Spatial reasoning with rectangular cardinal relations. Isabel Navarrete, Antonio Morales, Guido Sciavicco & M. Antonia Cardenas- Viedma

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1 Spatial reasoning with rectangular cardinal relations Isabel Navarrete, Antonio Morales, Guido Sciavicco & M. Antonia Cardenas- Viedma Annals of Mathematics and Artificial Intelligence ISSN Ann Math Artif Intell DOI /s

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3 Ann Math Artif Intell DOI /s Spatial reasoning with rectangular cardinal relations The convex tractable subalgebra Isabel Navarrete Antonio Morales Guido Sciavicco M. Antonia Cardenas-Viedma Springer Science+Business Media Dordrecht 2012 Abstract Qualitative spatial representation and reasoning plays a important role in various spatial applications. In this paper we introduce a new formalism, we name RCD calculus, for qualitative spatial reasoning with cardinal direction relations between regions of the plane approximated by rectangles. We believe this calculus leads to an attractive balance between efficiency, simplicity and expressive power, which makes it adequate for spatial applications. We define a constraint algebra and we identify a convex tractable subalgebra allowing efficient reasoning with definite and imprecise knowledge about spatial configurations specified by qualitative constraint networks. For such tractable fragment, we propose several polynomial algorithms based on constraint satisfaction to solve the consistency and minimality problems. Some of them rely on a translation of qualitative networks of the RCD calculus to qualitative networks of the Interval or Rectangle Algebra, and back. We show that the consistency problem for convex networks can also be solved inside the RCD calculus, by applying a suitable adaptation of the path-consistency algorithm. However, path consistency can not be applied to obtain the minimal network, contrary to what happens in the convex fragment of the Rectangle Algebra. Finally, we partially analyze the complexity of the consistency problem when adding non-convex A preliminary version of this paper appeared in [30]. There, only 240 out of the 400 convex relations of the RCD calculus were considered. In the present work, we also proposeadditional reasoning methods. I. Navarrete (B) A. Morales G. Sciavicco M. A. Cardenas-Viedma Department of Information Engineering, University of Murcia, Murcia, Spain inava@um.es A. Morales morales@um.es G. Sciavicco guido@um.es M. A. Cardenas-Viedma mariancv@um.es

4 I. Navarrete et al. relations, showing that it becomes NP-complete in the cases considered. This analysis may contribute to find a maximal tractable subclass of the RCD calculus and of the Rectangle Algebra, which remains an open problem. Keywords Qualitative spatial reasoning Cardinal direction relations Rectangle algebra Interval Algebra Qualitative constraint networks Constraint satisfaction problems Mathematics Subject Classifications (2010) 68T30 68T37 1 Introduction Qualitative spatial representation and reasoning plays a important role in various areas such as geographic information systems (GISs), spatial databases, document interpretation, robot navigation, and image retrieval, among others. Different formalisms distinguish between different aspects of space, such as direction, topology and distance (see [6] for a survey). It is unlikely that a single integrated framework for spatial reasoning will emerge, so the best one can do is to select or develop an appropriate calculus for a given application. For practical reasons, space is usually represented in two dimensions, and spatial entities may be represented by points, rectangles or polygons with a different shapes, depending on the required level of detail of a given application. Here, we focus our attention on qualitative constraint calculi with directional relations. Different orientational or directional spatial models have been proposed for different types of spatial entities and reference systems. Cardinal relations are direction relations, expressed as north, east, southwest and so on, for describing how spatial objects are placed relative to one another in a large-scale space using a fixed external frame of reference [11, 15, 46]. In a local space, such labels can be equally rephrased as, for instance, above, right, below and left, an so on. Cardinal relations are of particular interest for GIS and spatial and image databases, where, very often, the qualitative relationships among spatial objects are as important as the objects themselves [6, 11, 36, 43]. Our work is based on a previous calculus with cardinal direction relations between regular regions of the plane proposed by Goyal and Egenhofer [15, 16], and later formalized by Skiadopolous and Koubarakis [46] andliuetal.[26]. In this formalism, called here CD calculus, one can express the cardinal relation between a primary region a and a reference region b (e.g., a is to the northeast of b) by approximating b with its minimum bounding rectangle (mbr) and using the exact shape of a. The minimum bounding rectangle of a region is the smallest rectangle aligned to the axes of the plane that envelops the given region. The problem of deciding the consistency of a set of cardinal constraints is NP-complete [26, 46], and no tractable fragment of CD calculus, apart from the one with only basic relations, has been identified so far. Recently, Liu and Li [24] showed that even if only the universal relation (or unknown relation ) is added to the set of basic relations, the consistency problem in the resulting (small) fragment of the CD calculus is already NP-complete. Taking into account that those situations that present incomplete knowledge about spatial configurations are quite common in spatial reasoning, this result supposes a serious drawback of the CD calculus.

5 Spatial reasoning with rectangular cardinal relations In this paper we study a restricted variant of the CD calculus, that we name rectangular cardinal relation calculus (RCD calculus), containing only 36 basic relations, while there are 218 basic relations in the CD calculus over connected regions and 511 over possibly disconnected regions. In the RCD calculus the primary and the reference region are both represented by their mbrs. This solution, on the one hand, makes reasoning tasks to become more efficient, as we show in Section 6, while, on the other hand, implies a loss of accuracy of the representation of the relative direction between regions. However, rectangles allow us to describe the relative position between two regions with more precision than in a projective point-based approximation model, such as Frank [11]. For example, consider the Balearic Islands (BI, the reference region) and the Iberian Peninsula (IP, the primary region). Notice that IP is connected, while BI is a disconnected region. Using a point-based approximation one can express that IP is to the northwest of BI (see Fig. 1a), which does not describe the real situation. In our case, with regions approximated by rectangles (see Fig. 1b), we can express a more precise information: IP is partly to the southwest (sw), west (w), northwest (nw) and north (n) of BI (what is exact), but also partly to the south and partly inside in the rectangle of (b) of BI (which is imprecise). Finally, in the CD calculus a precise description of the spatial configuration of the example is possible, since one approximates only BI by a rectangle, while the exact shape of IP is used, as in Fig. 1c, which reflects that the Iberian Peninsula is partly to the southwest, west, northwest and north of the Balearic Islands. We investigate the connection between the RCD calculus and the Rectangle Algebra [3], which extends the temporal Interval Algebra [2] to a bidimensional Euclidean space and contains 169 basic relations. By exploiting such a connection, more precisely, by means of a suitable translation between relations, we identify a conve fragment of the RCD calculus which contains disjunctive cardinal relations, and so, it allows for representing and reasoning with incomplete or imprecise knowledge efficiently. For this subclass we propose several reasoning algorithms to solve the fundamental problem of deciding consistency of qualitative networks and other related problems, such as finding a consistent scenario or the minimal network equivalent to a given one. Most popular formalisms for temporal and spatial reasoning, such as the Interval Algebra and the RCC8 calculus [38] with topological relations, are algebraically simple, with a small number of basic relations, and contain tractable subsets with all basic relations together with a large number of disjunctive relations. Thegreat expressivenessof the CD calculus has a high computational cost, (a) IP NW BI (b) IP B:S:SW:W:NW:N BI (c) IP SW:W:NW:N BI Fig. 1 Cardinal relations at different levels of detail: a regions approximated by points; b by rectangles; c only the reference region is approximated by minimal bounded rectangle

6 I. Navarrete et al. since it seems there is no way of reasoning efficiently with disjunctive relations [26]. Hence, we think the RCD calculus is interesting for qualitative spatial reasoning with cardinal relations since it supposes a good compromise between simplicity, efficiency and expressiveness. Moreover, our formalism is not hard to extend with metric and topological relations in order to improve the expressiveness. The rest of the paper is organized as follows. In Section 2 we introduce a concise background on qualitative calculi, binary relations, constraint algebras and qualitative networks. In Section 3 we shortly recall Interval Algebra, Rectangle Algebra, and the CD calculus. In Section 4 we introduce the RCD calculus and its underlying constraint algebra, and we compare the RCD calculus with existing formalisms to handle cardinal directions relations. In Section 5 with we explore the connection of the RCD calculus with the Rectangle Algebra and analyze the consequences of such a connection. In particular, we define two mappings to translate networks between the two formalisms. In Section 6 we identify the convex subalgebra of the RCD calculus, we prove its tractability, and we propose several constraint-based reasoning methods. In Section 7 we deal with the computational problems raised by adding non-convex relations to the convex subclass of the RCD calculus, and we show how the inclusion of certain types of non-convex relations leads to intractable subsets, before concluding in Section 8. In Table 1 we summarize the major notational conventions used in this paper. We provide the utility package RCDsoftware [29], Table 1 Notation used in this paper Notation Short description JEPD B/2 B A Jointly Exhaustive and Pairwise Disjoint set of binary relations B/The full set of relations of a given calculus. (2 B,,, ) Constraint algebra with operators: intersection, converse and refers to composition ( ) orweakcomposition( w ). PC/Path consistency Weak path consistency./ N = (V, A ) A qualitative constraint network. NSAT(2 B )/NSAT(S) Consistency problem for qualitative networks, with relations in 2 B /A subproblem with relations in S 2 B. IA, RA, CD and RCD calculi Interval Algebra, Rectangle Algebra, Cardinal Direction Calculus and Rectangular Cardinal Direction Calculus. r, r 1, r 2 /R, R 1, R 2 Denote basic/arbitrary relations in different calculi. c, c 1, c 2 /C, C 1, C 2 Basic/arbitrary relations specifically in the RCD calculus. B ia /B ra /B cd /B rcd Sets of basic relations of the IA/RA/CD-andRCD calculus. H ia /H ra Sets of ORD-Horn relations of the IA/RA. C ia /C ra /C rcd Sets of convex relations of the IA/RA/RCD calculus. P x, P y Geometrical projections of rectangle P onto the axes. r (t, t ) Refers to a basic RA-relation r, with projections t, t B ia. π x (R), π y (R)/π x (N ), π y (N ) Projections of a RA-relation/Projections of a RA-network wcon Weak-converse operation. I(R) Convex closure of an interval or rectangle relation R. dim(r) Dimension of an interval or rectangle relation R. mbr(b) Minimum bounding rectangle of region b. a n:nw:w b Example of a CD-relation between regions a and b in tile notation. P n:nw:ne Q Example of a RCD-relation between rectangles P and Q. TR Translates a relation or network of the RCD calculus to the RA TC Cardinal closure (translates from the RA to the RCD calculus)

7 Spatial reasoning with rectangular cardinal relations that implements a CSP solver for qualitative networks of the Interval Algebra, Rectangle Algebra, and the RCD calculus; the software also implements several operations on relations of the RCD calculus, thus allowing one to verify some of the results presented in this paper. 2 Background on qualitative calculi There is a growing interest in binary qualitative constraint calculi as algebraic formalisms to represent and reason about temporal/spatial relations. Despite the great existing variety of qualitative calculi, they often share concepts and techniques, that we briefly introduce in this section. 2.1 Infinite CSPs and qualitative networks In what follows, we use the term qualitative calculus to refer exclusively to a calculus that, roughly speaking, consists of an underlying finite set of infinite binary relations, together with a constraint language to describe qualitative configurations, and reasoningmethodsadapted from the ones of finite (classic) constraint satisfaction problems (CSPs). A reasoning problem in a qualitative calculus is formulated as an inf inite CSP instance (V, D, ),wherev ={v i } i=1 n is a set of variables interpreted over an inf inite domain D (unlike a finite CSP instance where D is finite) containing the objets of interest, and is a set (conjunction) of constraints (predicates) with variables from V describing qualitative spatial/temporal configurations. A constraint is an atomic formula of the form v i R v j,wherev i,v j are variables in V and R denotes a binary relation over D which restricts the possible values for the pair of variables (v i,v j ).Aqualitative network may be considered a special infinite CSP instance that can be represented by a complete constraint-labeled digraph N = (V, A ),in which each node corresponds to a variable, and each arc from node i to node j is labeled with the expression denoting the binary relation that constrains the pair of variables (v i,v j ). In qualitative networks, binary relations are usually identified with constraints; thus, by overlapping notation, we write R ij to refer to the binary relation between v i and v j, as well as the constraint formula v i R ij v j.aninstantiation (or interpretation) of is a mapping representing an assignment of domain values to the variables appearing in. A relation (constraint) R ij in is said to be satisf ied by an instantiation ι if the pair (ι(v i ), ι(v j )) belongs to the binary relation over D represented by R ij.aconsistent instantiation or solution of is an assignment of domain values to variables satisfying all the constraints in. If such a solution exists, then the CSP instance (V, D, ) or, simply, the set of constraint, issaid to be consistent or satisf iable over D, otherwise it is inconsistent. If a CSP instance corresponds to a qualitative network then, in particular, we will say that the network is consistent or inconsistent, as the case. 2.2 Binary relations and constraint algebras The classic constraint-propagation methods involved in the resolution of finite CSPs [27, 28] are not directly applicable to qualitative networks, since binary relations contain infinite pairs of elements of the domain. Hence, these methods must be

8 I. Navarrete et al. adapted so that qualitative networks can be solved at a symbolic level, that is, by manipulating relation symbols with algebraic operators intended as set-theoretic relational operators. A common start point when developing a new qualitative calculus is to consider a finite family B of jointly exhaustive and pairwise disjoint (JEPD) binary relations over some domain D, which means that B is a partition of D D. Eachmemberof B is called basic relation and it is represented by a relation symbol in the constraint language of the calculus. Given a fixed background theory of binary relations (e.g. linear orderings, set-point topology, etc.), the semantic of relation symbols and thus, the satisfiability of constraint formulas, is defined by formal expressions specifying which pairs (X, Y) D D are represented by a particular relation symbol. Next, the set of allowed relations of the calculus is formed by considering all possible unions of basic relations, and such a set forms a Boolean algebra in which the basic relations are atoms, and is commonly represented by the powerset 2 B.AsetR 2 B is referred to as a relation, to indicate the fact that it represents the union of the basic relations it contains. The bottom element of 2 B w.r.t. set inclusion, is the empty relation while the top element, we denote as?, is the universal relation, which contains all basic relations. A singleton {r} stand for a basic relation from B and is used to write a basic constraint, suchasx {r} Y (or, simply, XrY), expressing definite knowledge about the relationship between the domain objects X and Y. A set containing two or more basic relations is named disjunctive relation, and it is used to write a disjunctive constraint, suchasx {r 1,...,r m } Y, which expresses indef inite or imprecise knowledge, since the formula X {r 1,...,r m } Y is equivalent to the disjunction Xr 1 Y Xr m Y w.r.t. satisfiability. The universal relation can be used to express that the relation between X an Y is unknown. Remark 1 With a little abuse of notation, hereafter we will use the letter r andthe singleton {r} indistinctly, since both refers to a basic relation of the partition B. Also, we will use the uppercase letter R to denote an arbitrary relation (basic or disjunctive). There is no agreement on which should be the ideal structure of the algebra of relations of a qualitative calculus. Ladkin and Maddux [19] propose the use of finite relation algebras in the sense of Tarski (see [9] for a brief tutorial on relationalgebra theory with applications in spatial reasoning), while Ligozat and Renz [23] are in favor of non-associative algebras, which subsume relation algebras. Nebel and Scivos [33] consider a substructure of a relation algebra, referred to as constraint algebra, namely (2 B,,, ), formed by the powerset 2 B, that arises from a finite set of JEPD relations B, together with the algebraic operations (see Table 2-right) necessary to apply constraint-based methods to qualitative networks at a symbolic level, namely, intersection ( ), converse ( ) and composition ( ). The algebraic operations, interpreted as the set-theoretic operations on binary relations shown in Table 2-left, are computed symbolically, that its, by manipulating sets of relation symbols representing relations from 2 B. We use the same notation for set-theoretic and algebraic operations. If the underlying set of binary relations of the calculus is not closed under composition, then the operation of weak composition ( w )[9, 23]isused instead of composition (see Table 2). Observe that, as a set of pairs, R R R w R and, as noticed in [23], the weak composition of two relations is the upper-best

9 Spatial reasoning with rectangular cardinal relations Table 2 Algebraic operations of a constraint algebra with carrier 2 B interpreted as set-theoretic operations on binary relations over a domain D Relational Set-theoretic definition Algebraic definition Operations x, y, z D, R, R D D R, R 2 B Intersection ( ) Converse ( ) Composition ( ) Weak comp. ( w ) R R def = (as usual) R R := {r r R, r R } R def ={(y, x) xry} R := {r r R} R R def ={(x, z) y : xry, yr z} R R := r R,r R (r r ) R w R def = R w R := r R,r R (r w r ) {S B (R R ) S = } approximation inside the calculus to (standard) composition. The symbolic composition of two basic relations r, r 2 B is obtained first from the semantics of the relation symbols and taking into account the transitivity rule: ifxry and Yr Z then r r must contain those basic relations r i such that Xr i Z,whereX, Y, Z are arbitrary objects of the domain. The symbol stands for composition or weak composition. The result of r r, expressed as a set of relation symbols, is stored in a composition table, which is thereafter used to compute the symbolic composition of disjunctive relations as in Table 2-right. Example 1 The simplest calculus for temporal reasoning is the point algebra (PA)[49], where B pa {<, =,>} is the set of basic relations that may hold between any two time points, where the relation symbols <, =, > are commonly interpreted over the reals (or the rational numbers) with the usual meaning. The full set of PArelations is 2 Bpa. The basic relation = is the identity relation, and > is the converse of <. Disjunctive relations such as {<, =} or {>, =} are usually abbreviated as and, respectively. The algebraic composition {<} {>} is equal to {<, =>}, and {<} {<} ={<}. If PA-relations are interpreted over a non-dense domain, e.g. the integers, then the set of binary relations of the PA is not closed under composition. For instance, 3 < 4 but there exists no integer y such that 3 < y < 4. Hence, according to the settheoretic definition of composition, the pair (3, 4) does not belong to the composition of the less than relation with itself. But (3, 4) belongs to the weak composition < w < since < w <=<, accordingtotable2. Nevertheless, the composition table contains the same relation symbols, no matter whether the domain of interpretation is dense or not. 2.3 The consistency problem The main problem that arises in a qualitative calculus given by D, B,(2 B,,, ), is the consistency (or satisf iability) problem, denoted by CSAT(2 B ), and it consist of deciding whether a given set of constraints with relations from 2 B and variables ranging over D is consistent. We are interested in a common variant of the consistency problem, we name NSAT(2 B ), which is stated for qualitative networks, so that there is exactly one constraint for each pair of variables in a network instance. We use the notation NSAT(S) to refer to a subproblem in which all relations in a network instance are taken from a subset S 2 B. Once the consistency subproblem with basic relations only, namely NSAT(B), has beenprovedto be decidable, then the

10 I. Navarrete et al. general problem can be always solved at a symbolic level, at worst, by backtracking over basic relations taken from disjunctive ones in the input network. In most cases the consistency problem is NP-complete for the full algebra 2 B, and thus, finding tractable subsets, that is, subsets T 2 B for which NSAT(T ) is polynomial, becomes a central issue. A subset M 2 B is said to be maximal w.r.t tractability, or M is a maximal tractable subset, if M cannot be extended with any other relation of 2 B without loosing tractability. Genuine qualitative consistency-checking algorithms are deductive, unlike the case of finite CSPs, where a constructive proof of the satisfiability problem is required in order to build a solution. The use of composition tables has become a key reasoning mechanism for inferring new information from existing one throughout constraint propagation, which achieves certain level of local consistency that prunes the search space and eventually guarantees the consistency of the network. The most prominent constraint propagation method for spatial and temporal reasoning is the path-consistency algorithm (PC-algorithm) which, in the case of a qualitative networks, refines relations by deleting some impossible basic relations. This is done in polynomial time by successively applying the following triangle operation for every triple of variables (v i,v k,v j ), until a stable network is reached: R ij R ij (R ik R kj ). If the empty relation is obtained during the process, then the input network is surely inconsistent; otherwise, the output network is not necessarily consistent, but is said to be path consistent. The notion of path consistency we adopt here is the symbolic version of the classical notion of path consistency (equivalent to 3- consistency [27]) initially stated for finite CSPs. A qualitative network N = (V, A ) is symbolically path-consistent or algebraically closed [39] if no relation is empty and R ij R ij (R ik R kj ), for every triple of variables (v i,v k,v j ) of the network. What (symbolic) path consistency always ensures is that every subnetwork of three variables in a (symbolic) path-consistent network is consistent, but not the global consistency of a network. When it is the case that for a specific qualitative calculus path consistency implies consistency, we say that the PC-algorithm is complete for consistency. As several authors as pointed out, it is highly recommended for a new calculus that the PC-algorithm is at least complete for consistency of basic networks, i.e., with basic relations only (non-disjunctive), so named scenarios. As it has been pointed out in [39], whether composition differs from weak composition in a qualitative calculus is not relevant for as much as consistency is concerned. 3 Related work Some of the contributions of this paper rely on several results borrowed from the IA, the RA, and the CD calculus. Therefore, we now briefly recall these formalisms. 3.1 Interval algebra and rectangle algebra Allen [2] introduced the so-called Interval Algebra (IA for short) to model the relative position between events or temporal intervals. An interval I can be interpreted as a closed interval of real numbers, denoted by [I, I + ],whereitsendpoints satisfy the relation I < I +.Wenamedom(IA) the domain of interval variables under

11 Spatial reasoning with rectangular cardinal relations Table 3 Basic relations of the IA defined in terms of PA-relations between their endpoints Relation Symbol Converse Pictorial examples Meaning IbeforeJ b bi I J I < I + < J < J + I overlaps J o oi I J I < J < I + < J + I during J d di I J J < I < I + < J + I meets J m mi I J I < I + = J < J + IstartsJ s si I J I = J < I + < J + I finishes J f fi I J J < I < I + = J + I equals J e e I J I = J < I + = J + this interpretation. 1 Let B ia be the set of basic relations of the IA denoted with the symbols b, o, d, m, s, f, e, bi, oi, di, mi, si, fi (see Table 3). This set is obtained by considering all possible ways to order the four endpoints of two intervals. Therefore, B ia is a JEPD set of 13 binary relations over dom(ia), which contains the converse of each basic relation (e.g., bi stands for b ) and the identity relation equals (e, which is the converse of itself). The semantics of the seven basic IA-relation symbols b, o, d, m, s, f, e can be defined in terms of their endpoint relations, in such a way that the constraint I r J is satisfied if the endpoints of I and J satisfies the ordering relations shown in Table 3 for the basic relation r. The other six basic relations bi, oi, di, mi, si, fi are simply defined by exchanging the roles of I and J in Table 3, since IrJ Jr I.TheIA is a constraint algebra (2 Bia,,, ),where2 Bia is the full set of interval relations, and the algebraic operations are computed symbolically as shownintable2. The composition table for the IA can found in [2]. Including the union and the complement operations, the IA becomes a relation algebra [19]. One extension of the Interval Algebra to a bidimensional space is called Rectangle Algebra (RA),proposedby Balbiani et al. [3], and later extended to a n-dimensional space [4]. The objects under consideration in this qualitative calculus are rectangles whose sides are parallel to axes of some orthogonal basis of a bidimensional Euclidean space, and are denoted by P, Q, etc. Without loss of generality, we assume here the Euclidean plane R 2 as a model of the space, and the Cartesian coordinate system as a reference system. Let dom(ra) be the domain of interpretation of rectangle variables. A rectangle P dom(ra) is totally characterized by a pair of intervals (P x, P y ) dom(ia) dom(ia), wherep x, P y are the projections of the rectangle P onto the x- andy-axis, respectively. The set of basic relations of the RA, denoted by B ra, is obtained considering the relative position between two rectangles throughout the IA-relations between their projections. A basic relation r B ra is represented 1 Q is another model of the time line frequently considered in the literature; however the distinction between the reals and the rational numbers is not significant for the issue of consistency.

12 I. Navarrete et al. by a pair r (t, t ) of basic IA-relations, where t = π x (r) and t = π y (r) are the x- and y-projections of r, respectively. A constraint P (π x (r), π y (r)) Q is satisfied iff, by definition, the IA-constraints P x π x (r) Q x and P y π y (r) Q y are satisfied. For example, P(s, mi)q def P x s Q x P y mi Q y (see Fig. 3a, p. 14). The relation (e, e) is the identity relation of the RA,andP (e, e) Q means that rectangles P, Q are equal. A pictorial illustration of all possible relations between rectangles is shown in Fig. 4, p. 18; in total, there are = 169 basic RA-relations. The full set of RA-relations is 2 Bra. In the following, we define two mappings in order to obtain the interval relations between the projections of two rectangles entailed by a rectangle relation. We also extend the mappings to obtain two networks of IA-relations from a given network of RA-relations. Definition 1 (Projections of relations/networks; adapted from [3, 4]) The projections of a RA-relation R 2 Bra are the IA-relations, denoted π x (R) and π y (R) and defined as π x (R) ={π def x (r) r R} π y (R) ={π def y (r) r R}, where, π x (r) ={t},π y (r) ={t } are the projections of the basic relation r. The mappings π x,π y : 2 Bra 2 Bia are extended to qualitative networks in such a way that the projections of a RA-network N = (V, A ) are the IA-networks π x (N ) and π y (N ) specified by the set of variables V interpreted over dom(ia), and by the set of constrains obtained replacing each relation R ij in A by π x (R ij ),inthecaseofπ x (N ), or by π y (R ij ),inthecaseofπ y (N ). The Rectangle Algebra, as a constraint algebra, is formed by the set 2 Bra of all RA-relations together with the fundamental operations of intersection, composition and converse. These operations are extensions of the respective operations of the Interval Algebra and can be computed symbolically using the mappings π x,π y,the composition table of the IA, and the operation of Cartesian product, namely: The composition of two basic relations r and r is computed as r r := (π x (r) π x (r )) (π y (r) π y (r )). The converse of a basic relation r is r := (π x (r),π y (r) ). For arbitrary relations from 2 B, intersection, composition and converse can be computed as indicated in Table 2-(right). Example 2 Let be the following RA-relations R ={(s, di), (o, mi)} (disjunctive) and R ={(f, oi)} (basic). We have π x (R) ={s, o},π y (R) ={di, mi}. The converse of R is {(si, d), (oi, m)} and the composition R R is (s f) (di oi) (o f) (mi oi) = {d} {oi, di, si} {o, s, d} {bi} ={(d, oi), (d, di), (d, si), (o, bi), (s, bi), (d, bi)}. TheRA extended with the operations of union and complement w.r.t. the universal relation becomes a relation algebra [4] Tractable subsets of the interval and rectangle algebra The problem NSAT(2 Bia ) is known to be NP-complete [49]. Several tractable fragments of the IA have been identified [8, 14, 21, 32, 48], and Nebel and Bürckert [32] proved that there exists only one maximal tractable subset of the IA that contains

13 Spatial reasoning with rectangular cardinal relations all basic relations, namely, the set H ia of the so-called ORD-Horn IA-relations. This set is formed by those relations R for which a constraint I R J can be equivalently expressed (w.r.t satisfiability) as a conjunction of ORD-Horn clauses between the endpoints of the intervals I and J. The largest tractable set of relations of the Rectangle Algebra identified so far is precisely that one containing the ORD-Horn RA-relations [4], which are those relations that can be encoded, as in the interval case, as conjunctions of ORD-Horn clauses between the endpoints of the projections of two rectangles. However, it has not been proved yet that the set H ra of ORD-Horn RA-relations is a subalgebra of the Rectangle Algebra, nor that H ra is a maximal tractable subset of the RA. Both consistency problems NSAT(H ia ) and NSAT(H ra ) can be solved with a PC-algorithm in O(n 3 ),wheren is the number of variables of the input network [4, 32]. In this paper we are mainly interested in convex IA-relations [21, 48] andconvex RA-relations [3], which are those relations that can be expressed as a set of constraints between, respectively, the endpoints of the interval variables and the endpoints of the projections of rectangle variables, using convex relations of the Point Algebra (all except = ) [48]. It is worth to mention that a convex RA-relation is equivalently characterized as a RA-relation which corresponds to the Cartesian product of two convex IA-relations [3]. It is known that C ia H ia [32] andc ra H ra [4], where C ia and C ra denote, respectively, the subsets (which in turn define subalgebras) containing all the convex relations of the IA and the RA.BeingNSAT(C ia ) and NSAT(C ra ) subproblems of NSAT(H ia ) and NSAT(H ra ), respectively, it follows that path consistency suffices to ensure consistency of convex IA- andra-networks, as well. 3.2 The cardinal direction calculus One of the most expressive formalisms for qualitative representation and reasoning with extended spatial objects is the so-called Cardinal Direction Calculus (CD calculus), based on the work by Goyal and Egenhofer [15, 16], which, in turn, overcomes some of the limitations of a previous model based on cardinal direction relations between points of the plane [11]. Several authors have investigated the CD calculus, formalizing concepts, developing reasoning algorithms and analyzing its computational properties [26, 45, 46]. Cardinal direction relations (or simply cardinal relations) describe the direction, based on the compass rose, of a primary region w.r.t a reference region of the plane, using the Cartesian coordinate system as a reference system. We assume here, following Liu et al. [26], that the domain of the CD calculus is the set of connected regions, where a connected region, denoted here with the letters a, b, etc., is as regular closed subset of the Euclidean plane R 2 with a connected interior (possibly with holes). In [16, 45] the considered domain is built over simple connected regions only, but the distinction between simply connected and connected regions seems not to be significant in terms of consistency [26]. The CD calculus has been extended in [46] (see also [26]) to consider a more comprehensive domain containing all regular regions of the plane, and, therefore, possibly disconnected, resulting in a variant of the cardinal direction calculus we name CD d calculus. Let b be a reference region, i.e., the object from which the cardinal relation is described. The minimum bounding rectangle of b, denoted by mbr(b), is the smallest rectangle, with sides parallel to the axes, that contains b. The straight lines forming mbr(b) divide the plane into nine zones, as shown in Fig. 2a. Cardinal tiles are subsets

14 I. Navarrete et al. (a): tiles w.r.t. to MBR (b) (b): a B:N:E b (c): a N:E b Fig. 2 Minimal bounding rectangle and tiles of the plane (a) example of a cardinal relation between connected regions (b) and a cardinal relation that holds only over disconnected regions (c) of R 2 to which set-theoretic operations such as the intersection may be applied. Notice that the intersection of two tiles is a space of dimension lower than two. Tiles correspond to the cardinal directions South, SouthWest, West, NorthWest, North, NorthEast, East, SouthEast of or in the Box of, denoted by the tile symbols from TS ={s, sw, w, nw, n, ne, e, se, b}. In Goyal and Egenhofer s approach, any cardinal relation between two regions is encoded in a direction matrix (see also [5, 26]), which is a 3 3 Boolean matrix with the following intuitive meaning: the positions on the matrix correspond to the tiles of the plane and an entry is true if the intersection between the exact shape of a primary region and the corresponding tile w.r.t a reference region is non-empty. A basic cardinal relation can be alternatively denoted using a more compact syntax (we call tile notation) introduced by Skiadopoulos and Koubarakis [45, 46], we name tile notation, as in the following definition. Definition 2 (Basic cardinal relation; adapted from [15, 26, 46]) A basic cardinal relation between a region a w.r.t. a region b, denoted by dir(a, b), is specified by the a direction matrix: d nw d n d ne dir(a, b) = d w d b d e d sw d s d se such that d τ = 1 a τ(b) =,wherea is the interior of a and τ TS.Alternatively, a basic cardinal relation is denoted by a tile string τ 1 :τ 2 :...:τ k,τ i TS, 1 i k, with the following meaning: a τ 1 :τ 2 :...:τ k b is satisfied iff a, b can be instantiated with connected regions in such a way that a τ(b) =, for each tile symbol τ {τ 1,τ 2,...,τ k },anda τ(b) = for each τ TS\{τ 1,τ 2,...,τ k }. Example 3 Figure 2b shows that the region a is partly to the north, partly to the east and partly in the rectangle of region b (i.e., a has nonempty intersection with n(b), e(b) and b(b)), and this information is expressed in tile notation by the basic constraint a b:n:e b. 2 If we take the region labeled by a as a reference region 2 Note the order of the tiles symbols in a tile string does not matter. For instance, b:n:e refers to the same basic relation that n:e:b.

15 Spatial reasoning with rectangular cardinal relations instead, then the relation of b to a is s:sw:w. The corresponding direction matrices are: [ ] 010 [ ] 000 dir(a, b) = dir(b, a) = Taking into account that the domain of the CD calculus contains only connected regions, it follows that exactly 218 different basic cardinal relations are valid (nonempty). When possibly disconnected regions are considered, the number of valid relations rises up to 512. For example, the relation b:n:e makes sense in both the connected and disconnected cases, but the constraint a n:e b can only be satisfied if a is a disconnected region (see Fig. 2c). A pictorial representation of the direction matrices of the 218 basic cardinal relations of the CD calculus can be found in [15]. We use B cd to denote the set of basic CD-relations, which forms a JEPD set, while 2 Bcd contains, as usual, all the allowed relations of the calculus. Disjunctive relations, such as {w:nw:n, e:se}, are used to describe incomplete information about the relative direction betweentwo regions.inorderto expressdefinite information, both the basic constraints a r b and br a must be specified; otherwise, from the constraint arb it cannot be always inferred which one is cardinal relation of b w.r.t. a. This is due to the (unusual, in the context of qualitative calculi) fact that the converse of a basic CD-relation may be not a basic relation, which implies that the underlying set of cardinal relations is not closed under converse; in fact, it is not even closed under composition [26, 45]. Deciding whether a qualitative network with cardinal relations between regions is consistent is a NP-complete problem [26, 46]. Liu et al. [26] proposed O(n 3 ) algorithms for deciding consistency of basic networks, both in the case of connected regions and regular regions in general. But these algorithms are constructive (i.e., they search for a concrete solution), since typical constraint propagation methods, such as the PC-algorithm or other methods based on local consistency, are not applicable, not even in the case of basic networks [24, 45]. Moreover, Liu and Li [24] recently showed that even the sole universal relation, when added to the set of basic CD-relations, makes the consistency problem already intractable, which can be considered a serious limitation when we want to reason with spatial configurations where several relations are unknown (which is a quite common case). Summarizing, the CD calculus, although very expressive, is rather complicated from an algebraic and computational point of view: it contains a very high number of basic relations, what affect negatively to the computation of the operations of the underlying constraint algebra, and the PC-algorithm is not complete for consistency of basic CD-networks; moreover, it presents computational difficulties when disjunctive relations are considered. Therefore, the question of if this model is practically applicable for spatial reasoning naturally arises. 4 The rectangular cardinal direction calculus In this section we are interested in the information about the cardinal direction relations that minimal bounding rectangles convey about the spatial regions they enclose. For that purpose we introduce a fragment of the CD calculus, we name RCD

16 I. Navarrete et al. calculus, 3 which deals with cardinal relations over the restricted domain of rectangles whose sides are parallel to the axes of the Euclidean plane, thus corresponding to mbrs of regular regions. From now on whenever, when we talk about rectangles we refer exclusively to rectangles of that type. Hence, dom(rcd) coincides with the domain assumed for the Rectangle Algebra in Section 3.1; sometimes we will just write domain of rectangles to refer to the domain of interpretation of both calculi, and we will omit that domain in formal expressions when there is no ambiguity. We call the relations of the RCD calculus rectangular cardinal relations and we use the tile notation (see Section 3.2) to specify constraints in the language of the RCD calculus. Next, we formally define the semantic of relation symbols (tile strings) as in the CD calculus, but over the domain of rectangles. Recall the set of tile symbols denoting cardinal directions is TS ={b, s, sw, w, nw, n, ne, e, se},andτ(q) refers to the subset of R 2 corresponding to the tile τ w.r.t. the reference rectangle Q. Definition 3 (Rectangular cardinal relation 4 ) A basic rectangular cardinal relation (basic RCD-relation) is a set of pairs of rectangles denoted by a tile string τ 1 :τ 2 :...:τ k, where τ i TS, 1 i k, with the following meaning: P τ 1 :τ 2 :...:τ k Q def P τ i (Q) =, τ i {τ 1,τ 2,...,τ k } and P τ(q) =, τ TS\{τ 1,τ 2,...,τ k },wherep is the interior of P. Arectangular cardinal relation (RCD-relation) is represented by a set {c 1,...,c m }, where each c i is a tile string denoting a basic relation. If C is not a singleton then the relation is said to be disjunctive.thesetofbasicrcd-relations is named B rcd and the powerset 2 Brcd is the full set of allowed relations of the RCD calculus. Remark 2 For better readability, from now on an arbitrary basic RCD-relation will be denoted with the letter c, while C will denote an arbitrary relation in 2 Brcd. Similarly, r, R and t, T will denote RA-relations and IA-relations, respectively. Sometimes we use D elsewhere to refer to a disjunctive relation of any type. If we fix a reference rectangle, there are 36 possibilities to a place primary rectangle so that different basic RCD-relations are satisfied. It follows that the set B rcd forms a JEPD set of binary relations over the domain of rectangles. The tile strings denoting basic relations are enumerated in the first column of Table 4, p. 19. As usual, indef inite or uncertain knowledge can be specified by means of disjunctive constraints, e.g., the constraint P {c 1,...,c m } Q expresses that the possible relations that hold between the pair of rectangles (P, Q) is one of the basic relations c i {c 1,...,c m }, i.e., P {c 1,...,c m } Q Pc 1 Q Pc m Q. If there is total ignorance about the cardinal direction of P w.r.t. Q, this situation is specified with the constraint P? Q, where? represents the universal relation of the RCD calculus, i.e., the relation that contains all basic relations. Since any two rectangles are related by exactly one 3 An earlier version of this calculus appeared in [30]. 4 Skiadopoulos and Koubarakis [45] introduced rectangular cardinal relations for the purpose of computing the weak composition of two basic CD-relations, but their definition is different from ours, since they consider the domain of simply connected regions of the plane.

17 Spatial reasoning with rectangular cardinal relations Table 4 Translation from basic RCD-relations to RA-relations via the mappingtr Basic RCD-relation RA-relation (I) Basic RCD-relation RA-relation (II) b {d, s, f, e} {d, s, f, e} w:nw {m, b} {si, oi} s {d, s, f, e} {m, b} e:se {mi, bi} {fi, o} n {d, s, f, e} {mi, bi} ne:e {mi, bi} {si, oi} e {mi, bi} {d, s, f, e} s:sw:se {di} {m, b} w {m, b} {d, s, f, e} nw:n:ne {di} {mi, bi} ne {mi, bi} {mi, bi} b:w:e {di} {d, s, f, e} nw {m, b} {mi, bi} b:s:n {d, s, f, e} {di} se {mi, bi} {m, b} sw:n:nw {m, b} {di} sw {m, b} {m, b} ne:e:se {mi, bi} {di} s:sw {fi, o} {m, b} b:s:sw:w {o, fi} {o, fi} s:se {si, oi} {m, b} b:w:nw:n {o, fi} {si, oi} nw:n {fi, o} {mi, bi} b:s:e:se {si, oi} {o, fi} n:ne {si, oi} {mi, bi} b:n:ne:e {si, oi} {si, oi} b:w {fi, o} {d, s, f, e} b:s:sw:w:nw:n {o, fi} {di} b:e {si, oi} {d, s, f, e} b:s:n:ne:e:se {si, oi} {di} b:s {d, s, f, e} {fi, o} b:s:sw:w:e:se {di} {fi, o} b:n {d, s, f, e} {si, oi} b:w:nw:n:ne:e {di} {si, oi} w:sw {m, b} {fi, o} b:s:sw:w:nw:n:ne:e:se {di} {di} basicrelation, basicconstraints canbe usedto representdefinite knowledge about the cardinal direction between anytwo rectangles P and Q; however both the constraints PcQand Qc P must be specified. This is due to the partition of the square of the domain of rectangles,generatedby the setof basicrcd-relations, is not refined enough to include the converse of some basic relations, contrary to what happen in the Rectangle Algebra. Moreover, the set of rectangular cardinal relations is not closed under converse or composition, analogously to what happens in the CD calculus. Example 4 On the one hand, from the definition of the relational operations in Table 2 it follows that the instantiation of the pair (P, Q) showed in Fig. 3a does not belong to the set-theoretic composition n n, because there is no space for (a): P N Q, Q S:SE P (b): P 2 N Q 2 ; Q 2 SW:S:SE P 2 (c): ane:nb; P 3 NE:NQ 3 ; c 1 NWc 2 P (s, mi)q, Q (si, m)p P 2 (d, mi)q 2, Q 2 (di, m)p 2 bse:sa; Q 3 SE:SP 3 ; c 2 SEc 1 Fig. 3 Illustration of basic relations between rectangles, regions and points of the plane

18 I. Navarrete et al. a third rectangle M such that P n M and M n Q. But(P, Q) is a member of n w n since there exist three rectangles, as P 2, Q 2, M depicted in Fig. 3b, for which P 2 n M, M n Q 2 and P 2 n Q 2. On the other hand, in (a) and (b) we have spatial realization or instantiations of the constraints P n Q, Q s:se P and P 2 n Q 2, Q 2 sw:s:se P 2, showing that the converse of the basic relation north is not a basic RCD-relation. 4.1 The constraint algebra of the rectangular cardinal relation calculus In spatial applications with cardinal relations between extended objects, e.g. rectangles, the situations in which the relation of an object w.r.t. another is known but the other way round is unknown or unspecified may be frequent. Suppose that P n Q but the direction of Q w.r.t. P is unknown, expressed as Q? P. What can be inferred about that unknown relation? Since we cannot turn to the converse operation, we define below a unary operation, named weak converse, to answer questions related with the converse relation and the property of pairwise consistency. Although this property has been formulated for basic cardinal relations between regions only [5] (see also [26, 46]), we extend that property to basic and disjunctive relations and to qualitative networks of an arbitrary spatial qualitative calculi with set relations 2 B : Definition 4 (Pairwise consistency) We say that two basic relations r and r over some domain D are pairwise consistent if there exists spatial objects X, Y D such that XrYand Yr X. Two disjunctive relations D, D 2 B are pairwise consistent if there exists r D, r D such that r and r are pairwise consistent. A qualitative network N = (V, A ) is said to be pairwise consistent if for any two variables v i,v j, the relations R ij and R ji in A are pairwise consistent. Remark 3 In a finite relation algebra, the converse of a basic relation r is a basic relation and r = (r ) [9]. Also, for disjunctive relation D, itisd = (D ),since converse distribute over union of basic relations. The algebra we propose for the RCD calculus consist of the structure (2 Brcd,,wcon, w ),where w refers to a weak composition operation (see Table 2), while wcon is anewalgebraicoperator, relatedwithconverseoperator, wedefine as: Definition 5 (Weak converse operation) Let c, c and D, D be basic and disjunctive RCD-relations, respectively. The weak converse algebraic operation, wcon : 2 Brcd 2 Brcd is defined as: 1. Case of basic relations wcon(c) := {c B rcd P, Q : PcQ, Qc P}, 2. Case of disjunctive relations wcon(d) := c D wcon(c). Intuitively, for a basic relation c, wcon(c) contains those basic relations c such that c and c are pairwise consistent. To compute wcon(c) manually we can make use of Fig. 4, that shows illustrations of all basic RCD-relations; the details of this figure will be explained in Section 5.1. For instance, wcon(n) ={s:se, sw:s:se,sw:s, s} and Fig. 3a, b illustrates instantiations of the constraints P n Q, Q s:se P and P 2 n Q 2, Q 2 sw:s:se P 2 showing why s:se, sw:s:se are two of the four possible basic