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A purely algebraic proof of McNaughton's theorem on infinite words

  • Bertrand Le Saec
  • Jean-Eric Pin
  • Pascal Weil
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 560)

Abstract

We give a new, purely algebraic proof of McNaughton's theorem on infinite words, which states that each recognizable set X of infinite words can be recognized by a deterministic Muller automaton. Our proof uses the semigroup approach to recognizability and relies on certain algebraic properties of finite semigroups. It also provides a simple solution to the problem of finding a deterministic automaton for X when one is given a semigroup recognizing X.

Keywords

Finite Automaton Testable Language Finite Semigroup Algebraic Proof Deterministic Automaton 
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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Bibliography

  1. [1]
    Brzozowski, J.A. and Simon, I., Characterization of locally testable languages, Discrete Math. 4 (1973), 243–271.CrossRefGoogle Scholar
  2. [2]
    Büchi, J.R., On a decision method in restricted second-order arithmetic, in Proc. Internat. Congr. on Logic, Methodology and Phil. of Science (E. Nagel et al., eds.), Stanford Univ. Press, Stanford, 1960.Google Scholar
  3. [3]
    Eilenberg, S., Automata, languages and machines, Vol. A, Academic Press, New York, 1974.Google Scholar
  4. [4]
    Eilenberg, S., Automata, languages and machines, Vol. B, Academic Press, New York, 1976.Google Scholar
  5. [5]
    Lallement, G., Semigroups and combinatorial applications, Wiley, New York, 1979.Google Scholar
  6. [6]
    Le Saec, B., Saturating right congruences, Informatique Théorique et Applications 24 (1990), 545–560.Google Scholar
  7. [7]
    Le Saec, B., A modular proof of McNaughton's theorem, in Logic and recognizable sets (Thomas, W., ed.), Kiel Universität, Kiel, 1991, 50–55.Google Scholar
  8. [8]
    Le Saec, B., Pin, J.-E. and Weil, P., Finite semigroups whose stabilizers are idempotent, to appear in Intern. Journ. Algebra and Computation.Google Scholar
  9. [9]
    McNaughton, R., Testing and generating infinite sequences by a finite automaton, Information and Control 9 (1966), 521–530.Google Scholar
  10. [10]
    McNaughton, R., Algebraic decision procedures for locally testable languages, Math. Systems Theor. 8 (1974), 66–76.Google Scholar
  11. [11]
    Muller, D., Infinite sequences and finite machines, in Switching Theory and Logical Design, Proc. 4th Annual Symposium IEEE, 1963, 3–16.Google Scholar
  12. [12]
    Pécuchet, J.-P., On the complementation of Büchi automata, Theoret. Comp. Sci. 47, 95–98.Google Scholar
  13. [13]
    Perrin, D. and Pin, J.-E., Mots infinis, to appear (LITP report 91-06).Google Scholar
  14. [14]
    Pin, J.-E., Variétés de langages formels, Masson, Paris, 1984; English translation: Varieties of formal languages, Plenum, New York, 1986.Google Scholar
  15. [15]
    Rabin, M. O., Automata on Infinite Objects and Church's Problem, American Mathematical Society, Providence, RI, 1972.Google Scholar
  16. [16]
    Safra, S., On the complexity of the ω-automata, Proc. 29th IEEE Symp. Found. Comp. Science (1988).Google Scholar
  17. [17]
    Thomas, W., A combinatorial approach to the theory of ω-automata, Information and Control 48 (1981), 261–283.CrossRefGoogle Scholar
  18. [18]
    Thomas, W., Automata on infinite objects, in Handbook of Theoretical Computer Science, vol. A (v. Leeuwen, J., ed.), Elsevier, Amsterdam, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Bertrand Le Saec
    • 1
  • Jean-Eric Pin
    • 2
  • Pascal Weil
    • 2
  1. 1.LaBRIUniversité Bordeaux ITalence CedexFrance
  2. 2.Institut Blaise PascalLITP-CNRSParis Cedex 05France

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