Advertisement

Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi

  • Benoît Valiron
  • Steve Zdancewic
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8687)

Abstract

In this paper we use finite vector spaces (finite dimension, over finite fields) as a non-standard computational model of linear logic. We first define a simple, finite PCF-like lambda-calculus with booleans, and then we discuss two finite models, one based on finite sets and the other on finite vector spaces. The first model is shown to be fully complete with respect to the operational semantics of the language. The second model is not complete, but we develop an algebraic extension of the finite lambda calculus that recovers completeness. The relationship between the two semantics is described, and several examples based on Church numerals are presented.

Keywords

Vector Space Operational Semantic Linear Logic Algebraic Extension Lambda Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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. and Comp. 163, 409–470 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arrighi, P., Díaz-Caro, A., Valiron, B.: A type system for the vectorial aspects of the linear-algebraic lambda-calculus. In: Proc. of DCM (2011)Google Scholar
  3. 3.
    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
  4. 4.
    Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models. Technical report, Cambridge U (1994)Google Scholar
  5. 5.
    Bierman, G.: On Intuitionistic Linear Logic. PhD thesis, Cambridge U (1993)Google Scholar
  6. 6.
    Bucciarelli, A., Ehrhard, T., Manzonetto, G.: A relational semantics for parallelism and non-determinism in a functional setting. A. of Pure and App. Logic 163, 918–934 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Groote, P.: Strong normalization in a non-deterministic typed lambda-calculus. In: Nerode, A., Matiyasevich, Y.V. (eds.) Logical Foundations of Computer Science. LNCS, vol. 813, pp. 142–152. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  8. 8.
    Díaz-Caro, A.: Du Typage Vectoriel. PhD thesis, U. de Grenoble (2011)Google Scholar
  9. 9.
    Ehrhard, T.: Finiteness spaces. Math. Str. Comp. Sc. 15, 615–646 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ehrhard, T., Pagani, M., Tasson, C.: The computational meaning of probabilistic coherence spaces. In: Proc. of LICS (2011)Google Scholar
  11. 11.
    Ehrhard, T., Regnier, L.: The differential lambda-calculus. Th. Comp. Sc. 309, 1–41 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Girard, J.-Y.: Linear logic. Th. Comp. Sc. 50, 1–101 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Girard, J.-Y., Lafont, Y., Taylor, P.: Proof and Types. CUP (1990)Google Scholar
  14. 14.
    Hillebrand, G.G.: Finite Model Theory in the Simply Typed Lambda Calculus. PhD thesis, Brown University (1991)Google Scholar
  15. 15.
    Hyland, M., Schalk, A.: Glueing and orthogonality for models of linear logic. Th. Comp. Sc. 294, 183–231 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    James, R.P., Ortiz, G., Sabry, A.: Quantum computing over finite fields. Draft (2011)Google Scholar
  17. 17.
    Lambek, J., Scott, P.J.: Introduction to Higher-Order Categorical Logic. CUP (1994)Google Scholar
  18. 18.
    Lang, S.: Algebra. Springer (2005)Google Scholar
  19. 19.
    Lidl, R.: Finite fields, vol. 20. CUP (1997)Google Scholar
  20. 20.
    Mac Lane, S.: Categories for the Working Mathematician. Springer (1998)Google Scholar
  21. 21.
    Milner, R.: Fully abstract models of typed lambda-calculi. Th. Comp. Sc. 4, 1–22 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Plotkin, G.: LCF considered as a programming language. Th. Comp. Sc. 5, 223–255 (1977)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pratt, V.R.: Re: Linear logic semantics (barwise). On the TYPES list (February 1992), http://www.seas.upenn.edu/~sweirich/types/archive/1992/msg00047.html
  24. 24.
    Pratt, V.R.: Chu spaces: Complementarity and uncertainty in rational mechanics. Technical report, Stanford U (1994)Google Scholar
  25. 25.
    Salvati, S.: Recognizability in the simply typed lambda-calculus. In: Ono, H., Kanazawa, M., de Queiroz, R. (eds.) WoLLIC 2009. LNCS, vol. 5514, pp. 48–60. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  26. 26.
    Schumacher, B., Westmoreland, M.D.: Modal quantum theory. In: Proc. of QPL (2010)Google Scholar
  27. 27.
    Scott, D.S.: A type-theoretic alternative to CUCH, ISWIM, OWHY. Th. Comp. Sc. 121, 411–440 (1993)CrossRefzbMATHGoogle Scholar
  28. 28.
    Selinger, P.: Order-incompleteness and finite lambda reduction models. Th. Comp. Sc. 309, 43–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Soloviev, S.: Category of finite sets and cartesian closed categories. J. of Soviet Math. 22(3) (1983)Google Scholar
  30. 30.
    Valiron, B.: A typed, algebraic, computational lambda-calculus. Math. Str. Comp. Sc. 23, 504–554 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Vaux, L.: The algebraic lambda-calculus. Math. Str. Comp. Sc. 19, 1029–1059 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Winskel, G.: The Formal Semantics of Programming Languages. MIT Press (1993)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Benoît Valiron
    • 1
  • Steve Zdancewic
    • 2
  1. 1.PPS, UMR 7126, Univ Paris Diderot, Sorbonne Paris CitéParisFrance
  2. 2.University of PennsylvaniaPhiladelphiaUS

Personalised recommendations