On the Complexity of k-Piecewise Testability and the Depth of Automata

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 Σa1Σ ∗ · · ·ΣanΣ for ai ∈ Σ and 0 ≤ n ≤ 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 on this problem and a detailed analysis for small k’s. The result can be use 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.