Spatial Reasoning with Rectangular Cardinal

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Spatial Reasoning with Rectangular Cardinal Direction Relations 1 Isabel Navarrete and Guido Sciavicco 2 Abstract. It is widely accepted that spatial reasoning plays a central role in various artificial intelligence applications. In this paper we study a recent, quite expressive model presented by Skiadopoulos and Koubarakis in [28, 29] for qualitative spatial reasoning with cardinal direction relations. We consider some interesting open problems of this formalism, mainly concerning finding tractable, expressive enough, subclasses of the full set (i.e., including disjunction) of relations. So far, no such subclass have been found except that of basic relations only. We focus on a small subset of cardinal relations, named rectangular cardinal relations. We investigate the connection between rectangular cardinal relations and Balbiani, Condotta and del Cerro’s Rectangle Algebra [2, 3]. By exploiting such a connection, we show that the set of basic rectangular cardinal relations is tractable but the case of disjunctive rectangular relations is not. Then, we introduce a tractable subset of the set of disjunctive rectangular cardinal relations, called saturated-convex rectangular, for which consistency can be decided in quadratic time, and the minimal network can be found in cubic time. Finally, we prove that the saturated-convex rectangular fragment is indeed a subclass of the general model for cardinal relations.

1

Introduction

It is widely accepted that spatial reasoning plays a central role in various artificial intelligence applications. As in the case of other qualitative reasoning formalisms (e.g., qualitative temporal reasoning [4, 26]), spatial reasoning can be viewed under different, somehow complementary, points of view. We distinguish between the algebraic level, that is, purely existential theories formulated as constraint satisfaction problems (CSP s) [31] over jointly exhaustive and mutually disjoint set of topological, directional based, or combined relations, the first-order level, that is, first-order theories of topological, directional based, or combined relations, and the modal logic level, where a (usually propositional) modal language is interpreted over opportune Kripke structures representing space. Moreover, spatial reasoning can be classified in topological, such as in the case of Region Connection Calculus [25] (RCC) or their specializations RCC8 [8], RCC5 [12], and RCC23 [5], directional, or combined (see, e.g., Sharma [27], and Sun and Li [30]), depending on the type of relations between basic objects that are used for reasoning about them (see also [6, 10]). It is worth to mention that, since common problems such as consistency checking of a set of constraints turns out to be (generally) computationally intractable in 1 2

This work was supported by the Spanish MEC under project MEDICI (TIC2003-09400-C04-01) University of Murcia, Murcia E-30100 - Espinardo (Murcia) SPAIN Campus de Espinardo, email: inava|[email protected]

the case of topological relations, much effort has been focused on finding tractable fragments of such systems (see, for example, Renz and Nebel [17]). As for topological relations, different directional formalisms have been studied for different classes of regions. In particular, it has been made an important distinction between regular-shaped regions and irregular-shaped ones. In the former case, beside the well-known Interval Algebra (IA) [1], which, to some extent, can be viewed as a spatial calculus for line-shaped regions in a one-dimensional space, we mention Ligozat’s work [19] on cardinal direction relations between points in a two-dimensional space, and the generalization of Allen’s interval algebra to space, by Guesgen [13], and Mukerjee and Joe [22], called Rectangle Algebra (RA). Rectangle Algebra has later been studied by Balbiani, Condotta, and Del Cerro [2, 3]. In RA, the considered objects are rectangles whose sides are parallel to the axes of some orthogonal basis in a bidimensional Euclidean space. So far, RA has been studied as the natural generalization of IA, but, at the best of our knowledge, no geographical interpretation has been given to it. One of the most expressive models for qualitative spatial (and geographical) reasoning with cardinal directions with irregular-shaped regions is that of Skiadopoulos and Koubarakis [28, 29]. Skiadopoulos and Koubarakis re-consider Goyal and Egenhofer’s model presented in [11]. A region in a twodimensional Euclidean space, which, according to the majority of the authors, can be defined as a non-empty bounded set of points in space which is closed, connected and has connected boundaries, or a finite union of such sets, is approximated by its minimum bounding box, and, given a reference region b and a referred region a, the relation between a and b is expressed by analyzing the 9 regions (tiles) formed by the axes of the minumum bounding box of b. For example, if a lies partially at the north and partially at the northeast of b, this relation is denoted by a N :N E b. Such a model (which we call here SK-model) for cardinal direction relations presents a number of advantages, since, to some extent, it overcomes the limitations of point-based and box-based approximation models. As a practical example, in a point-based approximation, Spain (as a country) lies at the northeast of Portugal, while in a model fully based on minimum bounding boxes Portugal is contained in Spain (since Spain’s region is convex); most people would agree that none of this representation is accurate, since Spain lies partially at the north, partially at the northeast, and partially at the east of Portugal [28, 29]. In this paper, we will consider some open problems in the SKmodel. Focusing on the set of all connected regions, called REG, we will consider a strict subset of the class of all basic cardinal relations, called rectangular cardinal relations. Such a fragment only contains 36 out of 218 basic SK-relations, but it is still expressive enough to formalize interesting spatial statements. The problem of

6 N W (b)

a+ y

W (b)

a a− y

SW (b)

N (b)

b

B(b)

S(b)

N E(b)

E(b)

SE(b)

a− x

a+ x Figure 2. Cardinal tiles w.r.t. mbb(b)

Figure 1. Axes-projections and mbb(a)

checking the consistency of a set of basic (only!) cardinal relations is still open for REG (in [29] such a problem has been solved for REG∗ by means of an ad-hoc, i.e., non composition-based, rather complex algorithm). We will explore the evident connection between rectangular cardinal relations and rectangle algebra, and, by exploiting such a connection, we will show that the set of basic-rectangular cardinal relations is tractable but the case of disjunctive rectangular relations is not. Then, we will introduce a tractable fragment of the set of disjunctive rectangular cardinal relations, whose relations will be called saturated-convex rectangular, for which consistency can be decided in quadratic time, and the minimal network can be decided in cubic time. This paper is organized as follows. In Section 2 we revise the SKmodel and its relations. In Section 3 we introduce rectangular cardinal relations and we explore the connection with Rectangle Algebra. Finally, in Section 4 we introduce saturated-convex rectangular relations, and we show how the problem of consistency checking and minimal network finding can be solved in the case of rectangular relations, before concluding.

2

The SK-Model

In this section, we shortly revise the formal model for qualitative spatial reasoning with cardinal direction relations proposed by Skiadopoulos and Koubarakis [28, 29], that we call here SK-model. Cardinal direction relations (or simply cardinal relations) are used to describe how regions of space are placed relative to one another. We consider regions which are homeomorphic to the closed unit disk ({(x, y) ∈ R2 | x2 + y 2 ≤ 1}). Such a set of regions, previously studied in [11, 24], is called REG, and it excludes disconnected regions, region with holes, points, lines and regions with emanating lines. Nevertheless regions in REG can be used to model areas in some interesting applications [9, 10]. The set of all finite unions of regions in REG is called REG∗ ; Skiadopoulos and Koubarakis extend in [29] the SK-model presented in [28] to consider regions in REG∗ , which may be disconnected and have holes. Definition 1 (axes-projections) Let a ∈ REG and consider the orthogonal axes of the space R2 . We call ax the interval that is obtained as the projection of region a on the x-axis, and ay the interval corresponding to the projection of a on the y-axis. Each interval is supposed to be closed, thus it is representable by its + − + endpoints. The symbols a− x (resp., ax ) and ay (resp., ay ) denote the infimum (resp., the supremum) of the intervals ax and ay . Definition 2 (minimal bounded box) The minimal bounded box of region a, denoted by mbb(a), is the box whose sides are parallel

to the axes, and it is formed by the straight lines x = a− x, x = − + a+ x , y = a y , y = ay . See Fig. 1 for an example.

2.1

Basic Cardinal Relations and Constraints

Let us consider two arbitrary regions a and b in REG. We are interested in how region a, called primary region, is related to region b through a cardinal relation (e.g., a is completely at the South of b). Region b is referred to as the reference region. By considering the axes of mbb(b), the space is divided into 9 areas named cardinal tiles (or tiles for short), as we can see in Fig. 2. Let CT = {B, S, SW, W, N W, N, N E, E, SE} be the set of symbols used to denote the cardinal tiles of some reference region. The symbol B corresponds to the central tile and denotes the relation belongs to the mbb of. The peripheral tiles correspond to the eight cardinal direction relations South, SouthWest, West, NorthWest, North, NorthEast, East, SouthEast. Given a, b ∈ REG and R ∈ CT , the expression a R b means that region a lies entirely on the tile R of region b (R(b) for short). Definition 3 (basic CR, tile CR, multitile CR) A basic cardinal relation (basic CR) is an expression R1 : · · · :Rk where: 1. 1 ≤ k ≤ 9; 2. R1 , . . . , Rk ∈ CT ; If k = 1 the relation is called tile CR; otherwise it is called a multitile CR; 3. Ri 6= Rj , ∀ 1 ≤ i, j ≤ k, i 6= j; 4. ∀ b ∈ REG, ∃ a1 , . . . , ak ∈ REG such that a1 ∈ R1 (b), . . . , ak ∈ Rk (b) and a1 ∪ . . . ∪ ak = a. If R1 : · · · :Rk is a basic CR and a, b ∈ REG, then a R1 : · · · :Rk b is a basic cardinal relation constraint. A basic CR constraint a R1 : · · · :Rk b is satisfiable if and only if: ∃ a1 , . . . ak ∈ REG such that (a1 R1 b, . . . , ak Rk b) ∧ (a1 ∪ . . . ∪ ak = a). When R is a tile CR we have that a B b ⇔ a ⊆ mbb(b). Such condition can be equivalently expressed as a set of order constraints between the endpoints of the intervals corresponding to the axesprojections of mbb(a) and mbb(b), as shown in Table 1. A (tile or multitile) basic cardinal relation R can also be defined using the set-theoretic notation for a binary relation, that is: R = {(a, b) ∈ REG2 | a R b is satisfied}

aBb aSb a SW b aW b a NW b aN b a NE b aEb a SE b

⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔ ⇔

− + + − − + + b− x ≤ ax ∧ ax ≤ bx ∧ by ≤ ay ∧ ay ≤ by − − − + + a+ ≤ b ∧ b ≤ a ∧ a ≤ b y y x x x x − + − a+ x ≤ bx ∧ ay ≤ by + − − − + ax ≤ bx ∧ by ≤ ay ∧ ay ≤ b+ y − + − a+ x ≤ bx ∧ by ≤ ay + − − − + by ≤ ay ∧ bx ≤ ax ∧ ax ≤ b+ x − + − b+ x ≤ ax ∧ by ≤ ay − − − + + b+ x ≤ ax ∧ by ≤ ay ∧ ay ≤ by − + − b+ x ≤ ax ∧ ay ≤ by

Table 1. Tile constraints and endpoints constraints

Example 1 In Figure 3 we show how regions a = a1 ∪a2 ∪a3 and b satisfies a W :N W :N b. It is worth observing that, for example, there are no regions c, d in REG such that c E:W d holds, so E:W is not a basic CR according to definition 3. The set of basic cardinal relations in the SK-model is named D and, when the considered ` ´set of regions is REG, it contains 218 elements out of the Σ9i=1 9i = 511 possible basic cardinal relations between any two regions in REG∗ . An enumeration and a pictorial representation for all relations in D can be found in [11]. The powerset 2D contains all possible disjunctions of basic cardinal relations. If R ∈ 2D then R is a cardinal relation (CR) and it can be used to represent indefinite information about the relative position of two regions. The semantics of a CR-constraint a R b with R ∈ 2D is given, as usual, by the logical disjunction between its components. Thus, given two region variables a, b over REG and a relation R = {R1 , R2 , . . . , Rn } ∈ 2D , then we have that a R b if and only if a R1 b ∨ a R2 b ∨. . . ∨ a Rn b.

2.2

Spatial Reasoning with Cardinal Relations

Once we have formally defined cardinal relations and constraints in the SK-model, we are able to deal with some qualitative spatial reasoning tasks within this model. For this purpose, one can use a binary constraint network [7] where nodes (variables) represents regions of REG and arcs (constraints) represent a set of allowed cardinal relations between them. We call such a network a CR-network and, as usual, the main problem is determining the consistency of the network. Definition 4 A CR-network N with variables a1 , . . . , an is consistent (or satisfiable) if an only if there exists a solution given by an n-uple (α1 , . . . , αn ) ∈ REGn such that all CR-constrains of N are satisfied by the assignment ai = αi , ∀ 1 ≤ i ≤ n. Such an algebraic approach to spatial reasoning requires the definition of several operations in the set of 2D cardinal relations. Beside the Boolean set-theoretic connectives, the natural operations on binary relations are inverse (or converse) and composition [14]. Although an algebra for cardinal relations has not been formally characterized in the SK-model, Skiadopoulos and Koubarakis [28] defines the operations of inverse and composition of cardinal relations. Definition 5 Let R ∈ 2D . The inverse of R, denoted by R−1 , is a cardinal relation which satisfies a R−1 b ⇔ b R a.

a2

a3

a1 b

Figure 3.

A representation of a W :N W :N b, where a = a1 ∪ a2 ∪ a3

As noticed in [28] the inverse of a basic CR may be not a basic CR. For instance, N −1 = {S:SW :SE, S:SW, SE:S, S}. This is not the case for several algebras in temporal reasoning, such as interval algebra [1] and point algebra [33]. An algorithm for computing the inverse is discussed in [29]. The composition operation is used in the general paradigm of constraint satisfaction problems (CSP s) [31] for inferring new information from existing ones throughout constraint propagation. For a qualitative calculi where variables have infinite domains (like temporal and spatial qualitative constraint formalisms), the path consistency algorithm [20, 21] plays an important role since it sometimes can be used for determining consistency and other reasoning tasks [16, 17, 23, 32]. In the SK-model the following definition of composition of two cardinal relation is used: Definition 6 Let R1 and R2 be cardinal relations. The composition of R1 and R2 is another CR, named R1 ¦R2 , is defined as R1 ¦R2 = {Q ∈ D | ∃ a, b, c ∈ REG : a R1 b ∧ b R2 c ∧ a Q c} It is worth noticing that such definition corresponds to a weak composition, in contrast to the so-called true composition, defined by R1 ◦ R2 = {(a, b) ∈ REG2 | ∃ c ∈ REG : (a, c) ∈ R1 ∧ (c, b) ∈ R2 }. The difference between weak and true composition with respect to their possible use for deciding consistency of a CSP in a qualitative calculus is studied by Renz and Ligozat in [15]. As shown in [28], weak composition of cardinal relations cannot be used for deciding consistency of a CR-network (not even with basic relations only) and true composition is not expressible in the SK-model because such operation is not internal. Thus, the SKmodel is an example (as in the cases reported in [15]) of a qualitative calculi where consistency of basic relations cannot be decided by an algebraic closure algorithm (essentially path consistency with weak composition). In [29], an ad-hoc algorithm for checking the consistency of a set of basic cardinal direction constraints but with variables ranging over REG∗ . In this case, the set of basic relations, denoted by D∗ contains 511 elements. The algorithm is quite complicated, and it works essentially by exploiting the following observation: each basic CR can be partially expressed as a set of constraints in the Point Algebra [33]. A constraint such as a R1 : · · · :Rk b is opportunely divided into a set of constraint on new components variables a1 , . . . , ak of a such that each new constraint only involves a single tile. Then, the new constraints are transformed into a set of order constraints on variables representing the x and y-axes projection of the minimum bounding box approximating each one of the new regions ai , aj . . .. Finally, if the resulting set of point-based constraints is consistent, a set of additional union constraints is checked in order to make sure that the division of a into a1 , . . . , ak (for every variable region a), is also consistent. The complexity of the algorithm is O(n5 ), where n is the number of region variables. Such an high complexity aims for finding new, more efficient, approaches.

reference interval: f inishes (f ): during (d): starts (s): overlaps (o):

bef ore (b):

Figure 4. Basic IA-relations

In [29] it has also been shown that the problem of checking the consistency of a network of cardinal relation constraints that belong ∗ to 2D is NP-complete. Three interesting open problems (pointed out in [29] and still open to our knowledge) within the SK-model are: 1. Are there tractable fragments of the SK-model which are more expressive than the one with only basic relations? 2. Is it possible to devise an algorithm for checking the consistency of a network of cardinal constraints which is entirely compositionbased? and related with this, is it possible to extend the model so that a true composition could be used? 3. How the restriction to the set REG of regions influences the complexity of the consistency problem for a network of CRconstraints? Moreover, no algorithm for deciding consistency of CR-networks with regions over REG has been found yet.

Rectangular Cardinal Relations and Rectangle Algebra

In this section, we concentrate our attention on a special case of cardinal relations, named rectangular cardinal relations (RCRs), and we investigate the connection between this type of cardinal relations and the rectangle algebra.

3.1

The set of all possible disjunctions of basic RA-relations is given by the power set 2Arec , and an RA-constraint a R b, where R ∈ 2Arec is satisfiable if and only if there exist two rectangles a, b satisfying one of the basic RA-relations in R. Beside the usual set-operations, the rectangle algebra is enriched with the operations of inverse and composition, which are stable for the set 2Arec :

meets (m):

3

Example 2 If rectangle a is entirely included into the rectangle b, and no side of a touches any of the sides of b, then the relations between a and b is (d, d).

Rectangle Algebra

Balbiani et al. [2] introduce the rectangle algebra as a natural generalization of the well-known Allen’s Interval Algebra [1] to the bidimensional space. The IA models the relative position between any two intervals as a suitable set of thirteen basic (or atomic) relations (see Fig.4), namely bef ore, meets, overlaps, starts, during, f inishes (b, m, o, s, d, f ) together with theirs inverses (bi, mi, oi, si, di, f i) and the basic relation equal (eq). Similarly, the domain considered in the rectangle algebra is the set of rectangles, called here REC, whose sides are parallel to the axes of some orthogonal basis in a bi-dimensional Euclidean space such as R2 . A basic relation between two rectangles (basic RA-relation) is a pair (Rx , Ry ) of basic IA-relations: the x-axis relation and the y-axis realiton. The set of basic RA-relations in called Arec . In this way, there are 132 = 169 possible basic relations between any two given rectangles. If a, b ∈ REC, then a (Rx , Ry ) b is a basic RA-constraint which is satisfied iff the IA-constraints ax Rx bx and ay Rx by are satisfied by the rectangle axes-projections ax , bx and ay , by , respectively.

• Composition: if (Ax , Ay ), (Bx , By ) ∈ Arec then (Ax , Ay ) ◦ (Bx , By ) S = ((Ax ◦ Bx ), (Ay ◦ By )), thus if R, S ∈ 2Arec then R ◦ S = {(Ax , Ay ) ◦ (Bx , By ) | (Ax , Ay ) ∈ R, (Bx , By ) ∈ S}; −1 • Inverse: if (Ax , Ay ) ∈ Arec then (Ax , Ay )−1 = (A−1 x , Ay ), −1 Arec −1 −1 thus if R ∈ 2 then R = {(Ax , Ay ) | (Ax , Ay ) ∈ R}. A RA-network is a structure given by a set of V of variables which represents rectangles and a set C of RA-constraints between variables. A RA-network N with variables a1 , . . . , an is consistent if an only if there exists a solution given by n rectangles (ρ1 , . . . , ρn ) ∈ REC n such that all RA-constrains of N are satisfied by the assignment ai = ρi , ∀ 1 ≤ i ≤ n. Being an extension of the IA, the rectangle algebra inherits the same complexity difficulty for what concerns the consistency problem of a RA-network. Some tractable subclasses has been identified in [2, 3]. When considering a subset of 2Arec , a fundamental property which has to be checked is the stableness for the operations of the algebra. Saturated RA-relations are those which are obtained though the cartesian product of two relations of the interval algebra, and the class of saturated relations in 2Arec is stable for the RA-operations. Example 3 The relation R = {(m, b), (m, m), (b, b), (b, m)} is a saturated RA-relation since R = {m, b} × {m, b}. On the contrary, the relation {(m, b), (b, m), (d, s)} is not saturated. Intuitively, a non-saturated relation cannot be viewed as the disjunction of all possible relative position between two given rectangles. Balbiani et al. [2] introduce the sets of saturated-convex relations and saturated-preconvex relations which coincide with the class of relations that are obtained through the cartesian product of two convex (respectively preconvex) relations of the IA [18, 23]. For both subclass, the consistency problem can be solved with a path consistency algorithm in polinomial time. We will refer again to these tractable fragments of the rectangle algebra in Section 4.

3.2

Rectangular Cardinal Relations

A special type of basic cardinal relation is defined within the SKmodel. Definition 7 A basic cardinal relation R is a basic-rectangular cardinal relation (or, simply, a basic-rectangular CR) if and only if there exists two rectangles (whose sides are parallel to the x and y-axes) a, b over REG such that a R b is satisfied. The set of basic-rectangular CRs, denoted here by Drec , contains 36 elements, shown in Table 2 and Table 3. Basic-rectangular CR are introduced in [28] with the only purpose of using them in the (complicated) process of computing the weak composition of two

R : Basic-rectangular CR (tile case) B S N E W NE NW SE SW

Tr (R) : RA-relation {d, s, f, e} × {d, s, f, e} {d, s, f, e} × {m, b} {d, s, f, e} × {mi, bi} {mi, bi} × {d, s, f, e} {m, b} × {d, s, f, e} {mi, bi} × {mi, bi} {m, b} × {mi, bi} {mi, bi} × {m, b} {m, b} × {m, b}

Table 2. The mapping between basic-rectangular CRs and RA-relations (tile case)

basic cardinal relations. There exists a natural connection between basic-rectangular CRs and relations in rectangle algebra. In what follows, we will explore such connection in order to re-use known results in rectangle algebra to find a tractable subclass of 2Drec in SK-model. The correspondence3 between a basic-rectangular CR and a RArelation is shown in Table 2 and Table 3. We define a mapping between rectangular cardinal relations and rectangle relations as a function Tr : 2Drec 7→ 2Arec as follows:  as in Table 2 and 3 if R ∈ Drec Tr (R) = Tr (R1 ) ∪ . . . ∪ Tr (Rk ) if R = R1 ∪ . . . ∪ Rk

R : Basic-rectangular CR (multitile case) S:SW S:SE N :N W N :N E B:W B:E B:S B:N W :SW W :N W E:SE E:N E S:SW :SE N :N W :N E B:W :E B:N :S W :N W :SW E:N E:SE B:S:SW :W B:W :N W :N B:S:E:SE B:N :N E:E B:S:SW :W :N W :N B:S:SE:E:N E:N B:S:SW :W :E:SE B:W :N W :N :N E:E B:S:SW :W :N W :N :N E:E:SE

Tr (R) : RA-relation {f i, o} × {m, b} {si, oi} × {m, b} {f i, o} × {mi, bi} {si, oi} × {mi, bi} {f i, o} × {d, s, f, e} {si, oi} × {d, s, f, e} {d, s, f, e} × {f i, o} {d, s, f, e} × {si, oi} {m, b} × {f i, o} {m, b} × {si, oi} {mi, bi} × {f i, o} {mi, bi} × {si, oi} {di} × {m, b} {di} × {mi, bi} {di} × {d, s, f, e} {d, s, f, e} × {di} {m, b} × {di} {mi, bi} × {di} {o, f i} × {o, f i} {o, f i} × {si, oi} {si, oi} × {o, f i} {si, oi} × {si, oi} {o, f i} × {di} {si, oi} × {di} {di} × {f i, o} {di} × {si, oi} {di} × {di}

Table 3. The mapping between basic-rectangular CRs and RA-relations (multitile case)

3

It is worth noticing that such a reduction had already been reported in [34], but this is practically their only contribution.

Lemma 1 If R is a basic-rectangular cardinal relation and a, b ∈ REG, then a R b ⇔ mbb(a) Tr (R) mbb(b). Proof. When R is consist of a single tile, the proof straightforwardly follows from the translation of each tile CR to point algebra relations between the endpoints of mbb(a) and mbb(b) given in Table 1 and the translation between P A-relations and IArelations given in [23, 32]. For multitile relations the idea is similar, directly considering the IA-relation satisfied by the projections mbb(a)x , mbb(b)x , mbb(a)y , and mbb(b)y . ¤ Figure 5 shows the relative position of two rectangles satisfying any of the 13 × 13 basic RA-relations. We have labeled each position of the table with the rectangular cardinal relation that is satisfied. By taking the union of pairs of basic IA-relations corresponding to rectangles that satisfies some basic-rectangular cardinal relation R we obtain the mapping Tr (R) showed in Table 3. Lemma 2 If R ∈ 2Drec is a rectangular cardinal relation and a, b range over REG then a R b ⇔ mbb(a) Tr (R) mbb(b) Proof. Suppose that R = R1 ∪ · · · ∪ Rk , where Ri is a basicrectangular CR, for 1 ≤ i ≤ k. So, the constraint a R b is satisfied if and only if a R1 b ∨ . . . ∨ a Rk b. But, by Lemma 1, a Ri b ⇔ mbb(a) Tr (Ri ) mbb(b). Hence a R b ⇔ mbb(a) Tr (R1 ) mbb(b) ∨ . . .∨mbb(a) Tr (Rk ) mbb(b). By definition of Tr and of satisfiability of RA-constraints, we have that a R b ⇔ mbb(a) Tr (R) mbb(b). ¤ Now we extend the mapping Tr to constraints networks. Let N = (C, V ) be a network with a set of variables V ranging over REG and a set of constraints C of 2Drec . The translation of N is an equivalent RA-network Tr (N ) = (Tr (C), Tr (V )), where: 1. Tr (V ) transforms any variable a over REG into a variable ar over REC. 2. Tr (C) transforms any rectangular CR-constraint a R b into a RA-constraint ar Tr (R) br Theorem 1 A network of rectangular cardinal constraints N is consistent iff Tr (N ) is consistent. Proof. (⇒) Suppose N is consistent. Then there is an assignment ai = αi , such that αi ∈ REG, ∀ 1 ≤ i ≤ n, which satisfies all rectangular CR-constraints in C. By Lemma 2, if αi R αj then mbb(αi ) Tr (R) mbb(αj ). So, the assignment ai r = mbb(αi ) satisfies all RA-constraints in Tr (N ). Hence (mbb(α1 ), . . . , mbb(αk )) is a solution to the RA-network and the network is thereafter consistent. (⇐) If Tr (N ) is consistent then there is an assignment ai r = ρi , such that each ρi is a rectangle in REC, which satisfies all RAconstraints in Tr (N ). If ρi Tr (R) ρj is satisfied in Tr (N ), it is the case that ρi R ρj is also satisfied in N , since R = Tr (R)−1 and Lemma 2 holds. Hence (ρ1 , . . . , ρk ) is also a solution to the network N. ¤

4 A Tractable Subclass of Cardinal Relations In this section, we will exploit the results about tractability of the consistency problem in the rectangle algebra in order to find tractable subclasses for the SK-model.

Figure 5. Basic RA-relations and basic-rectangular CRs

4.1

The Basic-Rectangular Cardinal Relation Fragment

Consider the mapping Tr applied to a basic-rectangular CR. From the translation shown in Table 2 and Table 3, it is easy to observe that for R ∈ Drec , only six different IA-relations are needed to generate Tr (R), namely: K1 = {d, s, f, e},

K2 = {m, b},

K3 = {mi, bi},

K4 = {f i, o},

K5 = {si, oi},

K6 = {di}.

It is simple to see that, for all 1 ≤ i ≤ 6, Ki is a convex IArelation (see Ligozat [18]), or, equivalently, it can be found in the list of continuous-endpoints IA-relations (whose class coincides with the class of convex IA-relations) provided in [23]. Consequently, Tr (R) is a saturated-convex RA-relation, being the cartesian product of two convex IA-relations, that is, of the form Ki × Kj , for some 1 ≤ i, j ≤ 6. Let N = (C, V ) be a network with a set of variables V ranging

over REG and a set C of constraints in Drec . We call BRC-SAT to the problem of deciding the consistency of such a network. Similarly, when the constraints in C range over 2Drec , we call such a problem RC-SAT. Theorem 2 BRC-SAT is polynomial. Proof. Let N = (C, V ) be a network with a set of variables V ranging over REG and a set of constraints C in Drec . Consider the translation Tr (N ) defined in section 3.2. By Theorem 1, we have that N is consistent if and only if Tr (N ) is consistent. Since the problem of deciding consistency of an RA-network with saturatedconvex RA-relations is polynomial, so is BRC-SAT. ¤ As stated in [2], the fact that relations in Tr (N ) are the form Ki ×Kj allows one to apply a path-consistency algorithm to the two IAnetworks independently, that is, one can apply a path-consistency algorithm to the network (Tr (N ))x of x-axis relations, and the network (Tr (N ))y of y-axis relations. Alternatively, a more efficient algorithm consists in solving each network (Tr (N ))x and (Tr (N ))y independently by translating each one to a point algebra network and checking for consistency within this model in O(n2 ) (see [32]).

K1 •

Hence, BRC-SAT is not only polynomial but also can be solved efficiently. Theorem 3 RC-SAT is NP-complete. Proof. Let N = (C, V ) be a network with a set of variables V ranging over REG and a set of constraints C in 2Drec . Deciding the consistency of N is in NP, since a nondeterministic algorithm first guesses a basic-rectangular CR-constraint upon each rectangular CR-constraint in N , obtaining a network N 0 , and then applies the polynomial algorithm to solve BRC-SAT for N 0 . In order to prove that RC-SAT is NP-complete, the reduction of the problem 3SAT to the problem of satisfiability of a set of cardinal constraints with variables over REG∗ , shown in [29], will suffice, since in such a reduction only regions in REG and rectangular CR are used. It is worth noticing that using Theorem 2 is essential here, since the algorithm in [29] cannot be applied when variables range over REG. ¤

4.2

The Saturated-Convex Rectangular Cardinal Relations Subclass

Once we know that solving a set of basic-rectangular CR-constraints is polynomial and that each one of these constraints can be translated to an equivalent saturated-convex RA-relation, the following question arises: how can we enlarge the set Drec in order to obtain a new tractable fragment? In this section, we will devise a subset of the set of rectangular CRs small enough to be totally included in the set of saturated-convex relations of RA. As we said in section 3.1, the set of saturated-convex relations defined for the rectangle algebra coincides with the class of relations that are obtained through the cartesian product of two convex relations of the IA. Hence, if R1 , . . . , Rk are RA-relations, the relation R1 ∪ . . . ∪ Rk is a convex RA-relation if and only if the following two conditions hold: 1. both disjunctions of x- and y- axis relations R1x ∪ . . . ∪ Rkx and R1y ∪ . . . ∪ Rky are convex IA-relations; 2. R1 ∪ . . . ∪ Rk is saturated. As we observed in the previous section, we have that each basicrectangular CR corresponds to a Cartesian product Ki × Kj of convex IA-relations, where 1 ≤ i, j ≤ 6. Definition 8 (saturated rectangular CR) Let R1 ∪ . . . ∪ Rk be a disjunction of basic-rectangular CRs, such that for each 1 ≤ m ≤ k, Tr (Rm ) = Kim × Kjm where 1 ≤ im , jm ≤ 6. If one of the following holds: 1. ∀ 1 ≤ m, n ≤ k : Kim = Kin ; 2. ∀ 1 ≤ m, n ≤ k : Kjm = Kjn , then we call R1 ∪ . . . ∪ Rk saturated rectangular CR. Lemma 3 For any saturated rectangular cardinal relation R = R1 ∪ . . . ∪ Rk , Tr (R) is a saturated RA-relation. Proof. By definition of Tr we have that Tr (R) = Tr (R1 ) ∪ · · · ∪ Tr (Rk ), and since each Rm is a basic-rectangular CR we have that Tr (Rm ) = Kim × Kjm , where 1 ≤ im , jm ≤ 6. So we have that Tr (R) = (Ki1 × Kj1 ) ∪ · · · ∪ (Kik × Kjk ). But since R is saturated and rectangular, by definition 8 it must be Tr (R) = Kis × Kj1 ∪ · · · ∪ (Kis × Kjk ) or Tr (R) = (Ki1 × Kjs ) ∪ · · · ∪ (Kik × Kjs ),

• K3

• K5

• K4

• K2

• K6

Figure 6. The lattice of axes relations.

for some 1 ≤ s ≤ k. So, by a simple set-theoretic property we have that Tr (R) = (Kis × (Kj1 ∪ · · · ∪ Kjk )) or Tr (R) = (Ki1 ∪ · · · ∪ (Kik × Kjs )). So we have that Tr (R) is a saturated RA-relation. ¤ Now, we restrict the definition of saturated rectangular CRs in order to characterize a set of cardinal relations which correspond with saturated-convex RA-relations. Definition 9 (saturated-convex rectangular CR) Let R1 ∪. . .∪Rk be a disjunction of basic-rectangular CRs, such that for each 1 ≤ m ≤ k, Tr (Rm ) = Kim × Kjm where 1 ≤ im , jm ≤ 6. If one of the following holds: 1. ∀ 1 ≤ m, n ≤ k : Kim = Kin , and Kj1 ∪ · · · ∪ Kjk is a convex IA-relation; 2. ∀ 1 ≤ m, n ≤ k : Kjm = Kjn , and Ki1 ∪ · · · ∪ Kik is a convex IA-relation, then we call R1 ∪ . . . ∪ Rk saturated-convex rectangular CR. As for example, the relation {N, N :E, N E} is saturated-convex, while neither {N, N E} or {N :N E, N :B} are so. Theorem 4 For any saturated-convex rectangular cardinal relation R = R1 ∪ . . . ∪ Rk , Tr (R) is a saturated-convex RA-relation. Proof. From lemma 3 we know that Tr (R) is a saturated RArelation of the form: Tr (R) = Kis × (Kj1 ∪ · · · ∪ Kjk ) or Tr (R) = (Ki1 ∪ · · · ∪ Kik ) × Kjs , for some 1 ≤ s ≤ k. Now, by definition 9 we have Kj1 ∪ · · · ∪ Kjk or Ki1 ∪ · · · ∪ Kik is a convex IA-relation. Hence, Tr (R) is the Cartesian product of two convex IA-relations and so Tr (R) is a saturated-convex RA-relation. ¤ With the help of Tables 2,3, and of diagram depicted in Figure 6, one can easily check whether a cardinal relation R = R1 ∪ . . . ∪ Rk is a saturated-convex rectangular CR: 1. obtain T (R) = Tr (R1 ) ∪ · · · ∪ Tr (Rk ), 2. check if condition 1 (or 2) of definition 8 holds, 3. if so, check if the list Kj1 , . . . , Kjk , if it is the case that condition 1 holds (or Ki1 , . . . , Kik , otherwise) corresponds to a path π in the diagram of Figure 6 such that π contains only nodes labeled with relations in the set {Kj1 , . . . , Kjk } (or {Ki1 , . . . , Kik }). The fact that point 3 above is correct can be justified as follows. Consider a saturated-convex cardinal relation R = R1 ∪ . . . ∪ Rk such that, for example, condition 1 of Definition 8 holds, and consider the translation Tr (R). This means that ∀ 1 ≤ m, n ≤ k : Kim = Kin (thus, by Lemma 3, R is a saturated RA-relation); then, in order to

check whether R is also convex, we have to make sure that the IArelation {Kj1 , . . . , Kjk } is convex. Since each Kjl (1 ≤ jl ≤ 6) is convex, and the union of convex IA-relations is associative, we have to prove only the case of two relations (i.e., k = 2). We have six different cases: 1. Kj1 = K3 and Kj2 = K5 . By definition, we have that K3 = {mi, bi} and K5 = {si, oi}, and the IA-relation {mi, bi, si, oi} is convex; 2. Kj1 = K5 and Kj2 = K1 . By definition, we have that K5 = {si, oi} and K1 = {d, s, f, e}, and the IA-relation {si, oi, d, s, f, e} is convex; 3. Kj1 = K5 and Kj2 = K6 . By definition, we have that K5 = {si, oi} and K6 = {di}, and the IA-relation {si, oi, di} is convex; 4. Kj1 = K6 and Kj2 = K4 . By definition, we have that K6 = {di} and K4 = {f i, o}, and the IA-relation {di, f i, o} is convex; 5. Kj1 = K1 and Kj2 = K4 . By definition, we have that K1 = {d, s, f, e} and K4 = {f i, o}, and the IA-relation {d, s, f, e, f i, o} is convex; 6. Kj1 = K4 and Kj2 = K2 . By definition, we have that K4 = {f i, o} and K2 = {m, b}, and the IA-relation {f i, o, m, b} is convex. Clearly, it is possible to show that the same holds when in the case of condition 2 of Definition 8. Now, let SCR be the subset of saturated-convex rectangular CRs from 2Drec . Let N = (C, V ) be a network with a set of variables V ranging over REG and a set of constraints based on relations in the set SCR. We call SCR-SAT the problem of deciding the consistency of such a network. Theorem 5 SCR-SAT is polynomial. Proof. We know that if R ∈ SCR then Tr (R) is a saturatedconvex RA-relation by Theorem 4. So Tr (N ) is a network of saturated-convex RA-relations, for which the consistency can be decided in polynomial time with a path-consistency algorithm [2]. Then, the theorem is proved by the equivalence of N and Tr (N ) with respect to the consistency problem, as shown in Theorem 1. ¤ Moreover, we can prove that SCR is a subclass of 2D , that is, SCR is closed under intersection, inverse and weak composition. To this end, we have to deal with the following problem: given a saturated-convex RA-relation R, we have to find the smallest saturated-convex rectangular CR R0 such that if a R b then a R0 b. We call Tc (R) such a relation and we refer to it as the cardinal closure of R. Before defining the mapping Tc we define an auxiliary function K from the set of convex IA-relations to the set 2{K1 ,...,K6 } in the following way:  K(R) =

Ki such that Ki ∩ R 6= Ø K(R1 ) ∪ . . . ∪ K(Rk )

if R is basic if R = R1 ∪ . . . ∪ Rk

Notice that function K is well-defined since any basic IA-relation appears in exactly one Ki for 1 ≤ 1 ≤ 6. Lemma 4 If R1 , R2 are two IA-relation and R1 × R2 is a saturated-convex RA-relation, then Tc (R1 × R2 ) = Tr−1 (K(R1 ) × K(R2 )).

Proof. If R1 × R2 is a basic RA, then, as shown in Figure 5, it is easy to see that if a (R1 × R2 ) b then there exists exactly one basic-rectangular cardinal relation R0 such that Tr (R0 ) ⊇ R1 × R2 and hence, a R0 b is satisfied. So R0 = Tc (R1 × R2 ) is precisely Tr−1 (K(R1 ) × K(R2 )). For the disjunctive case, K(R1 ) × K(R2 ) is a saturated-convex RA-relation which is the image under Tr of a relation R0 ∈ SCR, and there is no proper subset R00 of R0 such that Tr (R00 ) ⊃ R1 × R2 and a R00 b is satisfied. Again R0 = Tc (R1 × R2 ) = Tr−1 (K(R1 ) × K(R2 )) ¤ Now, recall that the set of saturated-convex RA-relations is a subclass of 2Arec , since the following lemma holds: Lemma 5 [see [3]] For every relation R1 , R2 , S1 , S2 ∈ 2Arec , • (R1 × R2 ) ∩ (S1 × S2 ) = (R1 ∩ S1 ) × (R2 ∩ S2 ); • (R1 × R2 ) ◦ (S1 × S2 ) = (R1 ◦ S1 ) × (R2 ◦ S2 ); • (R1 × R2 )−1 = R1−1 × R2−1 . Theorem 6 The set SCR is closed under intersection, inverse and weak composition. Proof. [Sketch] The case of intersection straightforwardly follows by construction of the set of SCR, which contains only relations R such that Tr (R) is a saturated-convex RA-relations. To see that, being R1 , R2 ∈ SCR, the compositon R1 ¦ R2 ∈ SCR, we observe that, by definition of ¦ in the SK-model, the composition of two rectangular CRs is a rectangular relation, and thus R1 ¦ R2 = Tc (Tr (R1 ) ◦ Tr (R2 )). Indeed, since Tr (R1 ) and Tr (R2 ) are convex-saturated RA-relations, by Lemma 5 Tr (R1 ) ◦ Tr (R2 ) is a saturated-convex RA-relation, whose cardinal closure is the composition R1 ¦ R2 . A similar argument can be given for the stability of inversion, since R−1 = Tc ((Tr (R))−1 ) by definition of inverse in the SK-model, Lemma 5, and definition of cardinal closure Tc . ¤ Hence, SCR is a tractable subclass of 2D . As a by-product of the proof the previous theorem we have also obtained an easy way of computing inverse and composition of rectangular cardinal relations, that can be used in a constraint propagation algorithm in order to prune the search space when dealing with relations in the set 2Drec − SCR (thus, when some of the relations are not in the tractable case just defined). Moreover, we argue that weak composition can be used to decide consistency of a set of relations in the SCR subclass. On the other hand, a method for deciding consistency of a network N with constraints in SCR can be outlined as follows. The idea is similar to the one used for networks whose constraints are in Drec : 1. obtain the equivalent RA-network Tr (N ), and 2. check the consistency of this network in the RA-model, or, more efficiently, obtain via further translation, a P A-network, and check it for consistency. For other reasoning tasks, such as finding the minimal network within the SCR subclass, a similar idea can be used. For IA, as well as for P A, the minimal network of a convex network can be found with by a path-consistency algorithm in O(n3 ). Therefore the minimal network corresponding to a network N in the SCR subclass can be found in O(n3 ) by first obtaining the minimal network of Tr (N ) and then applying the cardinal closure of every minimal RArelation. Finally, we consider the class of saturated-preconvex RArelations. Such class is strictly bigger than that of saturated-convex

RA-relations, and, thus, it is natural to ask whether there exists a subset of the set of cardinal relations which is not included in the class of saturated-convex CRs but such that its translation by means of Tr is included in the class of saturated-preconvex RA-relations. A natural way to do this is extending the class of saturated-convex CRs with new relations. It is simple to see that for any pair Ki and Kj such that Ki and Kj are not directly connected in the diagram in Figure 6, the IA-relation Ki ∪ Kj is not preconvex, and, thus, such an extension is not possible.

4.3

Discussion

Saturated-convex rectangular relations have a straightforward intuitive counterpart. Indeed, disjunctive constraints represent, in practise, uncertain information. Such information usually cames from physical observation, such as satellite images in geographical information systems, or from an abstraction/approximation phase of real data. Suppose that we are given of the following natural language statement: the region a lies on the North/NorthEast of the region b, and we know that this is a partially trustable information. Thus we partially know the mutual orientation of a and b, but we cannot place a in any tile with respect to b. A natural way to represent this information is to interpret it as: the region a lies on the North, or on the NorthEast, or partially on the North and partially on the NorthEast of region b. In this way, the representation through rectangular relation is a {N, N :N E, N E} b, which includes all the possibilities. It is easy to see that this relation is saturated-convex.

5

Conclusions and Future Work

In this paper we have considered an expressive model (SK-model) for qualitative spatial reasoning with cardinal relations presented by Skiadopoulos and Koubarakis in [28, 29]. We have presented the class of rectangular cardinal relations, and we showed that it is a subclass of the class of cardinal relations. By exploring the connection between such class and Balbiani, Condotta and del Cerro’s rectangle algebra we have showed that the class of basic-rectangular cardinal relations is tractable. Then, we have extended such class in order to include disjunction of constraint. On the one hand, the general case for rectangular cardinal relations turned out to be nontractable. We have identified a fragment, only containing what we call saturated-convex rectangular cardinal relations, for which the consistency problem can be solved in quadratic time, and the minimal network can be found in cubic time. Finally, we have proved that the saturated-convex rectangular fragment is indeed a subclass of the general model for cardinal relations. As for future work, we plan to extend the SK-model with new relations, which will allow us to devise a composition method strong enough to decide consistency. In this way, we plan to tackle the problem of the consistency checking of a network whose nodes represent regions in REG, which, in our knowledge, is still open, either for basic and non-basic cardinal relations.

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