Call-by-Value Non-determinism in a Linear Logic Type Discipline

  • Alejandro Díaz-Caro
  • Giulio Manzonetto
  • Michele Pagani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7734)


We consider the call-by-value λ-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent intersection types, we endow this calculus with a type system based on the so-called Girard’s second translation of intuitionistic logic into linear logic. We prove that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy call-by-value reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction.


λ-calculus linear logic non-determinism call-by-value 


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  1. 1.
    Amadio, R., Curien, P.L.: Domains and Lambda-Calculi. Cambridge Tracts in Theoretical Computer Science, vol. 46. Cambridge University Press (1998)Google Scholar
  2. 2.
    Arrighi, P., Díaz-Caro, A.: A System F accounting for scalars. Logical Methods in Computer Science 8(1:11) (2012)Google Scholar
  3. 3.
    Arrighi, P., Díaz-Caro, A., Valiron, B.: A type system for the vectorial aspects of the linear-algebraic λ-calculus. In: DCM 2011. EPTCS, vol. 88, pp. 1–15 (2012)Google Scholar
  4. 4.
    Arrighi, P., Dowek, G.: Linear-Algebraic λ-Calculus: Higher-Order, Encodings, and Confluence. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 17–31. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Barendregt, H.: The lambda calculus: its syntax and semantics. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  6. 6.
    Bernadet, A., Lengrand, S.: Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 88–107. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Boudol, G.: Lambda-calculi for (strict) parallel functions. Information and Computation 108(1), 51–127 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Breuvart, F.: On the discriminating power of tests in the resource λ-calculus (submitted), Draft available at
  9. 9.
    Bucciarelli, A., Ehrhard, T., Manzonetto, G.: A relational semantics for parallelism and non-determinism in a functional setting. APAL 163(7), 918–934 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coppo, M., Dezani-Ciancaglini, M.: A new type-assignment for λ-terms. Archiv für Math. Logik 19, 139–156 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    de Carvalho, D.: Execution time of lambda-terms via denotational semantics and intersection types. INRIA Report RR-6638,, To appear in Math. Struct. in Comp. Sci. (2008)
  12. 12.
    Dezani-Ciancaglini, M., de’Liguoro, U., Piperno, A.: Filter models for conjunctive-disjunctive lambda-calculi. Theor. Comp. Sci. 170(1-2), 83–128 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dezani-Ciancaglini, M., de’Liguoro, U., Piperno, A.: A filter model for concurrent lambda-calculus. SIAM J. Comput. 27(5), 1376–1419 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Ehrhard, T.: Collapsing non-idempotent intersection types. In: CSL 2012. LIPIcs, vol. 16, pp. 259–273 (2012)Google Scholar
  15. 15.
    Girard, J.Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Krivine, J.L.: Lambda-calcul: types et modèles. Études et recherches en informatique, Masson (1990)Google Scholar
  17. 17.
    Laurent, O.: Étude de la polarisation en logique. PhD thesis, Université de Aix-Marseille II, France (2002)Google Scholar
  18. 18.
    Manzonetto, G.: A General Class of Models of \(\mathcal{H}^*\). In: Královič, R., Niwiński, D. (eds.) MFCS 2009. LNCS, vol. 5734, pp. 574–586. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Maraist, J., Odersky, M., Turner, D.N., Wadler, P.: Call-by-name, call-by-value, call-by-need and the linear λ-calculus. Theor. Comp. Sci. 228(1-2), 175–210 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Pagani, M., Ronchi Della Rocca, S.: Linearity, non-determinism and solvability. Fundam. Inform. 103(1-4), 173–202 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Plotkin, G.D.: Call-by-name, call-by-value and the λ-calculus. Theor. Comp. Sci. 1(2), 125–159 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Sallé, P.: Une généralisation de la théorie de types en λ-calcul. RAIRO: Informatique Théorique 14(2), 143–167 (1980)zbMATHGoogle Scholar
  23. 23.
    Vaux, L.: The algebraic lambda calculus. Math. Struct. in Comp. Sci. 19(5), 1029–1059 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alejandro Díaz-Caro
    • 1
  • Giulio Manzonetto
    • 1
    • 2
  • Michele Pagani
    • 1
    • 2
  1. 1.Université Paris 13, Sorbonne Paris Cité, LIPNVilletaneuseFrance
  2. 2.CNRS, UMR 7030VilletaneuseFrance

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