Abstract
The analysis of the famous Kleene's theorem shows that it consists indeed in two different propositions that are better distinguished when one tries to generatize the result. The first one relates rational expressions and a suitable generalization of finite automata. It holds in any monoid or, even better, in the semiring of formal power series on any monoid. It is shown that several classical results in formal language theory, for instance Elgot and Mezei characterization of rational relations by transducers and Chomsky normal form for context-free grammars, can thus be seen as particular cases of this first half of Kleene's theorem.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J. Berstel, Transductions and Context free Languages, Teubner, 1979.
J. Berstel, Ch. Reutenauer, Les séries rationnelles et leurs languages, Masson, 1984.
J. H. Conway, Regular Algebra and Finite Machines, Chapman and Hall, 1971.
S. Eilenberg, Automata, languages, and Machines, Vol. A, Academic Press, 1974.
C. C. Elgot and G. Mezei, On relations defined by generalized finite automata, I.B.M. J. of Res. and Dev. 9, 1965, 47–65.
J. E. Hopcroft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison Wesley, 1979.
J. E. Hopcroft and J. D. Ullman, Formal Languages and their relation to Automata, Addison Wesley, 1969.
S. C. Kleene, Representation of Events in Nerve Nets and Finite Automata, in Automata Studies (C. E. Shannon and J. Mc Carty, Eds), Princeton University Press, 1956, 3–41.
M. Nivat, Transductions des languages de Chomsky, Ann. Inst. Fourier 18, 1968, 336–456.
J. Sakarovitch, Théorie des Automates, en préparation.
M. P. Schützenberger, Certain elementary families of automata, in Proceedings of Symposium on Mathematical Theory of Automata, Polytechnic Institute of Brooklyn, 1962, 139–153.
E. Shamir, A representation theorem for algebraic and context free power series in non commuting variables, Inform. and Control 11, 1967, 234–254.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sakarovitch, J. (1987). Kleene's theorem revisited. In: Kelemenová, A., Kelemen, J. (eds) Trends, Techniques, and Problems in Theoretical Computer Science. IMYCS 1986. Lecture Notes in Computer Science, vol 281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3540185356_29
Download citation
DOI: https://doi.org/10.1007/3540185356_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18535-2
Online ISBN: 978-3-540-48008-2
eBook Packages: Springer Book Archive