The Decidability Frontier for Probabilistic Automata on Infinite Words

TL;DR

This paper explores the boundaries of decidability for various decision problems in probabilistic automata operating on infinite words, providing a comprehensive map of what can and cannot be algorithmically determined.

Contribution

It offers a complete characterization of the decidability and undecidability frontier for both quantitative and qualitative problems in probabilistic automata on infinite words.

Findings

Decidability results for safety and reachability conditions.

Undecidability results for B"uchi and coB"uchi conditions.

Complete classification of decision problem boundaries.

Abstract

We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words.