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 $\Sigma^* a1 \Sigma^* \cdots \Sigma^* an \Sigma^*$ for $a_i\in\Sigma$ 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.