Stochastic Relational Presheaves and Dynamic Logic for Contextuality

TL;DR

This paper extends presheaf models to incorporate stochastic dynamics, enabling the modeling of quantum systems and contextuality through a dynamic modal logic framework.

Contribution

It introduces a novel extension of presheaf models combining category theory with stochastic processes to represent quantum contextuality.

Findings

Extended presheaf models to include stochastic dynamics

Provided a transitional formulation of sheaf-theoretic structures

Developed a dynamic modal logic for contextuality

Abstract

Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.