Logical Relations and Parametricity - A Reynolds Programme for Category Theory and Programming Languages

In his seminal paper on ''Types, Abstraction and Parametric Polymorphism,'' John Reynolds called for homomorphisms to be generalized from functions to relations. He reasoned that such a generalization would allow type-based ''abstraction'' (representation independence, information hiding, naturality or parametricity) to be captured in a mathematical theory, while accounting for higher-order types. However, after 30 years of research, we do not yet know fully how to do such a generalization. In this article, we explain the problems in doing so, summarize the work carried out so far, and call for a renewed attempt at addressing the problem.