A Survey of Compositional Signal Flow Theory

Signal flow graphs are combinatorial models for linear dynamical systems, playing a foundational role in control theory and engineering. In this survey, we overview a series of works [3, 10, 11, 13, 15–18, 31, 51, 63] that develop a compositional theory of these structures, and explore several striking insights emerging from this approach. In particular, the use of string diagrams, a categorical syntax for graphical models, allows to switch from the traditional combinatorial treatment of signal flow graphs to an algebraic characterisation. Within this framework, signal flow graphs may then be treated as a fully-fledged (visual) programming language, and equipped with important meta-theoretical properties, such as a complete axiomatisation and a full abstraction theorem. Moreover, the abstract viewpoint offered by string diagrams reveals that the same algebraic structures modelling linear dynamical systems may also be used to interpret diverse kinds of models, such as electrical circuits and Petri nets. In this respect, our work is a contribution to compositional network theory (see e.g., [1, 2, 4–6, 9, 12, 20, 21, 23, 24, 26, 28–30, 32, 37, 49, 59], ?), an emerging multidisciplinary research programme aiming at a uniform compositional study of different sorts of computational models.