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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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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
A Computational Interpretation of Modal Proofs
 Proof Theory of Modal Logics
, 1994
"... The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, ..."
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The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a twodimensional generalization of Gentzen's sequents (2sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...
Logical Modalities and MultiStage Programming
, 1999
"... . Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two log ..."
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Cited by 24 (13 self)
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. Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two logical modalities is well suited for multistage programming, and report the results from our study of categorical models for multistage languages. Keywords: Multistage programming, categorical models, semantics, type systems (multilevel typed calculi) , combination of logics (modal and temporal). 1 Introduction Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization [15, 13]. MetaML was the first language designed specifically to support this method. It provides a type constructor for "code" and staging annotations for building, combining, and executing code, thus allowing the programmer to have finer cont...
On the Fine Structure of the Exponential Rule
 Advances in Linear Logic
, 1993
"... We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on !prefixed formulas. The systems are based on a twodimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction ..."
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Cited by 14 (4 self)
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We present natural deduction systems for fragments of intuitionistic linear logic obtained by dropping weakening and contractions also on !prefixed formulas. The systems are based on a twodimensional generalization of the notion of sequent, which accounts for a clean formulation of the introduction/elimination rules of the modality. Moreover, the different subsystems are obtained in a modular way, by simple conditions on the elimination rule for !. For the proposed systems we introduce a notion of reduction and we prove a normalization theorem. 1. Introduction Proof theory of modalities is a delicate subject. The shape of the rules governing the different modalities in the overpopulated world of modal logics is often an example of what a good rule should not be. In the context of sequent calculus, if we want cut elimination, we are often forced to accept rules which are neither left nor right rules, and which completely destroy the deep symmetries the calculus is based upon. In the c...
A Categorical Analysis of MultiLevel Languages (Extended Abstract)
, 1999
"... We propose categorical models for fl, 2 , MetaML, and AIM. First, we focus on the underlying logical modalities and the interactions between them, then we investigate the interactions between logical modalities and computational monads. We give two examples of categorical model: one simpler but with ..."
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We propose categorical models for fl, 2 , MetaML, and AIM. First, we focus on the underlying logical modalities and the interactions between them, then we investigate the interactions between logical modalities and computational monads. We give two examples of categorical model: one simpler but with some limitations, the other more complex but able to model all features of AIM.
On a Modal \lambdaCalculus for S4*
 Proceedings of the Eleventh Conference on Mathematical Foundations of Programming Sematics
, 1995
"... We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and pr ..."
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Cited by 7 (0 self)
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We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for welltyped terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and bindingtime analysis for functional languages [6]. 1 Introduction Modal operators familiar from traditional logic have received renewed attention in computer science through their importance in linear logic. Typically, they are described axiomatically in the style of Hilbert or via sequent calculi. However, the CurryHoward isomorphism between proofs and terms is most poignant for natural deduction, so natural deduction formulations of modal and...
Causality, Modality, and Explanation
, 801
"... Abstract We start with Fodor’s critique of cognitive science in [8]: he argues that much mental activity cannot be handled by the current methods of cognitive science because it is nonmonotonic and, therefore, is global in nature, is not contextfree, and is thus not capable of being formalised by a ..."
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Abstract We start with Fodor’s critique of cognitive science in [8]: he argues that much mental activity cannot be handled by the current methods of cognitive science because it is nonmonotonic and, therefore, is global in nature, is not contextfree, and is thus not capable of being formalised by a Turinglike mental architecture. We look at the use of nonmonotonic logic in the Artificial Intelligence community, particularly with the discussion of the socalled “frame problem”. The mainstream approach to the frame problem is, we argue, probably susceptible to Fodor’s critique: however, there is an alternative approach, due to McCain and Turner, which is, when suitably reformulated, not susceptible. In the course of our argument, we give a proof theory for the McCainTurner system, and show that it satisfies cut elimination. We have two substantive conclusions: firstly, that Fodor’s argument depends on assumptions about logical form which not all nonmonotonic theories satisfy; and, secondly, that metatheory plays an important role
Abstract On a Modal λCalculus for S4 ⋆
"... We present λ →2, a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prov ..."
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We present λ →2, a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for welltyped terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and bindingtime analysis for functional languages [6]. 1
Logical Modalities and MultiStage Programming
"... Abstract. Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between ..."
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Abstract. Multistage programming is a method for improving the performance of programs through the introduction of controlled program specialization. This paper makes a case for multistage programming with open code and closed values. We argue that a simple language exploiting interactions between two logical modalities is well suited for multistage programming, and report the results from our study of categorical models for multistage languages. Keywords: Multistage programming, categorical models, semantics, type systems (multilevel typed calculi), combination of logics (modal and temporal). 1