Full Definability in a Profunctorial Model

2026-04-29

AI Summary

This study addresses the problem of full definability in profunctor-based semantic models over groupoids—namely, ensuring that every semantic element is denoted by a proof net of multiplicative linear logic (MLL). To this end, the work introduces stability into profunctor semantics for the first time as a key criterion for definability, combining logical relations with categorical semantics to fully characterize definable profunctors. The main contribution establishes that every stable and total family of logical profunctors can be precisely defined by MLL proof nets augmented with the MIX rule, thereby forging a rigorous correspondence between stability and proof nets. This result confirms the model’s capacity for highly refined expressiveness in capturing program semantics.

Abstract

A semantic model enjoys full definability if every semantic element in the model is a denotation of some proof or program. Full definability indicates that the model captures programs and proofs in a highly detailed manner. This paper studies full definability in a model based on the (bi)category of profunctors on groupoids, which is a proof-relevant variant of the relational model. Despite the fact that a profunctor is far more complicated than a relation, we show that a rather straightforward application of the ideas for the relational model, together with the notion of stability in profunctors, provides a complete characterisation of definable profunctors. More precisely, all logical families of stable and total profunctors are definable by proof-nets of multiplicative linear logic with MIX. As a part of the full definability proof, we show that the stability serves as a correctness criterion, which we think is of independent interest.

Problem

• full definability
• profunctor
• semantic model
• proof-relevant
• stability

Innovation

• full definability
• profunctor
• stability
• proof-net
• multiplicative linear logic