Higher-Order Behavioural Conformances via Fibrations

2025-07-24

AI Summary

This paper addresses the challenge of verifying program congruence for behavioral consistency—such as equivalence and behavioral distance—in higher-order languages. We propose a unified categorical framework based on fibrations. By modeling operational semantics via Abstract Higher-Order Specifications (AHOS), we decouple syntactic structure from behavioral modalities and, for the first time, categorify the Howe method—leveraging lifting properties and compatibility conditions to derive structural results. Our framework eliminates the need to re-engineer proofs for distinct languages or behavioral notions, uniformly supporting congruence verification for both bisimilarity and behavioral pseudometrics. Under natural assumptions, the greatest behavioral bisimulation is proven to be a congruence relation. The approach has been successfully instantiated for probabilistic higher-order languages and other cases, providing a single, formally verified foundation for qualitative and quantitative behavioral consistency across diverse higher-order calculi.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Coinduction is a widely used technique for establishing behavioural equivalence of programs in higher-order languages. In recent years, the rise of languages with quantitative (e.g.~probabilistic) features has led to extensions of coinductive methods to more refined types of behavioural conformances, most notably notions of behavioural distance. To guarantee soundness of coinductive reasoning, one needs to show that the behavioural conformance at hand forms a program congruence, i.e. it is suitably compatible with the operations of the language. This is usually achieved by a complex proof technique known as emph{Howe's method}, which needs to be carefully adapted to both the specific language and the targeted notion of behavioural conformance. We develop a uniform categorical approach to Howe's method that features two orthogonal dimensions of abstraction: (1) the underlying higher-order language is modelled by an emph{abstract higher-order specification} (AHOS), a novel and very general categorical account of operational semantics, and (2) notions of behavioural conformance (such as relations or metrics) are modelled via fibrations over the base category of an AHOS. Our main result is a fundamental congruence theorem at this level of generality: Under natural conditions on the categorical ingredients and the operational rules of a language modelled by an AHOS, the greatest behavioural (bi)conformance on its operational model forms a congruence. We illustrate our theory by deriving congruence of bisimilarity and behavioural pseudometrics for probabilistic higher-order languages.

Problem

• Extends coinductive methods to quantitative behavioral conformances
• Develops uniform categorical approach for Howe's method
• Proves congruence theorem for behavioral conformances in AHOS

Innovation

• General categorical approach to Howe's method
• Abstract higher-order specification (AHOS) modeling
• Fibrations model behavioral conformances