From Nondeterministic Büchi and Streett Automata to Deterministic Parity Automata (0705.2205v2)

Abstract: In this paper we revisit Safra's determinization constructions for automata on infinite words. We show how to construct deterministic automata with fewer states and, most importantly, parity acceptance conditions. Determinization is used in numerous applications, such as reasoning about tree automata, satisfiability of CTL*, and realizability and synthesis of logical specifications. The upper bounds for all these applications are reduced by using the smaller deterministic automata produced by our construction. In addition, the parity acceptance conditions allows to use more efficient algorithms (when compared to handling Rabin or Streett acceptance conditions).

• The paper introduces a novel dynamic naming method that directly transforms NBW and NSW into DPWs, significantly reducing state space complexity.
• It replaces Safra’s static naming with dynamic renaming, resulting in a more streamlined and computationally efficient determinization process.
• These advances improve formal verification and synthesis performance by lowering computational requirements in solving related parity and Rabin games.

From Nondeterministic Büchi and Streett Automata to Deterministic Parity Automata

In their paper, Nir Piterman explores the intricate processes of transforming nondeterministic Büchi and Streett automata (NBW and NSW, respectively) into deterministic parity automata (DPW). This research sheds light on pivotal methods in automata theory, specifically focusing on determinization—a foundational aspect crucial for the complementation of automata and for applications in formal verification, model checking, and synthesis problems.

Overview of Transformations

There are two primary transformations discussed: NBW to DPW and NSW to DPW. Traditionally, the determinization of NBW involved converting them into deterministic Rabin word automata (DRW) using Safra's method, which is inherently complex and results in a large increase in the number of states. Piterman's work innovatively substitutes Safra's static naming in trees with dynamic naming, allowing for a more streamlined construction of smaller state spaces and directly yielding DPWs. This reduces the state space magnitude from Safra's $(12)^n n^{2n}$ to $2n n!$ for NBWs and from $(12)^{n(k+1)}(n(k+1))^{n(k+1)}$ to $2n^{n(k+1)} (n(k+1))!$ for NSWs. This reduction notably simplifies the process due to the avoidance of intermediate Rabin or Streett automata in favor of a direct transition to parity automata.

Technical Specificity and Numerical Boundaries

Technically, Piterman's substitution of dynamic node labels enables nodes to be renamed in response to automaton transitions, allowing the construction to maintain a smaller active node set. This improves efficiency in the determinization process by adopting a parity condition acceptance, which carries a more uniform and computationally feasible approach for automata operations. As a direct result, the computational requirements for related problem-solving—like solving parity and Rabin games—are significantly lowered due to the inherently simpler nature of parity conditions as compared to Rabin or Streett conditions.

Implications of Research

The construction of smaller DPWs not only impacts theoretical computer science by advancing understanding of automata transformations but also ripples outward into practical verification tools, which benefit from reduced state spaces and improved execution times. In terms of complexity classifications, given that parity games belong to NP∩co-NP, while Rabin games lie in NP-complete territory, the results of this work facilitate more efficient implementations in solving such games, enhancing the performance of associated algorithms.

Future Considerations

While Piterman's innovations present significant improvements over Safra's constructions, further work is required to narrow the gap between existing lower bounds set by Michel and others, particularly in terms of state space size and optimal index requirements. Future advancements may lie in refining the bijection strategies further and exploring even more compact representations within the constraints of automata determinization.

In sum, this research provides a pivotal step forward in automata theory by offering more efficient structures and processes for determinization, which align closely with the needs of real-world applications in formal verification and logic synthesis. Through such advancements, the robustness, efficiency, and applicability of formal methods continue to grow, showcasing the potency of theoretical computer science when transposed into practical, usable technology.