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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
A short proof of the Strong Normalization of Classical Natural Deduction with Disjunction
 Journal of symbolic Logic
, 2003
"... We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1 ..."
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Cited by 23 (14 self)
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We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1
On the Intuitionistic Force of Classical Search
 THEORETICAL COMPUTER SCIENCE
, 1996
"... The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering ..."
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Cited by 19 (5 self)
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The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering this leastcommitment principle in the context of noninvertible rules. One successful approach is to view the nonclassical logic as a perturbation on search in classical logic and characterize when a leastcommitment (classical) search yields sufficient evidence for provability in the (nonclassical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a typetheoretic view of this approach for the case in which the nonclassical logic is intuitionistic. We develop a system of realizers (proofobje...
A Calculus of Substitutions for IncompleteProof Representation in Type Theory
, 1997
"... : In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the pr ..."
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Cited by 16 (1 self)
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: In the framework of intuitionnistic logic and type theory, the concepts of "propositions" and "types" are identified. This principle is known as the CurryHoward isomorphism, and it is at the base of mathematical formalisms where proofs are represented as typed lambdaterms. In order to see the process of proof construction as an incremental process of term construction, it is necessary to extend the lambdacalculus with new operators. First, we consider typed metavariables to represent the parts of a proof that are under construction, and second, we make explicit the substitution mechanism in order to deal with capture of variables that are bound in terms containing metavariables. Unfortunately, the theory of explicit substitution calculi with typed metavariables is more complex than that of lambdacalculus. And worse, in general they do not share the same properties, notably with respect to confluence and strong normalization. A contribution of this thesis is to show that the pr...
A games semantics for reductive logic and proofsearch
 GaLoP 2005: Games for Logic and Programming Languages
, 2005
"... Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive l ..."
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Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important examples of control, namely backtracking and uniform proof. 1 Introduction to reductive logic and proofsearch Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important
Some Pitfalls of LKtoLJ Translations and How to Avoid Them
 Proc CADE14, LNCS 1249
, 1997
"... . In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all m ..."
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. In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all minimal LJproofs are nonelementary but there exist short LKproofs of the same formula. Similar results are obtained even if some fragments of intuitionistic firstorder logic are considered. The size of the corresponding minimal search spaces for LK and LJ are also nonelementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule. 1 Introduction Characterizing classes of formulae for which classical derivability implies intuitionistic derivability was one topic in the Leningrad group around Maslov in the sixties. Such classes are called (complete) Glivenko classes which were extensively characterized by Orevkov [7]. More recently, people ar...
Correspondences between Classical, Intuitionistic and Uniform Provability
 Theoretical Computer Science
, 2000
"... Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that ..."
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Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that embodies a special form of goaldirectedness. We determine, first, the circumstances in which the former relations imply the latter. Using this result, we identify the richest versions of the socalled abstract logic programming languages in classical and intuitionistic logic. We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved. Our focus here is on understanding the situations in which this reduction is achieved. However, our discussions indicate the structure of a proof procedure based on the reduction, a matter also considered explicitly elsewhere.
On GoalDirected Proofs in MultipleConclusioned Intuitionistic Logic
"... A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multiple ..."
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A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multipleconclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goaldirected proofs for a multipleconclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both singleconclusioned and multipleconclusioned systems (although the latter are less well known than the former). In this paper we show that the language obtained for the multipleconclusioned system differs from that for the singleconclusioned case, and discuss the consequences of this result. Keywords: Multipleconclusioned intuitionistic logic, goaldirected proofs, logic programming languages, hereditary Harrop formu...
Notes Towards a Semantics for Proofsearch
"... Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proo ..."
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Algorithmic proofsearch is an essential enabling technology throughout informatics. Proofsearch is the prooftheoretic realization of the formulation of logic not as a theory of deduction but rather as a theory of reduction. Whilst deductive logics typically have a welldeveloped semantics of proofs, reductive logics are typically wellunderstood only operationally. Each deductive system can, typically, be read as a corresponding reductive system. We discuss some of the problems which must be addressed in order to provide a semantics of proofsearches of comparable value to the corresponding semantics of proofs. Just as the semantics of proofs is intimately related to the model theory of the underlying logic, so too should be the semantics of proofsearches. We discuss how to solve the problem of providing a semantics for proofsearches which adequately models both operational and logical aspects of the reductive system. 1
Unifying Classical and Intuitionistic Logics for Computational Control
"... Abstract—We show that control operators and other extensions of the CurryHoward isomorphism can be achieved without collapsing all of intuitionistic logic into classical logic. For this purpose we introduce a unified propositional logic using polarized formulas. We define a Kripke semantics for thi ..."
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Abstract—We show that control operators and other extensions of the CurryHoward isomorphism can be achieved without collapsing all of intuitionistic logic into classical logic. For this purpose we introduce a unified propositional logic using polarized formulas. We define a Kripke semantics for this logic. Our proof system extends an intuitionistic system that already allows multiple conclusions. This arrangement reveals a greater range of computational possibilities, including a form of dynamic scoping. We demonstrate the utility of this logic by showing how it can improve the formulation of exception handling in programming languages, including the ability to distinguish between different kinds of exceptions and constraining when an exception can be thrown, thus providing more refined control over computation compared to classical logic. We also describe some significant fragments of this logic and discuss its extension to secondorder logic. I.