Remarks on arbitrary multiple pattern interpretations

A word w is obtained by an arbitrary n-pattern interpretation of a word x if there are n homomorphisms h1,h2,…,hn and a positive integer k such that w=hi1(x)hi2(x)…hik(x) with 1⩽ij⩽n for all 1⩽j⩽k. Arbitrary multiple pattern interpretations of words are naturally extended to languages. We start with a short parallelism between pattern descriptions and arbitrary multiple pattern descriptions, and then investigate some closure properties of the families of languages obtained by arbitrary multiple pattern interpretations of finite, regular, and context-free languages, respectively. We show that the first of these families forms an infinite hierarchy and give a characterization of the arbitrary multiple pattern interpretations of finite languages. Two concepts of ambiguity and inherent ambiguity of arbitrary multiple pattern interpretations are defined. It is shown that both properties are decidable for arbitrary multiple pattern interpretations of finite languages but strong ambiguity is not decidable for arbitrary multiple pattern interpretations of context-free languages.