Power of Randomization in Automata on Infinite Strings

TL;DR

This paper investigates probabilistic Büchi automata on infinite strings, analyzing their computational power, decision problem complexities, and their relation to regular languages, including introducing a tractable subclass called Hierarchical PBAs.

Contribution

It provides new results on the decidability and complexity of key problems for PBAs, characterizes their recognized languages topologically, and introduces Hierarchical PBAs as a tractable subclass capturing all ω-regular languages.

Findings

Decidability and complexity results for emptiness, universality, and containment problems.

PBAs can recognize non-regular languages but are topologically equivalent to ω-regular languages.

Hierarchical PBAs are a syntactic restriction capturing exactly ω-regular languages.

Abstract

Probabilistic B\"uchi Automata (PBA) are randomized, finite state automata that process input strings of infinite length. Based on the threshold chosen for the acceptance probability, different classes of languages can be defined. In this paper, we present a number of results that clarify the power of such machines and properties of the languages they define. The broad themes we focus on are as follows. We present results on the decidability and precise complexity of the emptiness, universality and language containment problems for such machines, thus answering questions central to the use of these models in formal verification. Next, we characterize the languages recognized by PBAs topologically, demonstrating that though general PBAs can recognize languages that are not regular, topologically the languages are as simple as \omega-regular languages. Finally, we introduce Hierarchical PBAs, which are syntactically restricted forms of PBAs that are tractable and capture exactly the class of \omega-regular languages.