕ, n) �= ∅ forsome n ∈ N.

The sets of PSTL-formulas satisfiable in Kripkemodels and topological models turn out to be differ-ent. Of course, every Kripke model is equivalent tosome topological model. But the converse does nothold. A good example is provided by the formula♦+C p ↔ C♦+ p. This is valid in every Kripke PSTL-model because from any space-time point in such astructure any given other point is reachable by for-ward transition along the time line followed by tran-sition along the accessibility relation of F just in caseit is reachable by first moving along the accessibilityrelation and then forward along the time dimension(since the accessibility relation remains constant forall time points). However, in certain infinite topologi-cal spaces (e.g. R) one can construct infinite sequencesXn of closed sets such that

⋃n∈N Xn is not closed. In a

topological model based on such a space ♦+C p neednot have the same denotation as C♦+ p. This subtletyconcerning infinite unions is not accounted for by theKripke approach.

The divergence between topological and KripkePSTL-models is problematic because whereas thetopological models correspond to the desired spatialinterpretation, most currently known methods of de-termining decidability and complexity are based onKripke models. Fortunately, [69] have been able toshow that the two semantics agree on the satisfiabil-ity of ST+

2 -formulas as long as we adopt a reasonablynatural Finite State Assumption (FSA). The FSA re-quires that over the infinite sequence of time pointseach region that one can refer to can have only finitelymany distinct extensions (but it may change its exten-sion infinitely often). Although this restriction rulesout many mathematically interesting possibilities, it is

246 Bennett et al.

perfectly satisfactory for a wide range of practical ap-plications (for example planning tasks where we wantto get from an initial to a final situation—in a finitenumber of steps). To formalise the FSA we define thefollowing restricted class of tt-models:

Definition 5. Say that a tt-model M = 〈T, N, a〉satisfies FSA, or is an FSA-tt-model, if for every re-gion term t there are finitely many regular closed setsA1, . . . , Am ⊆ U such that {a(t, n) : n ∈ N} ={A1, . . . , Am}.

It can be shown that an ST+2 -formula is satisfiable in

an FSA-tt-model iff it is satisfiable in an FSA-modelbased on a finite topological space.

Because of the very restricted combinations of oper-ators that result from translating RCC-8 predicates andtheir region term arguments, the modal translations ofST+

2 -formulas form a rather special fragment of themodal language PSTL. As was mentioned above, [64]showed that an RCC-8 formula ϕ is satisfiable iff ϕ† issatisfiable in a Kripke model based on an S4-frame ofdepth ≤1 and width ≤2 (which means that it containsno chains of more than 2 distinct points, and no pointhas more than 2 distinct successors); and it turns outthat this result can be generalised to formulas of ST+

1and ST+

2 :

Theorem 2.

(a) An ST+2 -formula ϕ is satisfiable in a FSA-tt-model

iff ϕ† is satisfiable in a Kripke PSTL-model (whichalso satisfies FSA) whose underlying S4-frame isof depth ≤1 and width ≤2.4

(b) An ST+1 -formula ϕ is satisfiable in a tt-model iff ϕ†

is satisfiable in a Kripke PSTL-model of depth ≤1and width ≤2.

This result makes it possible to use the method ofquasi-models [45] to prove that the satisfiability prob-lem for all the languages ST+

i , i = 1, 2, in tt-models isdecidable. Given the rather weak interaction betweentime and space in ST0 it is not hard to show that thesatisfiability problem for ST0 formulas in tt-models isPSPACE-complete (the same as that of PTL). The sat-isfiability problem for ST1 formulas in tt-models is de-cidable in EXPSPACE (and becomes NP-complete forthe sublanguage of ST1 with only the temporal operator◦). If we restrict the admissible models to FSA-modelsthen the satisfiability problem for ST2 formulas is de-cidable in EXPSPACE. Moreover, all the complexity

results just given remain valid after replacing STi withST+

i .The topological temporal models we were consider-

ing above are based on the discrete flow of time N. Byreplacing N with Q, R, or any other strict linear orderwe can extend our semantics to cover different flowsof time. But then the question arises as to whether thedecidability and complexity results proved for N canbe extended to, say, the logic determined by the classof arbitrary strict linear orders, by the reals R, or by therationals Q. It turns out that for the language of ST0 wecan easily extend the decidability proof for N by usingthe fact that the propositional temporal logics basedon those flows of time are decidable [70]. As concernsST1, observe that the operator ◦ is meaningless fordense linear orders—thus for Q and R this languagereduces to ST0. Decidability of ST1 interpreted in ar-bitrary linear orders is an open problem. And nothingis known about the decidability of ST2 interpreted inflows of time different from N. We conjecture, how-ever, that the methods developed in [71] can be used toprove the decidability of ST2 based on any of the flowsof time mentioned above (under FSA).

5. RCC-8 and Interval Temporal Logic

We have seen how the RCC-8 theory can be tempo-ralised by combining it with a point-based temporallogic. However, since the region-based approach to spa-tial reasoning was inspired by and closely mirrors theinterval-based approach to temporal reasoning [5, 9](they both take extended entities, rather than points asprimitives) it would seem far more natural to tempo-ralise RCC-8 by combining it with an interval-basedlogic. In this section we show that this can indeed bedone; and moreover, that the resulting system can in factbe embedded by suitable syntactic definitions withinthe point-based logic PSTL defined above.

We remind the reader that Allen’s logic has thirteenbasic relations between time intervals—Before(i, j),Meets(i, j), Overlaps(i, j), During(i, j), Starts(i, j),Finishes(i, j), their inverses, and Equal(i, j). The setof formulas constructed using these predicates andthe Booleans can be regarded as a temporal ‘twin’ ofRCC-8.

Following [9], we write HOLDS(R, i) to say thatthe relation R holds during some time interval i . ThusHOLDS(PO(X, Y ), i) means that during interval i re-gions X and Y partially overlap. Let us call an ARCC-8formula any Boolean combination of basic temporal

Multi-Dimensional Modal Logic 247

predicates and formulas of the form HOLDS(ϕ, i)where ϕ is an RCC-8 formula. Here is a simple exampleof a valid entailment in this unsophisticated language:

Meets(i, j), During(i, k), During( j, k),

HOLDS(TPP(Hong Kong, UK)∧ EC(Hong Kong, China), i),

HOLDS(DC(Hong Kong, UK), j),

HOLDS(EC(UK, China) ∨ DC(UK, China), k)

|= HOLDS(EC(UK, China), i).

There are different views on how intervals should bemodelled in different time flows. A common interpre-tation is that the intervals are treated as ordered pairsof distinct points of the domain Q or R. Within ourframework we can in fact adopt any of these models;but, for simplicity of presentation it is convenient to em-ploy a semantics where intervals are arbitrary convexnon-empty subsets of the time points of an arbitrarytime flow. Thus we give the following semantics forARCC-8:

Definition 6. An it-model (interval topologicalmodel) is the triple M = 〈F, T, a〉, where: F =〈W, <〉 is a strict linear order (modelling the intendedflow of time); T = 〈U, I〉 is a topological space; andassignment a associates with every interval variable ia non-empty convex subset of W and with every regionvariable X and every moment of time u it associates aregular closed set a(X, u) in T.

The truth-relation is defined inductively as follows(clauses for the Booleans are standard):

• M |=a Before(i, j) iff (∃t ∈ W )(∀ti t j )[(ti ∈ a(i) ∧t j ∈ a( j)) → ti < t < t j ];

• M |=a Meets(i, j) iff (∀ti t j )[(ti ∈ a(i) ∧ t j ∈a( j)) → ti < t j ] and not M |=a Before(i, j);

• M |=a Overlaps(i, j) iff (∃t)[t ∈ a(i) ∩ a( j)] ∧(∃ti ∈ a(i))(∀t j ∈ a( j))[ti < t j ] ∧ (∃t j ∈a( j))(∀ti ∈ a(i))[ti < t j ]];

• The other interval relations are defined similarly;• M |=a HOLDS(ϕ, i) iff for every point u ∈ a(i) we

have T |=au ϕ.

We now show how by using some ideas of Blackburn[72] the language ARCC-8 can be directly embed-ded into PSTL. For convenience we define undirecteduniversal and existential modalities over time: �ϕ =ϕ ∧ �+ϕ ∧ �−ϕ (i.e., ϕ is true at every moment) and

♦ϕ = ¬�¬ϕ. We extend the translation function † de-fined above to encode the interval relations and HOLDSpredicate. For example, we replace in ϕ

Equals(i, j) with �(ti ↔ t j )

Before(i, j) with �(ti → (¬t j ∧ ♦+t j ))

∧ ♦(¬ti ∧ ¬t j ∧ ♦−ti ∧ ♦+t j )

Meets(i, j) with �(ti → ¬t j ∧ ♦+t j )

∧ ¬♦(¬ti ∧ ¬t j ∧ ♦−ti ∧ ♦+t j )

Overlaps(i, j) with ♦(ti ∧ t j ) ∧ ♦(ti ∧ ¬t j ∧ ♦−t j )

∧ ♦(t j ∧ ¬ti ∧ ♦+ti )

HOLDS(ψ, i)) with �(ti → ψ†)

and the other Allen relations can be encoded in similarfashion. Here the variables ti are just ordinary propo-sitional variables; however, we need to constrain theirinterpretation so that they can be employed to stand forintervals. Therefore, we add to the resulting formulathe conjuncts

♦ti ∧ �(♦−ti ∧ ♦+ti → ti ) ∧ (ti → ∀ti )

for every interval variable i occurring in ϕ, thus obtain-ing ϕ†. The first conjunct ensures that ti is non-empty;5

the second that it is a convex interval of the time se-ries; and the final conjunct means that the value of ti isconstant relative to the spatial dimension.

It can be shown that an ARCC-8 formula ϕ is satis-fiable in an it-model just in case its modal translationϕ† is satisfiable in a Kripke PSTL-model. Moreover,the satisfiability problem for ARCC-8 formulas in it-models is NP-complete. Thus ARCC-8 has the samecomputational complexity as both RCC-8 and the con-straint language of Allen’s temporal interval predicates[73].

6. How Far Can Multi-DimensionalModal Logic Take Us?

We have seen how the 2-dimensional language PSTLprovides an expressive formalism for representingspatio-temporal information, which encompasses bothtopological constraints and linear temporal logic. Wenow consider what other concepts one might want torepresent and note some difficulties that arise due toknown complexity results.

A concept that is very useful for describing real sit-uations is that of spatial convexity. The addition to

248 Bennett et al.

the first-order RCC formalism of a function giving the‘convex-hull’ of a region proposed in [54] (such a func-tion and a convexity predicate true of convex regions areinterdefinable). First-order languages including topo-logical relations and a convexity predicate have beenfound to be highly expressive [74]. Although satisfia-bility of a combination of RCC-8 relations and convex-ity predicates is known to be decidable is has also beenshown to be as hard as solving systems of non-linearconstraints over R [75]. It is unclear whether modallogics can contribute to reasoning about convexity.The obvious models of multi-dimensional space withinwhich convexity constraints could be specified involvecross products of linearly ordered infinite frames corre-sponding to coordinate axes of the space. However, itis known that all modal logics of such products areundecidable [50]. Balbiani et al. [76] have given amodal logic of incidence geometry within which onecan express collinearity and betweenness (and henceconvexity) but it is not known whether this logic isdecidable.

Another concept crucial to real situation descriptionsis that of connectedness of regions. The objects we nor-mally think and talk about are connected in the sensethat any two points within an object can be joined bya path that lies entirely within the object. One of themost important open problems in spatial reasoning iswhether there is a decision procedure for testing satis-fiability of sets of topological relations holding amongconnected planar regions. A lower exponential com-plexity bound follows from the results of [77, 78] con-cerning an analogous problem for planar graphs; andthis result applies even in the case where we deal withonly the basic non-disjunctive relations of RCC-8. Wespeculate that a modal analysis might shed some lighton this decision problem.

One might also want to introduce more expressivepower in the temporal dimension. In order to describecontinuous changes more adequately one may want toemploy a logic based on a dense model of time [79].Within such a time flow, we must consider whetherspatial relationships hold over open or closed intervals.This problem has been examined by Galton [80, 81],who argues that whether a spatial relation is true at thebounding point of an interval over which it holds de-pends on the nature of the relation in question. Thisanalysis may allow for more detailed description ofchanging spatial relationships than is possible withinAllen’s [9] treatment. Also, rather than treating timeas a linear ordering one might like to model alternative

possible histories in terms of some branching structure,such as is described by the logic CTL∗ [82]. Seman-tically it is straightforward to combine the spatial in-terpretation of S4u with any reasonable temporal logic;but so far nothing is known about the complexity ofcombinations with non-linear time-flows.

In considering decidability and complexity of multi-modal systems, logicians have almost always lookedat the problem for arbitrary formulas of the resultingmodal language. However, as we saw in the case ofPSTL, sub-languages with severely restricted syntax(e.g. ST(+)

i ) may be able to express a significant vocab-ulary sufficient for many applications. Thus we suggestthat looking for such sub-languages is a research areaof great potential.

In order to really demonstrate the utility of multi-dimensional logics one would need to develop andimplement reasoning algorithms capable of carryingout useful reasoning tasks. Automated reasoning withmulti-modal logics is still in its infancy; but recentlythere have been some successes in developing proofmethods [27, 83]. A tableau calculus for the local cu-bic logic LC2, which is closely related to S5 × S5, hasbeen implemented by Marx et al. [84].

7. Conclusion

We have outlined the general structure of a knowledgerepresentation formalism, based on multi-dimensionalmodal logics. This framework seems to be well suitedfor representing a large vocabulary of useful high-levelspatio-temporal relations. Our decidability and com-plexity results for the languages ST(+)

i and ARCC-8show that this approach enables one to construct veryexpressive yet effective spatio-temporal languages.

We hope that the reader will take from this paper notonly the particular details of the language PSTL buta wider appreciation of the possibilities of applyingmulti-dimensional modal logics to the development ofArtificial Intelligence. We would also like to suggestthat spatio-temporal reasoning is an area within whichcooperation between pure logicians and researcherstackling specific reasoning problems arising in applica-tions can lead to both interesting theorems and powerfulpractical algorithms.

Notes

1. A comprehensive account of this theory and its applications canbe found in [85].

Multi-Dimensional Modal Logic 249

2. Randell et al. [54] also considered an extended language incorpo-rating an additional primitive function conv(r ), which denotes theconvex-hull of region r . We will comment briefly on convexitylater (Section 6).

3. This means that I must satisfy the axioms I(X ) ⊆ X , II(X ) =I(X ), I(U ) = U and I(X ∩ Y ) = I(X ) ∩ I(Y ) [86].

4. Actually, since occurrences of �+ within the region terms ofST+

2 are always regularised by adding the prefix CI, one mightconjecture that the FSA is not necessary. However, at present wecannot envisage how to prove this.

5. If we replace this conjunct by ♦(ti ∧ ♦+ti ) we can also rule out‘intervals’ consisting of a single time point.

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