Aperiodic finite state automaton

An aperiodic finite-state 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.^[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 m ≥ n we have xy^mz 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]

References

↑ 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.
↑ 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.

McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph. 65. With an appendix by William Henneman. MIT Press. ISBN 0-262-13076-9. Zbl 0232.94024.
Sonal Pratik Patel (2010). An Examination of Counter-Free Automata (PDF) (Masters Thesis). San Diego State University. — An intensive examination of McNaughton, Papert (1971).
Thomas Colcombet (2011). "Green's Relations and their Use in Automata Theory". In Dediu, Adrian-Horia; Inenaga, Shunsuke; Martín-Vide, Carlos. Proc. Language and Automata Theory and Applications (LATA) (PDF). LNCS. 6638. Springer. pp. 1–21. ISBN 978-3-642-21253-6. — Uses Green's relations to prove Schützenberger's and other theorems.

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Abstract machines

Type-0 — Type-1 — — — — — Type-2 — — Type-3 — —	Unrestricted (no common name) Context-sensitive Positive range concatenation Indexed — Linear context-free rewriting systems Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular — Non-recursive	Recursively enumerable Decidable Context-sensitive Positive range concatenation^* Indexed^* — Linear context-free rewriting language Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular Star-free Finite	Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton restricted Tree stack automaton Embedded pushdown Nondeterministic pushdown Deterministic pushdown Visibly pushdown Finite Counter-free (with aperiodic finite monoid) Acyclic finite

Each category of languages, except those marked by a ^*, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.

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