The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits

TL;DR

This paper formalizes wiring diagrams as an operad to model hierarchical, self-similar structures like circuits and databases, enabling structured queries, plug-and-play components, and recursion within a unified graphical framework.

Contribution

It introduces an operadic framework for wiring diagrams, connecting graphical language with database queries, circuits, and recursive structures, advancing formal understanding of these systems.

Findings

Wiring diagrams form an operad capturing self-similarity.

Graphical language structures database queries and circuit diagrams.

Operadic framework supports plug-and-play devices and recursion.

Abstract

Wiring diagrams, as seen in digital circuits, can be nested hierarchically and thus have an aspect of self-similarity. We show that wiring diagrams form the morphisms of an operad $\mcT$, capturing this self-similarity. We discuss the algebra $\Rel$ of mathematical relations on $\mcT$, and in so doing use wiring diagrams as a graphical language with which to structure queries on relational databases. We give the example of circuit diagrams as a special case. We move on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well. Throughout we include many examples and figures.