T. Coquand. Computational content of classical logic. In A. Pitts and P. Dybjer, editor, Semantics and Logics of Computation. Cambridge University Press, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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. Computational content of classical logic. In A. Pitts and P. Dybjer, editor, Semantics and Logics of Computation. Cambridge University Press, 1997.

.... (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 inductively ....

Coquand, T., Computational content of classical logic. Semantics and logics of computation (Cambridge,

....partially formalised in Agda [Coq98] and can be executed to yield an n such that a n = 0 for coefficients a. 1 Introduction Zorn s lemma is a non constructive principle which is often used in algebra, for example in the proof that any ideal is contained in a prime ideal. In the work by Coquand [Coq96], it is argued that many such use of the axiom of choice and the excluded middle could be avoided by using constructive models based on pointfree topology. In particular, it is shown how to give non standard constructive models in which prime ideals exists effectively. These non standard models ....

....example which states that the coefficients of an invertible polynomial in a commutative ring with unit are nilpotent. This example has a simple nonconstructive proof which relies on the existence of enough prime ideals. The first model is based on Girard s phase semantics [Gir87] and developed in [Coq96]. We have formalised an interpretation of a particular proof with this model in Agda [Coq98] and used it to extract witnesses. Since these witnesses and the computation of them was unnecessarily complicated, we present a simpler model based on formal topology which gives much simpler witnesses. ....

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Th. Coquand. Computational content of classical logic. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation. Cambridge University Press, 1996.

.... for the possible computational content of a given classical proof of a given classical theorem (for example Van der Waerden s Theorem on arithmetic progressions) The extraction of a program has been carried out for some specic classical proofs, by Berger, Schwichtenberg, Coquand, and others (cf. [3]) 0.3 Further readings. This introduction can be complemented with the references below. Intended for a general audience are : Barendregt s survey on the impact of calculus in logic and computer science [2] Krivine s brief survey [16] which gives a clear idea of the present interest of the ....

T. Coquand, Computational content of classical logic, in A.Pitts ed., Semantics and logics of Computation, Cambridge University Press, Cambridge, p.470-517, 1997.

.... from such proofs Even better, is it possible to design a version of Peano Arithmetic in which proofs of every Pi 0 2 formula can be systematically transformed into an algorithm directly by a process of cut elimination An excellent treatment of this topic is Coquand s recent lecture notes [6]. See also the discussion in [11, x1.5] 3. A theory of functions or processes It is an old idea in proof theory that there is a deep connection between the Brouwer Heyting Kolmogorov interpretation of intuitionistic logic and a rule based theory of functions such as the calculus. Can the same ....

....there has been a great deal of interest in classical proofs. The following is a tentative (and incomplete) classification: ffl Algorithm extraction, control operators: Griffin [14] Murthy [22] Krivine [21] de 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 ....

T. Coquand. Computational content of classical logic. In Proc. Newton Institute CLiCS-II Summer Sch. on Semantics and Logics of Computation. Cambridge Univ. Press, 1996. To appear.

....Hilbert s basis theorem implies the existence of Grobner bases for noetherian and coherent rings [JL91] 6 3 Some Constructive Interpretations of Commutative Algebra Zorn s lemma is often used in algebra, for example in the proof of existence of prime ideals in ring theory. In the reference [Coq96] it is argued that many such use of the Axiom of Choice and the excluded middle could be avoided by using constructive models based on pointfree topology. This is exemplified here; in particular we want to show how to interpret the existence of maximal ideals, prime ideals and valuation rings. ....

Th. Coquand. Computational content of classical logic. In A. M. Pitts and P. Dybjer, editors, Semantics and Logics of Computation. Cambridge University Press, 1996. 9

....uy = 0: If follows that yzfflZ and so 0fflI (y) I (z) But we have yfflI (z) and hence I (y) I (z) Hence 0fflI (z) and so zfflZ: The positive content of this theorem is the following: if the theory of the maximal filter is inconsistent then we have 0 = 1 in the monoid M . In [2], we give examples explaining how such results can be used to make constructive sense of some classical arguments. Classically, the assumption that I preserves arbitrary suprema is not needed. We need only that there exists a proper filter F such that I(F ) F and that I is monotone. However, we ....

Th. Coquand. (1996) Computational Content of Classical Logic. In P. Dybjer and A. Pitts, ed., CLICS Cambridge Summer School.

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