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Modal Logics for Qualitative Spatial Reasoning
, 1996
"... Spatial reasoning is essential for many AI applications. In most existing systems the representation is primarily numerical, so the information that can be handled is limited to precise quantitative data. However, for many purposes the ability to manipulate highlevel qualitative spatial information ..."
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Cited by 93 (12 self)
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Spatial reasoning is essential for many AI applications. In most existing systems the representation is primarily numerical, so the information that can be handled is limited to precise quantitative data. However, for many purposes the ability to manipulate highlevel qualitative spatial information in a flexible way would be extremely useful. Such capabilities can be proveded by logical calculi; and indeed 1storder theories of certain spatial relations have been given [20]. But computing inferences in 1storder logic is generally intractable unless special (domain dependent) methods are known. 0order modal logics provide an alternative representation which is more expressive than classical 0order logic and yet often more amenable to automated deduction than 1storder formalisms. These calculi are usually interpreted as propositional logics: nonlogical constants are taken as denoting propositions. However, they can also be given a nominal interpretation in which the constants stand...
Computational Properties of Qualitative Spatial Reasoning: First Results
 KI95: ADVANCES IN ARTIFICIAL INTELLIGENCE
, 1995
"... While the computational properties of qualitative temporal reasoning have been analyzed quite thoroughly, the computational properties of qualitative spatial reasoning are not very well investigated. In fact, almost no completeness results are known for qualitative spatial calculi and no computati ..."
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Cited by 41 (5 self)
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While the computational properties of qualitative temporal reasoning have been analyzed quite thoroughly, the computational properties of qualitative spatial reasoning are not very well investigated. In fact, almost no completeness results are known for qualitative spatial calculi and no computational complexity analysis has been carried out yet. In this paper, we will focus on the socalled RCC approach and use Bennett's encoding of spatial reasoning in intuitionistic logic in order to show that consistency checking for the topological base relations can be done efficiently. Further, we show that pathconsistency is sufficient for deciding global consistency. As a sideeffect we prove a particular fragment of propositional intuitionistic logic to be tractable.
A modal walk through space
 JOURNAL OF APPLIED NONCLASSICAL LOGICS
, 2002
"... We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and ..."
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Cited by 40 (5 self)
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We investigate the major mathematical theories of space from a modal standpoint: topology, affine geometry, metric geometry, and vector algebra. This allows us to see new finestructure in spatial patterns which suggests analogies across these mathematical theories in terms of modal, temporal, and conditional logics. Throughout the modal walk through space, expressive power is analyzed in terms of language design, bisimulations, and correspondence phenomena. The result is both unification across the areas visited, and the uncovering of interesting new questions.
RegionBased Qualitative Geometry
, 2000
"... We present a highly expressive logical language for describing qualitative configurations of spatial regions. We call the theory Region Based Geometry (RBG). Our axiomatisation is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as pri ..."
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Cited by 34 (14 self)
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We present a highly expressive logical language for describing qualitative configurations of spatial regions. We call the theory Region Based Geometry (RBG). Our axiomatisation is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. We show that our theory is categorical: all models are isomorphic to a classical interpretation in terms of Cartesian spaces over R. We investigate
A Categorical Axiomatisation of RegionBased Geometry
, 2001
"... . Region Based Geometry (RBG) is an axiomatic theory of qualitative congurations of spatial regions. It is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. Whereas in Tarski's theory the combination of mereological and ..."
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Cited by 27 (8 self)
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. Region Based Geometry (RBG) is an axiomatic theory of qualitative congurations of spatial regions. It is based on Tarski's Geometry of Solids, in which the parthood relation and the concept of sphere are taken as primitive. Whereas in Tarski's theory the combination of mereological and geometrical axioms involves set theory, in RBG the interface is achieved by purely 1storder axioms. This means that the elementary sublanguage of RBG is extremely expressive, supporting inferences involving both mereological and geometrical concepts. Categoricity of the RBG axioms is proved: all models are isomorphic to a standard interpretation in terms of Cartesian spaces over R. 1. Introduction Many researchers in the eld of Qualitative Spatial Reasoning (QSR) have argued that it is useful to have representations in which spatial regions are the basic entities [10, 8]. This ontology contrasts with the approach of classical geometry, where lines, surfaces and regions are typically thought of as ...
Logical axiomatizations of spacetime. Samples from the literature
 In: NonEuclidean Geometries (J'anos Bolyai Memorial Volume
, 2005
"... Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and t ..."
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Cited by 25 (14 self)
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Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and then we experiment with generalizations in the direction of general relativity. The present paper gives samples from an ongoing broader research project which in turn is part of a research direction going back to Reichenbach and others in the 1920’s. We also try to give some perspective on the literature related in a broader sense. In the perspective of the present work, axiomatization is not a final goal. Axiomatization is only a first step, a tool. The goal is something like a conceptual analysis of relativity in the framework of logic. In section 1 we recall a complete FOLaxiomatization Specrel of special relativity from [5],[31]. In section 2 we answer questions from papers by Ax and Mundy concerning the logical status of faster than light motion (FTL) in relativity. We claim that already very small/weak fragments of Specrel prove “No FTL”. In section 3 we give a sketchy outlook for the possibility of generalizing Specrel to theories permitting accelerated observers (gravity). In section 4 we continue generalizing Specrel in the direction of general relativity by localizing it, i.e. by replacing it with a version still in firstorder logic but now local (in the sense of general relativity theory). In section 5 we give samples from the broader literature.
Grounding geographic categories in the meaningful environment
 CONFERENCE ON SPATIAL INFORMATION THEORY (COSIT 2009), VOLUME 5756 OF LNCS
, 2009
"... Ontologies are a common approach to improve semantic interoperability by explicitly specifying the vocabulary used by a particular information community. Complex expressions are defined in terms of primitive ones. This shifts the problem of semantic interoperability to the problem of how to ground p ..."
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Cited by 24 (15 self)
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Ontologies are a common approach to improve semantic interoperability by explicitly specifying the vocabulary used by a particular information community. Complex expressions are defined in terms of primitive ones. This shifts the problem of semantic interoperability to the problem of how to ground primitive symbols. One approach are semantic datums, which determine reproducible mappings (measurement scales) from observable structures to symbols. Measurement theory offers a formal basis for such mappings. From an ontological point of view, this leaves two important questions unanswered. Which qualities provide semantic datums? How are these qualities related to the primitive entities in our ontology? Based on a scenario from hydrology, we first argue that human or technical sensors implement semantic datums, and secondly that primitive symbols are definable from the meaningful environment, a formalized quality space established through such sensors.
Logical Patterns in Space
 University of Amsterdam
, 1999
"... In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra ..."
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Cited by 18 (5 self)
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In this paper, we revive the topological interpretation of modal logic, turning it into a general language of patterns in space. In particular, we define a notion of bisimulation for topological models that compares different visual scenes. We refine the comparison by introducing EhrenfeuchtFra iss'e style games between patterns in space. Finally, we consider spatial languages of increased logical power in the direction of geometry. Also, Intelligent Sensory Information Systems, University of Amsterdam 1 Contents 1 Reasoning about Space 3 2 Topological Structure: a Modal Approach 4 2.1 The topological view of space . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Special properties of topological spaces . . . . . . . . . . . 6 2.1.3 Structure preserving mappings . . . . . . . . . . . . . . . 7 3 Basic Modal Logic of Space 8 3.1 Topological language and semantics . . . . . . . . . . . . . . . . 8 3.2 Topologi...
A logic road from special relativity to general relativity. submitted
, 2010
"... Abstract. We present a streamlined axiom system of special relativity in firstorder logic. From this axiom system we “derive ” an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we h ..."
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Cited by 16 (8 self)
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Abstract. We present a streamlined axiom system of special relativity in firstorder logic. From this axiom system we “derive ” an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the nonspecialist.