Proceedings ArticleDOI
Full Abstraction for Signal Flow Graphs
Filippo Bonchi,Pawel Sobocinski,Fabio Zanasi +2 more
- Vol. 50, Iss: 1, pp 515-526
TLDR
This paper equips signal flow graphs with a structural operational semantics, and classifies the ways in which any graph can be realised -- rewritten, using the graphical theory, into an executable form where the operational behavior and the denotation coincides.Abstract:
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.read more
Citations
More filters
Journal ArticleDOI
Interacting Hopf algebras
TL;DR: The axioms of IHR are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid–comonoid pairs, instrumental in showing that IHR is isomorphic to the PROP of linear relations.
Posted Content
Seven Sketches in Compositionality: An Invitation to Applied Category Theory
Brendan Fong,David I. Spivak +1 more
TL;DR: In this article, a tour of advanced topics in category theory through concrete, real-world examples is presented, with the goal of giving a gentle, quick introduction to guide later exploration.
Book ChapterDOI
Quantomatic: A Proof Assistant for Diagrammatic Reasoning
TL;DR: This work briefly outlines the theoretical basis of Quantomatic's rewriting engine, then gives an overview of the core features and architecture and gives a simple example project that computes normal forms for commutative bialgebras.
Book ChapterDOI
Quantomatic: A Proof Assistant for Diagrammatic Reasoning
TL;DR: Quantomatic as discussed by the authors is a tool that supports the (semi-)automatic construction of equational proofs using string diagrams, which have applications in many areas including categorical algebra, programming language semantics, representation theory, algebraic quantum information, and quantum groups.
Journal Article
The algebra of open and interconnected systems
TL;DR: This thesis develops the theory of hypergraph categories and introduces the tools of decorated cospans and corelations, a more powerful version that permits construction of all hyper graph categories and hypergraph functors.
References
More filters
Journal ArticleDOI
On Network Theory
TL;DR: This paper analyzes two well-known network theories, Granovetter's strength of weak ties theory and Burt's structural holes theory, to identify characteristic elements of network theorizing and argues that both theories share an underlying theoretical model, which is labelled the network flow model, from which they derive additional implications.
Journal ArticleDOI
The geometry of tensor calculus, I
André Joyal,Ross Street +1 more
TL;DR: In this article, the correctness of appropriate string diagrams for various kinds of monoidal categories with duals has been proved for various classes of classes of subject classes, including algebra, geometry, physics, and astronomy.
Book ChapterDOI
A Survey of Graphical Languages for Monoidal Categories
TL;DR: In this article, a reference guide to various notions of monoidal categories and their associated string diagrams is presented, which is useful not only to mathematicians, but also to physicists, computer scientists, and others who use diagrammatic reasoning.
Journal ArticleDOI
Feedback Theory-Some Properties of Signal Flow Graphs
TL;DR: A study of the topological properties of such graphs leads to techniques which have proven useful, both for the discussion of the general theory of feedback and for the solution of practical analysis problems.
Journal ArticleDOI
Coherence for compact closed categories
G. M. Kelly,M.L. Laplaza +1 more
TL;DR: A compact closed category is a symmetric monoidal one whose internal-horn [A, C] has the form CBA' as mentioned in this paper, and is defined as a monoidal category d with tensor product @ and unit object I with a single O-cell, the l-cells of B being the objects of ti with @ as their composition.