# Computing the order of a locally testable automaton

## Abstract

A locally testable language is a language with the property that, for some nonnegative integer *j*, whether or not a word *x* is in the language depends on (1) the prefix and suffix of *x* of length *j*, and (2) the set of substrings of x of length *j*+1, without regard to the order in which these substrings occur or the number of times each substring occurs. This paper shows that computing the smallest *j* of a given locally testable deterministic automaton is NP-hard, and presents a polynomial-time *ε*- approximation algorithm for computing such *j*. It turns out that, for a fixed *j*, there is a polynomial time algorithm to decide whether a given automaton satisfies the above condition. In addition, we have obtained an upper bound of 2*n*^{2}+1 on the smallest such *j* for a locally testable automaton of *n* states.

## Keywords

Terminal Node Longe Path Finite Automaton Outgoing Edge Testable Language## Preview

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