Deciding Piecewise Testable Separability for Regular Tree Languages

Jean Goubault-Larrecq 1 Sylvain Schmitz 1, 2
2 DAHU - Verification in databases
CNRS - Centre National de la Recherche Scientifique : UMR8643, Inria Saclay - Ile de France, ENS Cachan - École normale supérieure - Cachan, LSV - Laboratoire Spécification et Vérification [Cachan]
Abstract : The piecewise testable separability problem asks, given two input languages, whether there exists a piecewise testable language that contains the first input language and is disjoint from the second. We prove a general characterisation of piecewise testable separability on languages in a well-quasi-order, in terms of ideals of the ordering. This subsumes the known characterisations in the case of finite words. In the case of finite ranked trees ordered by homeomorphic embedding, we show using effective representations for tree ideals that it entails the decidability of piecewise testable separability when the input languages are regular. A final byproduct is a new proof of the decidability of whether an input regular language of ranked trees is piecewise testable, which was first shown in the ranked case for a coarser ordering by Place [CSL 2008], and in the unranked case for the homeomorphic embedding relation by Bojańczyk, Segoufin, and Straubing [Log. Meth. in Comput. Sci., 8(3:26), 2012].
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Jean Goubault-Larrecq, Sylvain Schmitz. Deciding Piecewise Testable Separability for Regular Tree Languages. ICALP 2016, Jul 2016, Rome, Italy. pp.97:1--97:15, ⟨10.4230/LIPIcs.ICALP.2016.97⟩. ⟨hal-01276119v4⟩

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