An upper bound on the order of locally testable deterministic finite automata

Abstract

A locally testable language is a language with the property that for some nonnegative integer k, called the order of locality, 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. Recently, we introduced the first polynomial time algorithm for the local testability problem based on a simple characterization of locally testable deterministic automata. This paper investigates the upper bound on the order of locally testable automata. It shows that the order of a locally testable deterministic automaton is at most n⁴ + 1, where n is the number of states of the automaton.

Partial support for this research was provided by the Directorate of Computer and Information Science and Engineering of the National Science Foundation under Institutional Infrastructure Grant No. CDA-8805910.