On Bar Recursion and Choice in a Classical Setting

  • Valentin Blot
  • Colin Riba
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8301)

Abstract

We show how Modified Bar-Recursion, a variant of Spector’s Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Countable Choice in Parigot’s Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs.

We rely on Hyland-Ong innocent games. They provide a model for the instances of the axiom of choice usually used in the realization of classical choice with Bar-Recursion, and where, moreover, the standard datatype of natural numbers is in the image of a CPS-translation.

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References

  1. 1.
    Abramsky, S., McCusker, G.: Call-by-Value Games. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 1–17. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  2. 2.
    Amadio, R.M., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press (1998)Google Scholar
  3. 3.
    Berardi, S., Bezem, M., Coquand, T.: On the Computational Content of the Axiom of Choice. Journal of Symbolic Logic 63(2), 600–622 (1998)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Berger, U., Oliva, P.: Modified bar recursion and classical dependent choice. Lecture Notes in Logic 20, 89–107 (2005)MathSciNetGoogle Scholar
  5. 5.
    Blot, V.: Realizability for peano arithmetic with winning conditions in HON games. In: Hasegawa, M. (ed.) TLCA 2013. LNCS, vol. 7941, pp. 77–92. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  6. 6.
    Griffin, T.: A Formulae-as-Types Notion of Control. In: POPL 1990, pp. 47–58. ACM Press (1990)Google Scholar
  7. 7.
    Harmer, R.: Games and Full Abstraction for Nondeterministic Languages. PhD thesis, Imperial College, London (1999)Google Scholar
  8. 8.
    Hyland, J.M.E., Ong, C.-H.: On Full Abstraction for PCF: I, II, and III. Information and Computation 163(2), 285–408 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer (2008)Google Scholar
  10. 10.
    Krivine, J.-L.: Realizability in classical logic. In: Interactive Models of Computation and Program Behaviour. Panoramas et synthèses, vol. 27, pp. 197–229. Société Mathématique de France (2009)Google Scholar
  11. 11.
    Laird, J.: A Semantic analysis of control. PhD thesis, University of Edimbourgh (1998)Google Scholar
  12. 12.
    Miquel, A.: Existential witness extraction in classical realizability and via a negative translation. Logical Methods in Computer Science 7(2) (2011)Google Scholar
  13. 13.
    Oliva, P., Streicher, T.: On Krivine’s Realizability Interpretation of Classical Second-Order Arithmetic. Fundam. Inform. 84(2), 207–220 (2008)MathSciNetMATHGoogle Scholar
  14. 14.
    Parigot, M.: Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  15. 15.
    Selinger, P.: Control Categories and Duality: on the Categorical Semantics of the Lambda-Mu Calculus. Mathematical Structures in Computer Science 11, 207–260 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Simpson, S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Perspectives in Logic. Cambridge University Press (2010)Google Scholar
  17. 17.
    Streicher, T.: A Classical Realizability Model araising from a Stable Model of Untyped Lambda-Calculus (unpublished Notes, 2013)Google Scholar
  18. 18.
    Streicher, T., Reus, B.: Classical Logic, Continuation Semantics and Abstract Machines. J. Funct. Program. 8(6), 543–572 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Troelstra, A.S.: Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. LNM, vol. 344. Springer, Heidelberg (1973)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Valentin Blot
    • 1
  • Colin Riba
    • 1
  1. 1.ENS de LyonUniversité de Lyon, LIPFrance

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