When using the Curry-Howard correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem

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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are proof-theoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on non-collapsing hierarchies ( n -WKL+ ; n -WKL+ ) of principles which generalize (and for n = 0 coincide with) the so-called `weak' Konig's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context) Basic Research in Computer Science, Centre of the Danish National Research Foundation.

In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n -proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As

Abstract. The last section of “Lecture at Zilsel’s ” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in [5], we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied only to atomic formulas. ¬φ in general is represented by the complement φ of φ, obtained by interchanging ∧ with ∨, ∀ with ∃, and atomic sentences with their negations. The components of the sentences φ ∨ ψ and φ ∧ ψ are φ and ψ. The components of the sentences ∃xφ(x) and ∀xφ(x) are the sentences φ(¯n) for each numeral ¯n. A ∧- or ∀-sentence, called a �-sentence, is thus expressed by the conjunction of its components and a ∨- or ∃-sentence, called a �-sentence, is expressed by the disjunction of them; and so it follows that every sentence can be represented as an infinitary propositional formula built up from prime sentences— atomic or negated atomic sentences. Disjunctive and conjunctive sentences with the components φn (where the range of n is 1, 2 or ω) will be denoted respectively by

). In A. Tarlecki, editor, Proc. 16th International Symp. on Mathematical Foundations of Computer Science, MFCS '91, volume 520 of LNCS. Springer-Verlag, 1991. [BCHK93] G. Boudol, I. Castellani, M. Hennessy, and A. Kiehn. Observing localities. Theoretical Computer Science, 114(1):31-61, 1993. Full version of [BCHK91]. [BHR84] S.D. Brookes, C.A.R. Hoare, and A.W. Roscoe. A Theory of Communicating Sequential Processes. Journal of the ACM, 31(3):560-599, 1984. [BK84] J.A. Bergstra and J.W. Klop. Process Algebra for Synchronous Communication. Information and Control, 60:109-137, 1984. 12 REFERENCES [BK85] J.A. Bergstra and J.W. Klop. Algebra of communicating processes with abstraction. Theoretical Computer Science, 37(1):77{ 121, 1985. [BS94] M. Boreale and D. Sangiorgi. A fully abstract semantics for causality in the pi-calculus. Report ECS-LFCS-94-297, Laboratory for Foundations of Computer Science, Computer Science Department, Edinburgh University, 1994. An extract appeared in the ...

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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finite-type arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents

Le contenu computationnel des preuves: No-counterexample interpretation et spécification des théorèmes de l’arithmétique mémoire sous la direction de Jean-Louis Krivine

Dedicated to Grigori Mints in honor of his seventieth birthday. Almost from the inception of Hilbert’s program, foundational and structural efforts in proof theory have been directed towards the goal of clarifying the computational content of modern mathematical methods. This essay surveys various methods of extracting computational information from proofs in classical first-order arithmetic, and reflects on some of the relationships between them. Variants of the Gödel-Gentzen doublenegation translation, some not so well known, serve to provide canonical and efficient computational interpretations of that theory. 1