Interacting Hopf Algebras: the theory of linear systems

TL;DR

This paper develops a diagrammatic framework for linear algebra and signal processing circuits using interacting Hopf algebras, providing a canonical syntax and semantics with completeness and full abstraction results.

Contribution

It introduces a PROP of string diagrams for linear subspaces and applies it to formalize and analyze signal flow graphs with rigorous semantics.

Findings

Provides a faithful graphical representation of linear maps and subspaces.

Establishes completeness of IH equations for denotational equivalence.

Shows that any graph can be rewritten into an executable form matching operational and denotational semantics.

Abstract

As first main contribution, this thesis characterises the PROP SVk of linear subspaces over a field k - an important domain of interpretation for circuit diagrams appearing in diverse research areas. We present by generators and equations the PROP IH of string diagrams whose free model is SVk. IH stands for interacting Hopf algebras: its equations arise by distributive laws between Hopf algebras, which we obtain using Lack's technique for composing PROPs. The significance of the result is two-fold. First, it offers a canonical diagrammatic syntax for linear algebra: linear maps, kernels, subspaces, etc... are all faithfully represented in the graphical language. Second, the equations of IH describe familiar algebraic structures - Hopf algebras and Frobenius algebras - which are at the heart of graphical formalisms as seemingly diverse as quantum circuits, signal flow graphs, simple electrical circuits and Petri nets. Our characterisation enlightens the provenance of these axioms and reveals their linear algebraic nature. Our second main contribution is an application of IH to the semantics of signal processing circuits. We develop a formal theory of signal flow graphs, featuring a diagrammatic circuit syntax, a structural operational semantics and a denotational semantics. We prove completeness of the equations of IH for denotational equivalence. Also, we study full abstraction: it turns out that the purely operational picture is too concrete - two denotationally equal graphs may exhibit different operational behaviour. We classify the ways in which this can occur and show that any graph can be realised - rewritten, using the equations of IH, into an executable form where the operational behaviour and the denotation coincide. This realisability theorem suggests a reflection about the role of causality in the semantics of signal flow graphs and, more generally, of computing devices.