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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 20 (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.
A Tableau Algorithm for Reasoning about Concepts and Similarity
, 2003
"... We present a tableaubased decision procedure for the fusion (independent join) of the expressive description logic ALCQO and the logic MS for reasoning about distances and similarities. The resulting `hybrid' logic allows both precise and approximate representation of and reasoning about c ..."
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Cited by 16 (10 self)
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We present a tableaubased decision procedure for the fusion (independent join) of the expressive description logic ALCQO and the logic MS for reasoning about distances and similarities. The resulting `hybrid' logic allows both precise and approximate representation of and reasoning about concepts. The tableau algorithm combines the existing tableaux for the components and shows that the tableau technique can be fruitfully applied to fusions of logics with nominalsthe case in which no general decidability transfer results for fusions are available.
A logic for metric and topology
 Journal of Symbolic Logic
, 2005
"... Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the inten ..."
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Cited by 13 (11 self)
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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘εdefinitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIMEcompleteness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘wellbehaved ’ common denominator of logical systems constructed in temporal, spatial, and similaritybased quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of realtime systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial
Comparative similarity, tree automata, and Diophantine equations
 In Proceedings of LPAR 2005, volume 3835 of LNAI
, 2005
"... Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequ ..."
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Cited by 12 (9 self)
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Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional ’ logic with the binary operator ‘closer to a set τ1 than to a set τ2 ’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTimecomplete for the classes of all finite symmetric and all finite (possibly nonsymmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer ’ operator has the same expressive power as the standard operator> of conditional logic, these results may have interesting implications for conditional logic as well. 1
A logic for concepts and similarity
"... Categorisation of objects into classes is currently supported by (at least) two ‘orthogonal’ methods. In logicbased approaches, classifications are defined through ontologies or knowledge bases which describe the existing relationships among terms. Description logic (DL) has become one of the most ..."
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Cited by 6 (0 self)
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Categorisation of objects into classes is currently supported by (at least) two ‘orthogonal’ methods. In logicbased approaches, classifications are defined through ontologies or knowledge bases which describe the existing relationships among terms. Description logic (DL) has become one of the most successful formalisms for representing such knowledge bases, in particular because theoretically wellfounded and efficient reasoning tools have been readily available. In numerical approaches, classifications are obtained by first computing similarity (or proximity) measures between objects and then categorising them into classes by means of Voronoi tessellations, clustering algorithms, nearest neighbour computations, etc. In many areas such as bioinformatics, computational linguistics or medical informatics, these two methods have been used independently of each other: although both of them are often applied to the same domain (and even by the same researcher), up to now no formal interaction mechanism has been developed. In this paper, we propose a DLbased integration of the two classification methods. Our formalism, called SL + ALCQIO, extends the expressive DL ALCQIO by means of the constructors of the similarity logic SL which allow definitions of concepts in terms of both comparative and absolute similarity. In the combined knowledge base the user should declare the similarity spaces where the new operators are interpreted. Of course, SL + ALCQIO can only be useful if classifications with this logic are supported by automated reasoning tools. We lay theoretical foundations for the development of such tools by showing that reasoning problems for SL + ALCQIO can be decomposed into the corresponding problems for its DLpart ALCQIO and similarity part SL. Then we investigate reasoning in SL and prove that consistency and many other reasoning problems are ExpTimecomplete for this logic. Using this result and a recent complexity result of PrattHartmann for ALCQIO, we prove that reasoning in SL + ALCQIO is
Modal logics for metric spaces: Open problems
 We Will Show Them! Essays in Honour of Dov Gabbay, Volume Two
, 2005
"... The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \ ..."
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Cited by 5 (1 self)
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The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \Delta into the set R *0 (of nonnegative real numbers) satisfying the following
Topology, connectedness, and modal logic
 ADVANCES IN MODAL LOGIC
, 2008
"... This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of ..."
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Cited by 5 (3 self)
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This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (lowdimensional) Euclidean spaces.
From topology to metric: modal logic and quantification in metric spaces
 Proceedings of AiML– 2006
, 2006
"... abstract. We propose a framework for comparing the expressive power and computational behaviour of modal logics designed for reasoning about qualitative aspects of metric spaces. Within this framework we can compare such wellknown logics as S4 (for the topology induced by the metric), wK4 (for the ..."
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Cited by 5 (4 self)
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abstract. We propose a framework for comparing the expressive power and computational behaviour of modal logics designed for reasoning about qualitative aspects of metric spaces. Within this framework we can compare such wellknown logics as S4 (for the topology induced by the metric), wK4 (for the derivation operator of the topology), variants of conditional logic, as well as logics of comparative similarity. One of the main problems for the new family of logics is to delimit the borders between ‘decidable ’ and ‘undecidable. ’ As a first step in this direction, we consider the modal logic with the operator ‘closer to a set τ0 than to a set τ1 ’ interpreted in metric spaces. This logic contains S4 with the universal modality and corresponds to a very natural language within our framework. We prove that over arbitrary metric spaces this logic is ExpTimecomplete. Recall that over R, Q, and Z, as well as their finite subspaces, this logic is undecidable.