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39
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 26 (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.
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.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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Cited by 23 (12 self)
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
On the ternary spatial relation “between
 IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics
, 2006
"... Abstract—The spatial relation “between ” is a notion which is intrinsically both fuzzy and contextual, and depends, in particular, on the shape of the objects. The literature is quite poor on this and the few existing definitions do not take into account these aspects. In particular, an object that ..."
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Cited by 15 (4 self)
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Abstract—The spatial relation “between ” is a notion which is intrinsically both fuzzy and contextual, and depends, in particular, on the shape of the objects. The literature is quite poor on this and the few existing definitions do not take into account these aspects. In particular, an object that is in a concavity of an object 1 not visible from an object 2 is considered between 1 and 2 for most definitions, which is counter intuitive. Also, none of the definitions deal with cases where one object is much more elongated than the other. Here, we propose definitions which are based on convexity, morphological operators, and separation tools, and a fuzzy notion of visibility. They correspond to the main intuitive exceptions of the relation. We distinguish between cases where objects have similar spatial extensions and cases where one object is much more extended than the other. Extensions to cases where objects, themselves, are fuzzy and to threedimensional space are proposed as well. The original work proposed in this paper covers the main classes of situations and overcomes the limits of existing approaches, particularly concerning nonvisible concavities and extended objects. Moreover, the definitions capture the intrinsic imprecision attached to this relation. The main proposed definitions are illustrated on real data from medical images. Index Terms—Convex hull, fuzzy and threedimensional (3D) objects, mathematical morphology, relationship “between”, spatial reasoning, structural pattern recognition, visibility. I.
Spatial Logics with Connectedness Predicates
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2010
"... We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of thes ..."
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Cited by 11 (3 self)
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We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of these logics and show that the connectedness constraints can increase complexity from NP to PSpace, ExpTime and, if component counting is allowed, to NExpTime.
Modal Logics For Products Of Topologies
 STUDIA LOGICA
, 2004
"... We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 S4. We axiomatize t ..."
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Cited by 7 (1 self)
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We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 S4. We axiomatize the modal logic of products of topological spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers Q Q with the appropriate topologies.
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 6 (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
A Calculus for Shapes in Time and Space
 ICTAC 2004, VOLUME 3407 OF LNCS
, 2005
"... We present a spatial and temporal logic based on Duration Calculus for the specification and verification of mobile realtime systems. We demonstrate ..."
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Cited by 5 (1 self)
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We present a spatial and temporal logic based on Duration Calculus for the specification and verification of mobile realtime systems. We demonstrate
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.