title: Aperiodic finite state automaton
text: # Aperiodic finite state automaton
  
  An aperiodic finite-state automaton (also called a counter-free automaton) is a finite-state automaton whose transition monoid is aperiodic.
  
  ## Properties
  
  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. 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 which 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 m ≥ n we have $xy^mz$ in L if and only if $xy^nz$ in L. 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.
  
  