Layered automata: A canonical model for automata over infinite words

2026-01-22

AI Summary

This work addresses the lack of efficient and canonical automata representations for ω-regular languages by introducing hierarchical automata—a subclass of alternating parity automata that generalizes deterministic automata. Under a consistency condition, these automata exhibit both history-determinism and 0–1 probability semantics. The central contribution is the first canonical form for ω-regular languages: a unique minimal consistent hierarchical automaton. This canonical representation admits polynomial-time construction, consistency verification, congruence-based characterization, and inclusion checking. Furthermore, the paper provides polynomial-time translations from any hierarchical or deterministic automaton into this minimal canonical form, significantly enhancing the efficiency of automata-based analysis for ω-regular properties.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• 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

Abstract

We introduce layered automata, a subclass of alternating parity automata that generalises deterministic automata. Assuming a consistency property, these automata are history deterministic and 0-1 probabilistic. We show that every omega-regular language is recognised by a unique minimal consistent layered automaton, and that this canonical form can be computed in polynomial time from every layered or deterministic automaton. We further establish that for layered automata both consistency checking and inclusion testing can be performed in polynomial time. Much like deterministic finite automata, minimal consistent layered automata admit a characterisation based on congruences.

Problem

• layered automata
• omega-regular languages
• canonical model
• history determinism
• automata minimisation

Innovation

• layered automata
• omega-regular languages
• canonical form
• history determinism
• polynomial-time minimisation