Advertisement

Abstract

We describe a denotational semantics for a sequential functional language with random number generation over a countably infinite set (the natural numbers), and prove that it is fully abstract with respect to may-and-must testing.

Our model is based on biordered sets similar to Berry’s bidomains, and stable, monotone functions. However, (as in prior models of unbounded non-determinism) these functions may not be continuous. Working in a biordered setting allows us to exploit the different properties of both extensional and stable orders to construct a Cartesian closed category of sequential, discontinuous functions, with least and greatest fixpoints having strong enough properties to prove computational adequacy.

We establish full abstraction of the semantics by showing that it contains a simple, first-order “universal type-object” within which all types may be embedded using functions defined by (countable) ordinal induction.

Keywords

Operational Semantic Functional Language Denotational Semantic Stable Order Game Semantic 
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.

References

  1. 1.
    Apt, K.R., Plotkin, G.D.: Countable nondeterminism and random assignment. Journal of the ACM 33(4), 724–767 (1986)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berry, G.: Stable models of typed λ-calculi. In: Proceedings of the 5th International Colloquium on Automata, Languages and Programming. LNCS, vol. 62, pp. 72–89. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  3. 3.
    Cartwright, R., Felleisen, M.: Observable sequentiality and full abstraction. In: Proceedings of POPL 1992 (1992)Google Scholar
  4. 4.
    Curien, P.-L., Winskel, G., Plotkin, G.: Bistructures, bidomains and linear logic. In: Milner Festschrift, MIT Press, Cambridge (1997)Google Scholar
  5. 5.
    Di Gianantonio, P., Honsell, F., Plotkin, G.: Uncountable limits and the lambda calculus. Nordic Journal of Computing 2(2), 126–145 (1995)MathSciNetMATHGoogle Scholar
  6. 6.
    Harmer, R., McCusker, G.: A fully abstract games semantics for finite non-determinism. In: Proceedings of the Fourteenth Annual Symposium on Logic in Computer Science, LICS 1999, IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  7. 7.
    Laird, J.: Bistability: an extensional characterization of sequentiality. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Laird, J.: Sequentiality in bounded bidomains. Fundamenta Informaticae 65, 173–191 (2005)MathSciNetMATHGoogle Scholar
  9. 9.
    Lassen, S.B., Pitcher, C.: Similarity and bisimilarity for countable non-determinism and higher-order functions. Electronic Notes in Theoretical Computer Science 10 (1997)Google Scholar
  10. 10.
    Levy, P.B.: Infinite trace equivalence. In: Games for Logic and Programming Languages, pp. 195–209 (2005)Google Scholar
  11. 11.
    Longley, J.: Universal types and what they are good for. In: Domain Theory, Logic and Computation: Proceedings of the 2nd International Symposium on Domain Theory, Kluwer, Dordrecht (2004)Google Scholar
  12. 12.
    Plotkin, G.: Lectures on predomains and partial functions, Notes for a course given at the Center for the study of Language and Information, Stanford (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • James Laird
    • 1
  1. 1.Dept. of InformaticsUniversity of SussexUK

Personalised recommendations