University of Birmingham > Talks@bham > Theoretical computer science seminar > Diagrammatic mathematics and signal flow graphs

Diagrammatic mathematics and signal flow graphs

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If you have a question about this talk, please contact Neel Krishnaswami.

In recent years there have been a number of developments in the area of symmetric monoidal theories (SMTs), a generalisation of algebraic theories in which one considers operations of arbitrary arity and coarity together with equations, and in which linearity à la linear logic is default: variables cannot be copied and discarded.

String diagrams are an intuitive and powerful graphical syntax for the terms of SMTs. I will report on recent joint work with Filippo Bonchi and Fabio Zanasi, in which, building on the work of Steve Lack on composing SMTs via distributive laws and Yves Lafont on a string-diagrammatic theory of boolean circuits, we discovered the theory of Interacting Hopf Algebras. As well as being closely related to Coecke and Duncan’s ZX-calculus for quantum circuits, the associated string diagrams amount to a rigorous and intuitive diagrammatic universe for linear algebra: familiar concepts such as linear transformations and linear spaces appear as certain string diagrams.

As well as giving a new way to look at classical topics in linear algebra, string diagrams sometimes carry useful computational information. I will show how a special case of the theory of Interacting Hopf Algebras captures signal flow graphs, which are a classical circuit notation for linear dynamical systems. When signal flow graphs are considered as string diagrams, diagrammatic reasoning gives a novel, sound and complete technique to reason about them. I will explain how the underlying mathematics leads us to reconsider some popular assumptions about the roles of causality and direction of signal flow, which—-in contrast to the primary role traditionally given to them—-turn out to be secondary, derived notions: in this way, this work connects with recent trends in control theory, in particular the Behavioural Approach.

This talk is part of the Theoretical computer science seminar series.

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