Biautomata for k-Piecewise Testable Languages

Abstract

An effective characterization of piecewise testable languages was given by Simon in 1972. A difficult part of the proof is to show that if L has a $\mathcal J$-trivial syntactic monoid $\mathsf M(L)$ then L is k-piecewise testable for a suitable k. By Simon’s original proof, an appropriate k could be taken as two times the maximal length of a chain of ideals in $\mathsf M(L)$. In this paper we improve this estimate of k using the concept of biautomaton: a kind of finite automaton which arbitrarily alternates between reading the input word from the left and from the right. We prove that an appropriate k could be taken as the length of the longest simple path in the canonical biautomaton of L. We also show that this bound is better than the known bounds which use the syntactic monoid of L.