Probabilistic Finite Automaton Emptiness is Undecidable for a Fixed Automaton

TL;DR

This paper demonstrates that the emptiness problem for probabilistic finite automata is undecidable even with a fixed automaton, using reductions from the Post Correspondence Problem, and explores variations with different alphabet sizes and output values.

Contribution

It proves the undecidability of PFA emptiness for a fixed automaton with minimal states and alphabet, extending understanding of automata decision problems.

Findings

Undecidability of PFA emptiness with 7 states and 5-symbol alphabet.

Binary alphabet restriction increases the number of states needed.

Allowing rational outputs reduces the number of states to 6.

Abstract

We construct a probabilistic finite automaton (PFA) with 7 states and an input alphabet of 5 symbols for which the PFA Emptiness Problem is undecidable. The only input for the decision problem is the starting distribution. For the proof, we use reductions from special instances of the Post Correspondence Problem. We also consider some variations: The input alphabet of the PFA can be restricted to a binary alphabet at the expense of a larger number of states. If we allow a rational output value for each state instead of a yes-no acceptance decision, the number of states can even be reduced to 6.