Advertisement

Focalisation and Classical Realisability

  • Guillaume Munch-Maccagnoni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5771)

Abstract

We develop a polarised variant of Curien and Herbelin’s \(\bar{\lambda}\mu\tilde{\mu}\) calculus suitable for sequent calculi that admit a focalising cut elimination (i.e. whose proofs are focalised when cut-free), such as Girard’s classical logic LC or linear logic. This gives a setting in which Krivine’s classical realisability extends naturally (in particular to call-by-value), with a presentation in terms of orthogonality. We give examples of applications to the theory of programming languages.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Curien, P.L., Herbelin, H.: The duality of computation. ACM SIGPLAN Notices 35, 233–243 (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Herbelin, H.: Duality of computation and sequent calculus: a few more remarks (manuscript, 2008)Google Scholar
  3. 3.
    Andreoli, J.M.: Logic programming with focusing proof in linear logic. Journal of Logic and Computation 2, 297–347 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Girard, J.Y.: A new constructive logic: Classical logic. Math. Struct. Comp. Sci. (1991)Google Scholar
  5. 5.
    Wadler, P.: Call-by-value is dual to call-by-name. SIGPLAN Not. 38, 189–201 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dyckhoff, R., Lengrand, S.: LJQ: A strongly focused calculus for intuitionistic logic. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 173–185. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Zeilberger, N.: On the unity of duality. Ann. Pure and App. Logic 153, 1 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Krivine, J.L.: Realizability in classical logic. To appear in Panoramas et synthéses, Société Mathématique de France (2004)Google Scholar
  9. 9.
    Terui, K.: Computational Ludics. To appear in TCS (2008)Google Scholar
  10. 10.
    Danos, V., Joinet, J.B., Schellinx, H.: A new deconstructive logic: Linear logic. Journal of Symbolic Logic 62 (3), 755–807 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Laurent, O.: Etude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille II (2002)Google Scholar
  12. 12.
    Girard, J.Y.: Locus solum: From the rules of logic to the logic of rules. Mathematical Structures in Computer Science 11, 301–506 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Munch-Maccagnoni, G.: Étude polarisée du système L. Master’s thesis (2008)Google Scholar
  14. 14.
    Curien, P.L., Munch-Maccagnoni, G.: The duality of computation under focus (2009)Google Scholar
  15. 15.
    Nipkow, T.: Higher-order critical pairs. In: Proc. 6th IEEE Symp. Logic in Computer Science, pp. 342–349. IEEE Press, Los Alamitos (1991)Google Scholar
  16. 16.
    Krivine, J.L.: Lambda-calculus, types and models. Ellis Horwood (1993)Google Scholar
  17. 17.
    Lengrand, S., Miquel, A.: Classical Fω, orthogonality and symmetric candidates. Ann. Pure Appl. Logic 153(1-3), 3–20 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zeilberger, N.: Refinement types and computational duality. In: PLPV 2009 (2009)Google Scholar
  19. 19.
    Girard, J.Y.: Le Point Aveugle, Cours de logique, Tome II: Vers l’imperfection. Visions des Sciences. Hermann (2007)Google Scholar
  20. 20.
    Beffara, E., Danos, V.: Disjunctive normal forms and local exceptions. In: ACM SIGPLAN Int. Conf. Func, Prog., Uppsala, Sweden, pp. 203–211 (2003)Google Scholar
  21. 21.
    Krivine, J.L.: Structures de réalisabilité, RAM et ultrafiltre sur N (to appear) (2008)Google Scholar
  22. 22.
    Herbelin, H.: C’est maintenant qu’on calcule, au cœur de la dualité, Habilitation thesis (2005)Google Scholar
  23. 23.
    Girard, J.Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Guillaume Munch-Maccagnoni
    • 1
  1. 1.Université Paris 7France

Personalised recommendations