An algebraic theory of {omega}-regular languages, via {mu}{ u}-expressions

2025-05-15

AI Summary

This paper addresses the long-standing absence of an algebraic theory for alternating parity automata (APAs) by establishing the first complete equational axiomatization for APA language equivalence over ω-regular languages. Methodologically, it introduces μν-syntax—based on right-linear guarded expressions—as an algebraic representation of APAs, and leverages the completeness of linear-time μ-calculus to structurally derive the equational theory; duality of right-linear guarded expressions is further introduced to uniformly characterize infinite behaviors. The main contributions are: (1) the first sound and complete equational axiom system for APA language equivalence; (2) the revelation of a deep algebraic correspondence between APAs and linear-time μ-calculus; and (3) a novel paradigm for algebraic modeling of infinite behaviors in formal verification.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agreement, Argumentation & Negotiation

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Alternating parity automata (APAs) provide a robust formalism for modelling infinite behaviours and play a central role in formal verification. Despite their widespread use, the algebraic theory underlying APAs has remained largely unexplored. In recent work, a notation for non-deterministic finite automata (NFAs) was introduced, along with a sound and complete axiomatisation of their equational theory via right-linear algebras. In this paper, we extend that line of work, in particular to the setting of infinite words. We present a dualised syntax, yielding a notation for APAs based on right-linear lattice expressions, and provide a natural axiomatisation of their equational theory with respect to the standard language model of {omega}-regular languages. The design of this axiomatisation is guided by the theory of fixed point logics; in fact, the completeness factors cleanly through the completeness of the linear-time {mu}-calculus.

Problem

• Develop algebraic theory for ω-regular languages using μν-expressions
• Extend equational theory to alternating parity automata (APAs)
• Provide complete axiomatisation for APAs via fixed point logics

Innovation

• Dualised syntax for alternating parity automata
• Right-linear lattice expressions for APAs
• Axiomatisation based on fixed point logics