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 another 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 \(\mathcal{B}\Sigma_1(<)\) 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.
Supported by the Agence Nationale de la Recherche ANR 2010 BLAN 0202 01 FREC.
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Place, T., van Rooijen, L., Zeitoun, M. (2013). Separating Regular Languages by Piecewise Testable and Unambiguous Languages. In: Chatterjee, K., Sgall, J. (eds) Mathematical Foundations of Computer Science 2013. MFCS 2013. Lecture Notes in Computer Science, vol 8087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40313-2_64
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DOI: https://doi.org/10.1007/978-3-642-40313-2_64
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