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43
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 82 (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...
Operational Modal Logic
, 1995
"... Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact in ..."
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Cited by 65 (25 self)
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Answers to two old questions are given in this paper. 1. Modal logic S4, which was informally specified by Gödel in 1933 as a logic for provability, meets its exact provability interpretation. 2. BrouwerHeytingKolmogorov realizing operations (193132) for intuitionistic logic Int also get exact interpretation as corresponding propositional operations on proofs; both S4 and Int turn out to be complete with respect to this proof realization. These results are based on operational reading of S4, where a modality is split into three operations. The logic of proofs with these operations is shown to be arithmetically complete with respect to the intended provability semantics and sufficient to realize every operation on proofs admitting propositional specification in arithmetic.
Manyvalued logic
 Handbook of Philosophical Logic
, 1986
"... ABSTRACT. This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are “synonymous ” or “translationally equivalent. ” We discuss a proposed formal definition of translational equival ..."
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Cited by 54 (1 self)
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ABSTRACT. This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are “synonymous ” or “translationally equivalent. ” We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to wellknown examples. Some philosophical morals are drawn concerning the possibility of having two logical systems that are “empirically distinct ” but are both translationally equivalent to a common logic. KEY WORDS: modal logic, synonymy, translation 1.
MultiDimensional Modal Logic as a Framework for SpatioTemporal Reasoning
 APPLIED INTELLIGENCE
, 2000
"... In this paper we advocate the use of multidimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a twodimensional logic capable of describing topological relationships that change over time. This logic, ca ..."
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Cited by 35 (6 self)
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In this paper we advocate the use of multidimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a twodimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional SpatioTemporal Logic) is the Cartesian product of the wellknown temporal logic PTL and the modal logic S4u , which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both pointbased and interval based) of the spatial logic RCC8 can be embedded. We consider known decidability and complexity results that are relevant to computation with mulidimensional formalisms and discuss possible directions for further research.
On the Translation of Qualitative Spatial Reasoning Problems into Modal Logics
 In Proceedings of KI99
, 1999
"... . We introduce topological set constraints that express qualitative spatial relations between regions. The constraints are interpreted over topological spaces. We show how to translate our constraints into formulas of a multimodal propositional logic and give a rigorous proof that this translation p ..."
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Cited by 20 (0 self)
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. We introduce topological set constraints that express qualitative spatial relations between regions. The constraints are interpreted over topological spaces. We show how to translate our constraints into formulas of a multimodal propositional logic and give a rigorous proof that this translation preserves satisfiability. As a consequence, the known algorithms for reasoning in modal logics can be applied to qualitative spatial reasoning. Our results lay a formal foundation to previous work by Bennett, Nebel, Renz, and others on spatial reasoning in the RCC8 formalism. 1 Introduction An approach to qualitative spatial reasoning that has received considerable attention is the socalled Region Connection Calculus (RCC), which has been introduced by Randell, Cui, and Cohn [9]. A specialization of RCC is the calculus RCC8. Similar to Allen's calculus for temporal reasoning [1], which is based on 13 elementary relations that can hold between time intervals, in RCC8, there are eight element...
Intuitionistic Logic Redisplayed
, 1995
"... We continue the study of Belnap's Display Logic. Specifically, we show that the booleantensemodal setting of Wansing and Kracht not only allows us to "redisplay" intuitionistic logic but also allows us to display superintuitionistic (intermediate) logics by using the underlying Kripke semantics fo ..."
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Cited by 11 (9 self)
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We continue the study of Belnap's Display Logic. Specifically, we show that the booleantensemodal setting of Wansing and Kracht not only allows us to "redisplay" intuitionistic logic but also allows us to display superintuitionistic (intermediate) logics by using the underlying Kripke semantics for theselogics. The resulting Gentzen systems inherit Belnap's cutelimination and subformula properties, but also inherit the caveat that these two properties no longer immediately imply decidability. That is, although cutelimination now comes for free, we now have to work a lot harder to obtain decidability. The system pave one way for a prooftheoretical study of subintuitionistic logics, superintuitionistic logics and even hybrids like intuitionistic modal logics in one Display Logic framework. Contents 1 Introduction 2 2 Classical Modal Display Logic 2 2.1 Syntax: Formulae, Structures and Sequents : : : : : : : : : : : : 3 2.2 Structural Rules For Classical Modal Logic : : : : : : :...
Erdös Graphs Resolve Fine's Canonicity Problem
 The Bulletin of Symbolic Logic
, 2003
"... We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a b ..."
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Cited by 11 (8 self)
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We show that there exist 2^ℵ0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any firstorder definable class of relational structures. Using a variant of this construction, we resolve a longstanding question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any firstorder definable class of Kripke frames. The constructions use the result of Erd os that there are finite graphs with arbitrarily large chromatic number and girth.
A Pragmatic Interpretation Of Substructural Logics
"... Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from ..."
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Cited by 11 (8 self)
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Following work by Dalla Pozza and Garola [2, 3] on a pragmatic interpretation of intuitionistic and deontic logics, which has given evidence of their compatibility with classical semantics, we present sequent calculus system ILP formalizing the derivation of assertive judgements and obligations from mixed contexts of assertions and obligations and we prove the cutelimination theorem for it. For the formalization of reallife normative systems it is essential to consider inferences from mixed contexts of assertions and obligations, and also of assertions justi able relatively to a given state of information and obligations valid in a given normative system. In order to provide a formalization of the notion of causal implication and its interaction with obligations, the sequents of ILP have two areas in the antecedent, expressing the relevant and the ordinary intuitionistic consequence relation, respectively. To provide a pragmatic interpretation of reasoning with the lin...