On Boolean combinations forming piecewise testable languages

Abstract

A regular language is k-piecewise testable (k-PT) if it is a Boolean combination of languages of the form $L_{a_1{a}_2\dots a_n}={\mathrm{\Sigma }}^{⁎}{a}_1{\mathrm{\Sigma }}^{⁎}{a}_2{\mathrm{\Sigma }}^{⁎}\cdots {\mathrm{\Sigma }}^{⁎}{a}_n{\mathrm{\Sigma }}^{⁎}$, where $a_i\in \mathrm{\Sigma }$ and $0\le n\le k$. 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 $|u|\le k$, 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.