Local suffix substitution closure, Theory of Computation

The k-local Myhill graphs provide an easy means to generalize the suffix substitution closure property for the strictly k-local languages.

Lemma (k-Local Suffix Substitution Closure) If L is a strictly k-local language then for all strings u1, v1, u2, and v2 in Σ* and all strings x in Σk-1 :

u1xv1 ∈ L and u2xv2 ∈ L ⇒ u1xv2 ∈ L.

The justi?cation is essentially identical to that of our original suffix substitution closure lemma. If L ∈ SLk then it is recognized by an SLk automaton. In the k-local Myhill graph of that automaton, any path from ‘?' to the vertex labeled x can be put together with any path from that vertex to ‘?' to produce a path that represents a string in L.

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