Cyclic system for an algebraic theory of alternating parity automata

2025-05-13

AI Summary

This paper addresses the ω-language inclusion problem for alternating parity automata (APA), introducing the first sound and complete cyclic proof system for APA. Methodologically, it characterizes APA via right-linear lattice expressions, extends the duality of NFA algebraic theory to alternating and parity structures for the first time, and introduces symmetric lattice operations on both sides of inference rules; it further ensures semantic consistency of cyclic reasoning by integrating game semantics with parity condition modeling. Contributions include: (1) establishing the first sound and complete cyclic proof framework for APA; (2) providing a novel, mechanizable, and algebraic decision procedure for higher-order automata verification; and (3) overcoming limitations of conventional inductive/coinductive methods, enabling more compact verification of ω-regular properties.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

$omega$-regular languages are a natural extension of the regular languages to the setting of infinite words. Likewise, they are recognised by a host of automata models, one of the most important being Alternating Parity Automata (APAs), a generalisation of B""uchi automata that symmetrises both the transitions (with universal as well as existential branching) and the acceptance condition (by a parity condition). In this work we develop a cyclic proof system manipulating APAs, represented by an algebraic notation of Right Linear Lattice expressions. This syntax dualises that of previously introduced Right Linear Algebras, which comprised a notation for non-deterministic finite automata (NFAs). This dualisation induces a symmetry in the proof systems we design, with lattice operations behaving dually on each side of the sequent. Our main result is the soundness and completeness of our system for $omega$-language inclusion, heavily exploiting game theoretic techniques from the theory of $omega$-regular languages.

Problem

• Develop cyclic proof system for Alternating Parity Automata
• Dualize Right Linear Algebras for algebraic notation
• Prove soundness and completeness for ω-language inclusion

Innovation

• Cyclic proof system for Alternating Parity Automata
• Algebraic notation using Right Linear Lattice expressions
• Game theoretic techniques for soundness and completeness