  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.
