Abstract
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 characterization relates reversible automata to the profinite group topology of the free monoid.
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References
D. Angluin, Inference of reversible languages, Journal of the Association for Computing Machinery,29, (1982) 741–765.
C.J. Ash, Finite semigroups with commuting idempotents, J. Austral. Math. Soc. (Series A) 43, (1987) 81–90.
C.J. Ash, Inevitable sequences and a proof of the type II conjecture, in Proceedings of the Monash Conference on Semigroup Theory, World Scientific, Singapore, (1991) 31–42.
C.J. Ash, Inevitable Graphs: A proof of the type II conjecture and some related decision procedures, Int. Jour. Alg. and Comp. 1 (1991) 127–146.
J. Berstel, Transductions and Context Free Languages, Teubner Verlag, 1979.
J.C. Birget, Intersection and union of regular languages, and state complexity, to appear.
J.C. Birget, S.W. Margolis and J. Rhodes, Finite semigroups whose idempotents commute or form a subsemigroup, in Semigroups and Their Applications, edited by S.M. Goberstein and P.M. Higgins, Reidel, Dordrecht, 1987, 25–35.
J.M. Champarnaud and G. Hansel, A computing package for automata and finite semigroups, Journal of Symbolic Computation 12, 1991, 197–220.
S. Eilenberg, Automata, Languages and Machines, Academic Press, New York, Vol. A, 1974; Vol B, 1976.
M. Hall Jr., A topology for free groups and related groups, Ann. of Maths 52, (1950) 127–139.
T.E. Hall, Biprefix codes, inverse semigroups and syntactic monoids of injective automata, Theoretical Computer Science.
K. Henckell, S.W. Margolis, J.E. Pin and J. Rhodes, Ash's type II theorem, profinite topology and Malcev products, to appear in Int. Jour. Alg. and Comp.
G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979.
M. Lothaire, Combinatorics on words, Encyclopedia of Mathematics 17, Addison-Wesley, Reading, MA, 1983.
R. McNaughton, The loop complexity of pure-group events. Inf. Control 11, (1967) 167–176.
S.W. Margolis, Consequences of Ash's proof of the Rhodes Type II Conjecture, in Proceedings of the Monash Conference on Semigroup Theory, World Scientific, Singapore, (1991) 180–205.
S.W. Margolis and J.E. Pin, Languages and inverse semigroups, 11th ICALP, Lecture Notes in Computer Science 199, Springer, Berlin (1985) 285–299.
S.W. Margolis and J.E. Pin, Inverse semigroups and varieties of finite semigroups, Journal of Algebra 110 (1987) 306–323.
S.W. Margolis and J.E. Pin, New results on the conjecture of Rhodes and on the topological conjecture, to appear in J. Pure and Applied Algebra.
J.E. Pin, Finite group topology and p-adic topology for free monoids. 12th ICALP, Lecture Notes in Computer Science 199, Springer, Berlin, 1985, 285–299.
J.E. Pin, Variétés de langages formels, 160 p., Masson, Paris (1984). Varieties of formal languages, 138 p., North Oxford Academic (London), 1986 and Plenum (New York), 1986.
J.E. Pin, On the languages recognized by finite reversible automata, 14th ICALP, Lecture Notes in Computer Science 267 Springer, Berlin, (1987) 237–249.
J.E. Pin, A topological approach to a conjecture of Rhodes, Bulletin of the Australian Mathematical Society 38 (1988) 421–431.
J.E. Pin, On a conjecture of Rhodes, Semigroup Forum 39 (1989) 1–15.
J.E. Pin, Topologies for the free monoid, Journal of Algebra 137 (1991) 297–337.
J.E. Pin and Ch. Reutenauer, A conjecture on the Hall topology for the free group, to appear in the Notices of the London Math. Society.
Ch. Reutenauer, Une topologie du monoïde libre, Semigroup Forum 18, (1979) 33–49.
Ch. Reutenauer, Sur mon article “Une topologie du monoïde libre”, Semigroup Forum 22, (1981) 93–95.
L. Ribes and P.A. Zalesskii, On the profinite topology on a free group, to appear.
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© 1992 Springer-Verlag Berlin Heidelberg
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Pin, JE. (1992). On reversible automata. In: Simon, I. (eds) LATIN '92. LATIN 1992. Lecture Notes in Computer Science, vol 583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023844
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DOI: https://doi.org/10.1007/BFb0023844
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