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We present a new technique for obtaining decision procedures for modal logics of programs. The technique centers around a new class of finite automata on infinite trees for which the emptiness problem can be solved in polynomial time. The decision procedures then consist of constructing an automaton A_f for a given formula f, such that A_f accepts some tree if and only if f is satisfiable. We illustrate our technique by giving an exponential decision procedure for deterministic propositional dynamic logic and a variant of the μ-calculus of Kozen.