Piecewise and Local Threshold Testability of DFA

The necessary and sucient conditions for an automaton to be locally threshold testable are found. We introduce the polynomial time algorithm to verify local threshold testability of the automaton of time complexity O(n 5) and an algorithm of order O(n 3) for the local threshold testability problem for syntactic semigroup of the automaton. We modify necessary and sucient conditions for piecewise testability problem for deterministic nite automaton and improve the Stern algorithm to verify piecewise testability for the automaton. The time complexity of the algorithm is reduced from O(n 5) to O(n 2). An algorithm to verify piecewise testability for syntactic semigroup of the automaton of order O(n 2) is presented as well. The algorithms have been implemented as a C=C ++ package.