On the periodicity of morphisms on free monoids. (English) Zbl 0608.68065

It is shown that for a given endomorphism h on a given finitely generated free monoid there are only finitely many primitive words w for which \(h(w)=w^ n\) for some \(n\geq 2\). Moreover, all such words can be effectively found. Using this result, the D0L periodicity problem is shown to be decidable, that is, it is decidable whether there exist words v and w for a given u such that \(h^{\omega}(u)=vw^{\omega}\), where \(h^{\omega}(u)\) is the limit of the sequence \(u,h(u),h^ 2(u),... \). This latter result has also been solved by J. Pansiot [Decidability of periodicity for infinite words, RAIRO Inf. Théor. 20, 43-46 (1986)] using a different method.


68Q45 Formal languages and automata
20M35 Semigroups in automata theory, linguistics, etc.
20M05 Free semigroups, generators and relations, word problems
68Q42 Grammars and rewriting systems
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