Proof Identity and Categorical Models of BV

2026-04-28

AI Summary

This work addresses the long-standing absence of a precise notion of proof identity in BV logic and the lack of a coherence theorem for its categorical semantics (BV-categories), which has obscured the relationship between semantics and proof theory. The paper introduces atomic flows—specialized string diagrams—as a criterion for proof identity in BV logic and leverages this insight to refine the definition of BV-categories. By integrating atomic flows, string diagram representations, and categorical methods, the authors construct an enhanced notion of BV-category and rigorously establish its soundness with respect to BV logic. This development yields a clear and mathematically robust correspondence between semantic models and proofs, thereby filling a critical gap between the proof-theoretic and categorical approaches in the study of BV logic.

Abstract

BV-categories are a recent development that aims to give categorical semantics to proofs in the logic BV. However, due to the absence of a coherence theorem on one side and a well-defined notion of proof identity for BV on the other side, the precise relation between BV-categories and the logic BV is still not clear. To improve on this situation, we define in this paper a notion of proof identity for BV, based on the notion of atomic flows, which can be seen as a special form of string diagrams. Based on this notion of proof identity, we then strengthen the existing notion of BV-category and prove that it is sound with respect to the logic.

Problem

• proof identity
• BV logic
• categorical semantics
• coherence theorem
• atomic flows

Innovation

• proof identity
• atomic flows
• BV-categories
• categorical semantics
• string diagrams