  We introduce a general diagrammatic theory of digital circuits, based on
connections between monoidal categories and graph rewriting. The main
achievement of the paper is conceptual, filling a foundational gap in reasoning
syntactically and symbolically about a large class of digital circuits
(discrete values, discrete delays, feedback). This complements the dominant
approach to circuit modelling, which relies on simulation. The main advantage
of our symbolic approach is the enabling of automated reasoning about abstract
circuits, with a potentially interesting new application to partial evaluation
of digital circuits. Relative to the recent interest and activity in
categorical and diagrammatic methods, our work makes several new contributions.
The most important is establishing that categories of digital circuits are
Cartesian and admit, in the presence of feedback expressive iteration axioms.
The second is producing a general yet simple graph-rewrite framework for
reasoning about such categories in which the rewrite rules are computationally
efficient, opening the way for practical applications.
