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Probabilistic Regular Expressions and MSO Logic on Finite Trees

We introduce probabilistic regular tree expressions and give a Kleene-like theorem for probabilistic tree automata (PTA). Furthermore, we define probabilistic MSO logic. This logic is more expressive than PTA. We define bottom-up PTA, which are strictly more expressive than PTA. Using bottom-up PTA, we prove a Buchi-like theorem for probabilistic MSO logic. We obtain a Nivat-style theorem as an additional result.