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Convolution \(\bar\lambda\mu\)-Calculus

  • Lionel Vaux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)

Abstract

We define an extension of Herbelin’s \(\bar\lambda\mu\)-calculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of Ehrhard-Regnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambda-model in the category of sets and relations.

Keywords

Relational Semantic Classical Logic Intuitionistic Logic Linear Logic Reduction Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bucciarelli, A., Ehrhard, T.: An extensional model of the lambda-calculus in the category of sets and relations. Manuscript (2004)Google Scholar
  2. de Carvalho, D.: Execution time of lambda-terms via non uniform semantics and intersection types (2006) Revised version available at http://iml.univ-mrs.fr/c̃arvalho/Pub/computation.pdf
  3. Dougherty, D.J., Ghilezan, S., Lescanne, P.: Intersection and union types in the lambda-bar-mu-mu-tilde-calculus. Electr. Notes Theor. Comput. Sci. 136, 153–172 (2005)CrossRefGoogle Scholar
  4. Danos, V., Joinet, J.-B., Schellinx, H.: Sequent calculi for second order logic. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, pp. 211–224. Cambridge University Press, Cambridge (1995)Google Scholar
  5. Ehrhard, T.: On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science 12, 579–623 (2001)CrossRefGoogle Scholar
  6. Ehrhard, T.: Finiteness spaces. Mathematical. Structures in Comp. Sci. 15(4), 615–646 (2005)zbMATHCrossRefGoogle Scholar
  7. Ehrhard, T., Regnier, L.: The differential lambda-calculus. Theoretical Computer Science 309, 1–41 (2003)zbMATHCrossRefGoogle Scholar
  8. Ehrhard, T., Regnier, L.: Differential interaction nets. Electr. Notes Theor. Comput. Sci. 123, 35–74 (2005)CrossRefGoogle Scholar
  9. Herbelin, H.: Séquents qu’on calcule. Phd thesis, Université Paris 7 (1995)Google Scholar
  10. Lafont, Y.: From proof nets to interaction nets. In: Girard, J.-Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, pp. 225–247. Cambridge University Press, Cambridge (1995)Google Scholar
  11. [kLau02]
    Laurent, O.: Etude de la polarisation en logique. Thèse de doctorat, Université Aix-Marseille II (March 2002)Google Scholar
  12. Laurent, O.: Polarized proof-nets and λμ-calculus. Theoretical Computer Science 290(1), 161–188 (2003)zbMATHCrossRefGoogle Scholar
  13. Laurent, O., Regnier, L.: About translations of classical logic into polarized linear logic. In: Proceedings of the 18th annual IEEE symposium on Logic In Comp. Sci., pp. 11–20. IEEE Computer Society Press, Los Alamitos (2003)CrossRefGoogle Scholar
  14. Parigot, M.: λμ-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. Regnier, L.: Lambda-calcul et réseaux. PhD thesis, Université Paris 7 (1992)Google Scholar
  16. Schwartz, L.: Théorie des distributions. Hermann (1966)Google Scholar
  17. Vaux, L.: λ-calculus in an algebraic setting (2006) Research report, available at http://iml.univ-mrs.fr/Ṽaux/articles/alglam.ps.gz
  18. Vaux, L.: The differential λμ-calculus. Theoretical Computer Science (to appear, 2007) doi:10.1016/j.tcs.2007.02.028Google Scholar
  19. Vaux, L.: Polarized proof nets and differential structures. Unpublished manuscript (2007)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Lionel Vaux
    • 1
  1. 1.Institut de Mathématiques de Luminy, MarseilleFrance

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