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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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Cited by 23 (1 self)
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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 typetheoretic foundation of delimited continuations. Higher Order Symbol
 Comput
, 2009
"... Abstract. There is a correspondence between classical logic and programming language calculi with firstclass continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a finegrained analysis of control delimiters a ..."
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Cited by 14 (6 self)
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Abstract. There is a correspondence between classical logic and programming language calculi with firstclass continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a finegrained analysis of control delimiters and formalise that their addition corresponds to the addition of a single dynamicallyscoped variable modelling the special toplevel continuation. From a type perspective, the dynamicallyscoped variable requires effect annotations. In the presence of control, the dynamicallyscoped variable can be interpreted in a purely functional way by applying a storepassing style. At the type level, the effect annotations are mapped within standard classical logic extended with the dual of implication, namely subtraction. A continuationpassingstyle transformation of lambdacalculus with control and subtraction is defined. Combining the translations provides a decomposition of standard CPS transformations for delimited continuations. Incidentally, we also give a direct normalisation proof of the simplytyped lambdacalculus with control and subtraction.
Dual intuitionistic logic revisited
 Automated Reasoning with Analytic Tableaux and Related Methods, St
, 2000
"... Abstract. We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right ” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises fr ..."
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Cited by 11 (1 self)
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Abstract. We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right ” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises from intuitionistic and dualintuitionistic logic. We show that a previously reported generalised display framework does deliver the required cutfree display calculus. We also pinpoint precisely the structural rule necessary to turn this display calculus into one for classical logic. 1
Finitely Presented Heyting Algebras
 IN PREPARATION
, 1998
"... In this paper we study the structure of nitely presented Heyting algebras. Using algebraic techniques (as opposed to techniques from prooftheory) we show that every such Heyting algebra is in fact coHeyting, improving on a result of Ghilardi who showed that Heyting algebras free on a nite set ..."
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Cited by 7 (1 self)
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In this paper we study the structure of nitely presented Heyting algebras. Using algebraic techniques (as opposed to techniques from prooftheory) we show that every such Heyting algebra is in fact coHeyting, improving on a result of Ghilardi who showed that Heyting algebras free on a nite set of generators are coHeyting. Along the way we give a new and simple proof of the nite model property. Our main technical tool is a representation of nitely presented Heyting algebras in terms of a colimit of nite distributive lattices. As applications we construct explicitly the minimal joinirreducible elements (the atoms) and the maximal joinirreducible elements of a nitely presented Heyting algebras in terms of a given presentation. This gives as well a new proof of the disjunction property for intuitionistic propositional logic.
On logics with coimplication
 Journal of Philosophical Logic
, 1998
"... This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the God ..."
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Cited by 7 (1 self)
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This paper investigates (modal) extensions of HeytingBrouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We rst develop matrix as well as Kripke style semantics for those logics. Then, by extending the Godelembedding of intuitionistic logic into S4, itisshown that all (modal) extensions of HeytingBrouwer logic can be embedded into tense logics (with additional modal operators). An extension of the BlokEsakiaTheorem is proved for this embedding. 1
Nearly every normal modal logic is paranormal‘, Logique et Analyse
, 2005
"... The principal interest is philosophical: not to confine oneself to what is necessary for (current) practice, but to see what is possible by way of theoretical analysis. —Kreisel (1970). An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences h ..."
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Cited by 7 (3 self)
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The principal interest is philosophical: not to confine oneself to what is necessary for (current) practice, but to see what is possible by way of theoretical analysis. —Kreisel (1970). An overcomplete logic is a logic that ‘ceases to make the difference’: According to such a logic, all inferences hold independently of the nature of the statements involved. A negationinconsistent logic is a logic having at least one model that satisfies both some statement and its negation. A negationincomplete logic has at least one model according to which neither some statement nor its negation are satisfied. Paraconsistent logics are negationinconsistent yet nonovercomplete; paracomplete logics are negationincomplete yet nonovercomplete. A paranormal logic is simply a logic that is both paraconsistent and paracomplete. Despite being perfectly consistent and complete with respect to classical negation, nearly every normal modal logic, in its ordinary language and interpretation,
Proof Search and CounterModel Construction for Biintuitionistic Propositional Logic with Labelled Sequents
"... Abstract. Biintuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cutfree labelled sequent calculus for biintuitionistic propositional logic, BiInt, following S. Negri’s general method for ..."
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Cited by 6 (1 self)
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Abstract. Biintuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cutfree labelled sequent calculus for biintuitionistic propositional logic, BiInt, following S. Negri’s general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Kripke semantics, it is does not directly support proof search. To describe a proof search procedure, we develop a more algorithmic version that also allows for countermodel extraction from a failed proof attempt. 1
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
Categorical Proof Theory of CoIntuitionistic Linear Logic
"... Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of ..."
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Cited by 1 (1 self)
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Summary. To provide a categorical semantics for cointuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of coexponents as adjuncts of coproducts does not work in the category Set, where coproducts are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent!, we build models of cointuitionistic logic in symmetric monoidal closed categories with additional structure, using a variant of Crolard’s term assignment to cointuitionistic logic in the construction of a free category. 1