Th. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic 60 (1995), pp. 325--337.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This is very much related to the notion of observation at the basis of the study of the equivalence of programs and processes in computer science. Unfortunately, the dialogue games school remained at a static, descriptive level. The full avour comes when cut elimination is interpreted as a play [11] (see also below) Computation as interaction From the computer science perspective, the history of the computation asinteraction paradigm is inseparable from the study of sequentiality. Vuillemin, and Milner have given the rst denotational de nitions of a sequential function, which were later ....

T. Coquand, A semantics of evidence for classical arithmetic, Journal of Symbolic Logic 60, 325-337 (1995).

....cut free forms of or 0 and 0 . Thus any congruence (for instance, a denotational equivalence) generated by such a cut elimination procedure must equate and 0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction [9] and game semantics [4, 1, 5], that cut elimination is a process analogous to computation. Non determinism is both a standard property of computational processes and a key feature of many important algorithms and the physical systems on which programs are run. But can a connection between non deterministic computation and ....

.... Computationally, or corresponds to an erratic choice operator [15] showing in principal that many non deterministic algorithms may be extracted from proofs. A more difficult question is whether there are natural proofs which have computational content which is non deterministic. Coquand [5, 6] has described examples of symmetric classical existence proofs from which two different witnesses can be extracted by different double negation translations, but much work remains to be done. The structural dilemma can be seen as a problem of ambiguity classical proofs do not carry ....

T. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 60:325--337, 1995.

....was the minimum value theorem of every number theoretic function. This example was our guiding example through the investigation, and our proof of Proposition 1 is a variant of Berardi s proof. Berardi s interpretation was based on a game theoretic interpretation of classical proofs by Coquand [8, 3]. Although our work was done independently, it should be noted that a relationship of Coquand s game semantics to learning theory had been pointed out in [3] There are some resemblances between Coquand s game theoretic interpretation and our limiting realizability interpretation. It is desirable ....

T. Coquand, A Semantics of Evidence for Classical Arithmetic, the Journal of Symbolic Logic, 60(1995), pp.325-337.

.... proofs (proof mining) 95] 96] 97] 35] 103] 30] 7] 40] 74] 79] 84] 2) On Herbrand s theorem: 23] 40] 70] 113] 95] 102] 3) On the no counterexample interpretation: 40] 82] 90] 91] 119] 120] 111] For other approaches to proof mining not treated in these notes see [26] and [27] Chapter 2 Intuitionistic logic and arithmetic in all finite types In the following we formulate an axiomatic system for intuitionistic first order predicate logic IL. The particular axiomatization we choose is due to [45] and particular suited to carry out proof interpretations ....

Coquand, T., A semantics of evidence for classical arithmetic. J. Symbolic Logic 60, pp. 325-337 (1995).

....and to proofs of classical logic. The fact that features of classical reasoning can be identified in games, and are the same elements which model jumps in the flow of control also gives a way of understanding the computational content of classical reasoning: a similar distinction is used in [6] to extend a games semantics for intuitionistic arithmetic to the classical case by allowing backtracking. The prototypical language PCF has provided a framework in which to study the essential features of functional languages. So a reasonable place to examine the functional non functional ....

T. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 1995.

....is sceptical and probes more and more deeply into the reasoning behind the claim. Indeed, connections between proof systems and certain kinds of game or dialogue have had a long history in logic. Some important papers are: Novikov [20] Hintikka [12] Moschovakis [18] Aczel [2] Coquand, [5], Abramsky [1] Since the 4 part structure defined above has many applications, it isn t surprising that it should also admit applications in programming. One of these seems to be that the structure (S, C, R, represents an command response interface. 2 The Big Picture In this section we ....

T. Coquand. A semantics of evidence of classical arithmetic. Journal of Symbolic Logic, 6-:325--338, 1995. 16

....Groote [9] Nakano [23] Hirokawa [16] Schwichtenberg and Berger [4] Coquand [6] etc. ffl Formal systems and calculi: Girard [11, 12] Parigot [24] Berardi and Barbanera [2] Danos, Joinet and Schellinx [8] etc. ffl Proofs and semantics of cut elimination: Girard [11] Hofmann [17] Coquand [5], Pfenning [25] Herbelin [15] etc. Of these Parigot s ideas have provided much inspiration and, at an early stage of this work, the basis for believing that a categorical characterization of classical proofs was up for grabs . Girard s work on lc [11] presents, among other things, a ....

T. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 1995.

....certain sets of strategies. There have been two proposals: the first one is based on the idea of history free strategies [1] according to the second one players move depending on views of the play: let us call view games and view strategies games and strategies as defined in [5, 6] In [4] a different notion of game, originally introduced by Novikoff, is used to give an intuitionistic explanation of the classical notion of truth. As it will be apparent from sections 2 and 3 of the present paper, view games are a particular case of Novikoff Coquand games: the former can be obtained ....

....0 2 O 1:1 1 3 P 2 0 4 O 2:1 3 5 P 1:1:1 2 6 O v 1 5 7 P v 2 4 8 O v 3 3 9 P v 4 2 To the time moves are meaningless so that we forget about their actual form. We defer to section 3 and to example 3.3 the explanation about their interpretation. U cut free (pre) plays is the terminology of [4]. If a pre play has finite length then the previous definition is a generalization of [6] definition 3.1.3. Observe that in a U cut free pre play, U is the unique player allowed to play at limit points. Because of the finite depth of R and clauses (P 3) P 4) for all length(p) there exists ....

T. Coquand, "A Semantics of Evidence for Classical Arithmetic", JSL 60, 1995, 325-337.

....collapsing certain sets of strategies. There have been two proposals: the first one is based on the idea of history free strategies [3] according to the second one players move depending on views of the play: these are called dialog games and innocent strategies, as defined in [10, 11] In [7] an apparently different notion of game, originally introduced by Novikoff, is used to give an intuitionistic explanation of the classical notion of truth. As it will be explained in sections 2 and 3 of the present paper, dialog games and Novikoff Coquand games are closely related: the former can ....

....(called innocent in [10] having a simple inductive definition: view(U; p; i) ae fi Gamma 1g [ view(U; p; i Gamma 1) if turn(i) U fr(i)g [ view(U; p; r(i) if turn(i) 6= U . This is the standard notion of visibility in dialog games: it is defined in this way both in [10, 11] and in [7]. the case of plays of possibly transfinite length has been considered for the first time in [5] from which we borrow the axiomatic definition of Vis. Definition above does not tell, explicitly, who is the player on turn at a limit point 2 I, nor his views. The main theorem 4, however, states ....

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T. Coquand, "A Semantics of Evidence for Classical Arithmetic", Journal of Symbolic Logic 60, 1995, 325-337.

....provided one augments functions by appropriate control constructs. In particular he proposed the tautology : A ) A as the type for Felleisen s C operator. A spate of research into the semantics and computational contents of classical proofs ensued (some of which quite independently of Griffin s) [6, 13, 25, 28, 2, 21, 4, 7, 42], etc. Church s calculus is by now widely accepted as the logical basis of functional programming. A goal of our research is to find the calculus of functional computation with first class access to the flow of control, or functional computation with control, for short. In Sec tion 2 of ....

Th. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 60:325--337, 1995.

....a classical proof, the computational content of which, we conjecture, is the same as the computational content of the proof of IRT I given in subsection 2.2. Its second subsection gives another program for a classical proof which was obtained by Stefano Berardi by game interpretation (we refer to [2] for a description of game interpretation) Finally, appendix A shows the equivalence between IRT and the formulation of the Intuitionistic Ramsey Theorem given in [13] It justifies the unprovability of IRT in Type Theory. 2 The proofs All the proofs presented in subsections 2.1 and 2.2 have ....

Thierry Coquand. A semantics of evidence for classical arithmetic. Proceedings of the workshop on types for proofs and programs, Edinburgh, 1991.

.... [14, 59, 60, 61, 62] Barbanera and Berardi [2, 3, 4, 5, 6] Rezus [74, 75] Parigot [66, 67, 68, 69] de Groote [25, 27, 28, 29, 30] Krivine [57] Girard [42] Danos, Joinet, and Schellinx [18] Rehof and S rensen [73] Duba, Harper, and MacQueen[32] Harper and Lillibridge [46, 47] Coquand [15], Berardi, Bezem, and Coquand [11] Ong [65] Underwood [89] and Sato [78] Most of these lines of work study one specific classical natural deduction system or one specific typed calculus enriched with control operators. The authors [10] study a notion of classical pure type system (CPTS) ....

T Coquand. A semantics of evidence of classical arithmetic. Journal of Symbolic Logic, 60:230--260, 1995.

....following a rather pleasing interplay between theoretical computer science and practical computer science, there has been a renewed interest in CL and, in particular, the constructive content of classical proofs. This content appears to have links with, at the theoretical level, game theory [13] and at the practical level, certain control operators for functional programming languages [23] To some extent Girard s linear logic [21] has also renewed interest in game theory and functional programming languages. The refined connectives of linear logic have helped shed new light on work on ....

Th. Coquand. A semantics of evidence of classical arithmetic. Journal of Symbolic Logic, 60:325--337, 1995.

....di Torino Italy C. so Svizzera 185 10149 Italy e mail: stefano di.unito.it Thierry Coquand Computer Science Departement of Chalmers University Goteborg, S 41296 Sweden e mail: coquand cs.chalmers.se September 30, 1998 Introduction We generalise the notion of game introduced by [4] to possibly transfinite games. We prove in this setting a result that can be seen both as a generalisation and a refinement of the cutelimination result for Tait s infinitary logic of [4] Game models have been used to analyse what happens in computation at higher types [5, 7] Our result has ....

....e mail: coquand cs.chalmers.se September 30, 1998 Introduction We generalise the notion of game introduced by [4] to possibly transfinite games. We prove in this setting a result that can be seen both as a generalisation and a refinement of the cutelimination result for Tait s infinitary logic of [4]. Game models have been used to analyse what happens in computation at higher types [5, 7] Our result has been used for giving such a model [3] in which total functionals are exactly strategies with no divergent branch. Such a characterisation cannot hold in the models described in [7] We hope ....

Th. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic 60 (1995), pp. 325--337.

....p: Phi p 0 realizes the formula r 9n 8m r f(n) f(m) This program follows closely the previous strategy of 9loise, and the proof that p: Phi p 0 realizes the formula r 9n 8m r f(n) f(m) is similar to the argument showing that this strategy is winning. 7. 3 A strategy for the Axiom of Choice In [5] it was conjectured that it should be possible to extend this interpretation in the case of quantification over functions. The idea would be simply to allow as index set the set of all functions, and, apart from this single change, to keep the same notion of games and strategies. This suggested ....

Th. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, to appear, 1994.

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Th. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic 60 (1995), pp. 325--337.

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Thierry Coquand. A semantics of evidence for classical arithmetic. The Journal of Symbolic Logic, 60(1):325--337, 1995.

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Thierry Coquand. A semantics of evidence for classical arithmetic. The Journal of Symbolic Logic, 60(1):325--337, 1995.

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T. Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, (60), 1995.

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Thierry Coquand. A semantics of evidence for classical arithmetic. Journal of Symbolic Logic, 60(1):325--337, March 1995. 12

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T. Coquand. A semantics of evidence for classical arithmetic. J. Symbolic Logic 60, pp. 325-337 (1995).

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