Kleene Modules

  • Thorsten Ehm
  • Bernhard Möller
  • Georg Struth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3051)


We propose axioms for Kleene modules (KM). These structures have a Kleene algebra and a Boolean algebra as sorts. The scalar products are mappings from the Kleene algebra and the Boolean algebra into the Boolean algebra that arise as algebraic abstractions of relational image and preimage operations. KM is the basis of algebraic variants of dynamic logics. We develop a calculus for KM and discuss its relation to Kleene algebra with domain and to dynamic and test algebras. As an example, we apply KM to the reachability analysis in digraphs.


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  1. 1.
    Backhouse, R.C., van den Eijnde, J.P.H.W., van Gasteren, A.J.M.: Calculating path algorithms. Science of Computer Programming 22(1-2), 3–19 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brink, C.: Boolean modules. Journal of Algebra 71, 291–313 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Brunn, T., Möller, B., Russling, M.: Layered graph traversals and hamiltonian path problems–an algebraic approach. In: Jeuring, J. (ed.) MPC 1998. LNCS, vol. 1422, pp. 96–121. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  4. 4.
    Clenaghan, K.: Calculational graph algorithmics: Reconciling two approaches with dynamic algebra. Technical Report CS-R9518, CWI, Amsterdam (1994)Google Scholar
  5. 5.
    Conway, J.H.: Regular Algebra and Finite State Machines. Chapman&Hall, Sydney (1971)Google Scholar
  6. 6.
    Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. Technical Report 2003-07, Institut für Informatik, Universität Augsburg (2003)Google Scholar
  7. 7.
    Ehm, T.: Pointer Kleene algebra. Technical Report 2003-13, Institut für Informatik, Universität Augsburg (2003)Google Scholar
  8. 8.
    Ehm, T., Möller, B., Struth, G.: Kleene modules. Technical Report 2003-10, Institut für Informatik, Universität Augsburg (2003)Google Scholar
  9. 9.
    Harel, D., Kozen, D., Tiuryn, J.: AustraliaDynamic Logic. MIT Press, Cambridge (2000)Google Scholar
  10. 10.
    Hollenberg, M.: Equational axioms of test algebra. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 295–310. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Jacobson, N.: Basic Algebra, vol. I, II. Freeman, New York (1985)zbMATHGoogle Scholar
  12. 12.
    Kozen, D.: A representation theorem for ∗-free PDL. Technical Report RC7864, IBM (1979)Google Scholar
  13. 13.
    Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kozen, D.: Kleene algebra with tests. Trans. Programming Languages and Systems 19(3), 427–443 (1997)CrossRefGoogle Scholar
  15. 15.
    Leiß, H.: Kleenean semimodules and linear languages. In: Zoltán Ésik and Anna Ingólfsdóttir, editors, FICS’02 Preliminary Proceedings, number NS-02-2 in BRICS Notes Series, pp. 51–53. Univ. of Aarhus (2002)Google Scholar
  16. 16.
    Möller, B., Struth, G.: Greedy-like algorithms in Kleene algebra. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Németi, I.: Dynamic algebras of programs. In: Gecseg, F. (ed.) FCT 1981. LNCS, vol. 117, pp. 281–291. Springer, Heidelberg (1981)Google Scholar
  18. 18.
    Pratt, V.: Dynamic logic as a well-behaved fragment of relation algebras. In: Bergman, C.H., Pigozzi, D.L., Maddux, R.D. (eds.) Algebraic Logic and Universal Algebra in Computer Science. LNCS, vol. 425, pp. 77–110. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  19. 19.
    Pratt, V.: Dynamic algebras: Examples, constructions, applications. Studia Logica 50, 571–605 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    von Karger, B., Berghammer, R., Wolf, A.: Relation-algebraic derivation of spanning tree algorithms. In: Jeuring, J. (ed.) MPC 1998. LNCS, vol. 1422, pp. 23–43. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Ravelo, J.N.: Two graph algorithms derived. Acta Informatica 36, 489–510 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Trnkova, V., Reiterman, J.: Dynamic algebras with tests. J. Comput. System Sci. 35, 229–242 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Thorsten Ehm
    • 1
  • Bernhard Möller
    • 1
  • Georg Struth
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

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