Solvability in Resource Lambda-Calculus

  • Michele Pagani
  • Simona Ronchi della Rocca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6014)


The resource calculus is an extension of the λ-calculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a formal sum. Using the λ-calculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the non-determinism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability.


  1. 1.
    Boudol, G.: The Lambda-Calculus with Multiplicities. INRIA Report 2025 (1993)Google Scholar
  2. 2.
    Ehrhard, T., Regnier, L.: The Differential Lambda-Calculus. Theor. Comput. Sci. 309(1), 1–41 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Tranquilli, P.: Intuitionistic Differential Nets and Lambda-Calculus. Theor. Comput. Sci. (2008) (to appear)Google Scholar
  4. 4.
    de’Liguoro, U., Piperno, A.: Non Deterministic Extensions of Untyped Lambda-Calculus. Inf. Comput. 122(2), 149–177 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ehrhard, T., Regnier, L.: Böhm trees, Krivine’s Machine and the Taylor Expansion of Lambda-Terms. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 186–197. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Ehrhard, T., Regnier, L.: Uniformity and the Taylor Expansion of Ordinary Lambda-Terms. Theor. Comput. Sci. 403(2-3), 347–372 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Pagani, M., Tranquilli, P.: Parallel Reduction in Resource Lambda-Calculus. In: Hu, Z. (ed.) APLAS 2009. LNCS, vol. 5904, pp. 226–242. Springer, Heidelberg (2009)Google Scholar
  8. 8.
    Barendregt, H.: The Lambda-Calculus, its Syntax and Semantics, 2nd edn. Stud. Logic Found. Math., vol. 103. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  9. 9.
    Coppo, M., Dezani-Ciancaglini, M., Venneri, B.: Functional Characters of Solvable Terms. Zeitschrift für Mathematische Logik 27, 45–58 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hyland, J.M.E.: A Syntactic Characterization of the Equality in Some Models of the Lambda Calculus. J. London Math. Soc. 2(12), 361–370 (1976)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ronchi Della Rocca, S., Paolini, L.: The Parametric λ-Calculus: a Metamodel for Computation. EATCS Series. Springer, Berlin (2004)Google Scholar
  12. 12.
    de Carvalho, D.: Execution Time of λ-Terms via Denotational Semantics and Intersection Types (2009) (submitted for publication)Google Scholar
  13. 13.
    Bucciarelli, A., Ehrhard, T., Manzonetto, G.: Not Enough Points Is Enough. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 298–312. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Tranquilli, P.: Nets between Determinism and Nondeterminism. Ph.D. thesis, Università Roma Tre/Université Paris Diderot (Paris 7) (April 2009)Google Scholar
  15. 15.
    Boudol, G., Curien, P.L., Lavatelli, C.: A Semantics for Lambda Calculi with Resources. MSCS 9(5), 437–482 (1999)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Coppo, M., Dezani-Ciancaglini, M., Venneri, B.: Principal Type Schemes and Lambda-Calculus Semantics. In: Curry, To H.B. (ed.) Essays on Combinatory Logic, Lambda-calculus and Formalism, pp. 480–490. Academic Press, London (1980)Google Scholar
  17. 17.
    Kfoury, A.J.: A Linearization of the Lambda-Calculus and Consequences. Journal of Logic and Computation 10(3), 411–436 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wells, J.B., Dimock, A., Muller, R., Turbak, F.: A Calculus with Polymorphic and Polyvariant Flow Types. J. Funct. Program. 12(3), 183–227 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Neergaard, P.M., Mairson, H.G.: Types, Potency, and Idempotency: why Nonlinearity and Amnesia Make a Type System Work. In: ICFP, pp. 138–149. ACM, New York (2004)CrossRefGoogle Scholar
  20. 20.
    Coppo, M., Dezani-Ciancaglini, M., Zacchi, M.: Type Theories, Normal Forms and D  ∞ -Lambda-Models. Inf. Comput. 72(2), 85–116 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Girard, J.Y.: Interprétation Fonctionnelle et Élimination des Coupures de l’Arithmétique d’Ordre Supérieur. Thèse de doctorat, Université Paris 7 (1972)Google Scholar
  22. 22.
    Valentini, S.: An elementary proof of strong normalization for intersection types. Archive for Mathematical Logic 40(7), 475–488 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    de Carvalho, D., Pagani, M., Tortora de Falco, L.: A Semantic Measure of the Execution Time in Linear Logic. Theor. Comput. Sci. (2008) (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michele Pagani
    • 1
  • Simona Ronchi della Rocca
    • 1
  1. 1.Dipartimento di InformaticaUniversità di TorinoTorino(IT)

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