Precise estimation on the order of local testability of deterministic finite automaton

Abstract: A locally testable language L is a language with the property that for some non negative integer k, called the order or the level of local testable, whether or not a word u in the language L depends on (1) the prefix and the suffix of the word u of length k-1 and (2) the set of intermediate partial strings of length k of the word u. For given k the language is called k-testable. We give necessary and sufficient conditions for the language of an automaton to be k-testable in the terms of the length of paths of a related graph. Some estimations of the upper and of the lower bound of testable order follow from these results. We improve the upper bound on the testable order of locally testable deterministic finite automaton with n states to n(n-2)+1 This bound is the best possible. We give an answer on the following conjecture of Kim, McNaughton and Mac-CLoskey for deterministic finite locally testable automaton with n states: \Is the local testable order of no greater than n in power 1.5 when the alphabet size is two?" Our answer is negative. In the case of size two the situation is the same as in general case.