Separation and Gluing of Explanations on Sites of Dynamical Systems

TL;DR

This paper develops a new topological framework for explanations in dynamical systems using Grothendieck sites, demonstrating conditions under which local explanations can be uniquely combined into global explanations.

Contribution

It introduces a Grothendieck site for Mealy machines in o-minimal structures, defines presheaves of explanations, and characterizes when these presheaves form sheaves, revealing insights into the gluing of explanations.

Findings

Behavioral presheaf is separated, so global explanations are determined by local ones.

Gluing of explanations generally fails without additional conditions.

A topological criterion for the sheaf property is provided for stateless systems.

Abstract

We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge.