On countable RCC models

Fundamenta Informaticae XX (2006) 1–23 IOS Press

On countable RCC models ∗ Sanjiang Li† C State Key Laboratory of Intelligent Technology and Systems Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China [email protected]

Mingsheng Ying ‡ State Key Laboratory of Intelligent Technology and Systems Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China [email protected]

Yongming Li§ Department of Mathematics, Shaanxi Normal University, Xi’an 710062, China [email protected]

Abstract. Region Connection Calculus (RCC) is the most widely studied formalism of Qualitative Spatial Reasoning. It has been known for some time that each connected regular topological space provides an RCC model. These ‘standard’ models are inevitable uncountable and regions there cannot be represented finitely. This paper, however, draws researchers’ attention to RCC models that can be constructed from finite models hierarchically. Compared with those ‘standard’ models, these countable models have the nice property that regions where can be constructed in finite steps from basic ones. We first investigate properties of three countable models introduced by D¨untsch, Stell, Li and Ying, resp. In particular, we show that (i) the contact relation algebra of our minimal model is not atomic complete; and (ii) these three models are non-isomorphic. Second, for each n > 0, we construct a countable RCC model that is a sub-model of the standard model over the Euclidean unit n-cube; and show that all these countable models are non-isomorphic. Third, we ∗

Fundamenta Informaticae, 65(4), 2005, 329-351. Partly supported by National NSF of China (Grant No.: 60305005, 60496321). C Corresponding author ‡ Partly supported by National NSF of China (Grant No.: 60273003,60321002). § Partly supported by the Key Project of Fundamental Research of China (Grant No.: 2002CB312200). †

1

2

S. Li, M. Ying, Y. Li / On Countable RCC Models

show that every finite model can be isomorphically embedded in any RCC model. This leads to a simple proof for the result that each consistent spatial network has a realization in any RCC model. Keywords: Region Connection Calculus; Qualitative Spatial Reasoning; (Generalized) Boolean connection algebra; Countable RCC models; Hierarchical spatial reasoning.

1. Introduction Representing and reasoning about spatial knowledge is an active subfield of Artificial Intelligence and Geographic Information Sciences. Due to precise, quantitative spatial information is often inaccessible or undesirable, qualitatively representing and reasoning about spatial information is a pressing need. Known as Qualitative Spatial Reasoning (QSR), the particular research area has applications in areas such as GIS [31, 6, 30], spatial query languages [9], natural languages [1] and many other fields. For a detailed survey of QSR, we refer the reader to the work of Cohn and Hazarika [11]. This paper focuses on the best known formalism of QSR, viz. Region Connection Calculus (RCC), which is initially described in [23, 24] and studied by the Leeds group [12, 18, 3, 4, 5, 10] and other authors, e.g. [17, 28, 16, 14, 13, 22, 25, 27, 21, 20]. RCC is a first order theory that is based on the Whitehead-inspired system of Clarke [7, 8]. RCC provides an axiomatization of space that takes regions, rather than points, as primitive. It is well known that regular closed sets of connected regular spaces, e.g., the Euclidean plane, provide models of RCC [18]. In this paper we call these standard RCC models. We argue that these models are, practically speaking, not desirable for spatial representation and reasoning. Take the Euclidean plane for instance, regions in this model are non-empty regular closed sets with arbitrary complexity. They usually cannot be finitely constructed from basic regions such as squares, disks, triangles, etc. Notice that regular closed sets of a regular connected topological space forms a complete atomless Boolean algebra. These standard models are uncountable. Consequently regions in these models cannot be finitely constructed from basic regions in a presupposed countable collection. So, as far as spatial representation is concerned, these standard models are computing intractable. Furthermore, these models cannot avoid counterintuitive spatial configurations. For instance, there are three plane regions that share exactly the same boundary in the Euclidean plane.1 These deficiencies suggest that ‘standard’ RCC models contain too many regions. To obtain a computing feasible model for spatial representation and reasoning, we need to consider countable, therefore non-standard, RCC models. Note that by the L¨owenheim-Skolem Theorem (see e.g. [2, Theorem 5.2.8., p.172]), RCC, as a first order theory, must have a countable model. The first countable RCC model was constructed by Gotts [18]. Several years later, D¨untsch, Wang and McCloskey [16] and Stell [28] independently showed that each RCC model gives rise to a Boolean algebra. Stell [28] further gave an equivalent formulation of RCC models in terms of Boolean connection algebras (BCAs). A construction of BCAs from pseudo-complemented distributive lattices was also introduced in [28]. In particular, Stell constructed a countable BCA from the Heyting algebra of subgraphs of a certain graph. We write Bs for this model. Another example was given by D¨untsch [13] that is based on the interval algebra on [0, 1) ∩ Q, we write Bχ for this model. In [19], we proposed a general approach for constructing countable RCC models hierarchically. Compared with standard RCC models over regular connected spaces, these countable models have the 1

See for example http://www.cut-the-knot.org/do_you_know/brouwer.shtml.

S. Li, M. Ying, Y. Li / On Countable RCC Models

3

nice property that each region can be constructed in finite steps from basic regions. One interesting countable RCC model, Bω , was also given. This model is a minimal RCC model in the sense that it is a sub-model of any RCC model. In this paper, some basic properties of countable RCC models are investigated. In particular, we show that, on the one hand, each countable RCC model can be constructed hierarchically from a sequence of its finite sub-models; and on the other hand, each finite model can be isomorphically embedded in the minimal model Bω , hence in any RCC model. This latter result leads to a simple and more direct solution to the realization problem of consistent spatial network [25]. The properties of the minimal RCC model Bω , together with Bs and Bχ , are studied in detail. In particular, we show that the contact relation algebra (CRA) [15] of Bω is not atomic complete. Moreover, we show Bχ , Bs and Bω are non-isomorphic. Furthermore, for any n ≥ 1, we construct a sub-model of the standard model over [0, 1]n and show that these models are also non-isomorphic. The rest of this paper is structured as follows. Section 2 introduces (G)RCC models and RCC8 relations, then in Section 3 we recall several basic operations on GRCC models introduced in [19]. Section 4 first introduces a representation of the atomless countable Boolean algebra, written as B. Then we define three connections on B and therefore obtain three RCC models Bχ , Bψ , and Bω . The properties of these models are studied in Section 4 and Section 5. In Section 6, we introduce for each n ≥ 1 a countable RCC model that is a sub-model of the standard model over [0, 1]n and shows that these countable models are all non-isomorphic. Section 7 shows that every finite GRCC model can be isomorphically embedded in any RCC model. Based on this result, we give a more direct and simple solution to the realization problem of consistent spatial network. Conclusions are given in the final section. In this paper we write N for the set of non-negative integers and write N+ for the set of positive integers.

2. RCC models and RCC8 relations There are several equivalent formulations of the RCC theory, we here adopt the one using Boolean connection algebras given by Stell. Definition 2.1. ([28]) An RCC model is a Boolean algebra A containing more than two elements, together with a binary connection relation C on A − {⊥} that satisfies the following conditions: A1. C is reflexive and symmetric, A2. (∀x ∈ A − {⊥, >})C(x, x0 ), A3. (∀xyz ∈ A − {⊥})C(x, y ∨ z) ↔ C(x, y) or C(x, z), A4. (∀x ∈ A − {⊥, >})(∃z ∈ A − {⊥, >})¬C(x, z), where ⊥ and > are, respectively, the bottom and the top element of A, x0 is the complement of x in A, x ∨ z is the least upper bound (lub) of x and z in A. Such a construction is also known as a Boolean connection algebra (BCA for short) [28]. Note that, due to Condition A4, every RCC model is an atomless Boolean algebra. This condition also results in that each region is divisible and the model is therefore called ‘continuous’. But in practical applications, discrete spaces are assumed. With the intention of bridging this gap, we introduce a more general construction—GRCC models.

4

S. Li, M. Ying, Y. Li / On Countable RCC Models

Definition 2.2. ([19]) A GRCC model is a Boolean algebra A together with a binary connection relation C on A − {⊥} that satisfies Conditions A1-A3 in Definition 2.1. We also call such a model a generalized Boolean connection algebra (GBCA). We call a 2-tuple hA, Ci a connection structure if A is a Boolean algebra and C is a binary relation on A − {⊥} that satisfies Conditions A1 and A3 in Definition 2.1. We say a connection structure hA, CA i can be embedded in another connection structure hB, CB i if there exists an injective mapping f : A → B such that, for any a1 , a2 ∈ A, (1) f (a1 ∨ a2 ) = f (a1 ) ∨ f (a2 ) and f (a1 ∧ a2 ) = f (a1 ) ∧ f (a2 ); and (2) CB (f (a1 ), f (a2 )) if and only if CA (a1 , a2 ). Such an embedding is called isomorphic if f satisfies further (3) f (a0 ) = f (a)0 for any a ∈ A. In what follows, we call a non-zero element in a GRCC model hA, Ci simply a region in A. For a region a ∈ A, we say a is connected if a cannot be divided into two disconnected parts, i.e., ∀ b, c ∈ A − {⊥}, b ∨ c = a → C(b, c). The following result was proved by Gotts [18] for regular connected spaces. Proposition 2.1. Let X be a topological space, let RC(X) be the Boolean algebra of regular closed sets of X, and assume that RC(X) contains more than two nonempty regular closed sets. Define the relation C on RC(X) − {∅} by C(H, K) iff H ∩ K 6= ∅. Then hRC(X), Ci is a connection structure. Moreover, if X is connected (connected regular, respectively), then hRC(X), Ci is a model of the GRCC (RCC, respectively). For each topological space X, we call the relation defined above (for regular closed sets) the canonical connection on X and denote by hRC(X), CX i, or simply RC(X), the corresponding connection structure. In particular, we refer to the standard RCC model over Rn and [0, 1]n simply as RC(Rn ) and RC([0, 1]n ) respectively. Using C, a basic set of binary relations are defined [24]. Definitions and intended meanings of those used here are given in Table 1. The inverse relations of P, PP, TPP and NTPP are denoted by Pi, PPi, TPPi and NTPPi respectively. The eight relations DC, EC, PO, EQ, TPP, NTPP, TPPi, NTPPi (illustrated in Figure 1) are JEPD (Jointly Exhaustive and Pairwise Disjoint). This set of relations, known as RCC8, are of significant importance (see [26]).

3. Operations on GRCC models Formalisms of qualitative spatial representation such as RCC usually attend only to spatial models individually, relation between these models are rarely investigated. In this section we recall several operations on GBCAs introduced in [19]. These concepts, especially that of sub-models and direct limits will shed light on the relation between continuous models and discrete models.

S. Li, M. Ying, Y. Li / On Countable RCC Models

a

5

a a

b

b

DC(a, b) EC(a, b)

a

b

b

a b

a

b

a

b

a b

PO(a, b) TPP(a, b) TPPi(a, b) NTPP(a, b) NTPPi(a, b) EQ(a, b) Figure 1. Illustration of RCC8 topological relations

Table 1. Relation DC(x, y) P(x, y) PP(x, y) EQ(x, y) O(x, y) PO(x, y) EC(x, y) DR(x, y) TPP(x, y) NTPP(x, y)

Some relations definable in GRCC

interpretation x is disconnected from y x is a part of y x is a proper part of y x is identical with y x overlaps y x partially overlaps y x is externally connected to y x is discrete from y x is a tangential proper part of y x is a non-tangential proper part of y

Definition of R(x, y) ¬C(x, y) x≤y x⊥ x ∧ y > ⊥ & x 6≤ y & y 6≤ x C(x, y) & x ∧ y = ⊥ x∧y =⊥ x < y & C(x, y 0 ) ¬C(x, y 0 )

3.1. Sub-structures and local structures Given a GBCA hA, Ci, suppose B is a subalgebra of A with more than two elements. Then we have a binary relation, C|B , on B − {⊥} obtained by restricting C on B − {⊥}. Clearly hB, C|B i is also a GBCA [19]. This GBCA is called a sub-model, or a sub-GBCA of hA, Ci. If B happens to be a BCA, we say B is a sub-BCA of A. Given an element a in a Boolean algebra hA; ⊥, >,0 , ∨, ∧i with a 6= ⊥. For any x, y ≤ a, let x∗ = a ∧ x0 , and let x t y = x ∨ y, x u y = x ∧ y. Then h↓ a; ⊥, a,∗ , t, ui is also a Boolean algebra. Given a GBCA hA, Ci, for any a 6= ⊥, we have a binary relation on ↓ a−{⊥} obtained by restricting C on ↓ a − {⊥}. Write this binary relation as C|↓a . If a is a non-atomic connected region in hA, Ci, then h↓ a, C|↓a i is also a GBCA [19]. We call this GBCA the local model or local GBCA of A at a. Clearly, if hA, Ci happens to be a BCA and a is a connected region in A, then this local GBCA of A at a is indeed a BCA. By definition, the connection in a sub-GBCA coincides with the restriction of the original connection on the sub-algebra. Interestingly, this coincidence also applies to other RCC8 relations. Proposition 3.1. Suppose R is an RCC8 relation. If hB, CB i is a sub-GBCA of hA, CA i and x, y are two regions in B, then R(x, y) holds in B if and only if it holds in A.

6

S. Li, M. Ying, Y. Li / On Countable RCC Models

Proof: This follows from the definitions of the RCC8 relations given in Table 1 and the fact that CB = CA |B . u t One may further guess that this fact would hold for all binary relations in the relation algebra generated by C. This is, however, not true. Consider a sub-model of the standard RCC model RC(R2 ) that is isomorphic to Bω (see Example 4.3 of this paper). Notice that NTPP2 = NTPP is not true in Bω . There exist two regions a and b in Bω such that NTPP(a, b) and ¬NTPP2 (a, b). At the same time we have NTPP = NTPP2 in RC(R2 ), and this suggests that both NTPP(a, b) and NTPP2 (a, b) hold in RC(R2 ).

3.2.

Direct limits of GBCAs

Let A be a Boolean algebra. Suppose {hAk , Ck i}k∈N+ is a collection of GBCAs that satisfy the following conditions: (1) Ak is a subalgebra of A for k ∈ N+ ; (2) hAk , Ck i is a sub-GBCA of hAk0 , Ck0 i for k ≤ k 0 ∈ N+ . S Write Aω = k∈N+ Ak . Clearly it is also a subalgebra of A. Define a binary relation Cω on Aω − {⊥} as follows: ∀x, y ∈ Aω , Cω (x, y) iff ∃k ∈ N+ s.t. x, y ∈ Ak and Ck (x, y). S In other words, Cω = k∈N+ Ck . It is routine to check that hAω , Cω i is a GBCA and hAk , Ck i is its sub-GBCA for each k ∈ N+ . We call hAω , Cω i the direct limit of {hAk , Ck i}k∈N+ . The following theorem characterizes when is a direct limit a BCA. Theorem 3.1. ([19]) Let A be a Boolean algebra. Suppose hAω , Cω i is the direct limit of {hAk , Ck i}k∈N+ , where {hAk , Ck i}k∈N+ satisfies Conditions (1) and (2) given above. Then hAω , Cω i is a BCA if and only if for each k ∈ N and each x ∈ Ak − {⊥, >}, there exists some k 0 ≥ k and some y ∈ Ak0 − {⊥, >} such that ¬Ck0 (x, y). The following proposition shows that any countable BCA can be obtained as the direct limit of a sequence of its finite sub-GBCAs. Proposition 3.2. Suppose hB, Ci is a countable BCA. Then hB, Ci is the direct limit of a sequence of its finite sub-GBCAs. Proof: Suppose elements in B can be enumerated as ⊥, >, x1 , · · · , xn , · · · . Set Bk as the subalgebra of B generated by {x1 , · · · , xk }. Clearly hBk , C|Bk i is a finite sub-GBCA of hB, Ci. Now it’s straightforward S + to check S that {hBk , C|Bk i : k ∈ N } satisfies Conditions (1) and (2) given above and B = k∈N++ Bk , C = k∈N+ C|Bk . Consequently hB, Ci is the direct limit of its sub-models {hBk , C|Bk i : k ∈ N }. u t

S. Li, M. Ying, Y. Li / On Countable RCC Models

7

The above theorem shows that each countable model can be constructed hierarchically from a collection of finite models. This result can be further strengthened. We next show that, to determine the global connection, we need only to consider the “adjacency” relation between basic regions of the same granularity. a Boolean algebra A, a finite set X = {x1 , · · · , xn } ⊆ A − {>, ⊥} is called a resolution of A WGiven n if i=1 xi = > and xi ∧ xj = ⊥ for any two different i, j ∈ {1, 2, · · · , n}.2 For two resolutions X1 and X2 of A, we say X2 refines X1 if each elements in X1 is the sum of some elements in X2 . Notice that the connection of a finite GBCA is completely determined by the adjacency among its atoms. Proposition 3.3. ([19]) Suppose A is a finite Boolean algebra, X is the set of its atoms and C is a binary reflexive and symmetric relation C on A − {⊥}. Then hA, Ci is a GBCA if and only if C satisfies the following conditions: (1) For any two a, b ∈ A, C(a, b) if and only if there exist x, y ∈ X such that x ≤ a, y ≤ b and C(x, y); (2) For any two U, V ⊆ X with U ∪ V = X, there exist x ∈ U and y ∈ V such that C(x, y). The atoms set of a finite Boolean algebra is its finest resolution. Since a countable RCC model is atomless, we have no such a finest resolution. Nevertheless we can replace the finest resolution with a sequence of finer and finer resolutions. Definition 3.1. Suppose A is a countable Boolean algebra and {Xi }i∈N+ is a sequence of resolutions of A. We call {Xi }i∈N+ a resolution locus of A if it satisfies the following conditions: i) Xj refines Xi for i < j; ii) for any element a ∈ A, there exists some k ∈ N+ such that a is the lub (least upper bound) of some elements in Xk . Proposition 3.4. Given a countable Boolean algebra A and a resolution locus {Xi :}i∈N+ of A, suppose C is a binary reflexive and symmetric relation on A − {⊥}. Then hA, Ci is a BCA if and only if C satisfies the following conditions: W W 1) For any k ∈ N+ and any two U, V ⊆ Xk with a = U and b = V , C(a, b) if and only if there exist x ∈ U and y ∈ V such that C(x, y); 2) For any k ∈ N+ and any two U, V ⊆ Xk with U ∪ V = Xk , there exist x ∈ U and y ∈ V such that C(x, y); 3) For any k ∈ N+ and any x ∈ Xk , there exist some k 0 > k and some y ∈ Xk0 such that y < x and ¬C(y, z) for any z ∈ Xk with z 6= x. Proof: This follows directly from Theorem 3.1 and Proposition 3.3. 2

u t

The concept “resolution” defined here is a little general than the one given by Worboys [31], which is defined on powerset algebras.

8

S. Li, M. Ying, Y. Li / On Countable RCC Models

Suppose hAω , Cω i is the direct limit of {hAk , Ck i}k∈N+ . One may wonder whether a region a ∈ Aω would be connected in hAω , Cω i, provided that it is already connected in a sub-model hAk , Ck i? In general, a region that is connected in a small model is not necessarily connected in the big one. For example, suppose hA, Ci is a BCA and b ∈ A is a region that isn’t connected. Taking B as the subalgebra of A that is generated by the singleton {b}, then b is trivially connected in hB, C|B i. Nevertheless, if hAω , Cω i is the direct limit of {hAk , Ck i}k∈N+ , we have the following proposition: Proposition 3.5. Suppose hAω , Cω i is the direct limit of a sequence of GBCAs {hAk , Ck i}k∈N+ , and each atomic region in Xk is connected in hAω , Cω i. Then, for any k ∈ N+ , a region a ∈ Ak is connected in hAω , Cω i if and only if a is connected in hAk , Ck i. Proof: Suppose a ∈ Ak is connected inWhAk , Ck i. We now show a is also connected in hAω , Cω i. Take V = {x ∈ Xk : x ≤ a}, then a = V . Suppose b, c ∈ Aω are two regions such that b ∨ c = a. We now show Cω (b, c). Note that if both b ∧ x and c ∧ x are non-zero for some x ∈ V , then Cω (b ∧ x, c ∧ x) since x is connected in hAω , Cω i and x = (b ∧ x) ∨ (c ∧ x). This suggests Cω (b, c). Suppose for each x ∈ V , either b ∧ x = ⊥ or c ∧ x = ⊥. This suggests x ≤ b or x ≤ c for each x ∈ V . Therefore we have S b, c ∈ Ak . Recall that a is connected in hAk , Ck i. We have Ck (b, c), hence Cω (b, c) since Cω = i∈N+ Ci . u t

4. Three countable RCC models This section and the next section investigate some properties of three countable RCC models appeared in the literatures, i.e., the minimal RCC model given in [19], the RCC model constructed on a certain interval algebra given by D¨untsch [13], and the graph BCA given by Stell [28]. In this section we first recall a representation B of the countable atomless Boolean algebra, which was originally given in [21]. Second, we construct three connection structures on B, i.e., Bχ , which is an instance of D¨untsch’s construction; and Bψ , which is isomorphic to Stell’s graph BCA (the proof is given in Section 5); and Bω , the minimal RCC model. Third, we show that these three are non-isomorphic.

4.1.

A representation of the countable atomless Boolean algebra

Let {0, 1}∗ be the set of finite strings over {0, 1}. As usual, we write ² for the empty string. For a string s ∈ {0, 1}∗ , we denote by |s| its length. Now for each string s ∈ {0, 1}∗ , we associate a left-closed-andright-open sub-interval of [0, 1) as follows: Take x² = [0, 1); x0 = [0, 1/2), x1 = [1/2, 1); x00 = [0, 1/4), x01 = [1/4, 1/2), x10 = [1/2, 3/4), x11 = [3/4, 1); and so on. In general, suppose xs has been defined for a string s ∈ {0, 1}∗ , we define xs0 to be the first half left-closed-and-right-open sub-interval of xs , and xs1 the second half. Write Xn = {xs ∈ X : s ∈ {0, 1}∗ , |s| = n} for each n ≥ 1, and write Xω = {xs : s ∈ {0, 1}∗ }. Denote by Bn (B, resp.) the subalgebra of the powerset algebra 2[0,1) generated by Xn (Xω , resp.). Clearly Bn contains 2n atoms and B is a countable atomless Boolean algebra. Moreover, Bn is a

S. Li, M. Ying, Y. Li / On Countable RCC Models

9

x x0

x1

x00

x000

x01

x001

x010

x11

x10

x011

··· ··· ··· ··· ··· ··· ··· ···

x100

x101

x110

··· ··· ··· ··· ··· ···

x111

··· ···

Figure 2. The infinite complete binary tree Tω

S subalgebra of Bm for n < m and Bn is a subalgebra of B for each n. It is also clear that B = n∈N Bn = Sn−1 { i=1 xsi : si ∈ {0, 1}∗ , n ∈ N}. S Denote Yn = {xs ∈ X : s ∈ {0, 1}∗ , |s| ≤ n} for each n ≥ 1. Then Yn = nk=1 Xk . Moreover, for each n ∈ N, Yn (with the ordering of set inclusion) can be visualized as a complete binary tree Tn of height n. Similarly, Xω , with the ordering of set inclusion, can be visualized as an infinite complete binary tree Tω (See Figure 2). In what follows, we define three connections on B. Notice that Xn is a resolution of B for each n ∈ N+ and {Xn }n∈N+ is a resolution locus of B. Also recall that, by Proposition 3.4, to determine a connection on B, we need only to describe the relation among atomic regions in Xn for each n. Example 4.1. Suppose s, t, s0 are strings in {0, 1}∗ . Define Cχ to be the smallest binary relation on B that satisfies the following conditions: W W (Cχ 1) For any k ∈ N and any two nonempty U, V ⊆ Xk , set a = U and b = V . Then Cχ (a, b) if and only if there exist x ∈ U and y ∈ V such that Cχ (x, y); (Cχ 2) Cχ (xs , xs ); · · 0}}, where k ∈ N; (Cχ 3) Cχ (xs , xt ) if {s, t} = {0 1| ·{z · · 1}, 1 0| ·{z k

k

(Cχ 4) Cχ (xs0 s , xs0 t ) if Cχ (xs , xt ). Then, by Proposition 3.4, we can show that hB, Cχ i is a countable BCA. We write this BCA simply by Bχ . This BCA is indeed an instance of the RCC models constructed on certain interval algebras given by D¨untsch [13]. Proposition 4.1. For two strings s, t ∈ {0, 1}∗ with s 6= t and |s| = |t|, Cχ (xs , xt ) if and only if there exists s1 ∈ {0, 1}∗ such that {s, t} = {s1 0 1| ·{z · · 1}, s1 1 0| ·{z · · 0}} for some n ≥ 0. n

n

10

S. Li, M. Ying, Y. Li / On Countable RCC Models

Proof: This follows from the definition.

u t

Example 4.2. Suppose s, t, s0 are strings in {0, 1}∗ . Define Cψ to be the smallest binary relation on B that satisfies the following conditions: W W (Cψ 1) For any k ∈ N and any two nonempty U, V ⊆ Xk , set a = U and b = V . Then Cψ (a, b) if and only if there exist x ∈ U and y ∈ V such that Cψ (x, y); (Cψ 2) Cψ (xs , xs ); (Cψ 3) Cψ (xs , xt ) if {s, t} = {0 1| ·{z · · 1}, 1 0| ·{z · · 0}}, where k ∈ N; k

k

(Cψ 4) Cψ (xs0 s , xs0 t ) if Cψ (xs , xt ); (Cψ 5) Cψ (xs0 , xt0 ) if Cψ (xs , xt ). Then, by Proposition 3.4, we can show that hB, Cψ i is a countable BCA. We write this BCA simply by Bψ . Note that Cψ is clearly ‘finer’ than Cχ . Later, in Section 5, we shall show Bψ is isomorphic to the one constructed by Stell [28]. Proposition 4.2. For two strings s, t ∈ {0, 1}∗ with s 6= t and |s| = |t|, Cψ (xs , xt ) if and only if there exists s1 ∈ {0, 1}∗ such that {s, t} = {s1 0 1| ·{z · · 1} 0| ·{z · · 0}, s1 1 0| ·{z · · 0}} for some m, n ≥ 0. m

n

m+n

Proof: This follows from the definition.

u t

Example 4.3. ([19]) Suppose s, t, s0 are strings in {0, 1}∗ . Define Cω to be the smallest binary relation on B that satisfies the following conditions: W W (Cω 1) For any k ∈ N and any two nonempty U, V ⊆ Xk , set a = U and b = V . Then Cω (a, b) if and only if there exist x ∈ U and y ∈ V such that Cω (x, y); (Cω 2) Cω (xs , xs ); (Cω 3) Cω (xs0 , xs1 ) and Cω (xs1 , xs0 ); (Cω 4) Cω (xs0 s , xs0 t ) if Cω (xs , xt );3 (Cω 5) Cω (xs1 , xt1 ) if Cω (xs , xt ). Then, by Proposition 3.4, we can show that hB, Cω i is a countable BCA. We write Bω for this BCA. Proposition 4.3. ([19]) For two strings s, t ∈ {0, 1}∗ with s 6= t and |s| = |t|, Cω (xs , xt ) if and only if there exists s1 ∈ {0, 1}∗ · · 1}} for some n ≥ 0. such that {s, t} = {s1 0 1| ·{z · · 1}, s1 1 1| ·{z n 3

n

Note that (Cω 3) is indeed redundant, we include it here for an easy comparison.

S. Li, M. Ying, Y. Li / On Countable RCC Models

11

Proposition 4.4. For any s ∈ {0, 1}∗ , xs is connected in hB, Ci, where C is either Cχ , or Cψ , or Cω . Proof: Suppose there exists some s ∈ σ ∗ that is not connected. Write |s| = n. Then n ≥ 1 because s² is connected. Since s is not connected, there are two regions b, c ∈ B such that b ∨ c = xs and DC(b, c). For any m ∈ N, it’s clear that b ∈ Bm if and only if c ∈ Bm . So b, c ∈ Bk − Bk−1 for some k ∈ N. Now suppose t is a string such that |t| = k − 1 and s is an initial segment of t. We have xtl ⊆ b or xtl ∩ b = ∅ since xtl is an atom of Bk and b is a region in Bk (l = 0, 1). Moreover, if xtl ⊆ b, then we also have xtl0 ⊆ b (l, l0 = 0, 1). This is because that b and c are two disconnected subregions of xs and that C(xt0 , xt1 ). As a result, we have b ∈ Bk−1 , a contradiction. So xs is connected. u t By this proposition and Proposition 3.5, we have Corollary 4.1. For any k ∈ N+ and any region a ∈ Bk , a is connected in hB, Ci if and only if a is connected in hBk , C|Bk i, where C is either Cχ , or Cψ , or Cω .

x0

x1

x00

x000

x01

x001

x010

x10

x011

x100

x11

x101

x110

x111

Figure 3. Construction of connection Cχ

x0

x1

x00

x000

x001

x01

x010 Figure 4.

x10

x011

x100

x101

x11

x110

x111

Construction of connection Cψ

4.2. The relation algebra generated by Cω is not atomic complete The countable RCC model Bω has several interesting properties. In particular, Bω is a minimal BCA in the sense that each BCA contains Bω as a sub-BCA [19]. In what follows, we show (Theorem 4.1) a rather strange property of Bω . To begin with, we recall the following proposition given in [19].

12

S. Li, M. Ying, Y. Li / On Countable RCC Models

x0

x1

x00

x000

x001

x01

x010 Figure 5.

x10

x011

x100

x11

x101

x110

x111

Construction of connection Cω

Proposition 4.5. In the countable RCC model Bω , we have (i) For any string s and any n ≥ 1, NTPPω (xs0 , xs ) and NTPPω (xs0 , xs0 ∪ xs 1 · · · 1 ); | {z } n

(ii) For any nonempty a ∈ B and any string s 6= ², NTPPω (a, xs ) if and only if a ⊆ xs − xs 1 · · · 1 | {z } n

for some n ≥ 1.

Using the above proposition, we have shown in [21] that NTPPω ◦ NTPPω 6= NTPPω . Moreover, if we write inductively NTPPn+1 = NTPPω ◦ NTPPnω , then ω NTPPω , NTPP2ω , · · · , NTPPnω , · · · is a strict decreasing chain. Before proving this result, we first show the following lemma. Lemma 4.1. In the RCC model Bω , given a string t = t1 0 1| ·{z · · 1} and a region a ∈ B. If NTPPω (xt , a), m

then there exists some p ≥ 0 such that xt0 ⊆ a, where t0 = t1 1 1| ·{z · · 1}. m+p

Proof: Note that a ∈ Bn for some n ≥ 1. For any string s with |s| ≥ n, we have either xs ⊆ a or xs ∩ a = ∅. Suppose for some p bigger enough we have xt0 ∩ a = ∅ with t0 = t1 1 1| ·{z · · 1}. Then since xt0 is externally m+p

connected to xt , we shall have xt0 is also externally connected to a. This contradicts the assumption that NTPPω (xt , a). Therefore xt0 ⊆ a. u t Theorem 4.1. In the RCC model Bω , we have any positive integer n.

T n∈N+

NTPPnω = ∅, and NTPPnω 6= NTPPn+1 for ω

Proof: S S Notice that for any two regions a, b ∈ B, if we set a∗ = {x0s : xs ⊆ a} and b∗ = {x0s : xs ⊆ b}, then we have NTPPω (a, b) if and only if NTPPω (a∗ , b∗ ). T n Suppose there exist two regions a, b ∈ B with (a, b) ∈ n∈N+ NTPPω . By the above observation, T we also have (a∗ , b∗ ) ∈ n∈N+ NTPPnω . Take a string s = 0l1 l2 · · · lk (li ∈ {0, 1} for i = 1, · · · , k)

S. Li, M. Ying, Y. Li / On Countable RCC Models

13

T such that xs ⊆ a∗ ⊆ b∗ ⊆ x0 . We have (xs , x0 ) ∈ n∈N+ NTPPnω . We now show that a contradiction can be obtained by proving that (xs , x0 ) 6∈ NTPPk+1 ω . For a string t, set λ(t) as the total number of occurrences of 0 in t. By Lemma 4.1 we have, for t = 0t1 and a region a ⊆ x0 with NTPPω (xt , a), there exists another string t0 such that λ(t0 ) = λ(t) − 1 and xt0 ⊆ a. Now suppose there exist a1 , a2 , · · · , ak , ak+1 = x0 such that xs NTPPω a1 NTPPω a2 · · · ak NTPPω ak+1 = x0 , where s = 0l1 l2 · · · lk as above. Suppose λ(s) = m > 0. By the above observation, we shall have some s1 such that λ(si ) = λ(s) − 1 and xs1 ⊆ a1 . By assumption that a1 NTPPω a2 we shall have xs1 NTPPω a2 . Continuing this procedure, since 1 < m ≤ k + 1, we shall obtain a string t = 0 1| ·{z · · 1} k+p

(p ≥ 0) such that xt ⊆ ak , and therefore NTPP(xt , x0 ). This cannot be true since x1 is externally k+1 connected T to both xt nand x0 . As a result we have (xs , x0 ) 6∈ NTPPω , a contradiction. Hence we have n∈N+ NTPPω = ∅. Next we show NTPPnω 6= NTPPn+1 for any positive integer n. Set si = 0 0| ·{z · · 0} for i ≥ 0. By ω i

a similar argument as given above, we can show (xsk , x0 ) is not in NTPPk+1 for any k > 0. On the ω other hand, we have xsk NTPPω xsk−1 NTPPω xsk−2 · · · xs2 NTPPω xs1 NTPPω xs0 = x0 . This shows (xsk , x0 ) is in NTPPkω . Therefore we have NTPPkω 6= NTPPk+1 for any positive integer ω k. u t This result shows that the contact relation algebra (CRA) [14] of the countable RCC model Bω , namely the relation algebra generated by Cω , is not atomic complete, hence infinite.

4.3.

Bχ , Bψ and Bω are non-isomorphic

In an early paper [21], we identify three model constraints while determining compositional extensionality of RCC8 composition table.4 These conditions are: (∃z¬PP(z, y) ∧ NTPP(x, y)) → ∃z(NTPP(z, y) ∧ NTPP(x, z))

(1)

(∃z¬PP(z, y) ∧ NTPP(x, y)) → ∃z(NTPP(z, y) ∧ EC(x, z))

(2)

(∃z¬PP(z, y) ∧ NTPP(x, y)) → ∃z(TPP(x, z) ∧ TPP(z, y)).

(3)

These model constraints can be used to distinguish the three countable RCC models. Note that we have [21] Condition 1 ⇒ Condition 2 ⇒ Condition 3. In this subsection we prove that Bχ , Bψ , Bω are non-isomorphic by showing that Bχ satisfies Condition 1, Bψ satisfies Condition 3 but doesn’t satisfy Condition 2, and Bω doesn’t satisfy Condition 3. Table 4.3 summarizes the results. 4

Where we show that if an RCC model satisfies one of these conditions, the corresponding compositional triad has an extensional interpretation.

14

S. Li, M. Ying, Y. Li / On Countable RCC Models

Table 2.

Comparison of three countable RCC models

Bχ Bψ Bω

C1

C2

C3

YES NO NO

YES NO NO

YES YES NO

P We fix some notations here. Suppose s = l1 · · · ln is a string in {0, 1}∗ . Write ps = ni=1 2lii , then xs = [ps , ps + 21n ). Notice that for any a ∈ B, there exists some n ∈ N such that a ∈ Bk for any k ≥ n. Suppose aS∈ Bk , write Sk (a) = S {s ∈ {0, 1}∗ : |s| = k, xs ⊆ a}, Pk (a) = {ps ∈ [0, 1) : s ∈ Sk (a)}. Then a = {xs : s ∈ Sk (a)} = {[ps , ps + 21k ) : s ∈ Sk (a)}. We first show that Bχ satisfies Condition 1. 1 Proposition 4.6. For any a, b ∈ Bn , NTPPχ (a, b) if and only if [p − 21n , p + 2n−1 ) ∩ [0, 1) ⊆ b for 1 1 1 any p ∈ Pn (a). In particular, we have NTPPχ ([ps , ps + 2n ), [ps − 2n , ps + 2n−1 ) ∩ [0, 1)) for any s ∈ {0, 1}∗ with |s| = n.

Proof: By the definition of NTPP, we have NTPPχ (a, b) if and only if DCχ (a, x² − b). But by (Cχ 1) given in Example 4.1 and by Proposition 4.1, we know DCχ (a, x² − b) if and only if s− , the predecessor of s, and s+ , the successor of s, if they exist, are not in Sn (x² − b) for any s ∈ Sn (a). Consequently we have NTPPχ (a, b) if and only if for any s ∈ Sn (a), s, s− and s+ , if they exist, are all in Sn (b), or 1 equivalently, if and only if [ps − 21n , ps + 2n−1 ) ∩ [0, 1) ⊆ b. u t The following corollary then shows that Bχ satisfies Condition 1. Corollary 4.2. Suppose a, c ∈ B and NTPPχ (a, c) and c 6= [0, 1). Then there exists some b ∈ B such that NTPPχ (a, b) and NTPPχ (b, c). Proof: S 1 Suppose a, c ∈ Bn for some n > 0. Write b = {[p− 2n+1 , p+ 21n ) : p ∈ Pn+1 (a)}∩[0, 1). Note that for 1 any p ∈ Pn (a), we have [p, p+ 21n ) ⊆ a and, by Proposition 4.6, [p− 21n , p+ 2n−1 )∩[0, 1) ⊆ c. Moreover, 1 1 for any p ∈ Pn+1 (a), by Proposition 4.6 again, we have NTPPχ ([p, p+ 2n+1 ), [p− 2n+1 , p+ 21n )∩[0, 1)) 1 3 3 1 1 1 and NTPPχ ([p− 2n+1 , p+ 2n )∩[0, 1), [p− 2n , p+ 2n+1 )∩[0, 1)). Then by [p− 2n , p+ 2n+1 )∩[0, 1) ⊆ c, b fulfills the requirement. u t Next, we show that Bψ doesn’t satisfy Condition 2. j j+1 Proposition 4.7. Suppose a = [ 2in , i+1 2n ) and b = [ 2n , 2n ) are two base regions in B. Then Cψ (a, b) if j and only if either a = b or there exist m, k ∈ N such that m ≤ n and 2km = 2in , k+1 2m = 2 n .

Proof: This follows directly from Proposition 4.2.

u t

S. Li, M. Ying, Y. Li / On Countable RCC Models

15

Corollary 4.3. Bψ doesn’t satisfy Condition 2. Proof: Let a = [0, 1/2), c = [0, 3/4). Note that by DCψ (a, [3/4, 1)) we have NTPPψ (a, c). Suppose there n−1 exists some b = [ 2jn , j+1 2n ) such that ECψ (a, b) and NTPPψ (b, c). Clearly there exists some i < 2 j i such that ECψ ([ 2in , i+1 2n ), b). By Proposition 4.7 and 2n < 1/2, we have 2n ≤ 1/2. But by ECψ (a, b), j we have a ∩ b = ∅, and hence, 2n = 1/2. Then by Proposition 4.7 again, we shall have [3/4, 1) is externally connected to both b and c in Bψ . This contradicts the assumption that NTPPψ (b, c). Consequently Bψ doesn’t satisfy Condition 2. u t But the following proposition shows that Bψ satisfies Condition 3. Proposition 4.8. Suppose a, c ∈ B, NTPPψ (a, c), and c 6= [0, 1). Then there exists some b ∈ B such that TPPψ (a, b) and TPPψ (b, c). Proof: Suppose a, c ∈ Bn for some n > 2. By c 6= [0, 1), we have either c = [ 2kn , 1) (k ≥ 1) or there exists some p ∈ [0, 1 − 21n ) such that p ∈ Pn (c) but p + 21n 6∈ Pn (c). Suppose c = [ 2kn , 1). By aNTPPψ c, we have [ 2kn , k+1 2n ) ∩ a = ∅. This is because that, if otherwise, k−1 k [ 2n , 2n ) will externally connect to both a and c. k+1 k+2 k 2k+1 2k+1 k+1 ∗ ∗ 1. If k+1 2n ∈ a, i.e., [ 2n , 2n ) ⊆ a, set b = [ 2n , 2n+1 ) and let b = a ∨ b . Then since [ 2n+1 , 2n ) is externally connected to both b and a, we have aTPPψ b. That bTPPψ c follows directly from the k ∗ fact that [ k−1 2n , 2n ) externally connect to both b and c. k+1 k+2 i 2. If k+1 2n 6∈ a, i.e., [ 2n , 2n ) ∩ a = ∅, take m = min{i : 2n ∈ a}. Then m > k + 1. Set m 2k+1 k+1 ∗ b∗ = [ 2kn , 2k+1 ) ∨ [ k+1 2n , 2n ) and let b = a ∨ b . That bTPPψ c is clear. Note that [ 2n+1 , 2n ) is 2n+1 ∗ externally connected to both b and a. We have aTPPψ b.

Suppose there exists some p = 2jn ∈ [0, 1 − 21n ) such that p ∈ Pn (c) but p + 21n 6∈ Pn (c). Then j+1 j+2 j j+1 [ 2jn , j+1 2n ) ⊆ c and [ 2n , 2n ) ∩ c = ∅. Clearly p 6∈ a, i.e., [ 2n , 2n ) ∩ a = ∅. This is because that j+2 j j+1 2j+1 j+1 ∗ ∗ [ j+1 2n , 2n ) is externally connected to both [ 2n , 2n ) and c. Set b = [ 2n+1 , 2n ) and let b = b ∨ a. Then it is easy to prove that aTPPψ bTPPψ c. u t Finally we show that the minimal BCA Bω doesn’t satisfy Condition 3. Note that ECω (x00 , xs ) if and only if s = 01 1| ·{z · · 1} for some n ∈ N. Consequently, for a region n

a ∈ B, ECω (x00 , a) holds if and only if x00 ∩ a = ∅ and there exists some s = 01 1| ·{z · · 1} such that n

xs ⊆ a. Take a = x00 and c = x00 ∪ x011 . Then NTPPω (a, c) since there cannot exist another region b such that b is externally connected to both a and c. For any b with x00 ⊂ b ⊂ x00 ∪ x011 , if there exists some s = 01 1| ·{z · · 1} such that xs ⊆ b, then NTPPω (x00 , b) holds by Proposition 4.5 (i). Otherwise, n

we shall have some t = 01 1| ·{z · · 1} such that xt ∩ b = ∅. Hence we have b − a ⊆ x011 − xt . But by n

Proposition 4.5 (ii), this shows NTPPω (b − a, x011 ). Recall NTPPω (a, c), we have NTPPω (b, c). In a word, there cannot exist a region b such that TPPω (a, b) and TPPω (b, c) hold at the same time.

16

S. Li, M. Ying, Y. Li / On Countable RCC Models

Remark 4.1. We have shown above that the three countable BCAs are not isomorphic by investigating whether they satisfy Conditions 1-3. The results given above also suggest that Condition 2 is strictly stronger than Condition 3. But it is still not clear whether Condition 1 is strictly stronger than Condition 2. Table 4.3, together with Theorem 5.1, also shows that Remark 4.1 and Remark 4.2 in [21] are not right. In the next section we show that Bψ is isomorphic to Stell’s binary subdivision graph BCA.

5. Stell’s binary subdivision graph BCA Consider an infinite graph G = hGN , GA i, where GN , the set of nodes of G, contains all rational numbers m/2n with 0 ≤ m ≤ 2n − 1 and n ∈ N. There is an arc between two nodes a and b if and only if a and b can be expressed as m/2n and (m + 1)/2n . Write A as the Heyting algebra of all subgraphs of G. The operations in A are defined as follows: given subgraphs, H = hHN , HA i, and K = hKN , KA i, their meet is H ∧ K = hHN ∩ KN , HA ∩ KA i, and their join is H ∨ K = hHN ∪ KN , HA ∪ KA i, 0 , arcs(H 0 )i, where H 0 is the set complement of H and the pseudo-complement of H is H ∗ = hHN N N N 0 . 0 arcs(HN ) is the set of arcs with both ends in HN We write [a, b), where a, b ∈ GN or b = 1, as the subgraph with no arcs, and containing only those nodes of G that are in theSsemi-interval [a, b). Now consider all subgraphs of G that can be written as finite unions of the form ki=1 [ai , bi )∗∗ with ai , bi ∈ GN or bi = 1. Naturally these subgraphs form a subalgebra of the Heyting algebra A, we write this Heyting algebra A0 . Write Bs = {H ∈ A : H ∗∗ }. Notice that G and Φ, the empty graph, are both in Bs . For subgraphs H, K ∈ Bs , define H uK = H ∧K, H t K = (H ∨ K)∗∗ . Then Bs = hBs ; Φ, G,∗ , u, ti is a Boolean algebra. Note that Bs is countable since A0 is so. Define a binary connectedness relation Cs on Bs as follows: for non-empty subgraphs H, K ∈ Bs , Cs (H, K) if and only if H ∗ ∨ K ∗ 6= G. Stell [28] shows that Bs = hBs , Cs i is a BCA. We now show that this BCA is isomorphic to the one given in Example 4.2. ˜ the subgraph of G with no arcs, and containing only nodes in H. For each element H ∈ Bs , write H fs = {H fs is a ˜ : H ∈ Bs } is a subalgebra of the powerset algebra of GN . Clearly φ : Bs → B Then B fs , which mapping each a to a ∩ GN , is also a Boolean isomorphism. Moreover, the mapping ψ : B → B Boolean isomorphism, where B, defined in Section 4.1, is a subalgebra of the powerset algebra of [0, 1). Consequently, the mapping ϕ = ψ −1 ◦ φ : Bs → B is also a Boolean isomorphism. For any region [a, b)∗∗ ∈ Bs , we have ϕ([a, b)∗∗ ) = [a, b). In what follows, we show that Cs (H, K) if and only if Cψ (ϕ(H), ϕ(K)) for any two regions H, K ∈ ∗∗ Bs . To this end, we need only to show that this holds for all regions [ 2in , i+1 2n ) . The following proposition is a counterpart of Proposition 4.7 for Bψ . ∗∗ and K = [ j , j+1 )∗∗ are two regions in B with i ≤ j. Proposition 5.1. Suppose H = [ 2in , i+1 s 2n ) 2n 2n Then Cs (H, K) if and only if either H = K or there exist m, k ∈ N such that m ≤ n and 2km = 2in , j k+1 2m = 2n .

Proof: ∗∗ and K = [ j , j+1 )∗∗ are connected in B if and only By definition, two regions H = [ 2in , i+1 s 2n ) 2n 2n j j+1 ∗ i i+1 ∗ ∗ ∗ if H ∨ K = [ 2n , 2n ) ∨ [ 2n , 2n ) is not G. It is equivalent to saying that there exist two nodes

S. Li, M. Ying, Y. Li / On Countable RCC Models

17

j j+1 i i+1 x = 2km and y = k+1 2m such that x, y 6∈ [ 2n , 2n ) ∩ [ 2n , 2n ) but the arc α connecting x to y is not in arcs(H ∗ ) ∪ arcs(K ∗ ). Note that if H = K, Cs (H, K) holds. If there exist m, k ∈ N such that m ≤ n and 2km = 2in , j j j j+1 k+1 i i i+1 2m = 2n , set x = 2n and y = 2n . Clearly both x and y are not in [ 2n , 2n ) ∩ [ 2n , 2n ), which is the empty set, and the arc that connects x to y is not in arcs(H ∗ ) ∪ arcs(K ∗ ). We next show the ‘necessity’ part. The case that j = i + 1 is trivial. Now suppose Cs (H, K) and j > i + 1. By Cs (H, K), we have two nodes x, y ∈ GN and an arc α with ends x, y such that α is not i i+1 in arcs(H ∗ ) ∪ arcs(K ∗ ). Since α 6∈ arcs(H ∗ ), we have either x ∈ [ 2in , i+1 2n ) or y ∈ [ 2n , 2n ). Similarly, j j+1 j j+1 ∗ by α 6∈ arcs(K ), we have either x ∈ [ 2n , 2n ) or y ∈ [ 2n , 2n ). Note that i < j + 1 and x < y hold, j j+1 we must have x ∈ [ 2in , i+1 2n ) and y ∈ [ 2n , 2n ). 1 1 Suppose x = 2km and y = k+1 2m for some m, k ∈ N. Now, y − x = 2m > 2n holds since i < j + 1. j We must have m < n and x = 2in , y = 2n . In summary, if j > i + 1 and Cs (a, b), we must have some m < n such that x = 2km = 2in and j y = k+1 u t 2m = 2 n .

By this proposition and Proposition 4.7 we have the following theorem: Theorem 5.1. Bψ is isomorphic to Bs .

6. Countable RCC models over [0, 1]n For each n ∈ N+ , consider the standard RCC model RC([0, 1]n ) over the unit n-cube [0, 1]n . By definition, regions in this model are non-empty regular closed subsets of [0, 1]n . Write Ek = { 2ik : 0 ≤ i ≤ 2k } for each k ∈ N+ . Let Xk = {Πni=1 [ai , ai + 21k ] : ai ∈ Ek , ai 6= 1}. Clearly each element in Xk is a sub-cube of [0, 1]n , and hence a non-empty regular closed set. Let Bn S be the sub-algebra of RC([0, 1]n ) generated by these sub-cubes in {Xk : k ∈ N+ }. Clearly, Bn , with the connection inherited from [0, 1]n , is also a countable BCA. We now show that Bn and Bm are not isomorphic for any two different m, n > 0. For each k ≥ 2, we define a property p(k) as follows: a BCA hA, Ci has property p(k) if and only if it satisfies the following conditions: W (1) there exist k connected regions a1 , · · · , ak in A such that ki=1 ai = > and EC(ai , aj ) for any i 6= j; V (2) there exist a sequence of regions, v1 , v2 , v3 , · · · in A such that i∈N+ vi = ⊥, and NTPP(vi+1 , vi ), PO(vi , aj ) for any i ∈ N+ and any j ≤ k; (3) for any region v in A, if C(v, vi ) holds for each i, then C(v, aj ) holds for any j ≤ k. Now we show that, for each n ≥ 1, Bn has property p(2n ), but doesn’t have property p(k) for any k > 2n . This will show Bn and Bm are not isomorphic if n 6= m. Note that for each n ≥ 1, X1 contains 2n sub-cubes of [0, 1]n . Denote these sub-cubes by a1 , a2 , · · · , a2n respectively. Write o for the centroid of [0, 1]n . For each i ≥ 1, take vi as the sub-cube of [0, 1]n that is centered at o and the length of each edge is 21i . Then these ai and vj are regions in Bn that satisfy the above conditions. This shows Bn has property p(2n ).

18

S. Li, M. Ying, Y. Li / On Countable RCC Models

If there exists some k > 2n such that Bn also has property p(k), then by definition,Twe have regions a1 , · · · , ak and v1 , v2 , v3 , · · · in Bn that satisfy the above conditions. Note that V = {vi : i ≥ 1} is nonempty since each vi is a compact set. Take a point p in V . We claim that p ∈ aj for any j ≤ k. Suppose p 6∈ aj for some j ≤ k. Then there exists a region in Bn that contains p but is disconnected from aj . This contradicts the third condition since any region containing p is connected with each vi . We must have p ∈ aj for each j ≤ k. Note that there exists some m ≥ 1 such that all aj are regions composed of sub-cubes in Xm . For each j ≤ k, we have a sub-cube cj ⊆ aj containing p. These cj are pairwise different by the first condition. Since each point in [0, 1]n belongs to at most 2n many different sub-cubes in Xm , we have p 6∈ aj for some j ≤ k. This gives a contradiction. Therefore Bn doesn’t have property p(k) for any k > 2n . It is also worthy noting that B1 is isomorphic to Bχ . We omit details here.

7. Finite GRCC models are sub-models of Bω In Proposition 3.2, we have seen that a countable RCC model can be obtained as the direct limit of a sequence of its finite sub-models. In this section, however, we show that each finite GRCC model can be isomorphically embedded in the countable model Bω . Note that Bω can be isomorphically embedded in each RCC model [19], this means every finite GRCC model can be isomorphically embedded in any RCC model. Proposition 7.1. Suppose hAi , Ci i is a finite GBCA and Xi is its atoms sets (i = 1, 2). Then hA1 , C1 i and hA2 , C2 i are isomorphic if and only if there exists a bijection f : X1 → X2 such that EC1 (x, y) if and only if EC2 (f (x), f (y)) for any two x, y ∈ X1 . Proof: For any finite GBCA hA, Ci and any two regions a, b in A, by Proposition 3.3 we have C(a, b) if and only if there exist two atoms x, y such that x ≤ a, y ≤ b and C(x, y). Suppose hA1 , C1 i and hA2 , C2 i are isomorphic. Then there exists a Boolean isomorphism g : A1 → A2 such that C1 (a, b) in A1 if and only if C2 (g(a), g(b)) in A2 for any two a, b ∈ A1 . Set f as the restriction of g on X1 . Clearly f is a bijection from X1 to X2 with the desired property. On the other hand, suppose there exists a bijection f : X1 → X2 such that W EC1 (x, y) if and only if EC2 (f (x), f (y)) for any two x, y ∈ X1 . Define g : A1 → A2 as g(a) = {f (x) : x ≤ a} for any a ∈ A1 . Then it is straightforward to check that g is an isomorphism between the two finite GBCAs. u t Lemma 7.1. Suppose W hA, Ci is a finite GBCA and X = {x1 , x2 , · · · , xn } is its atoms set. For each 0 1 ≤ k ≤ n, set xk = {xi : 1 ≤ i ≤ n, i 6= k}, i.e., the complement of xk in A. Then there exists some k ≤ n such that x0k is connected. Proof: Suppose all x0k are disconnected. Then we have two disjoint subsets of X, written Fk and Gk , resp., such that Fk ∪ Gk = X \ {xk } and ¬C(f, g) for any W f ∈ Fk and any W g ∈ Gk . Since the universe region, >, in A is connected, we must have both x ∨ ( F ) and x ∨ ( Gk ) are connected regions in A. For k k k W example, if xk ∨ ( Fk ) is disconnected, then we have Hk ⊂ Fk such that atoms in Hk and atoms in

S. Li, M. Ying, Y. Li / On Countable RCC Models

19

{xk } ∪ (Fk \ Hk ) are disconnected. This suggests further that atoms in Hk and atoms in X \ Hk are disconnected. A contradiction. W W Take xi ∈ Fk . If xk ∈ Fi , then we must have {xk } ∪ Fk ⊂ Fi since xk ∨ ( Fk ) and xk ∨ ( Gk ) are connected regions; similarly, if xk ∈ Gi , then {xk }∪Fk ⊂ Gi . Similar results hold for the situation when xi ∈ Gk . This shows, for any k, there exists some i 6= k such that max{|Fk |, |Gk |} < max{|Fi |, |Gi |}. That’s obvious a contradiction. Consequently there exists at least one connected x0k in A. u t Proposition 7.2. Every finite GBCA is isomorphic to a sub-GBCA of Bω . Proof: Suppose hA, Ci is a finite GBCA and X is its atoms set. We prove the result by using induction on n = |X|. Notice that there is a unique connection on A when |X| = 2. The statement trivially holds for n = 2. Suppose for any m < n, if |X| = m, then hA, Ci can be isomorphically embedded in hBm , Cω |Bm i. Now consider X = {x1 , · · · , xn }. Since x0k is connected, we can consider the local GBCA h↓x0k , C|↓x0k i of A at x0k . This GBCA contains n − 1 atoms and therefore, by induction assumption, can be isomorphically embeddedSin hBn−1 , Cω |Bn−1 i. Write x∗i for the corresponding element of xi (i 6= k) in hBn−1 , Cω |Bn−1 i. Then {x∗i : 1 ≤ i ≤ n, i 6= k} = x² = [0, 1). Write T = {1 ≤ i ≤ n : i 6= k, EC(xi , xk )}. Clearly T is nonempty. Take for each i ∈ T a string si with length n − 1 such that ? ? ∗ xsi ⊆ x∗i . For 1 ≤ i ≤ S n, define xi as follows: if ?i ∈ T , ?take xi = xi − xsi 0 ; if i 6∈ T and i 6= k, take ? ∗ ? xi = xi ; take xk = {xsi 0 : i ∈ T }. Clearly X = {xi : 1 ≤ i ≤ n} ⊂ Bn is pairwise disjoint and jointly exhaustive. By Proposition 4.3, we can show (i) for i, j 6= k, ECω (x?i , x?j ) if and only if ECω (x∗i , x∗j ); (ii) for i 6= k, ECω (x?i , x?k ) if and only if i ∈ T . This is because that, take (ii) for instance, for any two strings s, t with |t| = |s| + 1, xs0 is externally connected to xt in Bω if and only if t = s1. Write A? as the subalgebra of Bn generated by X ? . We next show that hA? , Cω |A? i is isomorphic to hA, Ci. To this end, by Proposition 7.1, we need only to show that EC(xi , xj ) if and only if ECω (x?i , x?j ) for any i 6= j. Suppose i 6= j, we divide the proof in three cases: (1) if i, j 6= k, then by Proposition 7.1 and induction assumption, EC(xi , xj ) if and only if ECω (x∗i , x∗j ). Combining this with Conclusion (i) of the above paragraph, we have EC(xi , xj ) if and only if ECω (x?i , x?j ); (2) if i 6= k and j = k, notice that EC(xi , xk ) if and only if i ∈ T . By Conclusion (ii) of the above paragraph, we have EC(xi , xk ) if and only if ECω (x?i , x?k ); (3) if i = k and j 6= k, the proof is similar to that of (2). u t Note that Bω can be isomorphically embedded in any RCC model [19], we have the following: Theorem 7.1. Every finite GRCC model is isomorphic to a sub-model of an RCC model.

20

S. Li, M. Ying, Y. Li / On Countable RCC Models

Remark 7.1. Note that in the definition of GRCC models, we require the universe, viz. the top element, to be connected. This restriction is, however, not always necessary. For example, the regular closed algebra, RC(X), of a disconnected topological space X, with the canonical connection CX is a connection structure that is not a GRCC model. Another example is Galton’s adjacency spaces [17]. The above result, however, also holds for this general case. Note that a finite connection structure hA, Ci can be decomposed into k finite GRCC models, say hA1 , C1 i, · · · , hAk , Ck i, and each GRCC model can be isomorphically embedded in Bω . Now since Bω is isomorphic to its local model at xs for any s ∈ {0, 1}∗ , we can choose strings s1 , · · · , sk with same length such that they are pairwise disconnected in Bω . Now we can embed each hAi , Ci i in the local model of Bω at xsi . In this way we embed hA, Ci in Bω . In the rest of this section we show that how this result can be used to answer the “realization” problem of consistent RCC8 formulas. This problem was first studied by Renz. In [25], Renz showed that every consistent set of RCC8 formulas can be realized in any dimension where regions are (sets of) polytopes. Renz’s result was based on a modal encoding of RCC8 (Bennett 1995) and not directly involved with the RCC models.5 In this section, however, we show every consistent set of RCC8 formulas can be realized in any RCC model. An RCC8 formula R(x, y) is a relation between two spatial variables, where R is an RCC8 relation. A set of RCC8 formulas, or a network of RCC8 relations, Θ = {Rij (xi , xj ) : 1 ≤ i, j ≤ n} is consistent w.r.t. RCC if there exists an RCC model R such that every spatial variable xi can be instantiated by a region ai in R and Rij (ai , aj ) holds for any 1 ≤ i, j ≤ n. We now have the following theorem. Theorem 7.2. Suppose Θ is a consistent network of RCC8 relations and hB, Ci is a BCA. Then Θ can be realized in hB, CB i. Proof: Since Θ is consistent, there exist by definition a BCA hA, Ci and an assignment v that assigns to each variable xi a region v(xi ) in A with Rij (v(xi ), v(xj )). Consider the subalgebra A0 generated by {xi : n i = 1 · · · n} of A. Clearly |A0 | ≤ 22 . (This is because that F (X), the free Boolean algebra generated n by X, contains 22 elements, and that for every f : X → A0 , there exists a Boolean homomorphism f : F (X) → A0 with f ◦ i = f .) Naturally A0 with the connection inherited from hA, Ci is a finite GBCA. But by Proposition 7.2, we know this finite GBCA can be isomorphically embedded in Bω . Recall that Bω is a minimal BCA that can be isomorphically embedded in any BCA, hA0 , C|A0 i can therefore also be isomorphically embedded in hB, CB i. Consequently there exist an isomorphic embedding, say f , from hA0 , C|A0 i to hB, CB i. By Proposition 3.1, we have Rij (f (v(xi )), f (v(xj ))) for any 1 ≤ i, j ≤ n. This shows Θ can be realized in hB, CB i. u t Recall that the countable RCC model Bn (n ≥ 1) given in Section 6 is a sub-model of the standard RCC model RC([0, 1]n ), and regions in Bn are simply (finite sets of) sub-cubes of [0, 1]n . Theorem 7.2 then 5

Indeed, what Bennett [4], Nebel [22] and Renz [25] called “consistency” is w.r.t. topological spaces, i.e., a set of RCC8 formulas is called consistent if there exists a topological space (X, T ) such that every spatial variable xi can be instantiated by a non-empty regular closed subset of X and Rij (ai , aj ) holds for any 1 ≤ i, j ≤ n. Then one result of Renz [25] can be read as “a set of spatial formulas is consistent (w.r.t. topological spaces) if and only if it has a realization in the Euclidean plane.” As a result, our Theorem 7.2 indeed says that consistency w.r.t. topological spaces is the same as consistency w.r.t. RCC theory.

S. Li, M. Ying, Y. Li / On Countable RCC Models

21

shows every consistent set of RCC8 relations can be realized in any dimension n ≥ 1 where regions are (finite sets of) sub-cubes of [0, 1]n .

8. Conclusions and further work In this paper we have argued that standard RCC models have serious drawbacks and properties of countable RCC models were studied. We have shown that each countable RCC model can be constructed hierarchically from a sequence of finite GRCC models on the one hand, and every finite GRCC model can be isomorphically embedded in any RCC model on the other hand. Such an investigation is a contribution for bridging the gap between qualitative approaches for spatial information and quantitative ones. We also gave a simple and more direct proof for the realization problem of consistent spatial network. Results in this paper show that, using the approach based on direct limits, we can construct infinitely many non-isomorphic countable RCC models. One interesting question is how many different countable RCC models there are. This question has been recently answered by Xia.6 As a matter of fact, he shows that there are continuum many non-isomorphic minimal RCC models. Note that each countable RCC model has a hierarchical structure, one interesting question is how to use these models for hierarchical spatial reasoning. One more restrictive question is how to (approximately) determine the RCC8 relations between two arbitrarily shaped plane regions hierarchically. Note that the work of Winter (for RCC5) [29] and our early work (for RCC8) [19] are all limited to plane regions represented as quadtrees.

Acknowledgements We gratefully thank Prof. Ivo D¨untsch and the two anonymous referees for their helpful suggestions.

References [1] Asher, N., Vieu, L.: Toward a Geometry of Common Sense: A Semantics and a Complete Axiomatization of Mereotopology, Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI-95) (C. Mellish, Ed.), Montreal, 1995. [2] Bell, J., Machover, M.: A Course in Mathematical Logic, North-Holland, 1977. [3] Bennett, B.: Spatial Reasoning with Propositional Logics, Principles of Knowledge Representation and Reasoning: Proceedings of the 4th International Conference (KR94) (J. Doyle, E. Sandewall, P. Torasso, Eds.), Morgan Kaufmann, San Francisco, CA., 1994. [4] Bennett, B.: Modal Logics for Qualitative Spatial Reasoning, Bulletin of the Interest Group in Pure and Applied Logic (IGPL), 4(1), 1996, 23–45. [5] Bennett, B.: Logical Representations for Automated Reasoning about Spatial Relationships, Ph.D. Thesis, School of Computer Studies, University of Leeds, 1997. [6] Bittner, T., Stell, J.: A boundary-sensetive approach to qualitative location, Annals of Mathematics and Artificial Intelligence, 24, 1998, 93–114. 6

Xia, On minimal models of Region Connection Calculus, draft paper, 2004.

22

S. Li, M. Ying, Y. Li / On Countable RCC Models

[7] Clarke, B.: A calculus of individuals based on ‘connection’, Notre Dame Journal of Formal Logic, 22, 1981, 204–218. [8] Clarke, B.: Individuals and Points, Notre Dame Journal of Formal Logic, 26(1), 1985, 61–75. [9] Clementini, E., Sharma, J., Egenhofer, M.: Modeling topological spatial relations: strategies for query processing, Computers and Graphics, 18, 1994, 815–822. [10] Cohn, A., Bennett, B., Gooday, J., Gotts, N.: Qualitative Spatial Representation and Reasoning with the Region Connection Calculus, GeoInformatica, 1, 1997, 275–316. [11] Cohn, A., Hazarika, S.: Qualitative spatial representation and reasoning: An overview, Fundamenta Informaticae, 46, 2001, 2–32. [12] Cui, Z., Cohn, A., Randell, D.: Qualitative and Topological Relationships in Spatial Databases, in: Advances in Spatial Databases, Lecture Notes in Computer Sciences 692 (D. Abel, B. Ooi, Eds.), Springer Verlag, Berlin, 1993, 293–315. [13] D¨untsch, I.: Relation algebras and their application in qualitative spatial reasoning, 2003, Preprint. [14] D¨untsch, I., Schmidt, G., Winter, M.: A necessary relation algebra for mereotopology, Studia Logica, 69, 2001, 381–409. [15] D¨untsch, I., Wang, H., McCloskey, S.: Relation algebras in qualitative spatial reasoning, Fundamenta Informaticae, 39, 1999, 229–248. [16] D¨untsch, I., Wang, H., McCloskey, S.: A relation-algebraic approach to the Region Connection Calculus, Theoretical Computer Scinece, 255, 2001, 63–83. [17] Galton, A.: The mereotopology of discrete space, Spatial Information Theory. Cognitive and Computational Foundations of Geographic Information Science, International Conference COSIT’99 (C. Freksa, D. Mark, Eds.), Springer, Berlin, 1999. [18] Gotts, N.: An axiomatic approach to spatial information systems, Research Report 96.25, School of Computer Studies, University of Leeds, 1996. [19] Li, S., Ying, M.: Generalized Region Connection Calculus, Artificial Intelligence, In press. [20] Li, S., Ying, M.: Extensionality of the RCC8 composition table, Fundamenta Informaticae, 55(3-4), 2003, 363–385. [21] Li, S., Ying, M.: Region Connection Calculus: Its models and composition table, Artificial Intelligence, 145(1-2), 2003, 121–146. [22] Nebel, B.: Computational properties of qualitative spatial reasoning: First results, KI-95: Advances in Artificial Intelligence, Proceedings of the 19th Annual German Conference on Artificial Intelligence, SpringerVerlag, Berlin, Germany, 1995. [23] Randell, D., Cohn, A.: Modelling topological and metrical properties of physical processes, First International Conference on the Principles of Knowledge Representation and Reasoning (R. Brachman, H. Levesque, R. Reiter, Eds.), Morgan Kaufmann, Los Altos, 1989. [24] Randell, D., Cui, Z., Cohn, A.: A spatial logic based on regions and connection, Proceedings of the 3rd International Conference on Knowledge Representation and Reasoning (B. Nebel, W. Swartout, C. Rich, Eds.), Morgan Kaufmann, Los Allos, 1992. [25] Renz, J.: A canonical model of the Region Connection Calculus, Proceedings of the 6th International Conference on Knowledge Representation and Reasoning, Morgan Kaufmann, 1998.

S. Li, M. Ying, Y. Li / On Countable RCC Models

23

[26] Renz, J.: Qualitative spatial reasoning with topological information, vol. 2293 of Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, Germany, 2002. [27] Renz, J., Nebel, B.: On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus, Artificial Intelligence, 108, 1999, 69–123. [28] Stell, J.: Boolean connection algebras: A new approach to the Region-Connection Calculus, Artificial Intelligence, 122, 2000, 111–136. [29] Winter, S.: Topological relations in hierarchial partitions, Spatial Information Theory. Cognitive and Computational Foundations of Geographic information Science, International Conference COSIT’99 (C. Freksa, D. Mark, Eds.), Springer, Berlin, 1999. [30] Winter, S.: Topology in raster and vector representation, GeoInformatica, 4, 2000, 35–65. [31] Worboys, M.: Imprecision in finite resolution spatial data, GeoInfomatica, 2, 1998, 257–279.