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A Judgmental Reconstruction of Modal Logic
 Mathematical Structures in Computer Science
, 1999
"... this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deductio ..."
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Cited by 167 (39 self)
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this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as
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 56 (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
Monads and Effects
 IN INTERNATIONAL SUMMER SCHOOL ON APPLIED SEMANTICS APPSEM’2000
, 2000
"... A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeffects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of structu ..."
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Cited by 54 (6 self)
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A tension in language design has been between simple semantics on the one hand, and rich possibilities for sideeffects, exception handling and so on on the other. The introduction of monads has made a large step towards reconciling these alternatives. First proposed by Moggi as a way of structuring semantic descriptions, they were adopted by Wadler to structure Haskell programs, and now offer a general technique for delimiting the scope of effects, thus reconciling referential transparency and imperative operations within one programming language. Monads have been used to solve longstanding problems such as adding pointers and assignment, interlanguage working, and exception handling to Haskell, without compromising its purely functional semantics. The course will introduce monads, effects and related notions, and exemplify their applications in programming (Haskell) and in compilation (MLj). The course will present typed metalanguages for monads and related categorica...
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 23 (5 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...
Firstorder Lax Logic as a Framework for Constraint Logic Programming
, 1997
"... In this report we introduce a new prooftheoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic firstorder modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual exte ..."
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Cited by 12 (4 self)
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In this report we introduce a new prooftheoretic approach to the semantics of Constraint Logic Programming, based on an intuitionistic firstorder modal logic, called QLL. The distinguishing feature of this new approach is that the logic calculus of QLL is used not only to capture the usual extensional aspects of Logic Programming, i.e. "which queries are successful, " but also some of the intensional aspects, i.e. "what is the answer constraint and how is it constructed." It provides for a direct link between the modeltheoretic and the operational semantics following a formulasasprograms and proofsasconstraints principle. This approach makes use of logic in a different way than other approaches based on logic calculi. On the one side it is to be distinguished from the wellknown provability semantics which is concerned merely with what is derivable as opposed to how it is derivable, paying attention to the fact that it is the how that determines the answer constraint. ...
Ternary Simulation: A Refinement of Binary Functions or an Abstraction of RealTime Behaviour?
 PROCEEDINGS OF THE 3RD WORKSHOP ON DESIGNING CORRECT CIRCUITS (DCC96
, 1996
"... We prove the equivalence between the ternary circuit model and a notion of intuitionistic stabilization bounds. The results are obtained as an application of the timing interpretation of intuitionistic propositional logic presented in [12]. We show that if one takes an intensional view of the ternar ..."
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Cited by 10 (3 self)
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We prove the equivalence between the ternary circuit model and a notion of intuitionistic stabilization bounds. The results are obtained as an application of the timing interpretation of intuitionistic propositional logic presented in [12]. We show that if one takes an intensional view of the ternary model then the delays that have been abstracted away can be completely recovered. Our intensional soundness and completeness theorems imply that the extracted delays are both correct and exact; thus we have developed a framework which unifies ternary simulation and functional timing analysis. Our focus is on the combinational behaviour of gatelevel circuits with feedback.
Proof Search in Constructive Logics
 In Sets and proofs
, 1998
"... We present an overview of some sequent calculi organised not for "theoremproving" but for proof search, where the proofs themselves (and the avoidance of known proofs on backtracking) are objects of interest. The main calculus discussed is that of Herbelin [1994] for intuitionistic lo ..."
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Cited by 7 (2 self)
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We present an overview of some sequent calculi organised not for "theoremproving" but for proof search, where the proofs themselves (and the avoidance of known proofs on backtracking) are objects of interest. The main calculus discussed is that of Herbelin [1994] for intuitionistic logic, which extends methods used in hereditary Harrop logic programming; we give a brief discussion of some similar calculi for other logics. We also point to some related work on permutations in intuitionistic Gentzen sequent calculi that clarifies the relationship between such calculi and natural deduction. 1 Introduction It is widely held that ordinary logic programming is based on classical logic, with a Tarskistyle semantics (answering questions "What judgments are provable?") rather than a Heytingstyle semantics (answering questions like "What are the proofs, if any, of each judgment?"). If one adopts the latter style (equivalently, the BHK interpretation: see [35] for details) by regardi...
A Timing Refinement of Intuitionistic Proofs and its Application to the Timing Analysis of Combinational Circuits
 PROCEEDINGS OF THE 5TH INTERNATIONAL WORKSHOP ON THEOREM PROVING WITH ANALYTIC TABLEAUX AND RELATED METHODS
, 1996
"... Up until now classical logic has been the logic of choice in formal hardware verification. This report advances the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes at the same time. The modeltheoretic properties are e ..."
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Cited by 6 (3 self)
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Up until now classical logic has been the logic of choice in formal hardware verification. This report advances the application of intuitionistic logic to the timing analysis of digital circuits. The intuitionistic setting serves two purposes at the same time. The modeltheoretic properties are exploited to handle the secondorder nature of bounded delays in a purely propositional setting without need to introduce explicit time and temporal operators. The proof theoretic properties are exploited to extract quantitative timing information and to reintroduce explicit time in a convenient and systematic way. We present a natural Kripkestyle semantics for intuitionistic propositional logic, as a special case of a Kripke constraint model for Propositional Lax Logic [4], in which validity is validity up to stabilization. We show that this semantics is equivalently characterized in terms of stabilization bounds so that implication oe comes out as "boundedly gives rise to." An int...
Proof Search in Lax Logic
, 2000
"... This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic ..."
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This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic