Kleene algebra with tests: Completeness and decidability

  • Dexter Kozen
  • Frederick Smith
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1258)


Kleene algebras with tests provide a rigorous framework for equational specification and verification. They have been used successfully in basic safety analysis, source-to-source program transformation, and concurrency control. We prove the completeness of the equational theory of Kleene algebra with tests and-continuous Kleene algebra with tests over language-theoretic and relational models. We also show decidability. Cohen's reduction of Kleene algebra with hypotheses of the form r=0 to Kleene algebra without hypotheses is simplified and extended to handle Kleene algebras with tests.


Boolean Algebra Relational Model Regular Expression Equational Theory Concurrency Control 
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  1. 1.
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1975.Google Scholar
  2. 2.
    J. Berstel. Transductions and Context-free Languages. Teubner, 1979.Google Scholar
  3. 3.
    E. Cohen. Hypotheses in Kleene algebra., April 1994.Google Scholar
  4. 4.
    E. Cohen. Lazy caching., 1994.Google Scholar
  5. 5.
    E. Cohen. Using Kleene algebra to reason about concurrency control,, 1994.Google Scholar
  6. 6.
    E. Cohen, D. Kozen, and F. Smith. The complexity of Kleene algebra with tests. Tech. Rep. TR96-1598, Cornell University, July 1996.Google Scholar
  7. 7.
    J. H. Conway. Regular Algebra and Finite Machines. Chapman and Hall, 1971.Google Scholar
  8. 8.
    M. J. Fischer and R. E. Ladner. Prepositional dynamic logic of regular programs. J. Comput. Syst. Sci., 18(2):194–211, 1979.Google Scholar
  9. 9.
    A. Gibbons and W. Rytter. On the decidability of some problems about rational subsets of free partially commutative monoids. Theor. Comput. Sci., 48:329–337, 1986.Google Scholar
  10. 10.
    D. Harel. On folk theorems. Comm. Assoc. Comput. Mach., 23(7):379–389, July 1980.Google Scholar
  11. 11.
    K. Iwano and K. Steiglitz. A semiring on convex polygons and zero-sum cycle problems. SIAM J. Comput., 19(5):883–901, 1990.Google Scholar
  12. 12.
    S. C. Kleene. Representation of events in nerve nets and finite automata. In Shannon and McCarthy, editors, Automata Studies, pages 3–41. Princeton University Press, 1956.Google Scholar
  13. 13.
    D. Kozen. On induction vs. *-continuity. In Kozen, editor, Proc. Workshop on Logic of Programs, volume 131 of Lect. Notes in Comput. Sci., pages 167–176. Springer, 1981.Google Scholar
  14. 14.
    D. Kozen. On Kleene algebras and closed semirings. In Rovan, editor, Proc. Math. Found. Comput. Sci., volume 452 of Lect. Notes in Comput. Sci., pages 26–47. Springer, 1990.Google Scholar
  15. 15.
    D. Kozen. The Design and Analysis of Algorithms. Springer-Verlag, 1991.Google Scholar
  16. 16.
    D. Kozen. A completeness theorem for Kleene algebras and the algebra of regular events. Infor. and Comput., 110(2):366–390, May 1994.Google Scholar
  17. 17.
    D. Kozen. Kleene algebra with tests and commutativity conditions. In T. Margaria and B. Steffen, editors, Proc. Second Int. Workshop Tools and Algorithms for the Construction and Analysis of Systems (TACAS'96), volume 1055 of Lect. Notes in Comput. Sci., pages 14–33. Springer, March 1996.Google Scholar
  18. 18.
    W. Kuich and A. Salomaa. Semirings, Automata, and Languages. Springer, 1986.Google Scholar
  19. 19.
    G. Mirkowska. Algorithmic Logic and its Applications. PhD thesis, University of Warsaw, 1972. In Polish.Google Scholar
  20. 20.
    K. C. Ng. Relation Algebras with Transitive Closure. PhD thesis, University of California, Berkeley, 1984.Google Scholar
  21. 21.
    V. R. Pratt. Models of program logics. In Proc. 20th Symp. Found. Comput. Sci., pages 115–122. IEEE, 1979.Google Scholar
  22. 22.
    V. R. Pratt. Dynamic algebras and the nature of induction. In Proc. 12th Symp. Theory of Comput., pages 22–28. ACM, 1980.Google Scholar
  23. 23.
    V. R. Pratt. Dynamic algebras as a well-behaved fragment of relation algebras. In D. Pigozzi, editor, Proc. Conf. on Algebra and Computer Science, volume 425 of Lect. Notes in Comput. Sci., pages 77–110. Springer, June 1988.Google Scholar
  24. 24.
    L. J. Stockmeyer and A. R. Meyer. Word problems requiring exponential time. In Proc. 5th Symp. Theory of Computing, pages 1–9. ACM, 1973.Google Scholar
  25. 25.
    A. Tarski. On the calculus of relations. J. Symb. Logic, 6(3):65–106, 1941.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Dexter Kozen
    • 1
  • Frederick Smith
    • 1
  1. 1.Computer Science DepartmentCornell UniversityIthacaUSA

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