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Aperiodic finite state automaton

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An aperiodic finite-state automaton (also called a counter-free automaton) is a finite-state automaton whose transition monoid is aperiodic.

Properties

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A regular language is star-free if and only if it is accepted by an automaton with a finite and aperiodic transition monoid. This result of algebraic automata theory is due to Marcel-Paul Schützenberger.[1] In particular, the minimum automaton of a star-free language is always counter-free (however, a star-free language may also be recognized by other automata that are not aperiodic).

A counter-free language is a regular language for which there is an integer n such that for all words x, y, z and integers mn we have xymz in L if and only if xynz in L. For these languages, when a string contains enough repetitions of any substring (at least n repetitions), changing the number of repetitions to another number that is at least n cannot change membership in the language. (This is automatically true when y is the empty string, but becomes a nontrivial condition when y is non-empty.) Another way to state Schützenberger's theorem is that star-free languages and counter-free languages are the same thing.

An aperiodic automaton satisfies the Černý conjecture.[2]

References

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  1. ^ Schützenberger, Marcel-Paul (1965). "On Finite Monoids Having Only Trivial Subgroups" (PDF). Information and Control. 8 (2): 190–194. doi:10.1016/s0019-9958(65)90108-7.
  2. ^ Trahtman, Avraham N. (2007). "The Černý conjecture for aperiodic automata". Discrete Math. Theor. Comput. Sci. 9 (2): 3–10. ISSN 1365-8050. Zbl 1152.68461. Archived from the original on 2015-09-23. Retrieved 2014-04-05.