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A formulaeastypes interpretation of subtractive logic
 Journal of Logic and Computation
, 2004
"... We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural ..."
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We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for firstclass coroutines (a restricted form of firstclass continuations). Keywords: CurryHoward isomorphism, Subtractive Logic, control operators, coroutines. 1
Subtractive Logic
, 1999
"... This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any ..."
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Cited by 20 (1 self)
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This paper is the first part of a work whose purpose is to investigate duality in some related frameworks (cartesian closed categories, lambdacalculi, intuitionistic and classical logics) from syntactic, semantical and computational viewpoints. We start with category theory and we show that any bicartesian closed category with coexponents is degenerated (i.e. there is at most one arrow between two objects). The remainder of the paper is devoted to logical issues. We examine the propositional calculus underlying the type system of bicartesian closed categories with coexponents and we show that this calculus corresponds to subtractive logic: a conservative extension of intuitionistic logic with a new connector (subtraction) dual to implication. Eventually, we consider first order subtractive logic and we present an embedding of classical logic into subtractive logic. Introduction This paper is the first part of a work whose purpose is to investigate duality in some related ...
A TERM ASSIGNMENT FOR DUAL INTUITIONISTIC LOGIC.
"... Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatm ..."
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Cited by 5 (3 self)
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Abstract. We study the prooftheory of coHeyting algebras and present a calculus of continuations typed in the disjunctive–subtractive fragment of dual intuitionistic logic. We give a singleassumption multipleconclusions Natural Deduction system NJ � � for this logic: unlike the bestknown treatments of multipleconclusion systems (e.g., Parigot’s λ−µ calculus, or Urban and Bierman’s termcalculus) here the termassignment is distributed to all conclusions, and exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The present construction can be regarded as the construction of a free coCartesian
Games and WeakHead Reduction for Classical PCF
 Proceedings of TLCA 97, LNCS 1210
, 1997
"... . We present a game model for classical PCF, a finite version of PCF extended by a catch/throw mechanism. This model is build from Edialogues, a kind of twoplayers game defined by Lorenzen. In the Edialogues for classical PCF, the strategies of the first player are isomorphic to the Bohm trees of ..."
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Cited by 3 (0 self)
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. We present a game model for classical PCF, a finite version of PCF extended by a catch/throw mechanism. This model is build from Edialogues, a kind of twoplayers game defined by Lorenzen. In the Edialogues for classical PCF, the strategies of the first player are isomorphic to the Bohm trees of the language. We define an interaction in Edialogues and show that it models the weakhead reduction in classical PCF. The interaction is a variant of Coquand's debate and the weakhead reduction is a variant of the reduction in Krivine's Abstract Machine. We then extend Edialogues to a kind of games similar to HylandOng's games. Interaction in these games also models weakhead reduction. In the intuitionistic case (i.e. without the catch/throw mechanism), the extended Edialogues are HylandOng's games where the innocence condition on strategies is now a rule. Our model for classical PCF is different from Ong's model of Parigot's lambdamucalculus. His model works by adding new moves t...
A constructive restriction of the λµcalculus
, 1999
"... We define a very natural restriction of the λµcalculus which is stable under reduction and whose type system is a restriction of the Classical Natural Deduction to intuitionistic logic. However, we show that this system is in some sense degenerated unless we provide a native disjunction. We prove t ..."
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We define a very natural restriction of the λµcalculus which is stable under reduction and whose type system is a restriction of the Classical Natural Deduction to intuitionistic logic. However, we show that this system is in some sense degenerated unless we provide a native disjunction. We prove that the system with native disjunction is conservative over DISlogic and also that DISlogic is constructive. From a computational standpoint, this restriction on λµterms prevents a coroutine from accessing the local environment of another coroutine.
NATURAL DEDUCTION AND TERM ASSIGNMENT FOR COHEYTING ALGEBRAS IN POLARIZED BIINTUITIONISTIC LOGIC.
"... Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of ..."
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Abstract. We reconsider Rauszer’s biintuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized biintuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ⊃ ∩ and its dual LJ� � , extended with two negations partially internalizing the duality between LJ⊃ ∩ and LJ� �. Modal interpretations and Kripke’s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized biintuitionistic logic is interpreted over S4 rather than bimodal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJ� � exhibits several features of calculi for concurrency, such as remote capture of variable and remote substitution. The duality is extended from formulas to proofs and it is shown that every computation in our calculus is isomorphic to a computation in the simply typed λcalculus. §1. Preface. We present a natural deduction system for propositional polarized biintuitionistic logic PBL, (a variant of) intuitionistic logic extended with a connective of subtraction A � B, read as “A but not B”, which is dual to implication. 1 The logic PBL is polarized in the sense that its expressions are regarded as expressing acts of assertion or of conjecture; implications and conjunctions are assertive, subtractions and disjunctions are conjectural. Assertions and conjectures are regarded as dual; moreover there are two negations, transforming assertions into conjectures and viceversa, in some sense internalizing the duality. Our notion of polarity isn’t just a technical device: it is rooted in an analysis of the structure of speechacts, following the viewpoint of the