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13
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 restrict ..."
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Cited by 16 (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 15 (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 7 (4 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 3 (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.
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.
Intuitionistic Letcc via Labelled Deduction
"... Intuitionistic logic can be presented as a calculus of labelled deduction on multipleconclusion sequents. The corresponding natural deduction system constitutes a type system for programs using letcc, which capture the current program continuation, restricted to enforce constructive validity. This ..."
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Intuitionistic logic can be presented as a calculus of labelled deduction on multipleconclusion sequents. The corresponding natural deduction system constitutes a type system for programs using letcc, which capture the current program continuation, restricted to enforce constructive validity. This allows us to develop a rich dependent type theory incorporating letcc, which is known to be highly problematic for computational interpretations of classical logic. Moreover, we give a novel constructive proof for the soundness of labelled deduction, whose algorithmic content is a nondeterministic translation of programs that eliminates the restricted uses of letcc and is fully compatible with dependent types and therefore with program verification. This proof has been formally verified on the propositional fragment in the Twelf metalogical framework. 1.
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"... We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial ..."
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We give a categorial characterization of how labelled deduction systems for logics with a propositional basis behave under unconstrained fibring and under fibring that is constrained by symbol sharing. At the semantic level, we introduce a general semantics for our systems and then give a categorial characterization of fibring of models. Based on this, we establish the conditions under which our systems are sound and complete with respect to the general semantics for the corresponding logics, and establish requirements on logics and systems so that completeness is preserved by both forms of fibring.