ON TWO-DIMENSIONAL PATTERN-MATCHING LANGUAGES AND THEIR DECISION-PROBLEMS

We propose two new classes of two-dimensional array languages, called existential matching languages (EXMLs) and universal matching languages (UNMLs). These languages are closely related to two-dimensional pattern matching, and thus suited for studying them formally. In this paper, basic properties of these languages and decidability (or undecidability) of several problems about them are investigated. We show that, because of the two-dimensionality, several decision problems, such as the emptiness problem for UNMLs, the universe problem for EXMLs, and the equivalence problems for both languages, are undecidable. Thus we cannot decide, for example, whether two-dimensional pattern-matching tasks are equivalent.