An Axiomatic Approach to Topology For Spatial Information Systems

University of Leeds

SCHOOL OF COMPUTER STUDIES RESEARCH REPORT SERIES Report 96.25

An Axiomatic Approach to Topology For Spatial Information Systems 1 by

N M Gotts2

Division of Arti cial Intelligence August 1996

1 The support of the EPSRC under grant no. GR/H 78955, and helpful comments from Tony Cohn, John Gooday and Brandon Bennett, are gratefully acknowledged. 2 Address from September 1996: Department of Computer Science, University of Wales, Aberystwyth

Abstract

The paper reports work on the topological formalism `RCC', a region-based `calculus of connection' developed at Leeds university over the past several years. Speci cally, it is shown that the nonempty regular closed sets of a class of topological spaces (connected T3 -spaces) provide models for the RCC axiom-set. A brief assessment is made of RCC's potential as a formalism for applications in the area of spatial information systems (SIS). Two approaches to developing topological formalisms for SIS are compared, and a parallel is drawn with the two main parts of topology as understood by mathematicians.

1 Introduction

For the past 3 years, the author has been working on the development of a particular formal system for topological representation and reasoning, referred to here as RCC (RegionConnection Calculus) (Gotts 1994, Cohn and Gotts 1996, Gotts, Gooday and Cohn 1996). RCC itself has a longer history: the version to be discussed here was rst described in (Randell, Cui and Cohn 1992). RCC has non-topological aspects, described in (Cohn, Randell and Cui To appear), but these are not discussed here. The primary aim of this paper is to report on the progress made in exploring the topological aspects of RCC, particularly the tractability and expressiveness of the system, and its range of possible models. Secondary aims are to begin the process of assessing its potential in relation to spatial information systems (SIS); to discuss the relevance of topology as mathematicians understand it to topological formalisms for SIS, and to contrast two approaches to the development of such a topological formalism, which will be called `axiomatic' and `constructive'. By a `spatial information system' or SIS I mean a computer system designed for the storage and manipulation of large amounts of information with an important spatial component, referring to some entity or entities with a complex spatial structure. Currently, GIS are among the best known and most widely used kind of SIS: here, the entities dealt with are geographical regions, and are generally treated as 2-dimensional or less, although 3-dimensional GIS are likely to be increasingly important (Raper and Kelk 1991). Many of the other existing or potential application areas (e.g. architecture, human and animal anatomy) are inherently 3-dimensional. Desirable qualities of a topological formalism for use in SIS are considered in the nal section, where RCC is assessed in relation to them. It might seem initially that topology as mathematicians have developed it over the last century has relatively little to oer the SIS community. Although its earliest roots lie in readily comprehensible problems like that of the Konigsberg bridges, topology has shown the tendency, common to much of mathematics, towards increasing abstraction and generality; much of it is thus of little or no obvious relevance to those working on SIS. The RCC approach was indeed developed at least in part as an alternative to `conventional' topology. Nevertheless, mathematical topology provides many resources for the future development of SIS, and if we want to avoid reinventing too many wheels, we had better pay attention to it.

2 RCC: Background and Existing Work

This section is intended to give sucient information on RCC to give a context to the work reported in the subsequent section. By no means all the work done on the basis of the system is described. Mathematical topology de nes spaces in terms of sets of points. Euclidean spaces consist of uncountably in nite sets of points. A persistent minority within philosophy, logic, mathematics and arti cial intelligence has sought to construct what could be called an `alternative topology'. This takes spatial regions, or physical objects and their relations as fundamental, de ning points, if required, in terms of sets of regions. RCC forms part of this minority tradition, for which see also (Tarski 1956), (Clarke 1981, Clarke 1985) and (Simons 1987). Clarke's work has been taken as the basis of two approaches to region-based topology, although both have modi ed that basis somewhat (in dierent ways). RCC is one; the other is described in (Vieu 1993, Asher and Vieu 1995). Both Clarke's work and the approaches based on it axiomatise a fundamental relation of connection (C) between a pair of entities.

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RCC's topological calculus, set out in (Randell et al. 1992, section 4), is presented somewhat dierently here, but no change in content is intended. First, there are two axioms establishing that C is re exive and symmetric: (A1) 8xC(x; x) (A2) 8x; y[C(x; y) ! C(y; x)]; Additional pairwise region-relations are de ned in (Randell et al. 1992, section 4) in terms of C. De nitions and meanings of those used here are given below. DC(x; y) def :C(x; y) (x is disconnected from y) P(x; y) def 8z [C(z; x) ! C(z; y)] (x is part of y) PP(x; y) def P(x; y) ^ :P(y; x) (x is a proper part of y) EQ(x; y) def P(x; y) ^ P(y; x) (x coincides with y) O(x; y) def 9z [P(z; x) ^ P(z; y)] (x overlaps with y) PO(x; y) def O(x; y) ^ :P(x; y) ^ :P(y; x) (x partially overlaps y) EC(x; y) def C(x; y) ^ :O(x; y) (x externally connects with y) TPP(x; y) def PP(x; y) ^ 9z [EC(z; x) ^ EC(z; y)] (x is a tangential proper part of y) NTPP(x; y) def PP(x; y) ^ :9z [EC(z; x) ^ EC(z; y)] (x is a nontangential proper part of y ). Further axioms assert or (in one case) deny the existence of a region with particular C relations, unconditionally for the universal region u, conditional on the existence of another region or regions for the functions compl (complement of a region), and sum and prod (sum and product of a pair of regions). The functions compl and prod are partial over the domain of regions (compl(x) is only a region when x is not u, and prod(x; y) only when O(x; y) is true). Using the sorted logic LLAMA (Cohn 1992), the functions are rendered total by introducing a sort NULL, disjoint from the sort REGION; and specifying sortal restrictions on the functions' arguments. Here, we specify that these `quasi-boolean' functions3 , as well as C and the other relations de ned in terms of it, can only take arguments of sort REGION; any expression within which any of these functions or relations are apparently applied to an argument of sort NULL has no semantics. The axioms below depend on these. (A3) 8x[C(x; u)] (A4) 8x; y[[C(y; compl(x))  :NTPP(y; x)] ^ [O(y; compl(x))  :P(y; x)]] (A5) 8x; y; z [C(z; sum(x; y))  C(z; x) _ C(z; y)] (A6) 8x; y; z [C(z; prod(x; y))  9w[P(w; x) ^ P(w; y) ^ C(z; w)]] (A7) 8x; y[NULL(prod(x; y))  DR(x; y)]. Given these axioms, we can de ne another function, di: di(x; y)) =def prod(x; compl(y)). Finally, there is the `non-atomic axiom': (A8) 8x9y[NTPP(y; x)]. This suces to prove that all regions have an in nite number of NTPPs. Among the relations de ned in terms of C above, much of the existing work on RCC picks out a set of eight `jointly exhaustive and pairwise disjoint' (JEPD) relations: DC, EC, PO, EQ, TPP, NTPP, TPPi and NTPPi (the last two are inverses of the two before). This set is known as the `RCC-8' set, or the `base relations', and has been given considerable emphasis. A JEPD set is one such that one and only one of the set of relations will hold between any ordered pair of entities. Pictures representing intuitive interpretations of the RCC-8 are shown in gure 1. PO(a,b) a

b

TPP(a,b) NTPP(a,b) a a b b

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

ab

b a

EC(a,b) a

b

DC(a,b) a

b

Figure 1: The RRC-8 JEPD Relation Set (Randell et al. 1992) includes a `transitivity table' for the RCC-8 set of relations. This speci es which of the RCC-8 may hold between two regions, given the relation each of them has to a third. Such tables have potential uses in reasoning about consistency, and integrating old and new information. JEPD sets of relations can also be used in reasoning about change 3

`Quasi-boolean' because there is no null region.

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(Cohn, Gotts, Randell, Cui, Bennett and Gooday 1995): given certain assumptions (e.g., that entities change size and shape `smoothly'), one can specify the possible and impossible transitions from one relation to another. For some purposes (e.g. as a basis for representing `vague' spatial regions, without precise boundaries (Cohn and Gotts 1995)) a smaller JEPD set of topological relations, RCC-5, is useful. RCC-5 combines DC with EC as DR, TPP and NTPP as PP, and TPPi and NTPPi as PPi. As was shown in (Gotts 1994), RCC also permits ner topological distinctions between pairwise spatial relations to be made. It has recently emerged that the theory axiomatised by (A1)-(A8) above is undecidable. A proof sketch is included in (Gotts 1996c), but the result itself, as pointed out by (Davis 1996), can readily be derived from a result of (Grzegorczyk 1951).

3 Possible Models of the RCC Axiom-Set

If the RCC topological axioms set out above are to interest SIS researchers, it must be shown that they capture important aspects of what we want to say about spatial relations in SIS. To this end, work on determining the range of models permitted by the axioms is fundamental: we need to know which aspects of intuitive `spatiality' are captured by them and which are not. We also need to know whether the axioms rule out some of the kinds of topological model of spatial relations we might want to develop. As will be seen, the axioms can be criticised as too permissive in some respects, and too restrictive in others. However, they do have interesting topological models, and more specialised versions can be created by adding further axioms. A crucial point about the RCC axioms is the distinction between C(x; y) (`regions x and y connect'), and O (x; y ) (`regions x and y overlap'). A critic (Ho 1994) recently suggested an interpretation under which a `region' is an arbitrary set of points within a space such as the Euclidean plane, and C(x; y) just means that `regions' x and y share a non-empty set of points. In fact, this is ruled out by (A8), which demands that all regions have a PP (speci cally an NTPP), while a `region' under this interpretation could contain a single point, and so have no PP. However, it might appear the suggested interpretation could be modi ed to say that a region is any in nite set of points, and C(x; y) means x and y share an in nite set of points (which would of course itself be a region). If we take this approach, we nd no problem with the fundamental axioms, (A1) and (A2). The de nition of other relations in terms of C goes through | but we nd that C and O are equivalent: P(x; y) simply means that the points in x are a subset of those in y , and O(x; y ) that x and y share an in nite set of points | exactly what C means! The equivalence of C(x; y) and O(x; y) has the consequence that EC(x; y) is always false, and hence that TPP(x; y) is always false, while NTPP(x; y) is equivalent to PP(x; y). If we now consider (A4), we nd that, according to the rst conjunct on the right-hand side of the implication, C(z; compl(x)) should be true when and only when :NTPP(y; x) is true. Under the proposed interpretation, however, C is equivalent to O, and NTPP to PP, so this rst conjunct says that O(y; compl(x)) should be true when and only when :PP(y; x) is true. The second conjunct, however, speci es that O(y; compl(x)) should be true when and only when :P(y; x) is true. If y is EQ to x, therefore, the two conjuncts demand that y both overlap compl(y), and not overlap it. The compl axiom, therefore, rules out the proposed interpretation. The same axiom also rules out the possibility that u is not self-connected (one-piece). We de ne self-connectedness as follows: CON(x) def 8y8z [EQ(x; sum(y; z )) ! C(y; z )]. Imagine that u were some space such as `the surfaces of the Galilean moons of Jupiter' which consists of four separate pieces, one for each of Io, Ganymede, Callisto and Europa4 . Consider the surface of Callisto (calling it x) and the compl of x within the four-surface u. Intuitively, compl(x) should consist of the surfaces of the remaining three satellites. However, we nd that if x and compl(x) are DC, as is the case here, it cannot be the case that 8u[C(u; compl(x))  :NTPP(u; x)]: if u is EQ to x, it is certainly not an NTPP of x, but neither is it connected

4 I choose this example because it suggests that it may be useful to seek a way of extending RCC theory to deal with such cases: there is in this case no larger self-connected whole of the same dimensionality, of which this non-self-connected region could be considered a part, and it seems quite a `natural' region to talk about.

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to compl(x). Third, as noted in (Randell et al. 1992, section 5), RCC theory, and in particular the inclusion of the current compl axiom, makes it hard to construct atomic versions of the calculus: we cannot simply remove or replace (A8) and get a useful atomic version. Consider a version without (A8), and what the complement of an atomic region would be within it. Putting on one side the possibility that the atomic region, a, is u | which gives rise to a consistent but perhaps not very useful model in which there is just one region | the compl axiom says that the regions connected with compl(a) will be those that are not NTPPs of a | that is, all regions. The region compl(a) must therefore be EQ to u. However, all regions are parts of u (given the way P is de ned in terms of C), and therefore overlap (O) with u. Yet the last part of the compl axiom insists that a region overlaps with the compl of a given region, if and only if it is not part of that region. The atom a is certainly part of itself, so according to this part of the axiom, it should not overlap with its complement, u | but all regions must do so, as noted above. We conclude that the only possible atomic model for the set of RCC axioms excluding the non-atomic axiom is the single-region model. As for non-atomic models, the preferred model is not speci ed in (Randell et al. 1992). An informal proof is presented here that models of the RCC axioms are provided by certain sets of points within certain topological spaces. Using an interpretation expressed in terms of point-sets might seem inconsistent with the spirit of the work underlying the RCC approach. However, no alternative has been worked out in any detail, and as will be seen, linking RCC to `conventional' topology sheds considerable light on its properties, and points the way to future investigations of both the current axiom set and possible modi cations. Before setting out the proof, we review some elementary aspects of point-set topology. Topology as understood by the mathematician is divided into two main branches: general or point-set topology, and algebraic topology. In point-set topology, a `topological space' is de ned (Gaal 1964, p.21) simply as a set of points, together with a set of open subsets of that set. The set of points may be nite or in nite, and if in nite, countable (like the integers) or uncountable (like the real numbers). Any given set S with more than one member can be given dierent topologies. In any particular topology on S, the empty set (signi ed ;) and S itself are open; the intersection of any nite set of open sets is open, and so is the union of any set ( nite or in nite) of open sets. Any collection of subsets of S obeying these rules de nes a topology on S. The other basic terms of point-set topology can be de ned in terms of open sets. The complement within S of any open set is a closed set (so S itself and ; are both open and closed). The intersection of any set of closed sets, and the union of any nite set of closed sets, are closed (compare with the situation for open sets). The interior of a set is the largest open set contained in it (so an open set is equal to its interior). The closure of a set is the smallest closed set containing it (so a closed set is equal to its closure). A regular closed set is one which is equal to the closure of its interior. The closure of the interior of any set is regular closed (Requicha and Tilove 1978, p.16), (this technical report is the source of many of the properties of sets used here). A regular open set is one equal to the interior of its closure. When n-dimensional Euclidean space (the n-fold Cartesian product of the set of real numbers) is considered, we can say that an open set is any set of points such that any point in the set is surrounded by an n-dimensional `disc' of points which also belong to the set. Intuitively, if you draw an (in nitely thin) closed curve on a piece of paper the area within the line is an open set of points, the area on-or-within the line a closed set. If you add an isolated point, or an in nitely thin `spike' projecting from this area, the result will still be a closed set of points, but no longer a regular closed set: its interior will still be the area within the closed curve, and the closure of this interior will be the original closed set, without the isolated point or spike. We now show that the nonempty regular closed sets (henceforth NERC sets) of any connected T3 -space, other than a space containing only one NERC set, model the RCC axioms. We use [, \ and  for the set operations of union, intersection, and complementation (within the underlying set of a topological space), and x and x for the closure and interior of set x. As well as NERC sets it will be convenient to de ne a NERO (nonempty regular open) set: a nonempty set equal to the interior of its closure. We could in fact use these sets as a model instead, but choosing NERC sets simpli es our formulae. Every NERC set in any space has a corresponding NERO set (its interior) and every NERO set a corresponding NERC set (its

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closure); in a connected space, the only sets that are simultaneously open and closed are the empty set ;, and the complete set of points S itself; the latter is both NERC and NERO. A T3 -space S, (as de ned by (Gaal 1964, p.80), some other authors call this a `regular space') is one for which, for any point p 2 S and any open set G containing p, there is an open set H containing p such that H  G. A connected space is one which cannot be partitioned into two disjoint open sets. Within such a model of RCC, regions are NERC sets of a connected T3 -space, C(x; y) means that x \ y 6= ; (if we used NERO sets we would have to de ne C in terms of nonempty intersection of the closures of sets), and u is S itself. The predicate NULL is true of the empty set, false of any NERC set. The meanings of the functions sum, prod and compl are as follows: sum(x; y) means (x [ y) (the NERC sets' union); prod(x; y) means (x \ y) (the closure of the interior of the NERC sets' intersections); compl(x) means  x (the closure of the complement of the NERC set). Notice that we could, if we wished, use `the closure of the interior of the NERC sets' union' for sum, and `the closure of the interior of the NERC set's complement' for compl: these formulations always give the same results as those used, but although we lose in uniformity by using the chosen interpretations, the formulae produced are simpler and more readily understood. The `closure of the interior' formulation in the prod case does make a dierence: the intersection of two NERC sets may be nonempty but have no interior, in which case the closure of its interior is ;. Under this interpretation, it is clear that axioms A1 and A2 hold (intersection of nonempty sets is re exive and symmetric). A3 simply asserts that any NERC set intersects S. A4 we leave aside for the moment. A5, given the interpretation of sum above, asserts the following: 8x; y; z [z \ (x [ y) 6= ;  (z \ x 6= ; _ z \ y 6= ;)], with x; y; z ranging over the NERC sets of S. It clearly holds. Before turning to A6, it will be useful to consider what P(x; y) means under the suggested interpretation. Given the RCC de nition: P(x; y) def 8z [C(z; x) ! C(z; y)], we have, in our current model: P(x; y)  8z [z \ x 6= ; ! z \ y 6= ;]. Given that x; y; z range over NERC sets, this turns out to be equivalent to x  y. The implication from x  y to 8z [z \ x 6= ; ! z \ y 6= ;] is obvious: if one set is a subset of another, no set whatever can overlap the rst and not the second. The opposite implication: 8z [z \ x 6= ; ! z \ y 6= ;] ! x  y might also look obvious, but we have to ask if there could be a point of x, not in y, that falls outside any NERC set disjoint from y. This is where the characterisation of S as a T3 -space comes in. The implication above is of course equivalent to: x 6 y ! 9z [z \ x 6= ; ^ z \ y = ;]. If x 6 y, then there is a point p such that p 2 x ^ p 2 y. Now  y (call it G), as the complement of a closed set, is open. In a T3 -space, if we have a point p contained in an open set G, there is always an open set H such that p 2 H and H  G. The closure of an open set is always NERC, so H is a NERC set containing p (and so H \ x 6= ;), but H is disjoint from y (so H \ y = ;). The implication therefore holds. Having reduced the interpretation of P(x; y) to x  y, we are in a position to proceed with A6. This receives the interpretation: 8x; y; z [z \ (x \ y) 6= ;  9w[w  x ^ w  y ^ z \ w 6= ;]]. This can be simpli ed to: 8x; y; z [z \ (x \ y) 6= ;  9w[w  (x \ y) ^ z \ w 6= ;]]. First consider the left to right implication. Because x \ y is a closed set (the intersection of two closed sets is always closed), we know that (x \ y)  x \ y (because the closure of a set is the smallest closed set containing it). If we set w = (x \ y) , it is clear the implication holds. As for the right-to-left implication, for any pair of sets in any space, if u  v, then both u  v , and u  v. The rst of these facts gives us that w  (x \ y) , the second that w  (x \ y) , and since w is a NERC set, we deduce w  (x \ y) . Given that z \ w 6= ;, we conclude that z \ (x \ y) 6= ;, verifying A6. A7 translates as: 8x; y[(x \ y) = ;  :9z [z  x ^ z  y]]. The right-hand side of the equivalence can of course be rewritten as :9z [z  (x \ y)]. The expression (x \ y) always denotes either a NERC or ;. In the former case, let z be this NERC.

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As argued above, we know that (x \ y)  x \ y, so we deduce z  x \ y. If (x \ y) = ;, then (x \ y) = ;. For a set z such that z  x \ y to exist would therefore require that z have empty interior, but a NERC cannot do so. Returning to A4, we can divide it into two parts for convenience: (A4a)8x; y[C(y; compl(x))  :NTPP(y; x)], and (A4b)8x; y[O(y; compl(x))  :P(y; x)]. A4a translates as: 8x; y[y \  x 6= ;  :(y  x ^ :9w[(w \ y 6= ; ^ :9z [z  w ^ z  y]) ^ (w \ x 6= ; ^ :9z [z  w ^ z  x])]. Going from left to right, if y \  x 6= ; and y  x are both true, then  x itself has the properties required of w. It has points in common with both y and x, as required, but in both cases the points shared belong to the boundary of x | what is left of x when its interior is removed. These sets of points cannot have an interior (the boundary of a closed set can never do so). If, on the other hand, the left hand side of the equivalence is false, we have  y \  x = ; or equivalently, y  x . This last implies y  x (since x is closed and is not S, and S is connected). Now we need to prove that there could not be a w such as the rest of the right hand side speci es. Suppose there were such a w. We know it has a non-empty intersection with x, and with y, but it turns out that we can prove its intersection with x is empty. We show this by examining the implications of the assertion: :9z [z  w ^ z  x]. First, we show that it implies (w \ x) = ;. Suppose this were not the case. As (w \ x) is regular closed, if it is nonempty, it is a NERC, and so can be the very z whose existence is denied. Since it is empty, its interior is also empty, and since the interior of the intersection of two sets is equal to the intersection of their interiors, w \ x = ;. Since w is by hypothesis a NERC set, w \ x = ; as well (if an open set intersects a NERC set, it intersects its interior (Requicha and Tilove 1978, p.16)). However, since we already have w \ y 6= ; and y  x, no such w can exist. A4a therefore holds. Now for A4b: (A4b)8x; y[O(y; compl(x))  :P(y; x)]. This turns out to mean: 8x; y[9z [z  y ^ z   x]  y 6 x]. Going from left to right, z , which is contained in y, cannot be contained in x: as a NERC set, it has a nonempty interior, and so cannot be contained in the boundary of a closed set such as  x. It must therefore be partly in the interior of this set, and so partly outside x, as must y. Going from right to left, y 6 x is equivalent to y\  x 6= ;. Since, if an open (and in fact NERO) set such as  x intersects a NERC set, it intersects its interior, the set y \  x is nonempty. As the intersection of NERO sets, this implies it is NERO. Its closure therefore has the properties required of z . There now remains only A8. This, in our interpretation, says: 8x9y[y  x ^ :9w[(w \ y 6= ; ^ :9z [z  w ^ z  y]) ^ (w \ x 6= ; ^ :9z [z  w ^ z  x])]]. Here we use again some of the reasoning that we developed for A4a. If we can show that for any x, there is a y such that y  x , we are home and dry, since we have shown that for a w to meet the conditions laid down, w \ x = ; must be true. Any NERC set x has a nonempty interior, so there is a point p such that p 2 x . In a T3 -space, for any point and any open set G containing it, there is an open set H such that p 2 H and H  G. H, as the closure of an open set, is a NERC set, and if we take x as G, is contained in x . Unless x is open as well as closed, and hence equal to its own interior, this means H  x, so that H meets the conditions for y. If S is connected, the only set which is both open and closed is S itself. The only connected T3 -spaces in which A8 does not hold, therefore, will be those with just a single NERC set. If there is a NERC set other than S in such a space, S also will have an NTPP, namely that other NERC set. A single-NERC connected T3 -space, incidentally, models the single-atom version of RCC which is possible if A8 is omitted. Although in some ways the RCC axioms appear too restrictive (cutting out all potentially useful atomic spaces, and all disconnected ones), in other ways, as will be shown, they appear not restrictive enough. Consider gure 2. This shows the result of the rst few stages in buiding a rather strange set of regions which appear to conform to the RCC axioms. The rst stage in construction is to draw a single square, which is u. The second stage is to draw in an

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Figure 2: A Strange Model of the RCC Axioms NTPP of u, a square with a side one-third as long, and occupying the centre. The third stage

adds ve more square regions, with a side one-ninth that of the original; the fourth adds 25 more, smaller still. We can imagine this process continuing inde nitely. A region will be any square produced after some nite number of stages, any nite sum of such squares, and the dierence between any two such sums of squares. This set of regions obeys all the axioms, but has some strange properties. For example, each region has an in nite number of maximal CON NTPPs: CON NTPPs that are not a PP of any other CON NTPP of the original region. We could of course exclude this particular model by adding another axiom, for example: 8x; y[NTPP(x; y) ^ CON(x) ! 9z [NTPP(z; y) ^ CON(z ) ^ NTPP(x; z )]], which is a kind of `denseness' axiom on the NTPP relation. We can of course declare a restriction on the intended models of the axioms | for example, that u is a Euclidean manifold, and that the regions are all NERCs of that space. However, we may need a further axiom to restrict this set of NERCs. There are ways of dividing a circle or square into two connected NERC sets of points which do not overlap (in the sense of sharing a common NERC), but which do share every point on the boundary. (Basically, we start at a point in the centre, and divide the circle or square with two lines which spiral outward, circling the centre an in nite number of times before reaching the edge.) Every region EC to their sum would thus be EC to both of them. Whether RCC permits such regions is not entirely clear: (Randell et al. 1992) mentions a meta-theoretic restriction that `in nite unions' of regions cannot be allowed on pain of inconsistency, but it seems conceivable that a model of the axioms including such pairs of regions could be speci ed without using such in nite sums, by de ning a set of permitted boundary-curves for regions. Again, we could add another axiom, such as: 8x; y[C(x; y) ! 9z [CON(z ) ^ O(x; z ) ^ O(y; z ) ^ P(z; sum(x; y))]], (meaning that if x and y connect, there must be a self-connected z that overlaps both of them and is part of their sum. However, without a much deeper understanding of the `space' of possible axiom-sets for region-based topological systems, we have no way of knowing what other `weird' models may arise. Again, we could restrict the intended interpretations, for example by insisting that regions are some `nice' subset of NERCs. If we take this approach, however, we must concede that our axioms do not capture enough. Once we impose such a restriction, we can add further axioms to produce alternative versions of the calculus. For example, ways have recently been found of distinguishing 1dimensional, 2-dimensional, 3-dimensional and more-than-3-dimensional regions (Gotts 1996b). Space does not permit these to be explained in detail, but the basic idea is to exploit the fact that it is possible to de ne both simply-connected and multiply-connected regions within RCC. The way one or more multiply-connected regions can relate to an enclosing simplyconnected one are dierent in the 1-, 2-, 3- and more-than-3-dimensional cases: in one dimension, a simply-connected region cannot contain a multiply-connected one; in two dimensions the remaining part of the simply-connected region will be divided in two (or almost so, if the annulus is squeezed to a point anywhere; the `almost' can be made precise). In three dimensions, the remainder need not be so divided, but a pair of multiply-connected regions can be `linked' (like the links in a chain) within a simply-connected one. In more than 3 dimensions there is `too much room' for this to happen.

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4 Discussion

To a considerable extent, what will be required of a topological formalism in an SIS depends on the speci c application area, but some general points can be made. These are listed here, with brief indications of how RCC stands up, and comments linking these points with other issues discussed.  1. We will generally be concerned with entities which exist (or could exist) within the 3-dimensional space of our everyday experience. Entities of higher dimensionality, and ones such as the Klein bottle, which cannot be embedded in Euclidean 3-space (E 3 ) without self-intersection, are not of central interest. This is one of the main reasons that much of mathematical topology is of little relevance to SIS, as a great deal of it is designed to deal with higher-dimensional spaces.  2. We want to deal with both the topological properties of individual entities, and the topological relations between pairs, and larger sets, of entities. Mathematical topology has had rather little to say about the kinds of spatial relation discussed here, at least directly. It has tended to classify properties of spaces, and mappings between them, rather than relations in our sense. RCC, on the contrary, is relation-based; if we want to talk about topological properties of spaces or spatial entities, we must de ne them in terms of relations between their parts. While a surprising amount can be achieved in this way (Gotts 1996a), the de nitions arrived at are often neither simple nor intuitively appealing.  3. We will often want to be able to deal with time, change, and spatio-temporally extended processes (e.g. water ow, erosion). Although not described in any detail here, RCC does have resources for this.  4. We may want to model space either as `atomic' (having parts which cannot be further subdivided), or as non-atomic. As seen, the current RCC axioms have only one, not very useful atomic model.  5. Entities without determinate boundaries (clouds, conurbations, ranges of bird species) may require modelling. RCC has considerable promise here (Cohn and Gotts 1996).  6. We may want to model entities of dierent, or even mixed dimensionality (see for example (Worboys 1992). RCC does not look promising here, while algebraic topology provides, in simplicial complexes and CW-complexes (Munkres 1984), formalisms designed for the purpose. See (Gotts 1996a) for work on an RCC-like formalism that deals with entities of dierent dimensionalities.  7. We want to be able to deal with states of partial knowledge (where we do not have a full description of the topological properties and relations between a set of entities of interest). The ability to construct complex taxonomies of spatial properties and relations makes RCC promising in this regard.  8. Detecting inconsistencies, and combining old and new information, are important. Transitivity tables are intended for precisely these purposes, but their utility remains to be demonstrated in practice.  9. We want ecient inference procedures. As noted above, RCC is undecidable. Work is going on (Bennett 1994) in an attempt to isolate decideable parts of the theory. The RCC formalism, as can be seen above, is constructed by starting with a set of axioms and de nitions, xing the properties of a primitive relation (connection) and of the complete collection of entities (regions) to which this relation applies. So far as the formalism is concerned, the regions and the relation of connection have precisely the properties speci ed by the axioms, and nothing more. This `axiomatic' approach to developing a topological formalism can be contrasted with an alternative, `constructive' one, (e.g. (Egenhofer and Franzosa 1991, Egenhofer and Franzosa 1995, Worboys 1992, Worboys and Bofakos 1993)). The constructive approach begins by singling out a class of `elementary' objects, with topological properties which are speci ed as those of the topologically `simplest' kind of region that is of interest. More complex objects can then be constructed from these elementary ones. We will take a brief look at Egenhofer's approach here. Egenhofer begins with twodimensional, disclike regions, without holes or other complications, and classi es the relations

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between pairs of such regions in terms of the incidence relations between the interiors and boundaries of the pair (he calls this approach the `4-intersection', as it is based on considering the relations between the boundary of region A with both the boundary and interior of B, and of the interior of A with both boundary and interior of B). Remarkably, this approach produces the same set of eight basic relations between pairs of regions as RCC, although Egenhofer's set of relations was designed initially to apply only to pairs of discs embedded in Euclidean 2-space, while the RCC relations are intended to apply to any kind of regions whatever. Both approaches allow these relations to be subdivided further, but the strategies for doing this dier considerably. Egenhofer (Egenhofer and Franzosa 1995) introduces a collection of `invariants' (properties of relations which remain unchanged across topologically equivalent situations), which together are claimed fully to characterise each topologically distinct relation between two (topological) discs embedded in the plane. Relations between more complex 2-dimensional regions are handled (Egenhofer, Clementini and Di Felice 1994) by analysing them in terms of the relations between pairs of discs. How far this approach can be generalised, particularly when attention is turned to 3-dimensional regions, with their far more complex relations, is an interesting question. (For example, Egenhofer's techniques for dierentiating topologically distinct relations between pairs of discs depend upon the boundary-points of each disc having a circular ordering; in the 3-dimensional case, the points of the boundary of even the simplest type of region do not have such an ordering.) Using the axiomatic approach, it may be possible to construct a useful hierarchy of axiom systems, starting with a highly general system, such as RCC itself, and adding additional axioms to obtain greater speci city while retaining the advantages of a common formalism for dierent applications. By contrast, the constructive approach may nd that its chosen set of methods do not extend beyond a certain point, as noted. The corresponding advantage for constructive systems is that their methods can be precisely adapted to the entities and relations they deal with. The dierence between the two approaches parallels that between the two main branches of mathematical topology: point-set topology, with which we have largely been concerned here, and algebraic topology. Point-set topology de nes a complex hierarchy of types of spaces in terms of the axioms their point-sets obey; algebraic topology tends to build up its spaces from constituents which are topologically equivalent to n-dimensional discs, joined in various well-de ned ways. Future work planned includes both further comparisons of these two approaches, an examination of the possibility of combining them. Within the axiomatic approach, more work is planned on linking the RCC axioms and similar sets to classes of topological spaces.

References

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