Power of Randomization in Automata on Infinite Strings

Abstract

Probabilistic B\"{u}chi 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 precisely characterize the complexity of the emptiness, universality, and language containment problems for such machines,answering canonical questions central to the use of these models in formal verification. Next, we characterize the languages recognized by {\PBA}s topologically, demonstrating that though general {\PBA}s can recognize languages that are not regular, topologically the languages are as simple as $\omega$-regular languages. Finally, we introduce Hierarchical {\PBA}s, which are syntactically restricted forms of {\PBA}s that are tractable and capture exactly the class of $\omega$-regular languages.