Abstract

This talk illustrates the signal flow calculus, an algebraic and diagrammatic foundation of signal processing circuits. Signal flow graphs, a class of circuits that play a foundational role in control theory, are recovered via a Kleene’s theorem, as the rational fragment of the calculus. The high-point of our developments is a sound and complete axiomatisation for semantic equivalence, which we call the theory of interacting Hopf monoids (IH). The relevance of IH goes beyond the signal flow calculus: its equations describe the interplay of familiar structures, such as Frobenius monoids and Hopf monoids, in a way that appeared independently in other research threads, in quantum information theory and concurrency theory. Our approach gives a formal explanation for this ubiquity, by showing that the equations of IH present categories of linear subspaces — completeness for the signal flow calculus follows as a corollary. This characterisation passes through a modular account of IH : its axioms are explained in terms of composition of simpler algebraic theories, using distributive laws of PRO Ps as in the work of Steve Lack.

This talk is based on joint work with Filippo Bonchi and Pawel Sobocinski.