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.

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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

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