String diagrams - from control theory to concurrency and beyond

I will talk about two recent papers.

1) Graphical Diophantine Algebra

The wires of string diagrams in ordinary Graphical Linear Algebra carry elements of a field. In previous work we used these string diagrams as an algebra of open signal flow graphs --- the idea is that real numbers represent polarised quantities, e.g. the voltage or current of an electrical circuit. Replacing reals with naturals means that it is more useful to think of signals as discrete resources: indeed, string diagrams now become an algebra of open Petri nets. I will present the equational characterisation and focus on some of the interesting changes in the algebraic structure: e.g. the "white structure” is now a bialgebra instead of being Frobenius. These results were published in joint work with Bonchi, Holland, Piedeleu and Zanasi at POPL 19.

2) Graphical Affine Algebra

The move from linear to affine means to consider non-homogeneous equations. The language of ordinary graphical linear algebra is not expressive enough to capture such situations. To capture "affine relations" we extend the graphical syntax with a new generator that, intuitively, emits the constant 1. We have characterised the resulting structure equationally over fields and the naturals. The algebraic structure is again quite interesting, since, e.g. the empty relation is now expressible diagrammatically. Applications include open non-passive electrical circuits (given a continuous interpretation) and open *bounded* Petri nets (given a discrete interpretation). These results are joint work with Bonchi, Piedeleu and Zanasi and will be presented at LiCS 19.