Advertisement

Parsimonious Types and Non-uniform Computation

  • Damiano MazzaEmail author
  • Kazushige Terui
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)

Abstract

We consider a non-uniform affine lambda-calculus, called parsimonious, and endow its terms with two type disciplines: simply-typed and with linear polymorphism. We show that the terms of string type into Boolean type characterize the class L/poly in the first case, and P/poly in the second. Moreover, we relate this characterization to that given by the second author in terms of Boolean proof nets, highlighting continuous affine approximations as the bridge between the two approaches to non-uniform computation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramsky, S., Jagadeesan, R., Malacaria, P.: Full abstraction for PCF. Inf. Comput. 163(2), 409–470 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Baillot, P., Terui, K.: Light types for polynomial time computation in lambda calculus. Inf. Comput. 207(1), 41–62 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellantoni, S., Cook, S.A.: A new recursion-theoretic characterization of the polytime functions. Computational Complexity 2, 97–110 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Ehrhard, T., Regnier, L.: Differential interaction nets. Electr. Notes Theor. Comput. Sci. 123, 35–74 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ehrhard, T., Regnier, L.: Uniformity and the taylor expansion of ordinary lambda-terms. Theor. Comput. Sci. 403(2–3), 347–372 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gaboardi, M., Péchoux, R.: Upper bounds on stream I/O using semantic interpretations. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 271–286. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  7. 7.
    Ghica, D.R.: Geometry of synthesis: a structured approach to VLSI design. In: Proceedings of POPL, pp. 363–375 (2007)Google Scholar
  8. 8.
    Girard, J.Y.: Geometry of interaction I: Interpretation of system F. Proccedings of Logic Colloquium 1988, 221–260 (1989)MathSciNetGoogle Scholar
  9. 9.
    Girard, J.Y.: Light linear logic. Inf. Comput. 143(2), 175–204 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kfoury, A.J.: A linearization of the lambda-calculus and consequences. J. Log. Comput. 10(3), 411–436 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Leivant, D., Marion, J.Y.: Lambda calculus characterizations of poly-time. Fundam. Inform. 19(1/2) (1993)Google Scholar
  12. 12.
    Mazza, D.: An infinitary affine lambda-calculus isomorphic to the full lambda-calculus. In: Proceedings of LICS, pp. 471–480 (2012)Google Scholar
  13. 13.
    Mazza, D.: Non-uniform polytime computation in the infinitary affine lambda-calculus. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part II. LNCS, vol. 8573, pp. 305–317. Springer, Heidelberg (2014) Google Scholar
  14. 14.
    Mazza, D.: Simple parsimonious types and logarithmic space (2015), available on the author’s web pageGoogle Scholar
  15. 15.
    Melliès, P.A.: Asynchronous games 2: The true concurrency of innocence. Theor. Comput. Sci. 358(2–3), 200–228 (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Melliès, P.-A., Tabareau, N., Tasson, C.: An explicit formula for the free exponential modality of linear logic. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part II. LNCS, vol. 5556, pp. 247–260. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  17. 17.
    Ramyaa, R., Leivant, D.: Ramified corecurrence and logspace. Electr. Notes Theor. Comput. Sci. 276, 247–261 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Saurin, A.: Typing streams in the \(\Lambda \mu \)-calculus. ACM Trans. Comput. Log. 11(4) (2010)Google Scholar
  19. 19.
    Terui, K.: Proof nets and boolean circuits. In: Proceedings of LICS, pp. 182–191 (2004)Google Scholar
  20. 20.
    Vollmer, H.: Introduction to circuit complexity - a uniform approach. Texts in theoretical computer science. Springer (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CNRS, UMR 7030, LIPNUniversité Paris 13, Sorbonne Paris CitéVilletaneuseFrance
  2. 2.RIMSKyoto UniversityKyotoJapan

Personalised recommendations