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111
A Complete Classification of Tractability in RCC5
 Journal of Artificial Intelligence Research
, 1997
"... We investigate the computational properties of the spatial algebra RCC5 which is a restricted version of the RCC framework for spatial reasoning. The satisfiability problem for RCC5 is known to be NPcomplete but not much is known about its approximately four billion subclasses. We provide a compl ..."
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Cited by 29 (7 self)
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We investigate the computational properties of the spatial algebra RCC5 which is a restricted version of the RCC framework for spatial reasoning. The satisfiability problem for RCC5 is known to be NPcomplete but not much is known about its approximately four billion subclasses. We provide a complete classification of satisfiability for all these subclasses into polynomial and NPcomplete respectively. In the process, we identify all maximal tractable subalgebras which are four in total. 1. Introduction Qualitative spatial reasoning has received a constantly increasing amount of interest in the literature. The main reason for this is, probably, that spatial reasoning has proved to be applicable to realworld problems in, for example, geographical database systems (Egenhofer, 1991; Grigni, Papadias, & Papadimitriou, 1995) and molecular biology (Cui, 1994). In both these applications, the size of the problem instances can be huge, so the complexity of problems and algorithms is a highly...
When does a Composition Table Provide a Complete and Tractable Proof Procedure for a Relational Constraint Language?
 In Proceedings of the IJCAI97 Workshop on Spatial and Temporal Reasoning
, 1997
"... Originating in Allen's analysis of temporal relations, the notion of a composition table (CT) has become a key technique in providing an efficient inference mechanism for a wide class of theories. In this paper we challenge researchers working with CTs to give a general characterisation of the ..."
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Cited by 27 (4 self)
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Originating in Allen's analysis of temporal relations, the notion of a composition table (CT) has become a key technique in providing an efficient inference mechanism for a wide class of theories. In this paper we challenge researchers working with CTs to give a general characterisation of the class of theories and relational constraint languages for which a complete proof procedure can be specified by a CT. Several related problems and conjectures will be discussed. A secondary aim is to clarify the terminology used to describe CTs and to establish a general conceptual framework applicable to any CT whatever relations are involved. One of the main advantages of using CTs is that they can lead to tractable computation of significant classes of inference. An important aspect of our proposed research programme is to separate computational from purely logical issues and then to systematically investigate relationships between logical and computational complexity. 1 Reasoning about Relatio...
GQR – A Fast Reasoner for Binary Qualitative Constraint Calculi
"... GQR (Generic Qualitative Reasoner) is a solver for binary qualitative constraint networks. GQR takes a calculus description and one or more constraint networks as input, and tries to solve the networks using the path consistency method and (heuristic) backtracking. In contrast to specialized reasone ..."
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Cited by 26 (8 self)
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GQR (Generic Qualitative Reasoner) is a solver for binary qualitative constraint networks. GQR takes a calculus description and one or more constraint networks as input, and tries to solve the networks using the path consistency method and (heuristic) backtracking. In contrast to specialized reasoners, it offers reasoning services for different qualitative calculi, which means that these calculi are not hardcoded into the reasoner. Currently, GQR supports arbitrary binary constraint calculi developed for spatial and temporal reasoning, such as calculi from the RCC family, the intersection calculi, Allen’s interval algebra, cardinal direction calculi, and calculi from the OPRA family. New calculi can be added to the system by specifications in a simple text format or in an XML file format. The tool is designed and implemented with genericity and extensibility in mind, while preserving efficiency and scalability. The user can choose between different data structures and heuristics, and new ones can be easily added to the objectoriented framework. GQR is free software distributed under the terms of the GNU General Public License.
Combining Spatial and Temporal Logics: Expressiveness Vs. Complexity
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2004
"... In this paper, we construct and investigate a hierarchy of spatiotemporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic the spatial logics RCC8, BRCC8, S4 u and their fragments. The obtained results give ..."
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Cited by 25 (9 self)
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In this paper, we construct and investigate a hierarchy of spatiotemporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic the spatial logics RCC8, BRCC8, S4 u and their fragments. The obtained results give a clear picture of the tradeoff between expressiveness and `computational realisability' within the hierarchy. We demonstrate how di#erent combining principles as well as spatial and temporal primitives can produce NP, PSPACE, EXPSPACE, 2EXPSPACEcomplete, and even undecidable spatiotemporal logics out of components that are at most NP or PSPACEcomplete.
MODAL LOGICS OF TOPOLOGICAL RELATIONS
 ACCEPTED FOR PUBLICATION IN LOGICAL METHODS IN COMPUTER SCIENCE
, 2006
"... Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham’s modal logic of time intervals based on the Allen relations, we int ..."
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Cited by 24 (6 self)
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Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham’s modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the EgenhoferFranzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the twovariable fragment of firstorder logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to Π 1 1hard, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.
Undecidability of Plane Polygonal Mereotopology
 PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING: PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE (KR98
, 1998
"... This paper presents a mereotopological model of polygonal regions of the Euclidean plane and an undecidability proof of its firstorder theory. Restricted to the primitive operations the model is a Boolean algebra. Its single primitive predicate defines simple polygons as the topologically simplest p ..."
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Cited by 22 (0 self)
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This paper presents a mereotopological model of polygonal regions of the Euclidean plane and an undecidability proof of its firstorder theory. Restricted to the primitive operations the model is a Boolean algebra. Its single primitive predicate defines simple polygons as the topologically simplest polygonal regions. It turns out that both the relations usually provided by mereotopologies and more subtle topological relations are elementarily definable in the model. Using these relations, Post's correspondence problem, known as undecidable, can be reduced to the decision problem of the model.
Fast Algebraic Methods for Interval Constraint Problems
 Annals of Mathematics and Artificial Intelligence
, 1996
"... We describe an e#ective generic method for solving constraint problems, based on Tarski's relation algebra, using pathconsistency as a pruning technique. Weinvestigate the performance of this method on interval constraint problems. Time performance is a#ected strongly by the pathconsistenc ..."
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Cited by 22 (1 self)
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We describe an e#ective generic method for solving constraint problems, based on Tarski's relation algebra, using pathconsistency as a pruning technique. Weinvestigate the performance of this method on interval constraint problems. Time performance is a#ected strongly by the pathconsistency calculations, whichinvolve the calculation of compositions of relations. Weinvestigate various methods of tuning composition calculations, and also pathconsistency computations. Space performance is a#ected by the branching factor during search. Reducing this branching factor depends on the existence of `nice' subclasses of the constraint domain. Finally,we survey the statistics of consistency properties of interval constraint problems. Problems of up to 500 variables may be solved in expected cubic time. Evidence is presented that the `phase transition' occurs in the range 6 # n:c # 15, where n is the numberofvariables, and c is the ratio of nontrivial constraints to possible constra...
Spatial Logic and the Complexity of Diagrammatic Reasoning
 MACHINE GRAPHICS AND VISION
, 1997
"... Researchers have sought to explain the observed "efficacy" of diagrammatic reasoning (DR) via the notions of "limited abstraction" and inexpressivity [17, 20]. We argue that application of the concepts of computational complexity to systems of diagrammatic representation is neces ..."
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Cited by 22 (2 self)
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Researchers have sought to explain the observed "efficacy" of diagrammatic reasoning (DR) via the notions of "limited abstraction" and inexpressivity [17, 20]. We argue that application of the concepts of computational complexity to systems of diagrammatic representation is necessary for the evaluation of precise claims about their efficacy. We show here how to give such an analysis. Centrally, we claim that recent formal analyses of diagrammatic representations (DRs) (eg: [14]) fail to account for the ways in which they employ spatial relations in their representational work. This focus raises some problems for the expressive power of graphical systems, related to the topological and geometrical constraints of the medium. A further idea is that some diagrammatic reasoning may be analysed as a variety of topological inference [15]. In particular, we show how reasoning in some diagrammatic systems is of polynomial complexity, while reasoning in others is NP hard. A simple case study i...
An Axiomatic Approach to Topology For Spatial Information Systems
, 1996
"... The paper reports work on the topological formalism `RCC', a regionbased `calculus of connection' developed at Leeds university over the past several years. Specifically, it is shown that the nonempty regular closed sets of a class of topological spaces (connected T3 spaces) provide mode ..."
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Cited by 21 (1 self)
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The paper reports work on the topological formalism `RCC', a regionbased `calculus of connection' developed at Leeds university over the past several years. Specifically, it is shown that the nonempty regular closed sets of a class of topological spaces (connected T3 spaces) provide models for the RCC axiomset. A brief assessment is made of RCC's potential as a formalism for applications in the area of spatial information systems (SIS). Two approaches to developing topological formalisms for SIS are compared, and a parallel is drawn with the two main parts of topology as understood by mathematicians. 1 Introduction For the past 3 years, the author has been working on the development of a particular formal system for topological representation and reasoning, referred to here as RCC (RegionConnection Calculus) (Gotts 1994, Cohn and Gotts 1996, Gotts, Gooday and Cohn 1996). RCC itself has a longer history: the version to be discussed here was first described in (Randell, Cui and Cohn ...
Combining Topological and Size Information for Spatial Reasoning
 Artificial Intelligence
, 2000
"... Information about the size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we introduce four classes of qualitative and metric size constraints, and we study their integration with the Regi ..."
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Cited by 21 (8 self)
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Information about the size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we introduce four classes of qualitative and metric size constraints, and we study their integration with the Region Connection Calculus RCC8, a widely studied approach for qualitative spatial reasoning with topological relations. Reasoning about RCC8 relations is NPhard, but three large maximal tractable subclasses of RCC8, called b H8 , C8 and Q8 respectively, have been identied. We propose an O(n 3 ) time pathconsistency algorithm based on a novel technique for combining RCC8 relations and qualitative size relations forming a Point Algebra, where n is the number of spatial regions. This algorithm is correct and complete for deciding consistency when the topological relations are either in b H8 , C8 or Q8 , and has the same complexity as the best known method for deciding consistency...