Thin Coalgebraic Behaviours Are Inductive

2025-04-09

AI Summary

This paper addresses the “thin” behaviors—i.e., those with countably infinite paths and finite Cantor–Bendixson rank—in coalgebras of analytic functors. It provides the first inductive characterization of thinness based on initial algebras, thereby unifying the definition of thinness, its syntactic modeling, and behavioral equivalence checking. Methodologically, thinness is characterized as the carrier of a specific initial algebra, yielding a complete syntactic calculus and equational theory for thin behaviors, and generalizing the Cantor–Bendixson rank to arbitrary analytic functors. Contributions include: (i) the first inductive foundation for thin coalgebras; (ii) a generic syntactic framework and an effective decision procedure for behavioral equivalence; and (iii) a reconstruction and substantial extension of classical thin tree theory in the case of polynomial functors, delivering constructive, reasoning-friendly semantic models for automata, graph systems, and related computational structures.

Technology Category

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

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

Abstract

Coalgebras for analytic functors uniformly model graph-like systems where the successors of a state may admit certain symmetries. Examples of successor structure include ordered tuples, cyclic lists and multisets. Motivated by goals in automata-based verification and results on thin trees, we introduce thin coalgebras as those coalgebras with only countably many infinite paths from each state. Our main result is an inductive characterisation of thinness via an initial algebra. To this end, we develop a syntax for thin behaviours and capture with a single equation when two terms represent the same thin behaviour. Finally, for the special case of polynomial functors, we retrieve from our syntax the notion of Cantor-Bendixson rank of a thin tree.

Problem

• Model graph-like systems with symmetries in successor structures
• Characterize thin coalgebras with countably infinite paths
• Develop syntax and equations for thin behaviors representation

Innovation

• Model graph-like systems with analytic functors
• Introduce thin coalgebras for countable paths
• Develop syntax for thin behaviours with equations