Power of Randomization in Automata on Infinite Strings

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.

💡 Research Summary

The paper investigates the expressive power and decision‑problem complexity of Probabilistic Büchi Automata (PBA), a model of finite‑state machines that process infinite strings while making random choices at each transition. Two acceptance semantics are considered: the “probable” semantics L > 0(B), where a word is accepted if the probability of an accepting run is greater than zero, and the “almost‑sure” semantics L = 1(B), where acceptance requires probability one. The authors focus on rational‑probability PBAs (RatPBAs) and obtain a comprehensive picture of three central verification problems—emptiness, universality, and language containment—under both semantics.

For the almost‑sure semantics, both emptiness and universality are shown to be PSPACE‑complete. This improves earlier EXPTIME upper bounds and settles the exact complexity. For the probable semantics, emptiness and universality are Σ⁰₂‑complete, placing them at the second level of the arithmetical hierarchy rather than in the analytical hierarchy where many undecidable ω‑automata problems reside. The containment problems L = 1(B) ⊆ L = 1(B′) and L > 0(B) ⊆ L > 0(B′) are also Σ⁰₂‑complete, demonstrating that even inclusion checking is no easier than the basic emptiness questions.

The paper then turns to topological properties. Using the Cantor topology on Σ^ω, the authors prove that every language in L = 1(PBA) lies strictly within the G_δ class, mirroring the situation for deterministic Büchi automata. The class L > 0(PBA) is shown to be exactly the Boolean closure of G_δ, just as the full class of ω‑regular languages is. Closure results are refined: L = 1(PBA) is closed under union and intersection but not under complement, while every L > 0(PBA) language can be expressed as a Boolean combination of L = 1(PBA) languages. These findings illuminate the precise relationship between nondeterminism, probability, and determinism in the infinite‑word setting.

A major contribution is the introduction of Hierarchical PBAs (HPBAs), a syntactically restricted subclass. In an HPBA, states are partitioned into levels; from any state, at most one transition with non‑zero probability stays on the same level, and all other non‑zero transitions move to strictly higher levels. This structural restriction forces runs to eventually ascend to the highest level, yielding a form of “progress” that eliminates the ability to encode non‑regular ω‑languages. The authors prove two key characterizations: (i) under the probable semantics, HPBAs recognize exactly the ω‑regular languages; (ii) under the almost‑sure semantics, HPBAs capture precisely the ω‑regular languages that are recognizable by deterministic Büchi automata (i.e., the deterministic ω‑regular subclass).

Complexity for HPBAs matches that of classical Büchi automata: emptiness for the probable semantics is NL‑complete, while universality is PSPACE‑complete; for the almost‑sure semantics the roles are reversed (universality NL‑complete, emptiness PSPACE‑complete). Thus, the syntactic restriction restores tractability that is lost in general PBAs (where, for example, emptiness for probable semantics is Σ⁰₂‑complete).

The paper concludes with a discussion of related work, positioning its results alongside earlier studies of PBAs, probabilistic finite automata, and finite‑state monitors. By providing tight complexity bounds, topological classifications, and a natural syntactic restriction that aligns probabilistic automata with ω‑regularity, the work significantly advances our understanding of how randomization interacts with infinite‑word acceptance and offers practical avenues for verification of probabilistic reactive systems.