A Characterization for Decidable Separability by Piecewise Testable Languages

The separability problem for word languages of a class C by languages of a class S asks, for two given languages I and E from C, whether there exists a language S from S that includes I and excludes E, that is, I ⊆ S and S ∩E = ∅. In this work, we assume some mild closure properties for C and study for which such classes separability by a piecewise testable language (PTL) is decidable. We characterize these classes in terms of decidability of (two variants of) an unboundedness problem. From this, we deduce that separability by PTL is decidable for a number of language classes, such as the context-free languages and languages of labeled vector addition systems. Furthermore, it follows that separability by PTL is decidable if and only if one can compute for any language of the class its downward closure wrt. the scattered substring ordering (i.e., if the set of scattered substrings of any language of the class is effectively regular).