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Qualitative SpatioTemporal Representation and Reasoning: A Computational Perspective
 Exploring Artifitial Intelligence in the New Millenium
, 2001
"... this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science ..."
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Cited by 30 (11 self)
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this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fiction writers, and were of vital concern to our everyday life and commonsense reasoning. So whatever approach to AI one takes [ Russell and Norvig, 1995 ] , temporal and spatial representation and reasoning will always be among its most important ingredients (cf. [ Hayes, 1985 ] ). Knowledge representation (KR) has been quite successful in dealing separately with both time and space. The spectrum of formalisms in use ranges from relatively simple temporal and spatial databases, in which data are indexed by temporal and/or spatial parameters (see e.g. [ Srefik, 1995; Worboys, 1995 ] ), to much more sophisticated numerical methods developed in computational geom
Constructive and algebraic methods of the theory of rough sets
 Information Sciences
, 1998
"... This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtai ..."
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Cited by 21 (4 self)
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This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtained from different types of binary relations. In the algebraic approach, one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators. Various classes of rough set algebras are characterized by different sets of axioms. Axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators. 1
Dynamic topological logic
 ANNALS OF PURE AND APPLIED LOGIC
, 2005
"... Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological inter ..."
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Cited by 20 (3 self)
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Dynamic topological logic provides a context for studying the confluence of the topological semantics for S4, topological dynamics, and temporal logic. The topological semantics for S4 is based on topological spaces rather than Kripke frames. In this semantics, □ is interpreted as topological interior. Thus S4 can be understood as the logic of topological spaces, and □ can be understood as a topological modality. Topological dynamics studies the asymptotic properties of continuous maps on topological spaces. Let a dynamic topological system be a topological space X together with a continuous function f. f can be thought of in temporal terms, moving the points of the topological space from one moment to the next. Dynamic topological logics are the logics of dynamic topological systems, just as S4 is the logic of topological spaces. Dynamic topological logics are defined for a trimodal language with an S4ish topological modality □ (interior), and two temporal modalities, ○ (next) and ∗ (henceforth), both interpreted using the continuous function f. In particular, ○ expresses f ’s action on X from one moment to the next, and ∗ expresses the asymptotic behaviour of f.
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.
Knowledge in Multiagent Systems: Initial Configurations and Broadcast
 ACM TRANSACTIONS OF COMPUTATIONAL LOGIC
, 2000
"... ... this paper we study two special cases of this framework: full systems and hypercubes. Both model static situations in which no agent has any information about another agent's state. Full systems and hypercubes are an appropriate model for the initial congurations of many systems of interest ..."
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Cited by 12 (8 self)
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... this paper we study two special cases of this framework: full systems and hypercubes. Both model static situations in which no agent has any information about another agent's state. Full systems and hypercubes are an appropriate model for the initial congurations of many systems of interest. We establish a correspondence between full systems and hypercube systems and certain classes of Kripke frames. We show that these classes of systems correspond to the same logic. Moreover, this logic is also the same as that generated by the larger class of weakly directed frames. We provide a sound and complete axiomatization, S5WDn , of this logic, and study its computational complexity. Finally, we show that under certain natural assumptions, in a model where knowledge evolves over time, S5WDn characterises the properties of knowledge not just at the initial conguration, but also at all later congurations. In particular, this holds for homogeneous broadcast systems, which capture settings in which agents are initially ignorant of each others local states, operate synchronously, have perfect recall, and can communicate only by broadcasting.
Bimodal Logics for Reasoning About Continuous Dynamics
 Advances in Modal Logic
, 2000
"... We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme #a## # #a##. In the intended semantics, the plain is given the McKinseyTarski interpretation as the interior operator of a topology, while the labelled [a] is given the ..."
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Cited by 6 (1 self)
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We study a propositional bimodal logic consisting of two S4 modalities and [a], together with the interaction axiom scheme #a## # #a##. In the intended semantics, the plain is given the McKinseyTarski interpretation as the interior operator of a topology, while the labelled [a] is given the standard Kripke semantics using a reflexive and transitive binary relation Ra . The interaction axiom expresses the property that the Ra relation is lower semicontinuous with respect to the topology. The class of topological Kripke frames axiomatised by the logic includes all frames over Euclidean space where Ra is the positive flow relation of a di#erential equation. We establish the completeness of the axiomatisation with respect to the intended class of topological Kripke frames, and investigate tableau calculi for the logic, although decidability is still an open question. 1 Introduction We study a propositional bimodal logic consisting of two S4 modalities # and [a], to...
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
Propositional quantification in the topological semantics for S4
 Notre Dame Journal of Formal Logic
, 1997
"... quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S ..."
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Cited by 3 (1 self)
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quantifiers range over all the sets of possible worlds. S5π+ is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to secondorder logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub S4πt, isstrictly weaker than its Kripkean counterpart. I prove here that secondorder arithmetic can be recursively embedded in S4πt. Inthe course of the investigation, I also sketch a proof of Fine’s and Kripke’s results that the Kripkean system S4π+ is recursively isomorphic to secondorder logic. 1Introduction One way to extend a propositional logic to a language with propositional quantifiers is to begin with a semantics for the logic; extract from the semantics a notion of a proposition; and interpret the quantifiers as ranging over the propositions. Thus, Fine [4] extends the Kripke semantics for modal logics to propositionally quantified systems S5π+, S4π+, S4.2π+, and such: given a Kripke frame, the quantifiers range over all sets of possible worlds. S5π+ is decidable ([4] and Kaplan [14]). In later unpublished work, Fine and Kripke independently showed that S4π+, S4.2π+, K4π+, Tπ+, Kπ+, and Bπ+ and others are recursively isomorphic to full secondorder classical logic. (Fine informs me that he later proved this stronger result. Kripke informs me that he too proved this stronger result in the early 1970s. A proof of this result occurs in Kaminski and Tiomkin [13], who use techniques similar to those used in Kremer [16] and to those used below. These techniques do not apply to S4.3π+. But according to Kaminski and Tiomkin, work of Gurevich and Shelah ([9], [10], and [39]) implies that secondorder arithmetic is interpretable in S4.3π+ and furthermore that, under