Aperiodic finite state automaton

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An aperiodic finite-state automaton is a finite-state automaton whose transition monoid is aperiodic.


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]

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. A counter-free automaton is a finite-state automaton which accepts a counter-free language. A finite-state automaton is counter-free if and only if it is aperiodic.

An aperiodic automaton satisfies the Černý conjecture.[2]


  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.