Infinite Traces by Finality: a Sheaf-Theoretic Approach

2025-07-30

AI Summary

Final coalgebras cannot directly characterize infinite trace semantics in Kleisli categories. Method: This paper introduces an ordinal-layered construction framework based on “bounded-behavior functors,” integrating sheaf gluing mechanisms with bounded corecursion to systematically synthesize compatible families of finite approximations within Kleisli categories, thereby constructing final coalgebras that capture infinite traces. Contribution/Results: This framework achieves, for the first time in Kleisli categories, a categorical unification of infinite trace semantics, providing a rigorous constructive foundation for linear-time semantics. Under mild assumptions, the constructed final coalgebra is rigorously proven to precisely characterize the complete set of infinite traces, significantly extending the applicability of coalgebraic methods to modeling nonterminating behavior.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and 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: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Kleisli categories have long been recognised as a setting for modelling the linear behaviour of various types of systems. However, the final coalgebra in such settings does not, in general, correspond to a fixed notion of linear semantics. While there are well-understood conditions under which final coalgebras capture finite trace semantics, a general account of infinite trace semantics via finality has remained elusive. In this work, we present a sheaf-theoretic framework for infinite trace semantics in Kleisli categories that systematically constructs final coalgebras capturing infinite traces. Our approach combines Kleisli categories, sheaves over ordinals, and guarded (co)recursion, enabling infinite behaviours to emerge from coherent families of finite approximations via amalgamation. We introduce the notion of guarded behavioural functor and show that, under mild conditions, their final coalgebras directly characterise infinite traces.

Problem

• Final coalgebras lack general linear semantics in Kleisli categories
• No systematic framework exists for infinite trace semantics via finality
• Sheaf-theoretic approach needed to unify finite and infinite trace behaviors

Innovation

• Sheaf-theoretic framework for infinite traces
• Combines Kleisli categories and sheaves over ordinals
• Guarded behavioural functors characterize infinite traces