  String diagrams provide a convenient graphical framework which may be used
for equational reasoning about morphisms of monoidal categories. However,
unlike term rewriting, rewriting string diagrams results in shorter equational
proofs, because the string diagrammatic representation allows us to formally
establish equalities modulo any rewrite steps which follow from the monoidal
structure.
  Manipulating string diagrams by hand is a time-consuming and error-prone
process, especially for large string diagrams. This can be ameliorated by using
software proof assistants, such as Quantomatic.
  However, reasoning about concrete string diagrams may be limiting and in some
scenarios it is necessary to reason about entire (infinite) families of string
diagrams. When doing so, we face the same problems as for manipulating concrete
string diagrams, but in addition, we risk making further mistakes if we are not
precise enough about the way we represent (infinite) families of string
diagrams.
  The primary goal of this thesis is to design a mathematical framework for
equational reasoning about infinite families of string diagrams which is
amenable to computer automation. We will be working with context-free families
of string diagrams and we will represent them using context-free graph
grammars. We will model equations between infinite families of diagrams using
rewrite rules between context-free grammars. Our framework represents
equational reasoning about concrete string diagrams and context-free families
of string diagrams using double-pushout rewriting on graphs and context-free
graph grammars respectively. We will prove that our representation is sound by
showing that it respects the concrete semantics of string diagrammatic
reasoning and we will show that our framework is appropriate for software
implementation by proving important decidability properties.
