RENZ J., NEBEL B., "On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus", Artificial Intelligence, Journal of Applied Non-Classical Logics. Volume 11 - n  Home/Search   Document Details and Download   Summary   Related Articles   Check

....would be to have some combination of spatial and temporal logics. There exists a variety of spatial formalisms [14, 49, 15, 42] and a wide spectrum of temporal languages [3, 25, 55] Effective reasoning procedures have been developed and implemented for temporal [40] as well as spatial formalisms [6, 51]. For incorporation of time into space, the most logical step would be to have a combination of these two streams of reasoning. In fact, there have been attempts to have spatio temporal hybrids [30, 7] A recent spatio temporal representation and reasoning based on RCC 8 is [58] Motion can ....

J Renz and B Nebel, `On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus', Artificial Intelligence, 108(1-2), 69--123, (1999).

.... is a known technique used in modal logic, 12] In the spatial context, similar settings have been used initially in [7] to encode decidable fragments of the region connection calculus RCC (the fundamental and most widely used qualitative spatial reasoning calculi in the eld of AI, 14] then by [15] to identify maximal tractable fragments of RCC and, recently, by [16] Even though the logical technique is similar to that of [7, 15] there are two important di erences. First, in the proposed use of S4 u there is no commitment to a speci c de nition of connection (as RCC does by forcing the ....

.... fragments of the region connection calculus RCC (the fundamental and most widely used qualitative spatial reasoning calculi in the eld of AI, 14] then by [15] to identify maximal tractable fragments of RCC and, recently, by [16] Even though the logical technique is similar to that of [7, 15], there are two important di erences. First, in the proposed use of S4 u there is no commitment to a speci c de nition of connection (as RCC does by forcing the intersection of two regions to be non empty) Second, the stress is on model equivalence and model comparison issues, not only spatial ....

J. Renz and B. Nebel. On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus. Arti cial Intelligence, 108(1-2):69-123, 1999.

....PO ]4 aZ or PP ]4 aZ or PPi ]4 aZ or EQ ]4 aZ . If we define ; 9 ( 1 9 then the relations PO PP and EQ are indescernible with respect to ; The composition of the base relation often yields such relations of coarser level of resolution [24,15,37]. A similar effect occurred when reasoning about approximations. Consider the RCC5 relations. As a set these base relations are jointly exhaustive and pair wise disjoint and form the RCC5 lattice as depicted in Figure 1. Theorems 1 and 2 show that subsets of these relations form relations of ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

....but not to any proper subinterval of i. In both papers Allen s temporal logic is solely based on time intervals and not on time points. Galton [25] has extended Allen s approach to the treatment of temporally changing topological relationships. Topological relationships are based on the RCC model [10, 31] which comes to similar results as Egenhofer s 9 intersection model. In contrast to Allen, Galton also takes time points into account, as we do. In specifying changes of spatial situations he uses the notion of a fluent or state which corresponds to what we call a temporal function. A fluent can ....

J. Renz and B. Nebel. On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

....where constraints may be arbitrary disjunctions of RCC 8 relations) is not tractable. A maximal tractable subset (containing 148 relations) of the constraint language of 2 RCC 8 has been identi ed and furthermore have shown that path consistency is sucient for deciding consistency in this case [155]. If an appropriate size constraint is introduced between two regions then all reasoning in 2 RCC 8 e ectively becomes polynomial [89] More recently, a complete classi cation of the tractability of RCC 8 has been made [153] It turns out that there are two further maximal tractable subsets ....

Renz, J. and Nebel, B.: \On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus", Articial Intelligence, 108(1-2), 1999, pages 69-123.

....Having identified a tractable fragment of intuitionistic logic, several interesting questions arise. Nebel [27] uses his tractable class in order to obtain a tractable class for spatial reasoning, in the so called RCC 8 spatial algebra [28, 29] This class was later extended by Renz and Nebel [30] to a maximal tractable subclass of that spatial algebra. Thus, since this class is incomparable to Nebel s, is it possible to use it to obtain other tractable subclasses of RCC 8, by reducing their satisfiability problem to that of intuitionistic logic For the case of the RCC 5 spatial algebra, ....

.... class is incomparable to Nebel s, is it possible to use it to obtain other tractable subclasses of RCC 8, by reducing their satisfiability problem to that of intuitionistic logic For the case of the RCC 5 spatial algebra, all possible cases of tractable subclasses have already been characterised [30, 24]. Another relevant question is whether we can use our tractable class of set constraints for tractable inference in other logical systems. For instance, Renz and Nebel [30] use classes of modal logics in order to prove classes of the RCC 5 and RCC 8 spatial algebras tractable, so there is ....

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Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connected calculus. In Martha E. Pollack, editor, Proceedings of the 15th International Joint Conference on Artificial Intelligence (IJCAI '97), Nagoya, Japan, August 1997. Morgan Kaufmann.

....a logical framework with a precise semantics within which a variety of more practical representation languages might be embedded. For instance, a decision procedure for a significant set of toplogical relations was presented by Bennett (1994, 1996) and tractable subsets of these identified by Renz and Nebel (1997, 1999). Cristani et al. 2000) investigate the complexity of reasoning with a combination of mereological and morphological relations and proves tractability of a significant constraint language, which is a fragment of our formalism. 16 Recursive axioms such as that for MoveWithin, which is clearly of ....

Renz, J. and Nebel, B.: 1999, On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus, Artificial Intelligence 108(1--2), 69--123.

....as possible. The primary purpose of our formalism is to provide a secure ontological foundation (as advocated e.g. in (Guarino 1998) for theories of spatial information; but we also believe that it can serve as a framework within which more computationally oriented representations (e.g. that of (Renz and Nebel 1997)) can be embedded. Since our theory has a categorical interpretation in terms of Cartesian fields over R it is readily compatible with more traditional representations that employ this classical model of space. A precursor of the theory of Region Based Geometry (RBG) which we will present, was ....

....a logical framework with a precise semantics within which a variety of more practical representation languages might be embedded. For instance, a decision procedure for a significant set of toplogical relations was presented by Bennett (1994, 1996) and tractable subsets of these identified by Renz and Nebel (1997, 1999) Cristani et al. 2000) investigate the complexity of reasoning with a combination of mereological and morphological relations and proves tractability of a significant constraint language, which is a fragment of our formalism. 16 Recursive axioms such as that for MoveWithin, which is ....

Renz, J. and Nebel, B.: 1997, On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus, Proceedings of IJCAI-97.

....and geometrical concepts. The primary motivation for the development of RBG was to provide a secure ontological foundation (as advocated e.g. in [14] for theories of spatial information. It may also be of use as a framework within which more computationally oriented representations (e.g. [2, 19, 9, 23]) can be embedded. Since the theory theory has a categorical interpretation in terms of Cartesian elds over R, it is readily compatible with more traditional representations that employ this classical model of space. The formulation of RBG was in uenced by [6] which, drawing on [21] constructs ....

....nature of RBG poses severe problems for automated reasoning. For many practical applications one would much prefer a tractable or at least decidable formalism. Nevertheless, the theory may be useful as an interlingua within which more computationally e ective representations, such as those of [2, 19, 9, 23] can be embedded. Although RBG is extremely general, it does have the limitation that it can only deal with a domain of entities having a given xed dimension. For many applications it would be useful to have an even more comprehensive theory enabling one to refer to entities of di erent ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

....= i (R) or = i (R) 1 (R) Theta : Theta i Gamma1 (R) Theta i 1 (R) Theta : Theta n (R) Consequently, n (R) is a convex relation in 2 An Gamma1 . So, according to our opening hypothesis, i (R) is a strongly preconvex relation in 2 An Gamma1 . Thus, Lemma 12 For every point relation R in 2 An , if R is a convex relation in 2 An then R is a strongly preconvex relation in 2 An . 4 Fundamental Operations Composition of point relations is defined in the following way, for every point relation R; S in 2 An , R ffi S = S f(A 1 ffi B 1 ) Theta : ....

J. Renz, B. Nebel. On the complexity of qualitative spatial reasoning : a maximal tractable fragment of the region connection calculus. IJCAI-97, Proceedings of the Fifteenth International Joint Conference on Artificial Intellligence. 522--527, Morgan Kaufman, 1997.

....structural knowledge about adjacency graphs to recover the additional adjacency information. Related Work Topological spatial relations between regions have been studied mainly from two directions: the Region Connection Calculus (RCC) Randell, Cui, Cohn 1992; Bennett 1994; Cohn et al. 1997; Renz Nebel 1999) in AI, and the 9 intersection model (Egenhofer 1991; Engenhofer Mark 1995) in GIS. RCC adopts a region topology in which regions are primary objects and the connection relation is the primary relation. Other relations between regions are defined upon the connection relation with a set of ....

....in GIS. RCC adopts a region topology in which regions are primary objects and the connection relation is the primary relation. Other relations between regions are defined upon the connection relation with a set of axioms and Boolean functions using first order logic. RCC research (Bennett 1994; Renz Nebel 1999) studies the composition rules of different spatial relations and uses these rules to uncover unknown relations from known ones. The 9 intersection model adopts a point set topology in which points are primary objects and regions are defined as sets of points. A topological relation between two ....

Renz, J., and Nebel, B. 1999. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence 108:69--123.

.... others) has shown that some simple combinations of modalities together with very simple interaction axioms yield undecidable systems; on the positive side, there are a number of examples of quite expressive fragments of multi modal languages, whose decision procedures are polynomial, for example (Renz and Nebel, 1997). Thus, the viability of reasoning with combined modal logics depends very much on the particular combination of modalities and interaction axioms. An obvious way of reducing the complexity of a logical language is to restrict its syntax. We consider such an approach, called layering (Finger and ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

....tractable [98] in fact the satisfaction problem is solvable in polylogarithmic time since it is in the complexity class NC. However the constraint language of 2 RCC8 (i.e. where constraints may be arbitrary disjunctions of RCC8 relations) is not tractable, though some subsets are tractable [110] have identified a maximal tractable subset of the constraint language of 2 RCC8 and furthermore have shown that for path consistency is sufficient for deciding consistency in this case. As in the case of identifying the maximal tractable subset of Allen s interval calculus [99] the analysis ....

Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

....on spatial representations (cf. Glasgow and Papadias 1992, p. 356) It can be considered as an intermediating alternative to language oriented , logic based models and to perception oriented depictional models. Logic based models (e.g. Guesgen 1989, Hern andez 1991, Mittal and Mukerjee 1995, Renz and Nebel 1997) operate on qualitative information, i.e. symbolic expressions (like left of(x,y) Inference of relations between objects ( qualitative reasoning ) is performed by syntactical methods. A variant of purely logic approaches is to represent conceptual knowledge about spatial configurations by ....

Renz, J. and B. Nebel (1997). On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of RCC-8. In Proceedings of the 11th International Workshop on Qualitative Reasoning QR 97.

....EC , DC , U g with the intended meaning of Proper Overlap, Non Tangential Proper Part, Tangential Proper Part, EQual, Converse Tangential Proper Part, Converse Non Tangential Proper Part, External Connection, DisConnected, and Universal, respectively. Deciding satisfaction for RCC8 is NP complete [12]. Furthermore, RCC8 relations are closed under boolean operations. Hence D rcc8 is admissible. Given the NP results for RCC8, deciding satisfiability in ALC(D rcc8 ) with empty T boxes) is PSPACE [10] Definition 3 (ALC(D rcc8 ) Fix a signature = hC, R, F, CN, ANi (of Atomic Concepts, Roles, ....

B. Nebel and J. Renz. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1-2):69--123, 1999. 5

....as simply as possible. The primary purpose of our formalism is to provide a secure ontological foundation (as advocated e.g. in [13] for theories of spatial information; but we also believe that it can serve as a framework within which more computationally oriented representations (e.g. that of [17]) can be embedded. Since our theory has a categorical interpretation in terms of Cartesian fields over Rit is readily compatible with more traditional representations that employ this classical model of space. 2 MEREOLOGY We begin by presenting a formal theory of the parthood relation, P(x; y) ....

....2nd order nature of RBG poses severe problems for automated reasoning. However, our theory is not intended form the basis of a deductive mechanism; but rather to provide a logical framework with a precise semantics within which a variety of more practical representation languages (such as that of [17]) might be embedded. Although our theory is extremely general it does have the limitation that it can only deal with a domain of entities having a given fixed dimension. For many applications it would be useful to be able to refer to entities of different dimensionality [11] We would like to ....

J. Renz and B. Nebel, `On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus', in Proceedings of IJCAI-97, (1997).

.... for representing and reasoning about qualitative spatial information is now an active research area, both within AI, and within the field of geographical information systems [14] Much of the e#ort has been devoted to developing e#cient representations for reasoning about topological information [2, 25, 24, 20], although other aspects such as orientation [22, 19] distance [18] and qualitative morphology [12] have also been investigated. Qualitative representations have a natural facility to handle indefinite and imprecise information by abstracting away from metrical details. However, specific ....

....The main problem we can now investigate is the computational complexity of reasoning about the relations in the set L. Comparing the set with the four maximal tractable subclasses of RCC 5 described in [20] we can easily note that L # is a proper subset of the maximal tractable subclass which [25] had previously individuated and labelled as # H 5 . The immediate consequence of the above observation is the following theorem, where the notation RSAT(X) has to be read as the satisfiability problem in the subset X. Theorem 4 RSAT(L # ) is polynomial. Note that there is no analogous ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

....of the computation of the transitive closure of the temporal relations, and the covering step. Both these steps involve NP complete temporal constraint problems #van Beek Cohen, 1990#. However, it is possible to devise reasonable subsets of Allen s algebra for which the problem is polynomial #Renz Nebel, 1997#. The most relevant properties of a concept in CEF is that all the admissible interval temporal relations are explicit and the concept expression in each node is no more re#nable without changing the overall concept meaning; this is stated by the following proposition. Proposition 6.7 #Node ....

Renz, J., & Nebel, B. #1997#. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus. In Proc. of the 14 th IJCAI, pp. 522#527 Nagoya, Japan.

.... which is sometimes mentioned as tractability . Traditionally, tractability has been largely understood as polynomial time solvability of the reasoning problems. This resulted in a tremendous e ort on isolating the so called tractable fragments of various logics for knowledge representation [50, 56, 20, 57, 64]. The last years have seen a major shift from this paradigm, at least for two reasons. First, logics with polynomial time deductive problems have been criticized for their too limited expressive power [22] Altough systems with a limited but reliable KR R component have been successfully used in ....

Renz, J., and Nebel, B. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Articial Intelligence Journal 108, 1-2 (1999), 69-123.

....1 We do not expect to find many people to disagree with this statement as far as temporal constraints are concerned. Spatial constraints are trickier and will probably require our attention for some more time until we know enough about them. Also, some important advances in this area (e.g. (Renz Nebel 1999)) are very recent. where temporal or spatial reasoners are an important component. We do not claim to be the first to have raised this issue. It is explicit in (van Beek 1991) in most papers on the TMM system 2 (Dean McDermott 1987; Schrag, Boddy, Carciofini 1992) and possibly in other ....

Renz, J., and Nebel, B. 1999. On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus. Artificial Intelligence 1-2:95--149.

....were prohibitively complex. More recently some success has been achieved in nding much simpler tractable languages capable of representing signi cant aspects of the spatio temporal domain. One approach that has been fruitful so far is encoding into 0 order modal formalisms [ Bennett, 1996; Renz and Nebel, 1997 ] However, as yet such encodings provide a fairly limited vocabulary, whereas real applications typically require a large range of di erent concepts. In this exploratory paper we propose to generalise the modal approach by establishing a framework for spatio temporal representation and ....

.... with very simple interaction axioms yeild undecidable systems (for instance, the logic with three commuting S5 modalities, is such a system) On the positive side, there are a number of examples of quite expressive fragments of multi modal languages, whose decision procedures are polynomial (e.g. Renz and Nebel, 1997 ] An obvious way of reducing the complexity of a logical language is to restrict its syntax. We now consider two promising restrictions. 5.1 Layered Modal Logics A layered modal logic is a special kind of multi modal logic in which restrictions are placed on the nesting order of the di erent ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

....in the direction of restricted models, in such a way that at least for certain cases we may process a finite set of RCC constraints in polynomial fashion on deterministic machines. In particular, Nebel (1995) showed that reasoning with the basic relations of RCC 5 and RCC 8 are tractable problems. Renz and Nebel (1999) improved the results above, by showing that there exists a maximal tractable subclass of RCC 5, denoted by b H 5 , formed by 28 relations out of 32, which includes all the basic relations, and a maximal tractable subclass of RCC 8, denoted by b H 8 , formed by 148 relations out of 256 including ....

....tractable subclasses of RCC 5, one including the basic relations. The result is obtained in a similar fashion to (Drakengren Jonsson, 1997; Jonsson, Drakengren, Backstrom, 1999) A complete analysis of the RCC 8 maximal tractable subclasses including the basic relations has been provided by Renz in (1999). Our result is the analogous in MC 4 of Jonsson and Drakengren result for RCC 5. The MC 4 algebra, we describe in this paper, is structured in the same way as the Algebra of Partially Ordered Time (PO time algebra) studied by Anger, Mitra and Rodriguez. We would like to stress two main aspects ....

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Renz, J., & Nebel, B. (1999). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108 (12) , 69--123.

....University of Leeds, Leeds LS2 9JT, England fbrandon j agcg scs.leeds.ac.uk Abstract We present a model building algorithm for determining consistency of topological relations. This is a simpli cation of the modal logic approach proposed by [ Bennett, 1996 ] and subsequently developed in [ Renz and Nebel, 1997; 1998a; 1998b ] We construct a simple, nonmodal, topological representation whose models are comparatively easy to understand and visualise. They consist of a set of points satisfying certain classical relations. By associating these points with Cartesian coordinates it is ....

....robots, intelligent GIS systems) Amongst the most fundamental spatial relationships are those which are purely topological. Consequently much attention has been given to representing and reasoning about topological relations [ Randell et al. 1992; Bennett, 1994; Nebel, 1995; Bennett, 1996; Renz and Nebel, 1997; 1998b; 1998a ] Work on reasoning algorithms has focussed on sets of binary topological relations drawn from the set known as RCC 8 (see Fig. 1) and on disjunctions of these relations. However, for many applications it is clear that purely topological information is too weak to provide an ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

....On the other hand, the model of the regions has been brought in by Randell, Cui and Cohn [12, 13] who introduce a language for representation and reasoning about spatial relations between regions while adapting the usual network based framework of constraint satisfaction. Recently, Renz and Nebel [14] characterized a tractable subclass of region networks the consistency of which can be decided by means of the path consistency algorithm. This paper extends the concept of interval to the n dimensional Euclidean space over the field of real numbers. In actual fact it would be more accurate to ....

J. Renz, B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus. M. Polack (editor), IJCAI97. Proc. of the 15th International Joint Conf. on Artificial Intelligence, Volume one, 522--527, Morgan Kaufmann, 1997.

.... 1990; Freska, 1992; Randell et al. 1992 ] We can mention as an example the well known model of the regions proposed by Cohn, Cui and Randell [ Randell et al. 1992 ] whose objects are the regions of a topological space and relations are eight topological relations, and for which Renz and Nebel [ Renz et al. 1997 ] characterize a maximal tractable subclass of relations. Although this formalism is very attractive, it suffers from the impossibility of expressing orientation relations. An example which enables this is the rectangle algebra (RA) Gusgen, 1989; Mukerjee et al. 1990; Balbiani et al. 1998 ] ....

J. Renz, B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus. M. Polack (editor), IJCAI-97. Proc. of the 15th Int. Joint Conf. on Artificial Intelligence, Volume one, 522--527, Morgan Kaufmann, 1997.

....Y ) X Y TPP(X;Y ) X Y NTPP(X;Y ) Y X TPP(Y; X) Y X NTPP(Y;X) Fig. 1. A graphical representation of the RCC8 relations the correctness of Bennett s translation for granted and based their own work upon it. For instance, Renz and Nebel applied it to identify maximal tractable fragments of RCC8 [10], and Haarslev et al. 4] used it to develop algorithms for spatioterminological reasoning. In the present paper, we give for the first time a rigorous foundation for reasoning in RCC8. To represent topological relationships between regions, we introduce a language of topological set constraints, ....

....arise from a topological interpretation if and only if this conjunctive constraint is satisfiable. From this consideration, we infer the complexity of satisfiability for arbitrary set constraints. In addition, we obtain an an upper bound for RCC8 by an argument that is much simpler than the one in [10]. Proposition 6 (Complexity) Satisfiability of arbitrary topological set constraints is PSPACE complete, while satisfiability of RCC8 constraints systems is NP complete. Proof. Satisfiability of arbitrary set constraints is at least as hard as satisfiability in S4, which is PSPACE complete ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

....are considered) The core of the calculus can be expressed in terms of the composition table. Bennett [1] showed how to translate the basic relations in terms of intuitionistic logics. Using this translation, Nebel [8] proved that any path consistent atomic network is consistent. Renz and Nebel [11] extended this result to sub classes of the RCC 8 algebra, using an explicit construction of models (in terms of Kripke models) Along the same line, Renz [10] showed that any path consistent atomic network has a canonical model in the Euclidean n space, for any n 1, where the regions can be ....

....in Table 3. Table 3. Constraints associated to the basic relations in RCC 5 Possibility conditions Constraints DR u i , u j u i :u j , u j :u i PO u i u j ,u i :u j , u j :u i PP u i u j , u j :u i u i u j , u j :u i The following procedure yields a solution to the problem (cf. [11]) 1. Start with an empty set W . 2. For each pair of variables (u i ; u j ) i j, introduce an element (a world ) w for each possibility condition (hence two, or three elements are added to W for each pair, according to the corresponding relation) and instantiate u i , u j such that the ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. In Proc. of IJCAI-97, pages 522--527, 1997.

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J. Renz and B. Nebel, On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus, Artificial Intelligence 108 (1--2) (1999) 69--123.

....Furthermore we assume that for every pair of variables (x, y) such that xRy #, yR x #. Relation Converse Pictorial Example J J I m J J m I o J J o I d J J d I s J J s I f J J f I = J J = I I J Figure 2. The thirteen basic relations of the Interval Algebra [17, 18], and several maximal tractable fragments have been identified both for RCC 8 and IA. Nebel and B urckert [14] identified the unique maximal tractable sub algebra of IA containing all the basic relations, which is called ORD Horn class. ISAT for the ORD Horn class can be decided in cubic time by ....

.... is generated when enforcing path consistency) Other maximal tractable subclasses which do not contain all of the basic relations have been identified by Krokhin et al. 7] Regarding RCC 8, Renz and Nebel identified three maximal tractable subclasses of RCC 8 containing all the basic relations [17, 16]. In addition, Gerevini and Renz [6] showed (using a technique called BIPATH CONSISTENCY) that the relations in these maximal classes can be combined with qualitative size constraints without increasing the computational complexity. Finally, RSAT and ISAT for the full RCC 8 and IA can be solved ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1--2):69--123, 1999.

....direction: we introduce some types of qualitative and quantitative size information, and we study their integration with topological information. Our work is based on RCC 8 [54] a well known constraint language for topological spatial reasoning that is based on the Region Connection Calculus [5,6,14,19,35,36,55,57,59,64]. In this framework, regions are independent with respect to rotation, translation, and several other transformations of the underlying space which makes them very simple and natural. This has also been observed in cognitive evaluations [39,58] The topological distinctions made by RCC 8 are ....

....of eight relations, called basic relations, and by all the possible unions of them. In general, deciding consistency (satis ability) of a set of constraints in RCC 8 is NP complete, but for three large subsets of RCC 8, called H 8 , C 8 and Q 8 , this problem can be solved in polynomial time [57,56]. These classes are the only maximal tractable subclasses of RCC 8 that contain all the basic relations. In the rst part of the paper we consider a set of qualitative relations between region sizes forming a Point Algebra [42] which have been thoroughly studied in the context of temporal ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Arti cial Intelligence, 108(1-2):69-123, 1999.

.... a calculus for reasoning about qualitative directions [4] and a calculus for reasoning about topological relations [2, 21] For all of these calculi, their computational complexity has been analyzed, computationally tractable fragments have been identified, and algorithms have been specified [27, 13, 20, 19, 15, 9, 24, 23], which are variations of Ladkin and Reinefeld s [13] scheme of using backtracking employing the path consistency algorithm [17] as a forward checking technique. Another qualitative spatial calculus is Freksa s [5, 6, 28] calculus for reasoning about orientation. In contrast to the calculi ....

....below, however, this hope is unfounded. 4 Computational Complexity In almost all qualitative calculi that have been investigated so far, there exists some non trivial fragment containing all the base relations and the universal relation such that satisfiability can be decided in polynomial time [20, 15, 24]. Can we expect something similar for the Double Cross calculus As it turns out, this is not the case. Even if we consider only CSPs over sf and the universal relations , the satisfiability problem is already NP hard. The reason is that with sf we can relate three points on a line saying only ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69-- 123, 1999.

....document interpretation, and geographical information systems. The RCC 8 calculus (Randell, Cui, Cohn, 1992b) is well suited for representing topological relationships between spatial regions. Inference in the full calculus is, however, NP hard (Grigni, Papadias, Papadimitriou, 1995; Renz Nebel, 1999). While this means that it is unlikely that very large instances can be solved in reasonable time, this result does not rule out the possibility that we can solve instances up to a certain size in reasonable time. Recently, maximal tractable subsets of RCC 8 were identi ed (Renz Nebel, 1999; ....

....1995; Renz Nebel, 1999) While this means that it is unlikely that very large instances can be solved in reasonable time, this result does not rule out the possibility that we can solve instances up to a certain size in reasonable time. Recently, maximal tractable subsets of RCC 8 were identi ed (Renz Nebel, 1999; Renz, 1999) which can be used to speed up backtracking search for the general NP complete reasoning problem by reducing the search space considerably. In this paper we address several questions that emerge from previous theoretical results on RCC 8 (Renz Nebel, 1999; Renz, 1999) Up to which size is it ....

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Renz & Nebel Renz, J., & Nebel, B. (1999). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Articial Intelligence, 108 (12) , 69-123.

....of the Interval Algebra that DIA is closed under composition, intersection, converse, and reverse. 4 Reasoning over Directed Intervals The main reasoning problem in spatial and temporal reasoning is the consistency problem CSPSAT(S) where S is a set of relations over a relation algebra [Renz and Nebel, 1999] . Instance: A set V of variables over a domain D and a finite set of binary constraints xRy (R 2 S and x; y 2 V . Question: Is there a consistent instantiation of all n variables in with values from D which satisfies all constraints The consistency problem of the directed intervals ....

J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. AIJ, 108(1-2):69--123, 1999.

....spatial regions are subsets of topological space [18] Of particular interest for application purposes is RCC 8, a sub calculus of RCC that uses eight mutually exhaustive and pairwise disjoint 1 1 INTRODUCTION 2 base relations. The computational properties of RCC 8 have been studied thoroughly [17, 19] and efficient reasoning mechanisms were identified. Despite this, there are still some problems with representational aspects of RCC. As the calculus is based on topology, an obvious canonical model is the topological space. However, it appears to be difficult to represent such models in a ....

....3 Modal Encoding Canonical Models In this section we will introduce the modal encoding of RCC 8 and a canonical model for this encoding. An introduction to modal logics is given in Appendix A. 3. 1 Modal Encoding of RCC8 The modal encoding of RCC 8 was introduced by Bennett [2] and extended in [19]. In both cases the encoding is restricted to regular closed regions. The encoding is based on a set of model and entailment constraints for each base relation, where model constraints must be true and entailment constraints must not be true. Bennett encoded these constraints in modal logic by ....

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Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. In Proceedings of the 15th International Joint Conference on Artificial Intelligence, pages 522--527, Nagoya, Japan, August 1997. Extended version under review for journal publication.

.... [34, 14] compute vision, natural language processing (e.g. 34] visual languages (e.g. 22] and qualitative simulation of physical processes (e.g. 11, 39, 28] Previous work in spatial reasoning has been concentrated on various types of space representation, such as topology (e.g. [23, 40, 43, 25, 9]) direction (e.g. 33] distance or position (e.g. 8, 3, 4] without considering information on the size of spatial regions. Moreover, while most research on qualitative spatial reasoning has been focussed on single aspects of space, real world applications usually require more than just one ....

....that is a consistent re nement of all the constraints in the set to one of their basic relations. These problems are in general NP hard, but they can be decided in polynomial time for three large subsets of RCC 8 (denoted b H 8 , C 8 and Q 8 ) which are maximal tractable subclasses of RCC 8 [43, 42]. In particular, Renz and Nebel [43] and Renz [42] proved that the consistency of a set of constraints over these classes can be decided in O(n 3 ) time by using path consistency, where n is the number of variables involved. In this paper we study the combination of RCC 8 with qualitative and ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Articial Intelligence, 108(1-2):69-123, 1999. 36

....relations. We denote the standard operations composition, intersection and converse by ffi, and Gamma1 , respectively. Furthermore, we define the unary operation : such that :S = AnS for all S A. The consistency problem CSPSAT(S) for sets S 2 A over a domain D is defined as follows [16]: Instance: A set V of variables over a domain D and a finite set Theta of binary constraints xRy, where R 2 S and x; y 2 V . Question: Is there an instantiation of all variables in Theta such that all constraints are satisfied Naturally, a set of basic relations is to be interpreted as a ....

....between DC and EC and between TPP and NTPP. These relations are combined to the RCC 5 relations DR for DiscRete and PP for Proper Part, respectively. Thus, RCC 5 contains the five basic relations DR, PO, PP, PP Gamma1 and EQ. The consistency problem of both RCC 8 and RCC 5 is NP complete [16], but large maximal tractable subsets have been identified [16, 11, 15] In the following we demonstrate the usefulness of our method by identifying tractable disjunctive constraint classes of RCC 8 and RCC 5. We begin with RCC 5 which contains four maximal tractable subsets, R 28 (the only ....

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108(1--2):69--123, 1999.

....adequate. As already explained, this is due to disregarding the relationship of a reference object (RO) with a to be localized object (LO) We believe that when this distinction is explicitly emphasized in the instructions, subjects will group items accordingly. Furthermore, it follows from [RN99] that reasoning with RCC 8 relations is NP hard whereas reasoning with the eight RCC 8 relations is tractable [Neb95] Future work includes further lifting the restrictions on the shape of the presented regions. One question is whether subjects continue to distinguish the number of connections ....

Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

....Randell, Cui, and Cohn s [ 1992 ] Region Connection Calculus RCC 8. In the former case, the only maximal tractable subset containing all base relations has been identified [ Nebel and Burckert, 1995 ] in the latter case, only one maximal tractable subset has been identified so far [ Renz and Nebel, 1999 ] It was previously unknown whether there are others containing all base relations. For both subsets path consistency is sufficient for deciding consistency. Tractability and sufficiency of path consistency have been proven by reducing the consistency problem to a tractable propositional ....

....which makes Theta pathconsistent by eliminating all the impossible labels (base relations) in every subset of constraints involving three variables [ Mackworth, 1977 ] If the empty relation occurs during this process, then Theta is inconsistent, otherwise the resulting set is path consistent. Renz and Nebel [ 1999 ] identified a tractable subset of RCC 8 (denoted by b H 8 ) containing all base relations which is maximal with respect to tractability, i.e. if any other RCC 8 relation is added to b H 8 , the consistency problem becomes NP complete. They further showed that enforcing path consistency is ....

[Article contains additional citation context not shown here]

Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69-- 123, 1999.

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RENZ J., NEBEL B., "On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus", Artificial Intelligence, Journal of Applied Non-Classical Logics. Volume 11 - n

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Renz, J. and Nebel, B. (1997). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. In Proceedings of the 15th International Joint Conference on Arti cial Intelligence (IJCAI 97).

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

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J. Renz and B. Nebel. On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus. Arti cial Intelligence, 1-2:95-149, 1999.

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Renz, J. and Nebel, B. (1999). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1--2):69--123.

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Renz, J. and Nebel, B. (1997). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. In IJCAI 97, Proceedings of the 15th International Joint Conference on Artificial Intelligence.

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Renz, J. and Nebel, B. (1999). On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Artificial Intelligence, 108(1-2):69--123.

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J. Renz and B. Nebel. On the Complexity of Qualitative Spatial Reasoning: A Maximal Tractable Fragment of the Region Connection Calculus. Artificial Intelligence, 108(1-2):69--123, 1999.

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus. Arti cial Intelligence, 108(1{ 2):69-123, 1999.

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J. Renz and B. Nebel. On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the Region Connection Calculus. In Proceedings of IJCAI-97, 1997.

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J. Renz, B. Nebel, On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus, Artificial Intelligence 108 (1-2) (1999) 69--123.

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J. Renz, B. Nebel, On the complexity of qualitative spatial reasoning: a maximal tractable fragment of the region connection calculus, Artificial Intelligence, Vol. 108, pp. 69--123, 1999.

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