We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.

We give in this paper a short semantical proof of the strong normalization for full propositional classical natural deduction. This proof is an adaptation of reducibility candidates introduced by J.-Y. Girard and simplified to the classical case by M. Parigot.

In 1990 J-L. Krivine introduced the notion of storage operators. They are $\lambda$ -terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF2 type system for storage operators using Gödel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF2 to the classical logic. In his system (...) the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. (shrink)

In 1990, J.L. Krivine introduced the notion of storage operator to simulate, in $lambda$-calculus, the 'call by value' in a context of a 'call by name'. J.L. Krivine has shown that, using Gödel translation from classical into intuitionistic logic, we can find a simple type for storage operators in AF2 type system. In this present paper, we give a general type for storage operators in a slight extension of AF2. We give at the end (without proof) a generalization of this (...) result to other types. (shrink)

In 1990, J. L. Krivine introduced the notion of storage operator to simulate “call by value” in the “call by name” strategy. J. L. Krivine has showed that, using Gödel translation of classical into intuitionistic logic, one can find a simple type for the storage operators in AF2 type system. This paper studies the ∀-positive types and the Gödel transformations of TTR type system. We generalize by using syntactical methods Krivine's theorem about these types and for these transformations. We give (...) a proof of this result in the case of the type of recursive integers. (shrink)

The storage operators were introduced by J.L. Krivine ([6]); they are closed λ-terms which, for some fixed data type (the integers for example), allow to simulate “call by value” while using “call by name”. J.L. Krivine showed that such operators can be typed, in the type system, using Gödel's translation from classical to intuitionistic logic ([8]).This paper studies the existence of storage operators which give a normal form as result (strong storage operators) for recursive and iterative representation of data in (...) λ-calculus. We obtain the following result:We can find typed strong storage operators for the recursive representations of data type, but that is not the case for the iterative representations of an infinite data type.We give the proof of this result in the case of integers. (shrink)

Storage operators have been introduced by J. L. Krivine in [5] they are closed λ-terms which, for a data type, allow one to simulate a "call by value" while using the "call by name" strategy. In this paper, we introduce the directed λ-calculus and show that it has the usual properties of the ordinary λ-calculus. With this calculus we get an equivalent--and simple--definition of the storage operators that allows to show some of their properties: $\bullet$ the stability of the set (...) of storage operators under the β-equivalence (Theorem 5.1.1); $\bullet$ the undecidability (and semidecidability) of the problem "is a closed λ-term t a storage operator for a finite set of closed normal λ-terms?" (Theorems 5.2.2 and 5.2.3); $\bullet$ the existence of storage operators for every finite set of closed normal λ-terms (Theorem 5.4.3); $\bullet$ the computation time of the "storage operation" (Theorem 5.5.2). (shrink)

We prove the strong normalization of full classical natural deduction by using a translation into the simply typed λμ-calculus. We also extend Mendler’s result on recursive equations to this system.

In this paper, we extend the system AF2 in order to have the subject reduction for the $betaeta$-reduction. We prove that the types with positive quantifiers are complete for models that are stable by weak-head expansion.

In this paper we consider a type system with a universal type $omega$ where any term (whether open or closed, $beta$-normalising or not) has type $omega$. We provide this type system with a realisability semantics where an atomic type is interpreted as the set of $lambda$-terms saturated by a certain relation. The variation of the saturation relation gives a number of interpretations to each type. We show the soundness and completeness of our semantics and that for different notions of saturation (...) (based on weak head reduction and normal $beta$-reduction) we obtain the same interpretation for types. Since the presence of $omega$ prevents typability and realisability from coinciding and creates extra difficulties in characterizing the interpretation of a type, we define a class ${mathbb U}^+$ of the so-called positive types (where $omega$ can only occur at specific positions). We show that if a term inhabits a positive type, then this term is $beta$-normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for ${mathbb U}^+$ becomes interesting indeed since it establishes a perfect equivalence between typable terms and terms that inhabit a type. In other words, typability and realisability coincide on ${mathbb U}^+$. We give a number of examples to explain the intuition behind the definition of ${mathbb U}^+$ and to show that this class cannot be extended while keeping its desired properties. (shrink)

A numeral system is an infinite sequence of different closed normal -terms intended to code the integers in -calculus. Barendregt has shown that if we can represent, for a numeral system, the functions Successor, Predecessor, and Zero Test, then all total recursive functions can be represented. In this paper we prove the independancy of these three particular functions. We give at the end a conjecture on the number of unary functions necessary to represent all total recursive functions.

In this paper, we define a realizability semantics for the simply typed $lambdamu$-calculus. We show that if a term is typable, then it inhabits the interpretation of its type. This result serves to give characterizations of the computational behavior of some closed typed terms. We also prove a completeness result of our realizability semantics using a particular term model.

In this paper, we present three methods to give the value of a classical integer in $\lambda\mu$ -calculus. The first method is an external method and gives the value and the false part of a normal classical integer. The second method uses a new reduction rule and gives as result the corresponding Church integer. The third method is the M. Parigot's method which uses the J.L. Krivine's storage operators.

In this paper, we present an extension of λμ-calculus called λμ++-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or.

In this paper, we present an extension of $lambdamu$-calculus called $lambdamu^{++}$-calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallel-or.

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

In 1990, J. L. Krivine introduced the notion of storage operator to simulate, for Church integers, the “call by value” in a context of a “call by name” strategy. In the present paper we define for every λ-term S which realizes the successor function on Church integers the notion of S-storage operator. We prove that every storage operator is an S-storage operator. But the converse is not always true.

In 1990, J.L. Krivine introduced the notion of storage operator to simulate 'call by value' in the 'call by name' strategy. J.L. Krivine has shown that, using Gödel translation of classical into intuitionitic logic, we can find a simple type for the storage operators in AF2 type system. This paper studies the $forall$-positive types (the universal second order quantifier appears positively in these types), and the Gödel transformations (a generalization of classical Gödel translation) of TTR type system. We generalize, by (...) using syntaxical methods, the J.L. Krivine's Theorem about these types and for these transformations. We give a proof of this result in the case of the type of recursive integers. (shrink)

In this paper, we present the -calculus which at the typed level corresponds to the full classical propositional natural deduction system. The Church–Rosser property of this system is proved using the standardisation and the finiteness developments theorem. We also define the leftmost reduction and prove that it is a winning strategy.