Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Reasoning with !-Graphs   The aim of this thesis is to present an extension to the string graphs of
Dixon, Duncan and Kissinger that allows the finite representation of certain
infinite families of graphs and graph rewrite rules, and to demonstrate 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.
  String diagrams provide an intuitive method for reasoning about monoidal
categories. However, this does not negate the ability for those using them to
make mistakes in proofs. To this end, there is a project (Quantomatic) to build
a proof assistant for string diagrams, at least for those based on categories
with a notion of trace. The development of string graphs has provided a
combinatorial formalisation of string diagrams, laying the foundations for this
project.
  The prevalence of commutative Frobenius algebras (CFAs) in quantum
information theory, a major application area of these diagrams, has led to the
use of variable-arity nodes as a shorthand for normalised networks of Frobenius
algebra morphisms, so-called "spider notation". This notation greatly eases
reasoning with CFAs, but string graphs are inadequate to properly encode this
reasoning.
  This dissertation extends string graphs to allow for variable-arity nodes to
be represented at all, and then introduces !-box notation (and structures to
encode it) to represent string graph equations containing repeated subgraphs,
where the number of repetitions is abitrary. It then demonstrates how we can
reason directly about !-graphs, viewed as (typically infinite) families of
string graphs. Of particular note is the presentation of a form of graph-based
induction, allowing the formal encoding of proofs that previously could only be
represented as a mix of string diagrams and explanatory text.

11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
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The categories of this paper are: quant-ph