Probabilistic Finite Automaton Emptiness is undecidable

TL;DR

The paper proves that determining whether a probabilistic finite automaton recognizes any string (emptiness) is undecidable, and provides multiple proofs and strengthenings of this fundamental result.

Contribution

It offers three self-contained proofs of the undecidability of probabilistic finite automaton emptiness and extends the result to fixed automata with limited input.

Findings

Undecidability of probabilistic finite automaton emptiness confirmed

The problem remains undecidable for automata with only the initial distribution as input

Provides multiple proofs and strengthens the original theorem

Abstract

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.