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Right-Linear Lattices: An Algebraic Theory of ω-Regular Languages, with Fixed Points

Authors Anupam Das , Abhishek De

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ACM Subject Classification
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Logic and verification
  • Theory of computation → Modal and temporal logics
Keywords
  • omega-languages
  • regular languages
  • fixed points
  • Kleene algebras
  • right-linear grammars

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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 [Anupam Das and Abhishek De, 2024], 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.

Cite As Get BibTex

Anupam Das and Abhishek De. Right-Linear Lattices: An Algebraic Theory of ω-Regular Languages, with Fixed Points. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.39

Author Details

Anupam Das
  • School of Computer Science, University of Birmingham, UK
Abhishek De
  • School of Computer Science, University of Birmingham, UK

Funding

This work was supported by a UKRI Future Leaders Fellowship, "Structure vs Invariants in Proofs", project reference MR/S035540/1.

Acknowledgements

The authors thank Georg Struth, Mike Cruchten, Debam Biswas, Denis Kuperberg and Damien Pous for fruitful discussions.

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