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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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To see this, note that if there are any cycles in the Myhill graph of A then L(A) will be infinite, since any such cycle can be repeated arbitrarily many times. Conversely, if the
Another striking aspect of LTk transition graphs is that they are generally extremely ine?cient. All we really care about is whether a path through the graph leads to an accepting
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