Linearity in the Non-deterministic Call-by-Value Setting

  • Alejandro Díaz-Caro
  • Barbara Petit
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7456)

Abstract

We consider the non-deterministic extension of the call-by-value lambda calculus, which corresponds to the additive fragment of the linear-algebraic lambda-calculus. We define a fine-grained type system, capturing the right linearity present in such formalisms. After proving the subject reduction and the strong normalisation properties, we propose a translation of this calculus into the System F with pairs, which corresponds to a non linear fragment of linear logic. The translation provides a deeper understanding of the linearity in our setting.

Keywords

Linear Logic Unit Type Typing Rule Strong Normalisation Lambda Calculus 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boudol, G.: Lambda-calculi for (strict) parallel functions. Information and Computation 108(1), 51–127 (1994)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bucciarelli, A., Ehrhard, T., Manzonetto, G.: A relational semantics for parallelism and non-determinism in a functional setting. Annals of Pure and Applied Logic 163(7), 918–934 (2012)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Dezani-Ciancaglini, M., de’Liguoro, U., Piperno, A.: Filter models for conjunctive-disjunctive lambda-calculi. Theoretical Computer Science 170(1-2), 83–128 (1996)MathSciNetMATHGoogle Scholar
  4. 4.
    Dezani-Ciancaglini, M., de’Liguoro, U., Piperno, A.: A filter model for concurrent lambda-calculus. SIAM Journal on Computing 27(5), 1376–1419 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Arrighi, P., Dowek, G.: Linear-algebraic Lambda-calculus: Higher-order, Encodings, and Confluence. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 17–31. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Vaux, L.: The algebraic lambda calculus. Mathematical Structures in Computer Science 19(5), 1029–1059 (2009)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Assaf, A., Perdrix, S.: Completeness of algebraic cps simulations. In: Proceedings of the 7th International Workshop on Developments of Computational Models (DCM 2011), Zurich, Switzerland (2011), http://www.pps.univ-paris-diderot.fr/~jkrivine/conferences/DCM2011/DCM_2011.html
  8. 8.
    Díaz-Caro, A., Perdrix, S., Tasson, C., Valiron, B.: Equivalence of algebraic λ-calculi. In: Informal Proceedings of the 5th International Workshop on Higher-Order Rewriting, HOR 2010, Edinburgh, UK, pp. 6–11 (July 2010)Google Scholar
  9. 9.
    Hennessy, M.: The semantics of call-by-value and call-by-name in a nondeterministic environment. SIAM Journal on Computing 9(1), 67–84 (1980)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Girard, J.Y.: Linear logic. Theoretical Compututer Science 50, 1–102 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Cosmo, R.D.: Isomorphisms of Types: From Lambda-Calculus to Information Retrieval and Language Design. Progress in Theoretical Computer Science. Birkhäuser (1995)Google Scholar
  12. 12.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)CrossRefGoogle Scholar
  13. 13.
    Monroe, C., Meekhof, D.M., King, B.E., Itano, W.M., Wineland, D.J.: Demonstration of a fundamental quantum logic gate. Physical Review Letters 75(25), 4714–4717 (1995)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Plotkin, G.D.: Call-by-name, call-by-value and the λ-calculus. Theoretical Computer Science 1(2), 125–159 (1975)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jouannaud, J.P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM Journal on Computing 15(4), 1155–1194 (1986)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Barendregt, H.P.: The lambda calculus: its syntax and semantics. Studies in Logic and the Foundations of Mathematics, vol. 103. Elsevier (1984)Google Scholar
  17. 17.
    Díaz-Caro, A., Dowek, G.: Non determinism through type isomorphism. Draft (April 2012), http://diaz-caro.info/ndti.pdf
  18. 18.
    Barendregt, H.P.: Lambda calculi with types. In: Handbook of Logic in Computer Science, vol. 2. Oxford University Press (1992)Google Scholar
  19. 19.
    Girard, J.Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge University Press (1989)Google Scholar
  20. 20.
    Krivine, J.L.: Lambda-calcul: types et modèles. Études et recherches en informatique, Masson (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Alejandro Díaz-Caro
    • 1
  • Barbara Petit
    • 2
  1. 1.Université Paris 13, Sorbonne Paris Cité, LIPNVilletaneuseFrance
  2. 2.FOCUS (INRIA) – Università di BolognaItaly

Personalised recommendations