Separating regular languages by piecewise testable and unambiguous languages

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.

arXiv admin note: text overlap with arXiv:1303.2143

References in corpus (1)

• The separation problem for regular languages by piecewise testable languages

Cited by in corpus (7)

• Separating Regular Languages with First-Order Logic
• On Separation by Locally Testable and Locally Threshold Testable Languages
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• On the Height of Towers of Subsequences and Prefixes
• On Upper and Lower Bounds on the Length of Alternating Towers