Full Abstraction for Signal Flow Graphs
Résumé
Network theory uses the string diagrammatic language of monoidal
categories to study graphical structures formally, eschewing specialised
translations into intermediate formalisms. Recently, there
has been a concerted research focus on developing a network theoretic
approach to signal flow graphs, which are classical structures
in control theory, signal processing and a cornerstone in the study
of feedback. In this approach, signal flow graphs are given a relational
denotational semantics in terms of formal power series.
Thus far, the operational behaviour of such signal flow graphs
has only been discussed at an intuitive level. In this paper we equip
them with a structural operational semantics. As is typically the
case, the purely operational picture is too concrete – two graphs
that are denotationally equal 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 graphical theory,
into an executable form where the operational behavior and the
denotation coincides.