## Dependent choices, ‘quote’ and the clock (2003)

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Venue: | Th. Comp. Sc |

Citations: | 24 - 9 self |

### BibTeX

@ARTICLE{Krivine03dependentchoices,,

author = {Jean-louis Krivine},

title = {Dependent choices, ‘quote’ and the clock},

journal = {Th. Comp. Sc},

year = {2003},

pages = {259--276}

}

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### Abstract

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

### Citations

328 | Lambda-mu-calculus: An Algorithmic Interpretation of Classical natural Deduction - Parigot - 1992 |

246 | A Formulae-as-Types Notion of Control
- Griffin
(Show Context)
Citation Context ...h may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem has been solved for the axiom of excluded middle by T. Griffin=-=[5]-=-, who found that it can be interpreted by means of control instructions like call-with-current-continuation in SCHEME, catch and throw in LISP or try ... with ... in CAML. The solution for the compreh... |

228 |
Une extension de l’interpretation de Gödel a l’analyse, et son application a l’elimination des coupure dans l’analyse et la theorie des types
- Girard
- 1971
(Show Context)
Citation Context ...cond order logic was essentially given by G. Takeuti[14], who gave a formulation of this scheme by means of an elimination rule of the second order universal quantifier, and J.Y. Girard who showed in =-=[4]-=- that this rule can be interpreted by the identity instruction λxx. In [11], this problem is solved for the axioms of classical Zermelo-Fraenkel set theory, with the axiom of foundation, but without t... |

219 |
Mathematical logic
- Shoenfield
- 1967
(Show Context)
Citation Context ... opponent, in the game associated with the formula : Φ ≡∃x1∀y1 ...∃xk∀yk(φ(x1,y1,...,xk,yk) =0). This theorem is closely related to the no-counter-example interpretation of G. Kreisel[7, 8] (see also =-=[3, 6, 13]-=-) : Kreisel has shown that, if Φ is a theorem of first order Peano arithmetic, then there exists type recursive functionals in the sense of [13] Fi(f1,...,fk) (1 ≤ i ≤ k) such that : φ[ξ1,f1(ξ1), ξ2,f... |

114 |
Proof Theory
- Takeuti
- 1987
(Show Context)
Citation Context ...all-with-current-continuation in SCHEME, catch and throw in LISP or try ... with ... in CAML. The solution for the comprehension axiom scheme for second order logic was essentially given by G. Takeuti=-=[14]-=-, who gave a formulation of this scheme by means of an elimination rule of the second order universal quantifier, and J.Y. Girard who showed in [4] that this rule can be interpreted by the identity in... |

44 |
A semantics of evidence for classical arithmetic
- Coquand
- 1995
(Show Context)
Citation Context ... in classical analysis with choice, in the spirit of the nocounter-example interpretation of G. Kreisel. We use the ideas about game semantics for proofs of such formulas which have been developed in =-=[3]-=-. Other computational interpretations of the countable axiom of choice have been given in [1], and recently in [2]. It would be interesting to understand the relation with the present paper. I want to... |

34 | On the computational content of the axiom of choice, The - Berardi, Bezem, et al. - 1998 |

32 | Typed lambda-calculus in classical Zermelo-Fraenkel set theory - Krivine |

26 | On the interpretation of non-finitist proofs, part I - Kreisel - 1951 |

25 | Modified Bar Recursion and Classical Dependent Choice
- Berger, Oliva
- 2005
(Show Context)
Citation Context ...e ideas about game semantics for proofs of such formulas which have been developed in [3]. Other computational interpretations of the countable axiom of choice have been given in [1], and recently in =-=[2]-=-. It would be interesting to understand the relation with the present paper. I want to thank Thierry Coquand and the referee for their very pertinent observations ; and Vincent Danos for an uncountabl... |

18 | On the no-counterexample interpretation
- Kohlenbach
- 1999
(Show Context)
Citation Context ... opponent, in the game associated with the formula : Φ ≡∃x1∀y1 ...∃xk∀yk(φ(x1,y1,...,xk,yk) =0). This theorem is closely related to the no-counter-example interpretation of G. Kreisel[7, 8] (see also =-=[3, 6, 13]-=-) : Kreisel has shown that, if Φ is a theorem of first order Peano arithmetic, then there exists type recursive functionals in the sense of [13] Fi(f1,...,fk) (1 ≤ i ≤ k) such that : φ[ξ1,f1(ξ1), ξ2,f... |

18 | 1958b, ‘Mathematical Significance of Consistency Proofs - Kreisel |

16 |
On the interpretation of non-finitist proofs, part II: Interpretation of number theory, applications
- Kreisel
- 1952
(Show Context)
Citation Context ...y strategy of the opponent, in the game associated with the formula : Φ ≡∃x1∀y1 ...∃xk∀yk(φ(x1,y1,...,xk,yk) =0). This theorem is closely related to the no-counter-example interpretation of G. Kreisel=-=[7, 8]-=- (see also [3, 6, 13]) : Kreisel has shown that, if Φ is a theorem of first order Peano arithmetic, then there exists type recursive functionals in the sense of [13] Fi(f1,...,fk) (1 ≤ i ≤ k) such tha... |

15 | A general storage theorem for integers in call-by-name -calculus
- Krivine
- 1994
(Show Context)
Citation Context ...(n)λgg◦s)f0, wheresis a λ-term for the successor (e.g. s = λnλfλx(f)(n)fx). If φ ∈ Λ0 c is such that φ ?sn0.π ∈⊥ for all π ∈ kXk, then T φ k− Int[sn0] → X. Remark. T is called a storage operator (cf. =-=[10]-=-). We can understand intuitively what it does by comparing the weak head reductions of φν and T φν, whereφand ν are ordinary λ-terms, ν 'β λfλx(f) nx (a Church integer). Then T φν Â (φ)(s) n0, which m... |