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18
The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Cited by 102 (0 self)
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
Computational types from a logical perspective
 Journal of Functional Programming
, 1998
"... Moggi’s computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus ..."
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Cited by 54 (6 self)
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Moggi’s computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the CurryHoward correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbertstyle presentations of this logic and prove strong normalisation and confluence results. 1
Strongly Analytic Tableaux for Normal Modal Logics
, 1994
"... A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequentlike tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cu ..."
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Cited by 48 (13 self)
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A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequentlike tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequentlike tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbertstyle formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM + , OB + , OK4 + ,...
Formalizing action and change in modal logic I: the frame problem
, 1999
"... We present the basic framework of a logic of actions and plans defined in terms of modal logic combined with a notion of dependence. The latter is used as a weak causal connection between actions and literals. In this paper we focus on the frame problem and demonstrate how it can be solved in our fr ..."
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Cited by 47 (15 self)
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We present the basic framework of a logic of actions and plans defined in terms of modal logic combined with a notion of dependence. The latter is used as a weak causal connection between actions and literals. In this paper we focus on the frame problem and demonstrate how it can be solved in our framework in a simple and monotonic way. We give the semantics, and associate an axiomatics and a decision procedure to it. The decision procedure is based on a sound and complete tableau method with single step rules to treat dependence. We show how it can be used to generate plans. Our solution is formally assessed by a translation of Gelfond and Lifschitz' logic A. We briefly sketch the second part of the paper, showing how we can go beyond A by some examples involving nondeterminism and ramifications.
On an Intuitionistic Modal Logic
 Studia Logica
, 2001
"... . In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models ..."
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Cited by 19 (4 self)
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. In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4our formulation has several important metatheoretic properties. In addition, we study models of IS4, not in the framework of Kripke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also a means of modelling proofs as well as provability. 1. Introduction Modal logics are traditionally extensions of classical logic with new operators, or modalities, whose operation is intensional. Modal logics are most commonly justified by the provision of an intuitive semantics based upon `possible worlds', an idea originally due to Kripke. Kripke also provided a possible worlds semantics for intuitionistic logic, and so it is natural to consider intuitionistic logic extended with intensional modalities...
Modal Tableaux for Reasoning About Actions and Plans
, 1997
"... In this paper we investigate tableau proof procedures for reasoning about actions and plans. Our framework is a multimodal language close to that of propositional dynamic logic, wherein we solve the frame problem by introducing the notion of dependence as a weak causal connection between actions and ..."
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Cited by 15 (5 self)
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In this paper we investigate tableau proof procedures for reasoning about actions and plans. Our framework is a multimodal language close to that of propositional dynamic logic, wherein we solve the frame problem by introducing the notion of dependence as a weak causal connection between actions and atoms. The tableau procedure is sound and complete for an important fragment of our language, within which all standard problems of reasoning about actions can be expressed, in particular planning tasks. Moreover, our tableaux are analytic and provide thus a decision procedure.
Admissibility of Cut in Coalgebraic Logics
 CMCS
, 2008
"... We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional ..."
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Cited by 8 (7 self)
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We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional logics CK and CK Id.
A simple tableau system for the logic of elsewhere
 177– 192. LNAI 1071
, 1996
"... Abstract. We analyze different features related to the mechanization of von Wright’s logic of elsewhere E. First, we give a new proof of the NPcompleteness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes lin ..."
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Cited by 7 (3 self)
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Abstract. We analyze different features related to the mechanization of von Wright’s logic of elsewhere E. First, we give a new proof of the NPcompleteness of the satisfiability problem (giving a new bound for the size of models of the satisfiable formulae) and we show that this problem becomes lineartime when the number of propositional variables is bounded. Although E and the wellknown propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense to be specified). Second, we present a prefixed tableau system for E and we prove both its soundness and completeness. Two extensions of this system are also defined, one related to the logical consequence relation and the other related to the addition of modal operators (without increasing the expressive power). An example of tableau proof is also presented. Different continuations of this work are proposed, one of them being to implement the defined tableau system, another one being to extend this system to richer logics that can be found in the literature. 1
Semianalytic Tableaux For Propositional Normal Modal Logics with Applications to Nonmonotonicity
, 1991
"... The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in ..."
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Cited by 5 (4 self)
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The propositional monotonic modal logics K45, K45D, S4:2, S4R and S4F elegantly capture the semantics of many current nonmonotonic formalisms as long as (strong) deducibility of A from a theory \Gamma; \Gamma ` A; allows the use of necessitation on the members of \Gamma: This is usually forbidden in modal logic where \Gamma is required to be empty, resulting in a weaker notion of deducibility. Recently, Marek, Schwarz and Truszczi'nski have given algorithms to compute the stable expansions of a finite theory \Gamma in various such nonmonotonic formalisms. Their algorithms assume the existence of procedures for deciding (strong) deducibility in these monotonic modal logics and consequently such decision procedures are important for automating nonmonotonic deduction. We first give a sound, (weakly) complete and cutfree, semianalytic tableau calculus for monotonic S4R, thus extending the cut elimination results of Schwarz for monotonic K45 and K45D. We then give sound and complete semi...