A representation theorem for models of *-free PDL

  • Dexter Kozen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 85)


We introduce dynamic algebras and show how they can be used to give an algebraic interpretation to propositional dynamic logic (PDL). Dynamic algebras include all Kripke models, the standard interpretation of PDL. We give a simple algebraic condition on *-free dynamic algebras that is necessary and sufficient for representation by *-free Kripke models. In the presence of*, the condition is sufficient for representation by a nonstandard Kripke model. This result leads to a duality between certain topological Kripke models and dynamic algebras analogous to the duality between Boolean algebras and their Stone spaces.


Binary Relation Boolean Algebra Scalar Multiplication Relation Algebra Kripke Model 
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  1. [AHU]
    Aho A.V., J.E. Hopcroft, and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass., 1974.Google Scholar
  2. [Ba]
    Banachowski, L., A. Kreczmar, G. Mirkowska, H. Rasiowa, and A. Salwicki, "An introduction to Algorithmic Logic," in: Mazurkiewicz and Pawlak, eds., Math. Found. of Comp. Sci., Banach Center Publications, Warsaw, 1977.Google Scholar
  3. [Be]
    Berman, F., "A completeness technique for D-axiomatizable semantics," Proc. 11th ACM Symp. on Theory of Comp. (May 1979), 160–166.Google Scholar
  4. [BS]
    Bell, J.S. and A.B. Slomson, Models and Ultraproducts. North Holland, Amsterdam, 1971.Google Scholar
  5. [C]
    Conway, J.H. Regular Algebra and Finite Machines. Chapman-Hall, London, 1971.Google Scholar
  6. [CO]
    Constable, R.L. and M.J. O'Donnell. A Programming Logic. Winthrop, Cambridge, Mass., 1978.Google Scholar
  7. [EU]
    Everett, C.J. and S. Ulam, "Projective algebra I," Amer. J. Math. 68:1 (1946), 77–88.Google Scholar
  8. [FL]
    Fischer, M.J. and R.E. Ladner, "Propositional dynamic logic of regular programs," J. Comput. Syst. Sci. 18:2 (1979).Google Scholar
  9. [G]
    Gabbay, D., "Axiomatizations of logics of programs," manuscript, Nov. 1977.Google Scholar
  10. [H]
    Harel, D. First-Order Dynamic Logic. Lecture Notes in Computer Science 68, ed. Goos and Hartmanis, Springer-Verlag, Berlin, 1979.Google Scholar
  11. [JT]
    Jonsson, B. and A. Tarski, "Representation problems for relation algebras," abstract 89t, Bull. Amer. Math. Soc. 54 (1948), 80.Google Scholar
  12. [K1]
    Kozen, D., "A representation theorem for models of *-free PDL," Report RC7864, IBM Research, Yorktown Heights, New York, Sept. 1979.Google Scholar
  13. [K2]
    Kozen, D., "On the duality of dynamic algebras and Kripke models," Report RC7893, IBM Research, Yorktown Heights, New York, Oct. 1979.Google Scholar
  14. [K3]
    Kozen, D., "On the representation of dynamic algebras," Report RC7898, IBM Research, Yorktown Heights, New York, Oct. 1979.Google Scholar
  15. [L]
    Lyndon, R.C., "The representation of relation algebras," Ann. Math. 51:3 (1950), 707–729.Google Scholar
  16. [McK1]
    McKinsey, J.C.C., "Postulates for the calculus of binary relations," J. Sym. Logic 5:3 (1940), 85–97.Google Scholar
  17. [McK2]
    —, "On the representation of projective algebras," Amer. J. Math. 70 (1948), 375–384.Google Scholar
  18. [N]
    Nishimura, H., "Sequential Method in Propositional Dynamic Logic," Acta Informatica 12 (1979), 377–400.Google Scholar
  19. [Pa]
    Parikh, R., "A completeness result for PDL," Symp. on Math. Found. of Comp. Sci., Zakopane, Warsaw, Springer-Verlag, May 1978.Google Scholar
  20. [Pr1]
    Pratt, V.R., "Semantical considerations on Floyd-Hoare logic," Proc. 17th IEEE Symp. on Foundations of Comp. Sci. (Oct. 1976), 109–121.Google Scholar
  21. [Pr2]
    —, "A practical decision method for Propositional Dynamic Logic," Proc. 10th ACM Symp. on Theory of Computing (May 1978), 326–337.Google Scholar
  22. [Pr3]
    —, "Models of program logics," Proc. 20th IEEE Symp. on Foundations of Comp. Sci. (Oct. 1979), to appear.Google Scholar
  23. [Pr4]
    —, "Dynamic algebras: Examples, constructions, applications," manuscript, July 1979.Google Scholar
  24. [Se]
    Segerberg, K., "A completeness theorem in the modal logic of programs," Not. AMS 24:6 (1977), A-552.Google Scholar
  25. [SS]
    Salomaa, A. and M. Soittala. Automata Theoretic Aspects of Formal Power Series. Springer-Verlag, New York, 1978.Google Scholar
  26. [T]
    Tarski, A., "On the calculus of relations," J. Symb. Logic 6:3 (1941), 73–89.Google Scholar
  27. [vEB]
    van Emde Boas, "The connection between modal logic and algorithmic logics," report 78-02, Univ. of Amsterdam, May 1978.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • Dexter Kozen
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsNew York

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