A polynomial time algorithm for the local testability problem of deterministic finite automata

Abstract

We investigate the local testability problem of deterministic finite automata. A locally testable language is a language with the property that for some positive integer k, whether or not a word w is in the language depends on (1) the prefix and suffix of w of length k, and (2) the set of intermediate substrings of w of length k+1, without regard to the order in which these substrings occur. The local testability problem is, given a deterministic finite automaton, to decide whether it accepts a locally testable language or not. No polynomial time algorithm for this problem has appeared in the literature. We present an O(n2) time algorithm for the local testability problem based on two simple properties that characterize locally testable automata.