On the languages accepted by finite reversible automata

  • J. E. Pin
Formal Languages And Automata
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)


A reversible automaton is a finite (possibly incomplete) automaton in which each letter induces a partial one-to-one map from the set of states into itself. We give four non-trivial characterizations of the languages accepted by a reversible automaton equipped with a set of initial and final states and we show that one can effectively decide whether a given rational (or regular) language can be accepted by a reversible automaton. The first characterization gives a description of the subsets of the free group accepted by a reversible automaton that is somewhat reminiscent of Kleene's theorem. The second characterization is more combinatorial in nature. The decidability follows from the third — algebraic — characterization. The last and somewhat unexpected characterization is a topological description of our languages that solves an open problem about the finite-group topology of the free monoid.


Finite Group Inverse Semigroup Topological Description Membership Problem Finite Semigroup 
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  1. [1]
    D. Angluin, Inference of reversible languages, Journal of the Association for Computing Machinery 29 (1982) 741–765.Google Scholar
  2. [2]
    C.J. Ash, Finite semigroups with commuting idempotents, J. Austral. Math. Soc. Series A, to appear.Google Scholar
  3. [3]
    J.C. Birget, S.W. Margolis and J. Rhodes, Finite semigroups whose idempotents commute or form a subsemigroup, Proceedings of the Chico Conference on Semigroups (1986), to appear.Google Scholar
  4. [4]
    S. Eilenberg, Automata, Languages and Machines, Vol B, Academic Press, New-York (1976).Google Scholar
  5. [5]
    M. Hall Jr, A topology for free groups and related groups, Ann. Math. 52 (1950) 127–139.Google Scholar
  6. [6]
    T.E. Hall, Biprefix codes, inverse semigroups and syntactic monoids of injective automata, Theoretical Computer Science.Google Scholar
  7. [7]
    G. Lallement, Semigroups and Combinatorial Applications, Wiley, New-York (1979).Google Scholar
  8. [8]
    M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics 17, Addison Wesley, New-York (1983).Google Scholar
  9. [9]
    R. McNaughton, The loop complexity of pure-group events. Inf. Control 11 (1967) 167–176.Google Scholar
  10. [10]
    S.W. Margolis and J.E. Pin, Languages and inverse semigroups, 11th ICALP, Lecture Notes in Computer Science 199, Springer, Berlin (1985) 285–299.Google Scholar
  11. [11]
    S.W. Margolis and J.E. Pin, Finite inverse semigroups, varieties and languages, Journal of Algebra, to appear.Google Scholar
  12. [12]
    J.E. Pin, Finite group topology and p-adic topology for free monoids, 12th ICALP, Lecture Notes in Computer Science 194 (1985) 445–455.Google Scholar
  13. [13]
    J.E. Pin, Varieties of formal languages, Masson, Paris (1984), North Oxford Academic, London and Plenum, New-York (1986).Google Scholar
  14. [14]
    Ch. Reutenauer, Une topologie du monoide libre, Semigroup Forum 18, (1979), 33–49.Google Scholar
  15. [15]
    Ch. Reutenauer, Sur mon article "une topologie du monoide libre", Semigroup Forum 22, (1981), 93–95.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • J. E. Pin
    • 1
  1. 1.LITP, University Paris VI and CNRSParis Cedex 05France

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