Finite automata based on quantum logic and monadic second-order quantum logic

We introduce monadic second-order quantum logic and prove that the behaviors of finite automata based on quantum logic are precisely the quantum languages definable with sentences of our monadic second-order quantum logic. This generalizes Buchi's and Elgot's fundamental theorems to quantum logic setting. We also consider first-order quantum logic and show that star-free quantum languages and aperiodic quantum languages introduced here coincide with the first-order quantum definable ones. This generalizes Schutzenberger's fundamental theorems to quantum logic setting. The determinazation of finite automata based on quantum logic is studied by introducing the generalized subset construction method. Then the Kleene theorem in the frame of quantum logic is presented here.