Ontologies for Plane, Polygonal Mereotopology

Several authors have suggested that a more parsimonious and conceptually elegant treatment of everyday mereological and topological reasoning can be obtained by adopting a spatial ontology in which regions, not points, are the primitive entities. This paper challenges this suggestion for mereotopological reasoning in 2-dimensional space. Our strategy is to de ne a mereotopological language together with a familiar, point-based interpretation. It is proposed that, to be practically useful, any alternative region-based spatial ontology must support the same sentences in our language as this familiar interpretation. This proposal has the merit of transforming a vague, open-ended question about ontologies for \practical" mereotopological reasoning into a precise question in model theory. We show that (a version of) the familiar interpretation is countable and atomic, and therefore prime. We conclude that useful alternative ontologies of the plane are, if anything, less parsimonious than the one which they are supposed to replace. 1 The problem One of the many achievements of coordinate geometry has been to provide a conceptually elegant and unifying account of the nature of geometrical entities. According to this account, the one primitive spatial entity is the point, and the one primitive geometrical property of points is coordinate position. All other geometrical entities|lines, curves, surfaces and bodies|are nothing but collections of points; and all properties and relations involving these entities may be de ned in terms of the relative positions of the points which make them up. The success and power of this reduction is so great that the identi cation of spatial regions with the sets of points which they contain has come to seem virtually axiomatic. Yet various authors have sought to reverse this order of rational reconstruction, treating regions as primary, and admitting points, if at all, as logical constructions out of them. The best known of these approaches is perhaps Tarski's [36] axiomatization of Euclidean geometry, taking spheres to be the primitive entities. But the policy of taking regions as primitive is most attractive when considering problems involving mereological (part-whole) and topological notions|that is, where no metric information is to hand. If regions are rst class entities and points are logical constructions based on them, then who knows what interesting new ways of considering spatial entities and relations there might be? Clarke [12], [13], following an idea of Whitehead [40], sought to reconstruct mereotopology in terms of a primitive relation of connection holding between regions. Following this work, Biacino and Gerla [5] have studied models of Clarke's theory. More recently, and partially as a response to debates concerning temporal reasoning and knowledge representation (for example, Allen [1]), Clarke's mereotopology has received attention from several research groups in AI, working with the loosely de ned area of qualitative spatial reasoning, for example, Gotts, Gooday and Cohn [18], Asher and Vieu [2], and Borgo, Guarino, and Masolo [7]. Motivations for these developments vary, and we do not intend to provide a comprehensive account of them here. However, one common recurring theme is the suspicion that the familiar, point-based, view of space generates a richer ontology than is needed for mereotopological reasoning in \practical" situations. For example, Euclidean space contains not only the sorts of regions we want to recognize for everyday purposes, but also strange, physically unrealizable regions of the kind that populate point-set topology textbooks. Such regions seem to be mere artifacts of the Euclidean model of space|useless for describing, and reasoning about, the world we inhabit. If, on the other hand, we regard regions as primitive entities, perhaps we can be more selective as to what regions we take to exist and what mereotopological properties we take them to have. Perhaps|so some researchers in mereotopology suggest|treating regions as primary opens up the prospect of simpler and more parsimonious spatial ontologies than the familiar model based on points in the real plane. The present paper examines this suggestion for the special case of planemereotopology. We show that, under certain reasonable assumptions as to what practical mereotopological reasoning might involve, taking regions rather than points as primitive cannot lead to a more parsimonious spatial ontology. 1 a) Three polygons (each di erently shaded) b) Three non-polygons c) Three pairs of polygons and their respective sums Figure 1: Polygonal mereotopology 2 Polygonal mereotopology To get an idea of what \practical" mereotopological reasoning might involve, consider computer systems specialized for representing plane spatial data, such as, for example, Geographic Information Systems (GISs). Virtually all such systems represent regions of space by means of boundaries consisting of nitely many straight lines and straight-line segments. In e ect, then, all plane regions recognized by such systems are polygons. Experience has shown that such a spatial ontology, whatever its philosophical shortcomings, is certainly equal to the task of describing everyday planar spatial arrangements such as those found on maps and charts, since every arrangement of regions one is likely to encounter can be approximated by polygons with arbitrarily high accuracy. Suppose, then, we take as our spatial ontology the set P of polygons in the plane. We discuss the formal construction of P later; for the present, all that matters is that all members of P are the interiors of plane regions bounded by nitely many straight lines, as shown in gure 1a. Note that we allow polygons to consist of more than one piece, and to contain holes (as long as those holes have straight-line boundaries); however, polygons are not allowed to contain \cracks", as shown in gure 1b. In addition, we consider the empty set and the whole plane to be polygons. It turns out that P forms a Boolean algebra. In this Boolean algebra, the product of two polygons is their intersection; the negation of a polygon is that part of the plane lying outside it and its boundary; and the sum of two polygons is the polygon formed by taking their union and `rubbing out' any internal boundaries that result. Figure 1c illustrates the sum-operation. Accordingly, our mereotopological language will be equipped with functions-symbols , and + to denote these operations, as well as the constants 0 and 1 to denote the empty set and the whole plane, respectively. Note that the formula x:y = x states that x is a subset of y. Hence the Boolean functions can express various mereological properties and relations involving polygons. In point-set topology, it is usual to de ne an open set as being connected if it is not the union of 2 two disjoint, nonempty, open sets. Intuitively, connected sets are just that|they consist of one piece. Accordingly, our language will be equipped with a one-place predicate c(x) to express the property of being a connected polygon. (Incidentally, since we are concerned only with open sets, the right-most polygon in gure 1a is not connected.) If x and y are disjoint, connected and non-empty, then it is possible to show that the formula c(x + y) is satis ed if and only if x and y share one or more proper straight line segments on their boundaries. In other words, the formula c(x + y) can be used to express the relation of external contact along an edge. Hence, the predicate c(x), together with the Boolean functions, can express various topological properties and relations involving polygons. Thus, we take our mereotopological language L to be a rst-order language with equality and nonlogical constants +, , , 0, 1 and c(x). The set P of polygons will form the domain over which the variables ofL range, and the interpretation of the non-logical constants ofL given above de nes a modelP on the domainP . The sentences Th(P) true in this model represent, as it were, the facts of mereotopology according to the the polygonal ontology employed in most computer systems for representing plane spatial data. We propose to take Th(P) to be the facts of \practical" mereotopological reasoning. After all, the polygonal model P is relatively simple, admits no pathological regions, and yet is mereotopologically non-trivial and nds use in many practical applications without apparent loss of useful representational power. We further propose that an alternative spatial ontology for practical mereotopological reasoning is simply an alternative model of Th(P)|that is, a model A such that A P but A 6' P. The domain A of A will form the set of regions of space and the relations and properties needed to interpret the terms in L will give this space its mereotopological structure. Note that the domain P contains only polygonal regions, and not the points and lines of which they are made up. Thus, we employ a language which can talk, in the rst instance, only about regions, in keeping with the spirit of mereotopology. On the other hand, our model P is fundamentally Euclidean, in that polygons are objects in the Euclidean plane de ned in terms of the points they contain or the lines that bound them. Thus, P is, as it were, our familiar ontology|one constructed in the familiar way from points in the Euclidean plane. A general model of Th(P), by contrast, may have any sorts of objects in its domain, either primitive or constructed in some other way. The problem we face in the sequel is to identify such general models, and to determine whether any of them constitute a more elegant and parsimonious spatial ontology than P. It may be objected that this strategy is too conservative. After all, who says that the facts of practical mereotopological reasoning|the facts that we would want any alternative spatial ontology to support|are the facts that are true of the polygonal data-structures employed in many computer systems? Perhaps we could nd a better theory of space by revising this \computer-mereotopology". To some extent, this criticism is justi ed: we have little to say in favour of the polygonal theory, except that it is familiar, easily formalized and seems to be widely and successfully used in a vast range of practical applications. (Proponents of other theories of mereotopological reasoning should be so lucky.) Nevertheless, our strategy does have the virtue of transforming a vague and open-ended question about ontologies for \practical" mereotopological reasoning into a precise, technical question in model theory. To be sure, we do not regard the solution of this problem to be the last word on the metaphysics of space; but at least we now have a de nite question to address. 3 Regular sets Before any serious analysis begins, we must formalize the familiar model P and our mereotopological language L. We do this in two stages: the present section establishes some basic groundwork at a rather high level of generality; the next section goes on to present the details of P and L. Our rst task is to resolve the issue of whether regions include their boundary points. We adopt an approach, based on regular sets, which has become reasonably standard in discussions of spatial description languages. De nition 3.1 Let X be a topological space and x X. Then the set Sfy Xj y open, y \ x = ;g is an open set in X called the pseudocomplement of X, written x0. We say that x X is regular if 3 x = x00. Obviously, x \ x0 = ;. Hence, if x is open, then x x00. The following well-known theorem underlies the importance of the regular sets to mereotopology. We state it here without proof. (See, e.g. Johnstone [21], section 1.13.) Theorem 1 Let X be a topological space. Then the set of regular sets in X forms a Boolean algebra with top and bottom de ned by 1 = X and 0 = ;, and Boolean operations de ned by x:y = x \ y, x+ y = (x [ y)00 and x = x0. Accordingly, we shall sometimes use the term regular Boolean algebra of a topological space X to refer to the Boolean algebra of regular sets of X. We denote this algebra by M (X). When dealing with the elements of such a Boolean algebra, we shall write x:y, x+ y, x and instead of x\ y, (x [ y)00 and x0, respectively. There is an alternative characterization of the regular sets. If X is a topological space and y X is any set, then there will always be a largest open set contained in y, called the interior of y, and denoted by y0. Likewise, there will always be a smallest closed set containing y, called the closure of y, and denoted by [y]. The set [x] n ([x])0 is called the frontier of x, and is denoted by F(x). Lemma 3.1 Let X be a topological space and x X. Then x0 = X n [x]. Proof: X n [x] = X nTfu j u X closed and x ug =SfX n u j u X closed and x ug = Sfy X j y open and x \ y = ;g = x0. 2 Lemma 3.2 Let x be a subset of a topological space X. Then x00 = ([x])0. Proof: Certainly, ([x])0 is open and is a subset of [x] = X n x0. Then x0 \ ([x])0 = ; so ([x])0 x00. Conversely, x0 \ x00 = ; so x00 X n x0 = [x]. Since x00 is open and x00 [x], x00 ([x])0. 2 Hence, a set in a topological space is regular if and only if it is equal to the interior of its closure. Theorem 1 and lemma 3.2 show that the regular sets of a topological space form a natural domain of quanti cation for a mereotopological language. First, theorem 1 shows that the part-whole relationship, restricted to the regular sets, still obeys the axioms of a Boolean algebra, and so will be mathematically manageable. Second, lemma 3.2 guarantees that boundary points are ignored, in the sense that no two regular regions di er only with respect to their boundary points, and that all non-regular regions di er from regular regions only with respect to their boundary points. We conclude this section with two lemmas concerning regular sets: both will be used in the results derived below. As usual in topology, we say that an open set x is connected if there do not exist two nonempty, disjoint open sets whose union is x. Lemma 3.3 Let a1; a2; a3 be regular sets of a topological space X with a1 + a2; a2 + a3 connected and a2 6= 0. Then a1 + a2 + a3 is connected. Proof: By a standard result, if x is any connected set and x y [x] then y is connected. Since a2 6= 0, (a1 + a2)[ (a2 + a3) is certainly connected. And (a1 + a2)[ (a2 + a3) (a1 + a2) + (a2 + a3) = ((a1 + a2) [ (a2 + a3))00 [(a1 + a2) [ (a2 + a3)], by lemma 3.2 2 Lemma 3.4 Let X, Y be topological spaces and a homeomorphism from X onto Y . Let a; b be regular sets in X. Then (a) and (b) are regular sets in Y with: (i) (a:b) = (a): (b); (ii) ( a) = (a); and (iii) (a+ b) = (a) + (b). 4 For ease of notation, if x and y are open, we shall write x:y for x\ y and x for x0, even when x and y are not known to be regular. Proof: Equation (i) holds because is 1{1. Equation (ii) follows if we can show that ( a) (a) and (a) ( a). By equation (i), (a): ( a) = (;) = ;. Since 1 is continuous, (a) and ( a) are open. Hence ( a) (a). To prove the reverse inclusion, it su ces to show that, for any open set z Y satisfying z: (a) = ;, z ( a). Let z be such a set. Since z: (a) = ;, it follows that 1(z):a = ;. And since is continuous, 1(z) is open. Therefore 1(z) a, so z ( a). Equation (iii) follows from (i) and (ii). Finally, if a is regular, by equation (ii), (a) = ( a) = (a), so (a) is regular. 2 4 The polygonal models Having dealt with the very general notions of regular sets and their Boolean algebras, we return in this section to the polygonal domain. Any line in the Euclidean plane, IR2, cuts IR2 into two connected, open sets, called half-planes. It is easy to see that these sets are regular, with each being the pseudocomplement of the other. Hence, we can speak about the sums, products and complements of half-planes in M (IR2). De nition 4.1 A basic polygon is the intersection of nitely many half-planes in IR2. A polygon is the sum, in M (IR2), of any nite set of basic polygons. We denote the set of polygons by R, and will sometimes refer to it as the polygonal domain. Thus, the elements of R are simply polygons as introduced in the previous secion. Of course, R is not the only well-behaved spatial domain we might choose. If a line is de ned by an equation ax + by + c = 0, where a, b and c are rational numbers, we call it a rational line; and if a half-plane is bounded by a rational line, we call it a rational half-plane. It is obvious that a pair of rational lines can intersect only at points with rational coordinates. Now we de ne: De nition 4.2 A rational basic polygon is the intersection of nitely many rational half-planes. A rational polygon is the sum, in the Boolean algebra of regular sets of the plane, of any nite set of rational basic polygons. We denote the set of rational polygons by Q, and will sometimes refer to it as the rational polygonal domain. Thus, R, or perhaps, more modestly, Q, is the spatial ontology recognized by computer systems such as GISs|both domains provide a simple view of space from which any remotely pathological behaviour has been excluded. Clearly, R is uncountable, whereas Q is countable; so R and Q are di erent structures. Nevertheless, their ontologies are very similar, and share many basic properties. For brevity, we use the symbol P to denote either R or Q. Now that we have de ned the polygonal domain (or, more precisely, domains), P , we introduce our mereotopological language L. Let L be the rst-order language with equality having the following non-logical symbols: 1. one 1-place predicate c(x) 2. two constant symbols 0 and 1 3. one 1-place function symbol x 4. two 2-place function symbols x+ y and x:y Informally, the functions +, : and correspond to operations in the Boolean algebra M (IR2); and c(x) means that x is connected (in the usual topological sense). Thus, L has a mereological component 5 in the form of Boolean connectives representing operations on regular sets, and a topological component in the form of a connectedness predicate. Formally, we give L two `familiar' interpretations, R and Q, corresponding to the domains R and Q, respectively. First, we must check that the Boolean operations of M (IR2) can be applied freely in these domains. Theorem 2 P is a Boolean subalgebra of M (IR2). Proof: We need only show that P is closed under the Boolean operations. But this is obvious given the distribution laws for M (IR2). 2 De nition 4.3 We de ne the polygonal modelR to have the domain R and the following interpretations of the predicate, constant and function symbols in L: 1. cR(x) = x is connected 2. 0R = ;; 1R = IR2 3. For all a 2 R: R(a) = a 4. For all a; b 2 R: +R(a; b) = a + b and :R(a; b) = a:b We de ne the rational polygonal model Q exactly as for R but with R and R replaced throughout by Q and Q respectively. Again, in view of the similarities between R and Q, we write P to refer indeterminately to either. Thus, the domain of P is P . Anticipating a result of the next section, it turns out|unsurprisingly|that R and Q make the same sentences of L true. That is, the ontologies Q and R are indistinguishable for the mereotopological language L. Hence we may write Th(P) to denote Th(R) = Th(Q). Our main task in this paper is to nd alternative models of Th(P). We nish this section on the familiar models for L with an example to show that the pains we took to de ne our domain of interpretation were not in vain. Consider the following formula of L: 8x18x28x30@0@ ^ 1 i 3 c(xi) ^ c(x1 + x2 + x3)1A! (c(x1 + x2) _ c(x1 + c3))1A : (1) This formula asserts that, if the sum of three connected regions is connected, then the rst must be connected to at least one of the other two. It is true in the model P; but it would be false in a model whose domain extended to all regular sets of the plane. For consider the regions a1, a2 and a3 de ned by a1 = f(x; y)j 1 < x < 0 ; 1 x < y < 1 + xg a2 = f(x; y)j0 < x < 1 ; 1 x < y < sin(1=x)g a3 = f(x; y)j0 < x < 1 ; sin(1=x) < y < 1 + xg ; and depicted in gure 2. (Note: in this gure, the x-axis has been dilated.) It is not di cult to show that a1, a2 and a3 are regular, that a1 + a2 + a3 is the interior of the large triangle in gure 2 and so is connected, but that neither a1 + a2 nor a1 + a3 is connected. Thus, for the purposes of practical mereotopological reasoning, (1) is quite a sensible formula to have as a theorem of our calculus, since|as it turns out|the only counterexamples are rather pathological and seem to be artifacts of the Euclidean model of space rather than anything that one could come across in real life. (Adopting such a stance is very much within the spirit of mereotopological research in AI, 6 a3 a1 a2 Figure 2: Three regular sets in the plane where the ability to reason e ciently in everyday situations is in focus.) But this case does demonstrate the importance of having a precise characterization of the regions our mereotopological language talks about. When looking for models elementarily equivalent to P as alternative ontologies for practical mereotopological reasoning, we are making some very speci c choices about the facts of mereotopology that we want to support. 5 Topological analysis Our next step is to establish some basic topological properties of P . The most important of these properties are described by theorem 4 and lemmas 5.9 and 5.10. All other results in this section are ancillary. We begin with a lemma on which much of the subsequent analysis depends. Lemma 5.1 Any element of P is the sum of nitely many connected elements of P . Proof: Since half-planes are convex, basic polygons are convex, and so are certainly connected. 2 It easy to see that this property does not hold for all Boolean sub-algebras of M (IR2), even where the elements are relatively well-behaved. For example, even when x and y are Jordan regions (i.e. topologically equivalent to the unit circle), the intersection x:y may have in nitely many disconnected parts. It is precisely to prevent this possibility that we shall restrict attention to the domains R and Q. To be sure, the polygons are not the only set of regions satisfying lemma 5.1, but they are the simplest. As usual, we take a component of a set to be a maximal, nonempty connected subset of that set. Lemma 5.2 Let a 2 P and let c be a component of a. Then c 2 P . Moreover, a equals the sum of its components. Proof: By lemma 5.1, let c1; : : : ; cn be connected elements of P such that a = c1 + : : :+ cn. For all i (1 i n), if c:ci 6= 0 then, by lemma 3.3, ci + c is connected. If, in addition, ( c):ci 6= 0, then c < c + ci, contradicting the maximality of c. Thus, if c:ci 6= 0, then ( c):ci = 0. Hence c can be expressed as the sum of various ci (1 i n), and c 2 P . The remainder of the lemma is trivial. 2 Let M be any Boolean subalgebra of M (X). If A is a nite subset of M and the elements of A are pairwise disjoint, nonempty and sum to a 2 M , we call A a partition of a in M . If, in addition, every element of A is connected, we call A a connected partition of a in M . In the case a = 1, we refer to A, simply, as a (connected) partition in M . The following (rather technical) lemma will be useful later. Lemma 5.3 Let X be a topological space, M a Boolean subalgebra of M (X) and a1; : : : ; an a partition in M . Let m be such that 1 m n. Then a1 + : : :+ am = a1 [ : : : am [ fpj p 2 F(ai) for some i (1 i m), p 62 F(aj) for any j (m < j N)g. 7 Figure 3: A graph* with two nodeless edges Proof: Denote the right hand side of the above equation by x. Suppose p 2 [aj] for some j (m < j n). Then p 2 F(aj) or p 2 ([aj])0 = aj by lemma 3.2. If p 2 aj, then p 62 [ai] for any i (1 i m) by the disjointness of a1; : : : ; an. Either way, then, p 62 x. Suppose p 62 [aj] for any j (m < j n). Then certainly p 62 F(aj) for any j (m < j n). Moreover, a1; : : : ; an sum to 1, so [a1] [ : : :[ [an] = 1. Hence p 2 [ai] for some i (1 i m), so, again, p 2 F(ai) or p 2 ([ai])0 = ai by lemma 3.2. Either way, p 2 x. Hence x = (1 n [am+1]): : : : :(1 n [an]). By lemma 3.1, x = ( am+1): : : : :( an) = a1 + : : :+ am. 2 Connected partitions play an important role in understanding R and Q. In particular, we have: Lemma 5.4 Let A be a nite subset of P . Then there exists a connected partition C of P such that each a 2 A is expressible as a sum of various c 2 C. Proof: If A = fa1; : : : ; ang, let C be the set of all components of all non-zero products of the form a1: : : : : an. By lemma 5.2, these components are elements of P , and form a connected partition such that every ai can be expressed as a sum of various C. 2 Furthermore, it should come as no surprise that we can picture connected partitions in P by thinking in terms of plane graphs. De nition 5.1 A graph* G is a plane graph in the closed real plane having no nodes of degree 0, together with a (possibly empty) set of nodeless edges. These nodeless edges are all Jordan curves intersecting no other edge of G (nodeless or otherwise). A graph* is piecewise linear if all of its edges lie on nitely many straight lines; a graph* is rational piecewise linear if all of its edges lie on nitely many rational straight lines. Hence, all plane graphs in the closed plane are graphs*. Figure 3 shows a piecewise linear graph* (where the page represents the whole plane) with two nodeless edges. This particular specimen also has no isthmuses and no nodes of degree 2. We note also that Euler's formula for a k-component graph, namely n e+f = k+1, applies also to a k-component graph*, where nodeless edges do not count as components. If G is a graph*, we denote by jGj the set of points in the edges and vertices of G, ignoring the point at in nity. If G is a graph*, then it makes sense to talk about the faces of G in the open plane|that is, the components of IR2 n jGj. Henceforth, if G is a graph*, when we speak of `the faces of G', we mean `the faces of G in the open plane.' The following basic lemma establishes the importance of piecewise linear and rational piecewise linear graphs. Theorem 3 Let A be a connected partition in R; then there exists a nite piecewise linear graph* with no isthmuses whose faces are precisely A. Conversely, let G be a nite piecewise linear graph* with no isthmuses; then the faces of G form a nite connected partition of R. The above equivalence also holds if \R" is replaced by \Q" and \piecewise linear" by \rational piecewise linear". 8 Proof: Suppose that a1; : : : ; an form a connected partition in R. Consider all the half-planes involved in the construction of elements a1; : : : ; an. The lines bounding these half-planes form a nite graph* G in the obvious way, and the faces of G must form a connected partition of basic polygons, say, b1; : : : ; bN . Moreover, each ai (1 i n) can certainly be expressed as a sum of various bj (1 j N ). By renumbering if necessary, let a1 = b1 + : : :+ bm for some m (1 m N ). Now remove from G all nodes p such that p 62 SfF(bk)jm < k Ng and all edges e such that e 6 SfF(bk)jm < k Ng. The result will be a graph* G1 in which the faces b1; : : : ; bm are merged into a number of faces f1; : : : ; fl for some l (1 l m). The union of these faces will then be the set b1 [ : : : bm [ fp 2 jGj :p 2 F(bi) for some i(1 i m), p 62 F(bj) for any j(m < j N )g By lemma 5.3 this set is just b1 + : : :+ bm = a1. Since a1 is connected, l = 1 and G1 contains the face f1 = a1. Proceeding in the same way for a2; : : : ; an yields a graph* G = Gn with faces a1; : : : ; an. That G has no isthmuses follows from the fact that each face of G is regular. Conversely, suppose that G is a nite piecewise linear graph*; then the edges of G lie on nitely many straight lines l1; : : : ; ln. Consider the graph G made up of all of these lines (extended in both directions). Each face of G is a basic polygon; hence each face fi of G will be divided into a nite number of basic polygons, say, bi;1; : : : ; bi;mi by a nite number of straight lines. Since G has no isthmuses, f is a regular set, and it is easy to check that no smaller regular open set contains bi;1; : : : ; bi;mi. In other words, f = bi;1 + : : :+ bi;mi 2 R. The corresponding proof for Q is identical except for the obvious changes. 2 Next we come to some topological results concerning P which, as we shall see in the next section, will have a signi cant e ect on possible alternative models of Th(P). Lemma 5.5 There exists a function e : IN ! IN such that, for all n > 0, if a1; : : : ; an are disjoint, connected elements of P , then there exist at most e(n) points lying on the boundaries of more than two of these regions. Proof: We suppose that p1; p2; p3 are distinct points all lying on the boundaries of a1, a2, and a3 and derive a contradiction. The result then follows by putting e(n) = n(n 1)(n 2)=3. Let p1; p2; p3 be as described. Choose points q1; q2; q3 such that qi 2 ai (i = 1; 2; 3). Since a1, a2, and a3 are polygons, it is clear that for i = 1; 2; 3, we can draw three end-cuts in ai, say i;1, i;2, and i;3 from the point qi to the points p1, p2, and p3, respectively. (An end-cut in an open set x is a Jordan arc lying in x except for one end-point, which lies on F(x).) Since we can choose i;1, i;2 and i;3 so that they intersect only at qi, this gives us a planar embedding of the graph K3;3, which is well-known to be non-planar (see, e.g. Bollob as [6], p.19). 2 Lemma 5.6 There exists a function f : IN! IN such that, for all n > 0, if A is any connected partition in P with n members and G is a piecewise linear graph* with no nodes of degree 2 whose faces in the open plane are A, then the size of G is bounded by f(n). Proof: Assuming the result that any node of a plane graph of degree greater than 2 must lie on the boundary of at least 3 faces, by lemma 5.5, the number of nodes in G is bounded by a function of n. The lemma then follows from Euler's formula. 2 We then have: Theorem 4 There exists a function g : IN ! IN such that, for all n > 0, there exist at most g(n) n-element connected partitions in P up to homeomorphism. 9 Proof: By theorem 3, any such partition is the set of faces of some piecewise linear graph* with no isthmuses, hence of some piecewise linear graph* with no isthmuses and no nodes of degree 2. By lemma 5.6, all such graphs* are of size bounded by f(n). Assuming the result that every abstract graph can be embedded in the closed plane in only nitely many homeomorphically distinct ways, the result follows immediately. 2 We note in passing that theorem 4 is false for Euclidean spaces of higher dimension than 2. It is also false for arbitrary partitions of M (IR2). The following lemmas are concerned with showing that P is, in a sense that will become clear below, topologically homogeneous. It is well-known that every nite plane graph G in the open plane can be continuously deformed into piecewise linear plane graph G0. (See, e.g. Bollob as [6], p.16.) Indeed, this can be done in such a way that piecewise linear edges in G are una ected. In e ect, nite plane graphs can have their edges `straightened out' by a homeomorphism, without a ecting any points in those faces whose boundaries involve only straight edges. These results can easily be extended to nite graphs*. If is a homeomorphism of the open plane onto itself and a a subset of the open plane, we write ja to denote the restriction of to a. Then we have: Lemma 5.7 Let a; b be connected elements of P such that there is a homeomorphism of the open plane onto itself taking a to b. Let a1; : : : ; an be a connected partition of a in P . Then there exists a connected partition b1; : : : ; bn of b in R and a homeomorphism of the open plane onto itself such that j a = j a and (bi) = bi for all i (1 i n). Proof: We give the proof for P = R; the case P = Q is similar. Let the components of a be t1; : : : ; tm. Since t1; : : : ; tm; a1; : : : ; an is a connected partition, theorem 3 guarantees that we can nd a piecewise linear graph* G with no isthmuses having these elements as faces. Now maps a to b, hence the components of a to the components to b, hence G to a graph* G0 with faces u1; : : : ; um; f1; : : : ; fn, say, where f1 + : : : ; fm = s. But then we can continuously deform G0 to a piecewise linear graph* G00 without a ecting any points in b or its frontier. Hence, the faces of G00 will be u1; : : : ; um; b1; : : : ; bn, say. Thus, there is a homeomorphism 0 of the closed plane onto itself which is the identity mapping outside b and which maps fi to bi, for all i (1 i n). Since G00 clearly contains no isthmuses, theorem 3 guarantees that the faces of G00 will be in S, so that = 0 is the required homeomorphism. 2 Lemma 5.8 Let a; b be connected elements of P such that there is a homeomorphism of the open plane onto itself taking a to b. Let a0 2 P satisfying a0 a. Then there exists b0 2 P and a homeomorphism of the open plane onto itself such that j a = j a and (a0) = b0. Proof: By lemma 5.4, we can nd a nite connected partition of a in P some of whose elements sum to a0. The result then follows from lemma 5.7. 2 The following notation will prove useful. Let a1; : : : ; an; b1; : : : ; bn be elements of P . Then we write a1; : : : ; an b1; : : : ; bn if there is a homeomorphism mapping the open plane on to itself such that (ai) = bi for all i (1 i n). Now we can state the lemma guaranteeing homogeneity of P : Lemma 5.9 Let a1; : : : ; an; b1; : : : ; bn; a 2 P such that a1; : : : ; an b1; : : : ; bn. Then there exists b 2 P such that a1; : : : ; an; a b1; : : : bn; b. Proof: Let be a homeomorphism of the closed plane onto itself mapping a1; : : : ; an to b1; : : : ; bn. Let c1; : : : ; cN be all the components of all products of the form a1: : : : : an and let d1; : : :dN be all the 10 components of all products of the form b1: : : : : bn. Then, by lemma 5.2, c1; : : : cN and d1; : : : ; dN are connected partitions in P , and by renumbering if necessary, maps c1; : : : ; cN to d1; : : : ; dN . It su ces to nd a b 2 P such that c1; : : : ; cN ; a d1; : : : ; dN ; b. For all j (1 j N ), let c0j = a:cj. By lemma 5.8, there exists a d0j 2 P and a homeomorphism j mapping c0j to d0j and equal to outside cj. Then the mapping =[f jjcj : 1 j Ng [ jF(c1)[:::[F(cN ) is a homeomorphism of the open plane onto itself mapping cj to dj for all j (1 j N ) and mapping a = c01 + : : :+ c0N to b = d01 + : : :+ d0N 2 P as required. 2 Finally we turn to the embedding ofQ inR. Again, we show that this embedding satis es a topological homogeneity property. Lemma 5.10 Let a1; : : : ; an 2 Q and b 2 R. Then there exists a 2 Q such that a1; : : : ; an; b a1; : : : ; an; a. Proof: The result follows from theorem 3 and the fact that any piecewise linear graph can be homeomorphically `tweaked' so that all line-segments satisfy equations with rational coe cients. The details are routine. 2 6 Model-theoretic analysis This section contains the main technical result of this paper, theorem 8. As we shall see, this theorem has negative consequences for the search for alternative spatial ontologies. Throughout this section, we use the notation a to denote an ordered n-tuple a1; : : : ; an. Let us begin by establishing the promised elementary equivalence of Q and R. For the remainder of this paper, we assume that the rst order language in question is L as de ned in section 4. First, a reminder from model theory. A type ( x) in variables x is a maximal consistent set of formulae having x as their only free variables. Given a model A, we say that a tuple a (of the right arity) belongs to type if A j= [ a] for every ( x) 2 ( x). Lemma 6.1 Let a and b be tuples in P such that a b. Then a and b are of the same type in P. Proof: We prove by induction on the complexity of ( x) that P j= [ a] i P j= [ b]. If ( x) is c(t), where t is some Boolean combination of the variables x, then the result is guaranteed by lemma 3.4 and the fact that connectedness is a topological property. The sole non-trivial recursive case involves showing that, if ( x) is 9y ( x; y) andP j= [ a], thenP j= [ b]. But if P j= [ a], there exists a 2 P such that P j= [ a; a]. By lemma 5.9, there exists a b 2 P such that a; a b; b. By inductive hypothesis, P j= [ b; b]. Therefore P j= [ b]. 2 Lemma 6.1 yields a straightforward result on the relationship between R and Q: Lemma 6.2 Q R. Proof: According to the Tarski-Vaught lemma (Mendelson [28], proposition 2.38), if Q R and, for any n-tuple a of Q and any formula ( x) of the form 9y ( x; y) such that R j= [ a], there exists b 2 Q such that R j= [ a; b], then Q R. By construction, Q R. 11 Let a be an n-tuple of elements of Q, and let ( x) be any formula of L of the form 9y ( x; y) such that R j= [ a]. Then there exists a 2 R such that R j= [ a; a]. By lemma 5.10, there exists b 2 Q such that a; a a; b. By lemma 6.1 applied to R, R j= [ a; b]. 2 It follows from lemma 6.2 that Th(Q) = Th(R). We have already agreed to denote this set of formulae by Th(P). The language L allows us to formalize wide a range of topological properties and relationships over the domainsQ and R. However, for present purposes, we require only very minimal observations concerning the expressive power of L. Here is one of them: L contains a formula N ( x) expressing the notion of being an N -element connected partition. Lemma 6.3 For all N > 0, let N (z1; : : : ; zN ) be the formula ^ 1 i N(c(zi) ^ zi 6= 0) ^ ^ 1 icannot be atomic, and so cannot be prime. Thus Q is, in a strong sense, strictly minimal.However, it turns out that Th(P) satis es a weakened form of of !-categoricity. The next theoremshows that the only alternative models to P are those containing regions comprising, as we might putit, in nitely many pieces.Theorem 10 Any two countable models of Th(P) omitting the set of formulae(x) =8<::9z1; : : : ; 9zN 0@ ^1 i N c(zi) ^ x = X1 i N zi1A N 2IN9=;are isomorphic.Proof: Let A be countable such that A j= Th(P) and A omits (x). We show that A ' Q.We show that, since A omits (x), for every n-tuple a in A, there exists an N -tuple c satisfying N inA such that the elements of a are expressible according to A as sums of various elements of c. That is,lemma 6.4 holds with A substituted for Q.To see this, simply take all non-zero atoms of the form a1: : an and, using the fact that A omits(x), express each such atom as a sum of elements of A, each of which satis es c(x) in A. By lemma 3.3,Th(P) j= 8x8y((c(x) ^ c(y) ^ x:y 6= 0)! c(x+ y)):So we may sum together any non-disjoint pairs of these elements until we have elements c1; : : : ; cNsatisfying N in A. Since the atoms a1: : an are expressible as the sums of the c1; : : : ; cN , so arethe a1; : : : ; an, as required.Now, using the notation of lemma 6.5, let i be complete formulae in the type i, (1 i k) butinconsistent with j for j 6= i. Then any N -tuple in Q satisfying N in Q also satis es one of the iin Q. Hence, Th(P) j= N ! ( 1 _ : : : _ k). It follows that any N -tuple in A satisfying N in A alsosatis es one of the i in A. Assume without loss of generality that A j= N [c] ^ 1[c]. It follows thatN ^ 1 is a complete formula in Th(P) = Th(A) satis ed by c.Lemma 6.6 certainly holds with A substituted for Q. Hence the proof of theorem 8 goes through withA substituted for Q, and A is atomic. By theorem 6, A ' Q.27 Related workThe results presented here not only have rami cations for mereotopological theories [10, 9, 37, 18, 2, 7],but they have connections with more practical disciplines. Various logicians have sought to give deductivetheories of space and space-time [8, 4, 20, 17, 23], many in terms of modal logics [31, 32, 39, 33, 35, 3, 27].Recent interest in the analysis of visual languages, such as maps and diagrams [29, 26, 19, 24], has led tothe exploration of planar mereotopology in relation to qualitative spatial reasoning, since it is theorizedthat an important aspect of the semantics of such representations may be given by analysis of spatialrelations between representational tokens in the plane. Another more practical area in which ontologicalissues about the plane are raised is in the construction of computational spatial representations forrobots, and in Geographical Information Systems eg: [14, 38, 25]. As we have mentioned, GISs useplanar polygonal regions to represent geographic objects. In mobile robotics too, it is common torepresent a robot's information about its environment as a planar arrangement of places together withtheir connection relations, eg:[22, 15, 16, 34].15 8 ConclusionIn this paper, we have investigated the possibility of alternative spatial ontologies for \practical"mereotopo-logical reasoning. In order to constrain the problem, we insisted that any such ontology provide a modelelementarily equivalent to the `familiar' polygonal model P. Our motivation for taking P as our point ofdeparture was that many computer packages designed to manipulate spatial data, such as GISs, restrictthemselves to piecewise linear objects, without any apparent loss of useful representational power.We identi ed rational and real `versions' of P, namely Q and R, with the former being countable.The main technical results of this paper state that, althoughQ is not the only countable countable modelof Th(P), it is, in the sense of elementary embedding, the minimal such model. Thus, the countablealternatives to Q all contain a copy of Q|the `familiar' regions, plus some `non-familiar' regions whichmake no di erence to any properties of the familiar regions expressible in L. Thus, in a strong sense,they are less parsimonious. Moreover, we found a simple condition on models of Th(P) which determinesQ up to isomorphism, and provides a useful characterization of the other models of Th(P). Apparently,revisions to our ontology of the plane which do not violate the facts of polygonal mereotopology|to theextent they exist at all|must be less parsimonious than the one we started with.The corresponding problem for the 3-dimensional case, or for domains in which lemma 5.1 fails, isopen. A complete axiomatization of Th(P) has been developed in Pratt and Schoop [30].References[1] J. F. Allen. An interval-based representation of temporal knowledge. In The 7th International JointConference on Arti cial Intelligence (IJCAI '81), 1981.[2] Nicholas Asher and Laure Vieu. Toward a Geometry of Common Sense: a semantics and a completeaxiomatizationof mereotopology. In International Joint Conference on Arti cial Intelligence (IJCAI'95), 1995.[3] Philippe Balbiani, Luis Farinas del Cerro, Tinko Tinchev, and Dimiter Vakarelov. GeometricalStructures and Modal Logic. In Gabbay and Ohlbach, editors, Practical Reasoning, Lecture notesin Arti cial Intelligence 1085. 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