A Note on Decidable Separability by Piecewise Testable Languages

  • Wojciech Czerwiński
  • Wim MartensEmail author
  • Lorijn van Rooijen
  • Marc Zeitoun
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)


The separability problem for word languages of a class \(\mathcal {C}\) by languages of a class \(\mathcal {S}\) asks, for two given languages I and E from \(\mathcal {C}\), whether there exists a language S from \(\mathcal {S}\) that includes I and excludes E, that is, \(I \subseteq S\) and \(S\cap E = \emptyset \). 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 with 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 languages defined by labeled vector addition systems and the 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 in I and E.


Regular Language Factorization Pattern Boolean Combination Finite Alphabet Definite Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank Tomáš Masopust for pointing us to [16] and Thomas Place for pointing out to us that determining if a given context-free language is piecewise testable is undecidable. We are also grateful to the anonymous reviewers for many helpful remarks that simplified proofs. We are much indebted to Georg Zetzsche for many useful remarks and most of all for sending us a simple proof that showed that, for full trios, separability by PTL implies decidability of the diagonal problem, thereby turning Theorem 2 into an equivalence. We plan to incorporate his proof in the full version of this paper.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Wojciech Czerwiński
    • 1
  • Wim Martens
    • 2
    Email author
  • Lorijn van Rooijen
    • 3
  • Marc Zeitoun
    • 3
  1. 1.University of WarsawWarsawPoland
  2. 2.University of BayreuthBayreuthGermany
  3. 3.Bordeaux UniversityTalenceFrance

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