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The Kleene Equality for Graphs

  • Arnaud Carayol
  • Didier Caucal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

In order to generalize the Kleene theorem from the free monoid to richer algebraic structures, we consider the non deterministic acceptance by a finite automaton of subsets of vertices of a graph. The subsets accepted in such a way are the equational subsets of vertices of the graph in the sense of Mezei and Wright. We introduce the notion of deterministic acceptance by finite automaton. A graph satisfies the Kleene equality if the two acceptance modes are equivalent, and in this case, the equational subsets form a Boolean algebra. We establish that the infinite grid and the transition graphs of deterministic pushdown automata satisfy the Kleene equality and we present families of graphs in which the free product of graphs preserves the Kleene equality.

Keywords

Boolean Algebra Cayley Graph Free Product Colored Vertex Free Monoid 
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.

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References

  1. [Büc64]
    Büchi, J.: Regular canonical systems. Arch. Math. Logik Grundlag. 6, 91–111 (1964)MATHGoogle Scholar
  2. [Car05]
    Carayol, A.: Regular sets of higher-order pushdown stacks. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 168–179. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. [Cau96]
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. In: Meyer auf der Heide, F., Monien, B. (eds.) ICALP 1996. LNCS, vol. 1099, pp. 194–205. Springer, Heidelberg (1996)Google Scholar
  4. [Cay78]
    Cayley, A.: On the theory of groups. Proc. London Math. Soc. 9, 126–133 (1878)CrossRefGoogle Scholar
  5. [Cou89]
    Courcelle, B.: On recognizable sets and tree automata. In: Resolution of Equations in Algebraic Structures. Academic Press, London (1989)Google Scholar
  6. [ES69]
    Eilenberg, S., Schützenberger, M.: Rational sets in commutative monoids. J. Algebra 13, 344–353 (1969)Google Scholar
  7. [GS64]
    Ginsburg, S., Spanier, E.: Bounded algol-like languages. Trans. Amer. Math. Soc. 113, 333–368 (1964)MATHMathSciNetGoogle Scholar
  8. [MS85]
    Muller, D., Schupp, P.: The theory of ends, pushdown automata, and second-order logic. TCS 37, 51–75 (1985)MATHCrossRefMathSciNetGoogle Scholar
  9. [MW67]
    Mezei, J., Wright, J.: Algebraic automata and context free sets. Information and Control 11, 3–29 (1967)MATHCrossRefMathSciNetGoogle Scholar
  10. [Sak87]
    Sakarovich, J.: On regular trace languages. TCS 52, 59–75 (1987)CrossRefGoogle Scholar
  11. [Sén96]
    Sénizergues, G.: On the rational subsets of the free group. Acta Informatica 33, 281–296 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Arnaud Carayol
    • 1
  • Didier Caucal
    • 1
  1. 1.IrisaRennes CedexFrance

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