Optimal estimation on the order of local testability of finite automata

A locally testable language L is a language with the property that for some nonnegative integer k, called the order of local testability, whether or not a word u is in the language L depends on (1) the prefix and suffix of the word u of length k - 1 and (2) the set of subwords of length k of the word u. For given k the language is called k-testable. We improve the upper bound on the order of local testability of a locally testable deterministic finite automaton with n states to 1/2(n² - n) + 1. This bound is the best possible. We give an answer to the following conjecture of Kim, McNaughton and McCloskey for deterministic finite locally testable automata with n states: "Is the order of local testability no greater than O(n^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: the order of local testability is Ω(n²). The necessary and sufficient conditions for the language of an automaton to be k-testable are given in terms of the length of paths of a related graph. Some estimates of the bounds on the order of local testability follow from these results.