Right-Linear Lattices: An Algebraic Theory of ω-Regular Languages, with Fixed Points
Das, Anupam, De, Abhishek · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2025
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 [10], 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 to the setting of infinite words. In particular, 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 ω-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 µ-calculus.