Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Thomas Place (1) , Lorijn van Rooijen (1) , Marc Zeitoun (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 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.

Mots clés: Separation problem; Regular language; Piecewise testable language; Unambiguous language

Domaines: Théorie et langage formel [cs.FL]; Logique en informatique [cs.LO]; Mathématique discrète [cs.DM]

Thomas Place, Lorijn van Rooijen, Marc Zeitoun. Separating Regular Languages by Piecewise Testable and Unambiguous Languages. Mathematical Foundations of Computer Science, Aug 2013, Austria. pp.729-740, ⟨10.1007/978-3-642-40313-2_64⟩. ⟨hal-00948943⟩