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Some applications of topology to program semantics

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Abstract

The relationship between programs and the set of partial correctness assertions that they satisfy, constitutes a Galois connection. The topology resulting from this Galois connection is closely related to the Lindenbaum baum topology for the language in which these partial correctness assertions are stated. This relationship provides us with a tool for understanding the incompleteness of Hoare Logics and for answering certain natural questions about the connection between the relational semantics and the partial correctness assertion semantics for programs, especially in connection with the question of modularity of programs. Two questions to which we shall find topological answers in this paper are “When is a language expressive for a program?”, and “When can we have rules of inference which are adequate to infer the properties of the complex program ±#β from those of its components ±,β?” We also obtain a natural answer to the question “What can the set{(A, B)|{A}±{B} is true) look like for arbitraryα?”.

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References

  1. 1.

    G. Birkhoff,Lattice Theory, Amer. Math. Soc. Colloq. Publications vol. 25 (1940).

  2. 2.

    J. Bergstra, J. Tiuryn and J. Tucker, Correctness Theories and Program Equivalence,Stichting Mathematisch Centrum, Amsterdam (1979).

  3. 3.

    P. M. Cohn,Universal Algebra, Harper and Row (1965).

  4. 4.

    E. Clarke, Programming Language Constructs for which it is Impossible to Obtain Good Hoare-like Axiom Systems,Proc. 4th ACM Symposium on Principles of Programming Languages 1977, 10–20.

  5. 5.

    S. Cook, Soundness and Completeness of an Axiom System for Program Verification,SIAM J. Comp 7 (1978).

  6. 6.

    A. Ehrenfeucht, An Application of Games to the Completeness Problem for Formalised Theories,Fundamenta Mathematica 49 (1961) 129–141.

  7. 7.

    I. Greif and A. Meyer, Specifying the Semantics of While Programs,ACM TOPLAS 3 (1981) 484–507.

  8. 8.

    D. Harel,First Order Dynamic Logic, Springer Lecture Notes in Computer Science #68.

  9. 9.

    A. Hoare and P. Lauer, Consistent and Complementary Formal Theories of the Semantics of Programming Languages,Acta Informatica 3 (1974) pp. 135–155.

  10. 10.

    D. Kfoury and D. Park, On the Termination of Program Schemas,Information and Control, 29 (1975) 243–251.

  11. 11.

    D. Luckham, D. Park and M. Paterson, On Formalised Computer Programs,JCSS 3 (1970) pp. 220–249.

  12. 12.

    A. Meyer and J. Halpern, Axiomatic Definitions of Programming Languages: A Theoretical Assessment,Jour. Assoc. Comp. Mach. 29 (1982) 555–576.

  13. 13.

    A. Meyer and R. Parikh, Definability in Dynamic Logic,Proc 12th Annual ACM Symposium on Theory of Computation (1980) pp. 1–7. To appear inJCSS.

  14. 14.

    O. Ore, Galois Connexions,Trans. Amer. Math Soc. 55 (1944) pp. 493–513.

  15. 15.

    R. Parikh, Propositional Logics of Programs-Systems, Models and Complexity,7th Annual Symposium on Principles of Programming Languages, ACM (1980), pp. 186–192.

  16. 16.

    R. Parikh, Some Applications of Topology to Program Semantics, (an earlier version of the present paper).Logics of Programs (Ed. D. Kozen), Springer LNCS # 131 (1981) 375–86.

  17. 17.

    R. Parikh, Propositional Dynamics Logics of Programs: A Survey,Logic of Programs, (Ed. E. Engeler) Springer LNCS # 125, fall 1981, 102–144.

  18. 18.

    R. Parikh, Models for Programs,Proc. Bangalore Conference on Software Technology and Theoretical Computer Science, Published by TIFR, Bombay 1981.

  19. 19.

    M. Wand, A New Incompleteness Result for Hoare's System,Proc 8th ACM Symposium on the Theory of Computing, (1976) 87–91.

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Research supported in part by NSF grant MCS79-10261.

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Parikh, R. Some applications of topology to program semantics. Math. Systems Theory 16, 111–131 (1983). https://doi.org/10.1007/BF01744573

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Keywords

  • Computational Mathematic
  • Natural Question
  • Relational Semantic
  • Complex Program
  • Partial Correctness