A Linear-non-Linear Model for a Computational Call-by-Value Lambda Calculus (Extended Abstract)

  • Peter Selinger
  • Benoît Valiron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4962)

Abstract

We give a categorical semantics for a call-by-value linear lambda calculus. Such a lambda calculus was used by Selinger and Valiron as the backbone of a functional programming language for quantum computation. One feature of this lambda calculus is its linear type system, which includes a duplicability operator “!” as in linear logic. Another main feature is its call-by-value reduction strategy, together with a side-effect to model probabilistic measurements. The “!” operator gives rise to a comonad, as in the linear logic models of Seely, Bierman, and Benton. The side-effects give rise to a monad, as in Moggi’s computational lambda calculus. It is this combination of a monad and a comonad that makes the present paper interesting. We show that our categorical semantics is sound and complete.

References

  1. 1.
    Abramsky, S.: Computational interpretations of linear logic. Theoretical Computer Science 111, 3–57 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of LICS 2004, pp. 415–425 (2004)Google Scholar
  3. 3.
    Barendregt, H.P.: The Lambda-Calculus, its Syntax and Semantics. North Holland, Amsterdam (1984)MATHGoogle Scholar
  4. 4.
    Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models (extended abstract). In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 121–135. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. 5.
    Benton, N., Bierman, G., de Paiva, V.C.V., Hyland, M.: A term calculus for intuitionistic linear logic. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 75–90. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  6. 6.
    Benton, N., Bierman, G., Hyland, M., de Paiva, V.C.V.: Linear lambda-calculus and categorical models revisited. In: Martini, S., Börger, E., Kleine Büning, H., Jäger, G., Richter, M.M. (eds.) CSL 1992. LNCS, vol. 702, Springer, Heidelberg (1993)Google Scholar
  7. 7.
    Benton, N., Wadler, P.: Linear logic, monads and the lambda calculus. In: Proceedings of LICS 1996, pp. 420–431 (1996)Google Scholar
  8. 8.
    Bierman, G.: On Intuitionistic Linear Logic. PhD thesis, Computer Science Department, Cambridge University, Cambridge (1993)Google Scholar
  9. 9.
    Coecke, B.: Quantum information-flow, concretely, abstractly. In: Selinger, P., (ed.) Proceedings of QPL 2004. TUCS General Publication No. 33, Turku Centre for Computer Science pp. 57–73 (2004) Google Scholar
  10. 10.
    Coecke, B., Pavlovic, D.: Quantum measurements without sums. In: Chen, G., Kauffman, L., Lomonaco, S.J. (eds.) Mathematics of Quantum Computation and Technology, pp. 559–596. Chapman & Hall, Boca Raton (2007)Google Scholar
  11. 11.
    Gay, S.J., Nagarajan, R.: Communicating quantum processes. In: Proceedings of POPL 2005, ACM Press, New York (2005)Google Scholar
  12. 12.
    Lalire, M., Jorrand, P.: A process algebraic approach to concurrent and distributed computation: Operational semantics. In: Selinger, P. (ed.) Proceedings of QPL 2004. TUCS General Publication No. 33, Turku Centre for Computer Science, pp. 109–126 (2004)Google Scholar
  13. 13.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1998)MATHGoogle Scholar
  14. 14.
    Moggi, E.: Notions of computation and monads. Information and Computation 93, 55–92 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Schalk, A.: What is a model for linear logic. Manuscript (2004)Google Scholar
  16. 16.
    Seely, R.A.G.: *-autonomous categories and cofree coalgebras. Contemporary Mathematics 92 (1989)Google Scholar
  17. 17.
    Selinger, P. (ed.): Proceedings of QPL 2004. TUCS General Publication No. 33, Turku Centre for Computer Science (2004)Google Scholar
  18. 18.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14, 527–586 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Selinger, P., Valiron, B.: A lambda calculus for quantum computation with classical control. Mathematical Structures in Computer Science 16, 527–552 (2006)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Selinger, P., Valiron, B.: On a fully abstract model for a quantum linear functional language. In: Preliminary proceedings of QPL 2006, pp. 103–115 (2006)Google Scholar
  21. 21.
    van Tonder, A.: A lambda calculus for quantum computation. SIAM Journal of Computing 33, 1109–1135 (2004)MATHCrossRefGoogle Scholar
  22. 22.
    Wadler, P.: There’s no substitute for linear logic. Manuscript, presented at MFPS 1992 (1992)Google Scholar
  23. 23.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Peter Selinger
    • 1
  • Benoît Valiron
    • 2
  1. 1.Dalhousie University 
  2. 2.University of Ottawa 

Personalised recommendations