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

Abstract

A locally testable language L is a language with the property that for some nonnegative integer k, called the order or the level of local testability, whether or not a word u in the language L depends on (1) the prefix and suffix of the word u of length k - 1 and (2) the set of intermediate substrings 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 order of testability follow from these results.

We improve the upper bound on the order of testability of locally testable deterministic finite automaton with n states to $\frac{{n^2 - n}}{2} + 1$. This bound is the best possible.

We give an answer on the following conjecture o£ Kim, McNaughton and McCloskey for deterministic finite locally testable automaton with n states: “Is the order of local testability no greater than Ω(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(n ²).