Kleene algebra with tests (KAT) is a decidable equational system for program verification, that uses both Kleene and Boolean algebras. In spite of KAT?s elegance and success in providing theoretical solutions for several problems, not many efforts have been made towards obtaining tractable decision procedures that could be used in practical software verification tools. The main drawback of the existing methods relies on the explicit use of all possible assignments to boolean variables. Recently, Silva introduced an automata model that extends Glushkov's construction for regular expressions. Broda et al. extended also Mirkin's equation automata to KAT expressions and studied the state complexity of both algorithms. Contrary to other automata constructions from KAT expressions, these two constructions enjoy the same descriptional complexity behaviour as their counterparts for regular expressions, both in the worst case as well as in the average case. In this paper, we generalize, for these automata, the classical methods of subset construction for nondeterministic finite automata, and the Hopcroft and Karp algorithm for testing deterministic finite automata equivalence. As a result, we obtain a decision procedure for KAT equivalence where the extra burden of dealing with boolean expressions avoids the explicit use of all possible assignments to the boolean variables. Finally, we specialize the decision procedure for testing KAT expressions equivalence without explicitly constructing the automata, by introducing a new notion of derivative and a new method of constructing the equation automaton.