For nondeterministic weighted automata over Markov chains, exact expected value and distribution can be irrational or uncomputable, but both can be approximated in exponential time in the automaton and polynomial time in the Markov chain and precision.
Non-deterministic weighted automata on random words
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Non-deterministic weighted automata evaluated over Markov chains
For nondeterministic weighted automata over Markov chains, exact expected value and distribution can be irrational or uncomputable, but both can be approximated in exponential time in the automaton and polynomial time in the Markov chain and precision.