A regular language is k-piecewise testable (k-PT) if it is a Boolean combination of languages of the form L a 1 a , where a i ∈ Σ and 0 ≤ n ≤ k. Given a finite automaton A , if the language L(A) is piecewise testable, we want to express it as a Boolean combination of languages of the above form. The idea is as follows. If the language is k-PT, then there exists a congruence ∼ k of finite index such that L(A) is a finite union of ∼ k-classes. Every such class is characterized by an intersection of languages of the from L u , for |u| ≤ k, and their complements. To represent the ∼ k-classes, we make use of the ∼ k-canonical DFA. We identify the states of the ∼ k-canonical DFA whose union forms the language L(A) and use them to construct the required Boolean combination. We study the computational and descriptional complexity of related problems.