Wojciech Czerwiński ; Wim Martens ; Lorijn van Rooijen ; Marc Zeitoun ; Georg Zetzsche - A Characterization for Decidable Separability by Piecewise Testable Languages

dmtcs:1335 - Discrete Mathematics & Theoretical Computer Science, December 11, 2017, Vol. 19 no. 4, FCT '15 - https://doi.org/10.23638/DMTCS-19-4-1
A Characterization for Decidable Separability by Piecewise Testable Languages

Authors: Wojciech Czerwiński ; Wim Martens ; Lorijn van Rooijen ; Marc Zeitoun ; Georg Zetzsche

    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 $S\cap E = \emptyset$. In this work, we assume some mild closure properties for $\mathcal{C}$ 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). The obtained decidability results contrast some undecidability results. In fact, for all (non-regular) language classes that we present as examples with decidable separability, it is undecidable whether a given language is a PTL itself. Our characterization involves a result of independent interest, which states that for any kind of languages $I$ and $E$, non-separability by PTL is equivalent to the existence of common patterns in $I$ and $E$.


    Volume: Vol. 19 no. 4, FCT '15
    Section: special issue FCT'15
    Published on: December 11, 2017
    Accepted on: December 11, 2017
    Submitted on: November 20, 2015
    Keywords: Computer Science - Formal Languages and Automata Theory,68Q45,F.4.3

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    Source : ScholeXplorer IsCitedBy DOI 10.4230/lipics.icalp.2020.109
    Source : ScholeXplorer IsCitedBy ARXIV 2004.12187
    Source : ScholeXplorer IsCitedBy DOI 10.48550/arxiv.2004.12187
    • 10.4230/lipics.icalp.2020.109
    • 2004.12187
    • 10.48550/arxiv.2004.12187
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