Abstract
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.
This work was supported by DFG grant MA 4938/2-1, by Poland’s National Science Centre grant no. UMO-2013/11/D/ST6/03075, and Agence Nationale de la Recherche ANR 2010 BLAN 0202 01 FREC.
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Notes
- 1.
Emptiness of L over alphabet A can be decided by taking the \(\{x\}\)-upward closure of L, where \(x \notin A\), intersecting the resulting language with the regular language \((A\cup \{x\})^*A(A\cup \{x\})^*\), and then taking the \(\{x\}\)-projection. In the resulting language, the diagonal problem returns true iff L is nonempty [36].
- 2.
Of course, one could also immediately obtain \(L_3\) from \(L_1\) by performing a single intersection with a regular language.
- 3.
A simple proof of this fact can be found in [11].
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Acknowledgments
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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Czerwiński, W., Martens, W., van Rooijen, L., Zeitoun, M. (2015). A Note on Decidable Separability by Piecewise Testable Languages. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_14
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