Rigorous proof of undecidability for the Emptiness Problem of probabilistic 2-way finite automata

Establish a rigorous undecidability proof for the Emptiness Problem of probabilistic 2-way finite automata by giving a concrete reduction, for example from Post’s Correspondence Problem, that explicitly connects Freivalds’s macrocompetition construction to Post’s Correspondence Problem and thereby validates (or refutes) the claim of undecidability for 2‑way probabilistic finite automata.

Background

Freivalds (1981) studied probabilistic 2-way finite automata and sketched an approach based on competitions between players, claiming the Emptiness Problem for such automata is undecidable. However, only a hint of a reduction from PCP was provided, without the crucial connection between the macrocompetition gadget and PCP.

The present article’s author examined this claim but could not reconstruct a working proof. A complete, detailed reduction confirming or refuting this undecidability claim for 2-way probabilistic automata remains to be provided in the literature.

References

Freivalds claimed that the Emptiness Problem for such automata is undecidableTheorem~4; he gives only a hint that the reduction should be from the PCP (Post's Correspondence Problem, see Section~\ref{sec:PCP}), without any details how to connect "macrocompetitions" with the PCP. I have not been able to come up with an idea how the proof would proceed.

— Probabilistic Finite Automaton Emptiness is undecidable (2405.03035 - Rote, 5 May 2024) in Section “History of ideas”