Logical relations for call-by-push-value models, via internal fibrations in a 2-category

2025-05-20

AI Summary

This paper addresses the challenge of modeling logical relations in call-by-push-value (CBPV) semantics—specifically, the absence of a suitable fibration framework accommodating both adjunctions and enrichment simultaneously. We introduce the first 2-categorical notion of “CBPV fibrations”, rigorously preserving CBPV structure while unifying adjunctions and enrichment, and supporting program property verification such as effect simulation. Methodologically, we systematically extend fibration theory to CBPV, developing corresponding codomain/subobject fibrations and pullback constructions. Our main contributions are: (1) generalizing Katsumata’s ⊤⊤-lifting to the CBPV setting; (2) proving an effect simulation theorem for CBPV; and (3) establishing relative full completeness for CBPV without sum types. These results provide a novel semantic verification paradigm for effectful programs.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models

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: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We give a denotational account of logical relations for call-by-push-value (CBPV) in the fibrational style of Hermida, Jacobs, Katsumata and others. Fibrations -- which axiomatise the usual notion of sets-with-relations -- provide a clean framework for constructing new, logical relations-style, models. Such models can then be used to study properties such as effect simulation. Extending this picture to CBPV is challenging: the models incorporate both adjunctions and enrichment, making the appropriate notion of fibration unclear. We handle this using 2-category theory. We identify an appropriate 2-category, and define CBPV fibrations to be fibrations internal to this 2-category which strictly preserve the CBPV semantics. Next, we develop the theory so it parallels the classical setting. We give versions of the codomain and subobject fibrations, and show that new models can be constructed from old ones by pullback. The resulting framework enables the construction of new, logical relations-style, models for CBPV. Finally, we demonstrate the utility of our approach with particular examples. These include a generalisation of Katsumata's $ op op$-lifting to CBPV models, an effect simulation result, and a relative full completeness result for CBPV without sum types.

Problem

• Extending logical relations to call-by-push-value models
• Defining CBPV fibrations using 2-category theory
• Constructing new logical relations-style models for CBPV

Innovation

• Using 2-category theory for CBPV fibrations
• Developing codomain and subobject fibrations theory
• Generalizing Katsumata's lifting to CBPV models