Open-graphs and monoidal theories†

@article{Dixon2013OpengraphsAM,
  title={Open-graphs and monoidal theories†},
  author={Lucas Dixon and Aleks Kissinger},
  journal={Mathematical Structures in Computer Science},
  year={2013},
  volume={23},
  pages={308 - 359}
}
  • L. Dixon, A. Kissinger
  • Published 2013
  • Mathematics, Computer Science
  • Mathematical Structures in Computer Science
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open… Expand
Reasoning with !-graphs
TLDR
An extension to the string graphs of Dixon, Duncan and Kissinger is presented that allows the finite representation of certain infinite families of graphs and graph rewrite rules, and it is demonstrated that a logic can be built on this to allow the formalisation of inductive proofs in the string diagrams of compact closed and traced symmetric monoidal categories. Expand
Pattern Graph Rewrite Systems
TLDR
This extended abstract shows how the power of graph rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs on both sides of a rule can be copied any number of times or removed. Expand
Pictures of processes : automated graph rewriting for monoidal categories and applications to quantum computing
TLDR
The introduction of a discretised version of a string diagram called a string graph is introduced, and it is shown how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature. Expand
!-graphs with Trivial Overlap Are Context-free
String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be doneExpand
Equational reasoning with context-free families of string diagrams
TLDR
A class of context-free grammars is defined that are suitable for defining entire families of string graphs, and crucially, of string graph rewrite rules, and it is shown that the language-membership and match-enumeration problems are decidable for these Grammars, and hence that there is an algorithm for rewriting string graphs according to B-ESG rewrite patterns. Expand
!-Logic : first order reasoning for families of non-commutative string diagrams
Equational reasoning with string diagrams provides an intuitive method for proving equations between morphisms in various forms of monoidal category. !-Graphs were introduced with the intention ofExpand
Tensors, !-graphs, and non-commutative quantum structures.
Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited forExpand
Rewriting modulo symmetric monoidal structure
TLDR
This paper interprets diagrams combinatorially as typed hypergraphs and establishes the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hyper graphs, subject to a soundness condition called convexity, on the other. Expand
Synthesising Graphical Theories
TLDR
This extended abstract adapts a technique called conjecture synthesis for automatically generating conjectured term equations to feed into an inductive theorem prover to diagrammatic theories, expressed as graph rewrite systems, and demonstrates its application by synthesising a graphical theory for studying entangled quantum states. Expand
Rewriting Context-free Families of String Diagrams
TLDR
A mathematical framework for equational reasoning about infinite families of string diagrams which is amenable to computer automation is designed and it is shown that the framework is appropriate for software implementation by proving important decidability properties. Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 39 REFERENCES
Open Graphs and Computational Reasoning
TLDR
A form of algebraic reasoning for computational objects which are expressed as graphs which allows equations and rewrite rules to be specified between graphs and which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information. Expand
Pure bigraphs: Structure and dynamics
  • R. Milner
  • Computer Science, Mathematics
  • Inf. Comput.
  • 2006
TLDR
This paper is a devoted to pure bigraphs, which in turn underlie various more refined forms, and it is shown that behavioural analysis for Petri nets, π-calculus and mobile ambients can all be recovered in the uniform framework ofbigraphs. Expand
Graph-Grammars: An Algebraic Approach
TLDR
An algebraic theory of graph-grammars is presented using homomorphisms and pushout-constructions to specify embeddings and direct derivations constructively and allows simplification of the proofs and pregnant formulation of concepts like "parallel composition" and "translation of grammars". Expand
Interacting Quantum Observables: Categorical Algebra and Diagrammatics
TLDR
The ZX-calculus is introduced, an intuitive and universal graphical calculus for multi-qubit systems, which greatly simplifies derivations in the area of quantum computation and information and axiomatize phase shifts within this framework. Expand
Proceedings Sixth Workshop on Developments in Computational Models: Causality, Computation, and Physics, Edinburgh, Scotland, 9-10th July 2010
TLDR
A form of algebraic reasoning for computational objects which are expressed as graphs which allows equations and rewrite rules to be specified between graphs and which can be formalised in this way include traditional electronic circuits as well as recent categorical accounts of quantum information. Expand
A Survey of Graphical Languages for Monoidal Categories
This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but alsoExpand
Abstract and Concrete Categories - The Joy of Cats
TLDR
This chapter discusses Categories and Functors, Topological Categories, Partial Morphisms, Quasitopoi, and Topological Universes, as well as partial Morphisms in Abstract Categories and Cartesian Closed Categories. Expand
PROOF-NETS : THE PARALLEL SYNTAX FOR PROOF-THEORY
The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linearExpand
Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories
TLDR
Under which conditions additional properties initial pushouts, binary coproducts compatible with a special morphism class $\cal{M}$ and a pair factorization are preserved by the categorical constructions in order to avoid checking these properties explicitly is analyzed. Expand
Graphical reasoning in compact closed categories for quantum computation
  • L. Dixon, R. Duncan
  • Mathematics, Computer Science
  • Annals of Mathematics and Artificial Intelligence
  • 2009
TLDR
A generic proof system with a fixed logical kernel for reasoning about compact closed categories that provides a formal and declarative account of derived results that can include ‘ellipses’-style notation. Expand
...
1
2
3
4
...