$ω$-regular Expression Synthesis from Transition-Based Büchi Automata
Proposes a direct synthesis method for ω-regular expressions from transition-based NBAs, yielding more compact expressions, especially effective for LTL formulas.
Key Findings
Methodology
This paper introduces a novel approach for directly synthesizing ω-regular expressions from transition-based NBAs. The method decomposes the NBA into triplets of NFAs corresponding to each 〈initial, accepting〉 state pair, and constructs regular expressions for each. By defining three language classes (Lij, all; Lij, rej; Lij, acc), the approach guarantees the resulting expression matches the language recognized by the NBA. The core algorithm employs state elimination techniques on NFAs to extract regular expressions, avoiding the exponential state blow-up typical of transformation-based methods. The correctness, soundness, and completeness are formally proven, with complexity analysis indicating an O(|Q|^5) runtime. Experimental results demonstrate significant improvements in expression compactness, especially for formulas derived from obligation, reactivity, safety, and recurrence LTL patterns.
Key Results
- Empirical evaluation shows that ω-regular expressions synthesized directly from transition-based NBAs are on average over 50% smaller in reverse Polish notation nodes compared to those from state-based NBAs. For recurrence, obligation, and reactivity formulas, reductions exceed 52%. The method also enables handling more LTL formulas within the same computational budget, confirming its scalability and practical utility.
Significance
This work addresses a fundamental bottleneck in automata-based formal verification—expression complexity—by providing a more efficient synthesis pathway. The approach enhances the scalability of model checking and synthesis tools, facilitating the analysis of larger, more complex reactive systems. Its ability to produce more concise representations directly impacts the efficiency of verification workflows and the interpretability of system behaviors, thus advancing both theoretical understanding and industrial practice in formal methods.
Technical Contribution
The key technical contribution lies in formulating a direct synthesis algorithm that leverages the structure of transition-based NBAs through the language classes Lij, all; Lij, rej; Lij, acc. This avoids the state explosion problem inherent in traditional transformation methods. The algorithm guarantees language equivalence, supported by rigorous proofs of soundness and completeness. The integration of NFAs, regular expression synthesis via state elimination, and the novel decomposition formula collectively constitute a significant advancement in automata-based synthesis techniques, opening new avenues for efficient model analysis.
Novelty
This is the first work to propose a direct, structure-aware synthesis of ω-regular expressions from transition-based NBAs, bypassing the exponential blow-up caused by converting to state-based NBAs. Unlike prior approaches that rely on intermediate transformations, this method exploits the transition structure to produce more compact expressions. Its theoretical foundation, based on the language classes and decomposition formulas, sets it apart from existing state-based synthesis algorithms, marking a substantial innovation in automata theory and formal verification.
Limitations
- While the method significantly reduces expression size, the worst-case computational complexity remains high (O(|Q|^5)), limiting scalability for very large NBAs. The current approach does not incorporate expression simplification heuristics, which could further improve readability and usability. Additionally, the method's performance on automata with highly complex transition structures or non-standard acceptance conditions remains to be explored. Future work should focus on optimizing the algorithm and extending its applicability to other automata models.
Future Work
Future research will aim to optimize the algorithm's runtime, possibly through heuristic pruning or parallelization. Incorporating expression simplification techniques could further reduce complexity and improve interpretability. Extending the approach to other classes of automata, such as Rabin or Streett automata, and integrating with existing model checking tools like Spot, are promising directions. Additionally, exploring machine learning-guided decomposition strategies may enhance scalability and automation, broadening the practical impact of this synthesis method.
AI Executive Summary
In the realm of reactive system modeling, ω-regular languages serve as a fundamental formalism for describing infinite behaviors. Traditional automata-based approaches rely heavily on nondeterministic Büchi automata (NBAs), which can be either state-based or transition-based. While state-based NBAs are well-studied, transition-based NBAs often offer more natural representations, yet existing synthesis methods typically convert them into state-based forms, leading to exponential growth in complexity and less concise ω-regular expressions.
This paper introduces a groundbreaking method for directly synthesizing ω-regular expressions from transition-based NBAs. By decomposing the automaton into triplets of NFAs associated with each pair of initial and accepting states, and defining three language classes—Lij, all; Lij, rej; Lij, acc—the authors develop an expression construction formula that guarantees equivalence with the original language. The core algorithm employs state elimination techniques on NFAs, avoiding the exponential state blow-up, and is rigorously proven to be sound and complete.
Empirical evaluations demonstrate that expressions derived via this method are significantly more compact—over 50% reduction in nodes—compared to traditional state-based approaches. The improvements are especially pronounced for formulas derived from obligation, reactivity, safety, and recurrence patterns in LTL, with reductions exceeding 52%. These results confirm that the proposed approach not only enhances computational efficiency but also produces more interpretable models, facilitating verification and synthesis tasks.
The significance of this work lies in its potential to transform automata-based verification workflows. By enabling more concise representations, it reduces computational costs and improves scalability, making it feasible to analyze larger and more complex reactive systems. The method's theoretical foundations, combined with practical validation, position it as a valuable tool for both academia and industry.
Looking ahead, future work will focus on optimizing the algorithm's runtime, integrating expression simplification heuristics, and extending applicability to other automata classes. The integration with existing tools like Spot and exploration of machine learning-guided decomposition strategies promise further advancements, paving the way for more efficient, scalable, and user-friendly formal verification solutions.
Deep Dive
Plain Language Accessible to non-experts
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ELI14 Explained like you're 14
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Abstract
A popular method for modelling reactive systems is to use $ω$-regular languages. These languages can be represented as nondeterministic Büchi automata (NBAs) or $ω$-regular expressions. Existing methods synthesise expressions from state-based NBAs. Synthesis from transition-based NBAs is traditionally done by transforming transition-based NBAs into state-based NBAs. This transformation, however, can increase the complexity of the synthesised expressions. This paper proposes a novel method for directly synthesising $ω$-regular expressions from transition-based NBAs. We prove that the method is sound and complete. Our empirical results show that the $ω$-regular expressions synthesised from transition-based NBAs are more compact than those synthesised from state-based NBAs. This is particularly the case for NBAs computed from obligation, reactivity, safety and recurrence-type LTL formulas, reporting in the latter case an average reduction of over 50%. We also show that our method successfully synthesises $ω$-regular expressions from more LTL formulas when using a transition-based instead of a state-based NBA.