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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Büchi, J.: Regular canonical systems. Arch. Math. Logik Grundlag. 6, 91–111 (1964)
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)
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)
Cayley, A.: On the theory of groups. Proc. London Math. Soc. 9, 126–133 (1878)
Courcelle, B.: On recognizable sets and tree automata. In: Resolution of Equations in Algebraic Structures. Academic Press, London (1989)
Eilenberg, S., Schützenberger, M.: Rational sets in commutative monoids. J. Algebra 13, 344–353 (1969)
Ginsburg, S., Spanier, E.: Bounded algol-like languages. Trans. Amer. Math. Soc. 113, 333–368 (1964)
Muller, D., Schupp, P.: The theory of ends, pushdown automata, and second-order logic. TCS 37, 51–75 (1985)
Mezei, J., Wright, J.: Algebraic automata and context free sets. Information and Control 11, 3–29 (1967)
Sakarovich, J.: On regular trace languages. TCS 52, 59–75 (1987)
Sénizergues, G.: On the rational subsets of the free group. Acta Informatica 33, 281–296 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carayol, A., Caucal, D. (2006). The Kleene Equality for Graphs. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_19
Download citation
DOI: https://doi.org/10.1007/11821069_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37791-7
Online ISBN: 978-3-540-37793-1
eBook Packages: Computer ScienceComputer Science (R0)