Multiple Pattern Interpretations

Abstract

A word w is obtained by an ordered n-pattern interpretation of a word x if there are n homomorphisms h1,h2,ċċċ, hn such that w=h1(x)h2(x)ċs hn(x). This ordered multiple pattern interpretation is naturally extended to languages. We show a strong relationship between the family of languages obtained by ordered multiple pattern interpretations of regular, linear, and context-free languages and the family of regular, linear, and context-free simple matrix languages. Concepts of ambiguity and inherent ambiguity of ordered multiple pattern interpretation are defined and it is shown that these properties are not decidable on the class of context-free languages. Then, we investigate arbitrary multiple pattern interpretations of the same classes of languages in the Chomsky hierarchy. We show that the classes of languages obtained in this way are recognizable in polynomial time provided that all components of the pattern interpretation are injective homomorphisms. We also present a series of open problems.