A Note on Decidable Separability by Piecewise Testable Languages

The separability problem for languages from a class C by languages of a class S asks, for two given word languages I and E from C, whether there exists a language S from S which includes I and excludes E, that is, I ⊆ S and S ∩ E = ∅. It is known that separability for context-free languages by any class containing all definite languages (such as regular languages) is undecidable. We show that separability of context-free languages by piecewise testable languages is decidable. This contrasts the fact that testing if a context-free language is piecewise testable is undecidable. We generalize this decidability result by showing that, for every full trio (a class of languages that is closed under rather weak operations) which has decidable diagonal problem, separability with respect to piecewise testable languages is decidable. Examples of such classes are the class of languages defined by labeled vector addition systems and the class of languages accepted by higher order pushdown automata of order two. The proof goes through a result which is of independent interest and shows that, for any kind of languages I and E, separability can be decided by testing the existence of common patterns