Summary: A regular language is \(k\)-piecewise testable (\(k\)-PT) if it is a Boolean combination of languages of the form \(L_{a_1 a_2 \ldots a_n} = \Sigma^\ast a_1 \Sigma^\ast a_2 \Sigma^\ast \cdots \Sigma^\ast a_n \Sigma^\ast\), where \(a_i \in \Sigma\) and \(0 \leq n \leq k\). Given a finite automaton \(\mathcal{A}\), if the language \(L(\mathcal{A})\) 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 \(\sim_k\) of finite index such that \(L(\mathcal{A})\) is a finite union of \(\sim_k\)-classes. Every such class is characterized by an intersection of languages of the from \(L_u\), for \(| u | \leq k\), and their complements. To represent the \(\sim_k\)-classes, we make use of the \(\sim_k\)-canonical DFA. We identify the states of the \(\sim_k\)-canonical DFA whose union forms the language \(L(\mathcal{A})\) and use them to construct the required Boolean combination. We study the computational and descriptional complexity of related problems.