Skip to main content
SpringerLink
Log in

A Semantic Proof of Polytime Soundness of Light Affine Logic

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We define realizability semantics for Light Affine Logic ( \(\mathsf{LAL}\) ) which has the property that denotations of functions are polynomial time computable by construction of the model. This gives a new proof of polytime-soundness of \(\mathsf{LAL}\) which is considerably simpler than the standard proof based on proof nets and is entirely semantical in nature. The model construction uses a new instance of a resource monoid; a general method for interpreting systems based on Linear Logic introduced earlier by the authors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amadio, R.M.: Max-plus quasi-interpretations. In: Proc. of the 7th International Conference on Typed Lambda Calculi and Applications. LNCS, vol. 2701, pp. 31–45. Springer, Berlin (2003)

    Chapter  Google Scholar 

  2. Asperti, A., Roversi, L.: Intuitionistic light affine logic. ACM Trans. Comput. Log. 3(1), 137–175 (2002)

    Article  MathSciNet  Google Scholar 

  3. Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics. North Holland, Amsterdam (1984)

    MATH  Google Scholar 

  4. Bellantoni, S., Niggl, K.H., Schwichtenberg, H.: Higher type recursion, ramification and polynomial time. Ann. Pure Appl. Logic 104, 17–30 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cook, S., Urquhart, A.: Functional interpretations of feasible constructive arithmetic. Ann. Pure Appl. Logic 63(2), 103–200 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Coppola, P., Martini, S.: Typing lambda terms in elementary logic with linear constraints. In: Proc. of the 6th International Conference on Typed Lambda Calculi and Applications. LNCS, vol. 2044, pp. 76–90. Springer, Berlin (2001)

    Chapter  Google Scholar 

  7. Crossley, J., Mathai, G., Seely, R.: A logical calculus for polynomial-time realizability. J. Methods Logic Comput. Sci. 3, 279–298 (1994)

    MathSciNet  Google Scholar 

  8. Dal Lago, U., Hofmann, M.: Quantitative models and implicit complexity. In: Proc. Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 3821, pp. 189–200. Springer, Berlin (2005)

    Chapter  Google Scholar 

  9. Dal Lago, U., Martini, S.: Phase semantics and decidability of elementary affine logic. Theor. Comput. Sci. 318(3), 409–433 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dal Lago, U., Martini, S.: The weak lambda calculus as a reasonable machine. Theor. Comput. Sci. 398(1–3), 32–50 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Girard, J.-Y.: Light linear logic. Inf. Comput. 143(2), 175–204 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Girard, J.-Y., Lafont, Y., Taylor, P.: Proof and Types. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  13. Hofmann, M.: Linear types and non-size-increasing polynomial time computation. In: Proc. of the 14th IEEE Syposium on Logic in Computer Science, pp. 464–473 (1999)

  14. Hofmann, M.: Safe recursion with higher types and BCK-algebra. Ann. Pure Appl. Logic 104, 113–166 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hofmann, M., Scott, P.: Realizability models for BLL-like languages. Theor. Comput. Sci. 318(1–2), 121–137 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kreisel, G.: Interpretation of analysis by means of constructive functions of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, pp. 101–128. North-Holland, Amsterdam (1959)

    Google Scholar 

  17. Lafont, Y.: Soft linear logic and polynomial time. Theor. Comput. Sci. 318, 163–180 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Murawski, A.S., Ong, C.-H.L.: Discreet games, light affine logic and ptime computation. In: Proc. of 14th International Workshop on Computer Science Logic. LNCS, vol. 1862, pp. 427–441. Springer, Berlin (2000)

    Chapter  Google Scholar 

  19. Roversi, L.: A P-time completeness proof for light logics. In: Proc. of 13th International Workshop on Computer Science Logic. LNCS, vol. 1683, pp. 469–483. Springer, Berlin (1999)

    Chapter  Google Scholar 

  20. van Emde Boas, P.: Machine models and simulation. In: Handbook of Theoretical Computer Science. Algorithms and Complexity, vol. A, pp. 1–66. MIT Press, Cambridge (1990)

    Google Scholar 

  21. Wadsworth, C.: Some unusual λ-calculus numeral systems. In: Seldin, J.P., Hindley, J.R. (eds.) To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, San Diego (1980)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze dell’Informazione, Università di Bologna, Bologna, Italy

    Ugo Dal Lago

  2. Institut für Informatik, LMU München, Munich, Germany

    Martin Hofmann

Authors
  1. Ugo Dal Lago
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Hofmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ugo Dal Lago.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dal Lago, U., Hofmann, M. A Semantic Proof of Polytime Soundness of Light Affine Logic. Theory Comput Syst 46, 673–689 (2010). https://doi.org/10.1007/s00224-009-9210-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9210-x

Keywords

Search

Navigation