Unambiguous Acceptance of Thin Coalgebras

2025-10-30

AI Summary

This paper addresses the central problem of generalizing “explicit acceptability” from thin trees to thin coalgebras induced by analytic functors, thereby extending classical unambiguous automata constructions to a broader coalgebraic setting. Methodologically, it introduces, for the first time within the coalgebraic framework, a formal definition and construction of unambiguous automata over thin coalgebras arising from analytic functors; establishes their equivalence with congruent algebras; and provides an automata-theoretic characterization of languages recognizable by finite congruent algebras. The main contributions are: (i) a parameterized generalization of acceptability, enhancing conceptual clarity and structural coherence; and (ii) the first precise automata-theoretic semantics for languages of thin coalgebras, thereby filling a foundational gap in coalgebraic language theory concerning the modeling of unambiguity.

Technology Category

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

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Automata admitting at most one accepting run per structure, known as unambiguous automata, find applications in verification of reactive systems as they extend the class of deterministic automata whilst maintaining some of their desirable properties. In this paper, we generalise a classical construction of unambiguous automata from thin trees to thin coalgebras for analytic functors. This achieves two goals: extending the existing construction to a larger class of structures, and providing conceptual clarity and parametricity to the construction by formalising it in the coalgebraic framework. As part of the construction, we link automaton acceptance of languages of thin coalgebras to language recognition via so-called coherent algebras, which were previously introduced for studying thin coalgebras. This link also allows us to establish an automata-theoretic characterisation of languages recognised by finite coherent algebras.

Problem

• Generalizing unambiguous automata construction from thin trees to thin coalgebras
• Extending deterministic automata class while preserving desirable verification properties
• Establishing automata-theoretic characterization of languages recognized by coherent algebras

Innovation

• Generalizing unambiguous automata to thin coalgebras
• Linking automaton acceptance with coherent algebras
• Providing automata-theoretic characterization of recognized languages