Reducing the time complexity of testing for local threshold testability

A locally threshold testable language L is a language with the property that for some nonnegative integers k and l, 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 intermediate substrings of length k of the word u where the sets of substrings occurring at least j times are the same, for j ≤ l. For given k and l the language is called l-threshold k-testable. A finite deterministic automaton is called threshold locally testable if the automaton accepts a l-threshold k-testable language for some l and k. New necessary and sufficient conditions for a deterministic finite automaton to be locally threshold testable are found. On the basis of these conditions, we modify the algorithm to verify local threshold testability of the automaton and reduce the time complexity of the algorithm. The algorithm is implemented as a part of C/C⁺⁺ package TESTAS (testability of automata and semigroups).