International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 12 Issue: 12 | Dec 2025
p-ISSN: 2395-0072
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A Structured Approach to Büchi Automata Construction from ω-Regular Expressions Shruti Mishra1, Rahul Gupta2 1 M.Tech. (CSE) Scholar, Department of Computer Science and Engineering, S. R. Institute of Management and
Technology Lucknow, Uttar Pradesh, India
2 Assistant Professor, Department of Computer Science and Engineering, S. R. Institute of Management and
Technology Lucknow, Uttar Pradesh, India ---------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - ω-regular languages play a central role in the
Büchi automata are a canonical automaton model for ωregular languages and are widely used in automata-based model checking, particularly in linear temporal logic (LTL) verification. The automata-theoretic approach to verification typically involves translating a logical specification into a Büchi automaton and then checking language emptiness or inclusion against a system model. As a result, the efficiency and structure of Büchi automata construction directly impact the scalability and practicality of verification tools.
formal specification and verification of non-terminating systems such as operating systems, communication protocols, and reactive controllers. Büchi automata are the most widely used automaton model for recognizing ω-regular languages and form the backbone of automata-theoretic model checking. While Büchi automata can be constructed from various specification formalisms, ω-regular expressions offer a compact and algebraic representation of infinite-word behaviors. However, direct construction of Büchi automata from ω-regular expressions often leads to complex transition structures and significant state explosion. This paper presents a structured approach to Büchi automata construction from ω-regular expressions, emphasizing syntactic decomposition, semantic preservation, and systematic automaton synthesis. The proposed method decomposes ω-regular expressions into finite and infinite components, constructs intermediate automata, and integrates them using well-defined acceptance conditions. The approach improves clarity, modularity, and extensibility while maintaining correctness. Formal definitions, construction algorithms, illustrative examples, and complexity analysis are provided. The paper contributes a rigorous framework that can serve as a foundation for optimization, symbolic implementation, and advanced verification tools.
ω-regular expressions constitute an algebraic formalism for specifying ω-regular languages using concatenation, union, Kleene star, and the ω-iteration operator. They provide an expressive and concise way to specify infinite behaviors, often more intuitive than temporal logic for certain classes of specifications. However, transforming ω-regular expressions into Büchi automata is non-trivial due to the inherent infiniteness of the accepted words and the need to enforce Büchi acceptance conditions. Existing construction techniques often rely on ad hoc transformations or direct translations that obscure the semantic structure of the original expression. These approaches may lead to automata with unnecessary states and transitions, reducing interpretability and increasing verification cost. There is therefore a strong motivation to develop a structured and systematic approach to Büchi automata construction from ω-regular expressions.
Key Words: ω-regular expressions, Büchi automata, infinite words, automata theory, formal verification, model checking etc.
This paper proposes such a structured approach. By decomposing ω-regular expressions into finite and infinite components and constructing intermediate automata that reflect this structure, the proposed method ensures semantic clarity and correctness while laying a foundation for future optimizations.
1. INTRODUCTION Modern computing systems increasingly rely on components that exhibit non-terminating behavior. Examples include operating systems, network protocols, embedded controllers, and cyber-physical systems. Such systems are often reactive, continuously interacting with their environment rather than halting after producing an output. Formal verification of these systems requires models capable of expressing infinite executions. ω-regular languages provide a mathematically robust framework for describing sets of infinite words and have become fundamental in theoretical computer science and formal methods.
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2. Preliminaries This section introduces the fundamental concepts and definitions required for the remainder of the paper. 2.1 Infinite Words and ω-Languages Let Σ be a finite alphabet. An infinite word over Σ is an infinite sequence w=a0a1a2…w = a0 a1 a2 , where each ai ∈ Σ.
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