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Normal Multimodal Logics With Interaction Axioms: A Tableau Calculus and Some (Un)Decidability Results
, 2000
"... In this paper we present a prefixed analytic tableau calculus for a wide class of normal multimodal logics; the calculus can deal in a uniform way with any logic in this class. To achieve this goal, we use a prefixed tableau calculus a la Fitting, where we explicitly represent accessibility relation ..."
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Cited by 18 (8 self)
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In this paper we present a prefixed analytic tableau calculus for a wide class of normal multimodal logics; the calculus can deal in a uniform way with any logic in this class. To achieve this goal, we use a prefixed tableau calculus a la Fitting, where we explicitly represent accessibility relations between worlds by means of a graph and we use the characterizing axioms as rewriting rules. Such rules create new paths among worlds in the countermodel construction. The prefixed tableau method is, then, used to prove (un)decidability results about certain classes of multimodal logics.
Sequent Calculi for Nominal Tense Logics: A Step Towards Mechanization?
, 1999
"... . We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule ..."
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Cited by 15 (4 self)
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. We define sequentstyle calculi for nominal tense logics characterized by classes of modal frames that are firstorder definable by certain \Pi 0 1 formulae and \Pi 0 2 formulae. The calculi are based on d'Agostino and Mondadori's calculus KE and therefore they admit a restricted cutrule that is not eliminable. A nice computational property of the restriction is, for instance, that at any stage of the proof, only a finite number of potential cutformulae needs to be taken under consideration. Although restrictions on the proof search (preserving completeness) are given in the paper and most of them are theoretically appealing, the use of those calculi for mechanization is however doubtful. Indeed, we present sequent calculi for fragments of classical logic that are syntactic variants of the sequent calculi for the nominal tense logics. 1 Introduction Background. The nominal tense logics are extensions of Prior tense logics (see e.g. [Pri57, RU71]) by adding nomina...
Bringing them all Together
, 2001
"... this paper, Jerry Seligman takes us on an interesting journey. The satisfaction denition of most modal operators is specied in terms of rstorder conditions. Hence we can always obtain a complete calculus for the basic logic characterizing any collection of such operators by appealing to a calculus ..."
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Cited by 14 (0 self)
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this paper, Jerry Seligman takes us on an interesting journey. The satisfaction denition of most modal operators is specied in terms of rstorder conditions. Hence we can always obtain a complete calculus for the basic logic characterizing any collection of such operators by appealing to a calculus which is complete for the full rstorder language. Seligman shows here that by making use of the expressiveness provided by the hybrid apparatus, we can, step by step, transform a rstorder sequent calculus into an internalized sequent calculus specically tailored for a particular hybrid fragment
Extended CurryHoward Correspondence for a Basic Constructive Modal Logic
 In Proceedings of Methods for Modalities
, 2001
"... this paper. This calculus satises cutelimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary f ..."
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Cited by 10 (2 self)
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this paper. This calculus satises cutelimination, as for instance shown (in a more complicated form) in [Wij90]. This calculus is dierent from what is usually taken as the basic constructive system K, as we do not assume the distribution of possibility (3) over disjunctions neither in its binary form 3(A _ B) ! (3A _ 3B) nor in its nullary form 3? ! ? The sequent calculus above corresponds to an axiomatic formulation given by axioms for intuitionistic logic, plus axioms: 2(A ! B) ! (2A ! 2B) 2(A ! B) ! (3A ! 3B) 2A3B ! 3(A B) together with rules for Modus Ponens and Necessitation: ` A ! B ` A ` B MP ` A ` 2A Nec Wijesekera proved a Craig interpolation theorem, one of the usual consequences of syntactic cutelimination and produced Kripke, algebraic and topological semantics for a calculus very similar to the one above. The only dierence is that he does assume 3? ! ?. From our \wish list" for logical systems only a natural deduction formulation and a categorical semantics are missing. These we proceed to discuss
Automated Natural Deduction for Propositional Lineartime Temporal Logic ∗
"... We present a proof searching technique for the natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. This opens the prospect to apply our technique as an automated reasoning tool in a number of emerging computer science applications and in a deliberat ..."
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Cited by 4 (1 self)
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We present a proof searching technique for the natural deduction calculus for the propositional lineartime temporal logic and prove its correctness. This opens the prospect to apply our technique as an automated reasoning tool in a number of emerging computer science applications and in a deliberative decision making framework across various AI applications. 1
Implementing Modal and Relevance Logics in a Logical Framework
, 1996
"... We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, ..."
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Cited by 2 (2 self)
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We present a framework for machine implementation of both partial and complete fragments of large families of nonclassical logics such as modal, relevance, and intuitionistic logics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework. 1 INTRODUCTION The origins of natural deduction (ND) are both philosophical and practical. In philosophy, it arises from an analysis of deductive inference in an attempt to provide a theory of meaning for the logical connectives [24, 33]. Practically, it provides a language for building proofs, which can be seen as providing the deduction theorem directly, rather than as a derived result. Our interest is on this practical side, and a development of our work on ap...
TruthValues as Labels: A General Recipe for Labelled Deduction
"... We introduce a general recipe for presenting nonclassical logics in a modular and uniform way as labelled natural deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truthvalues. ..."
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Cited by 1 (1 self)
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We introduce a general recipe for presenting nonclassical logics in a modular and uniform way as labelled natural deduction systems. Our recipe is based on a labelling mechanism where labels are general entities that are present, in one way or another, in all logics, namely truthvalues.
Deductive Verification of Component Model: Combining Computation Tree Logic and Deontic Logic in Natural Deduction Style Calculus.
"... systems, reconfiguration. Abstract. In this paper we present a natural deduction calculus for branchingtime deontic logic, a combination of the branchingtime logic CTL and deontic modalities. We show how this new proof technique can be applied as a reasoning tool in modeling dynamic normative sys ..."
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systems, reconfiguration. Abstract. In this paper we present a natural deduction calculus for branchingtime deontic logic, a combination of the branchingtime logic CTL and deontic modalities. We show how this new proof technique can be applied as a reasoning tool in modeling dynamic normative systems, namely, Grid Component System. We formally define behaviour of stateful components and a concept of a reconfiguration showing how these can be specified in our language and then apply the natural deduction verification technique defined for this new formalism. We believe that, due to the specifics of the natural deduction proof technique, this new framework not only gives us more transparent proofs but that it also leads us to a lower complexity of the overall verification scenario. 1 Introduction. In this paper we continue our systematic investigation of natural deduction construction for various logics, this time considering a combination of temporal and