It is undecidable whether the language recognized by a probabilistic finite automaton is empty. Several other undecidability results, in particular regarding problems about matrix products, are based on this important theorem. We present three proofs of this theorem from the literature in a self-contained way, and we derive some strengthenings. For example, we show that the problem remains undecidable for a fixed probabilistic finite automaton with 11 states, where only the starting distribution is given as input.
The results about have undecidability of the emptiness problem probabilistic finite automata with a small number of states have been superseded by improved results of the paper Probabilistic Finite Automaton Emptiness is undecidable for a fixed automaton, where a smaller number of states was achieved with different techniques.
Last update: August 21, 2025.