Diagrammatic Semantics for Digital Circuits

TL;DR

This paper develops a diagrammatic, categorical framework for reasoning about digital circuits, enabling symbolic and automated analysis of circuits with feedback and delays, complementing simulation-based methods.

Contribution

It introduces a foundational diagrammatic theory for digital circuits based on monoidal categories and graph rewriting, facilitating symbolic reasoning and potential partial evaluation.

Findings

Categories of digital circuits are Cartesian and support iteration axioms.

A simple, efficient graph-rewrite framework for circuit reasoning is proposed.

The approach enables automated symbolic analysis of circuits with feedback.

Abstract

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.