On $k$-piecewise testability (preliminary report)

For a non-negative integer $k$, a language is $k$-piecewise test\-able ($k$-PT) if it is a finite boolean combination of languages of the form $Σ^* a_1 Σ^* \cdots Σ^* a_n Σ^*$ for $a_i\inΣ$ 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.

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