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arXiv · 1304.6734

Separating regular languages by piecewise testable and unambiguous languages

Thomas Place·Lorijn van Rooijen·Marc Zeitoun
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Abstract

Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from a smaller class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a $B{\Sigma}1(<)$ sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

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Thomas Place, Lorijn van Rooijen, Marc Zeitoun. 2013-04-24. Separating regular languages by piecewise testable and unambiguous languages. https://arxiv.org/abs/1304.6734

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