Abstract
For a non-negative integer k, a language is k-piecewise testable (k-PT) if it is a finite boolean combination of languages of the form \(\varSigma ^* a_1 \varSigma ^* \cdots \varSigma ^* a_n \varSigma ^*\) for \(a_i\in \varSigma \) and \(0\le n \le k\). We study the following problem: Given a DFA recognizing a piecewise testable language, decide whether the language is k-PT. We provide a complexity bound and a detailed analysis for small k’s. The result can be used to find the minimal k for which the language is k-PT. We show that the upper bound on k given by the depth of the minimal DFA can be exponentially bigger than the minimal possible k, and provide a tight upper bound on the depth of the minimal DFA recognizing a k-PT language.
T. Masopust—Research supported by the DFG in grant KR 4381/1-1.
M. Thomazo—Research supported by the Alexander von Humboldt Foundation.
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Masopust, T., Thomazo, M. (2015). On the Complexity of k-Piecewise Testability and the Depth of Automata. In: Potapov, I. (eds) Developments in Language Theory. DLT 2015. Lecture Notes in Computer Science(), vol 9168. Springer, Cham. https://doi.org/10.1007/978-3-319-21500-6_29
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DOI: https://doi.org/10.1007/978-3-319-21500-6_29
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