Categorical semantics of digital circuits

Abstract

This paper proposes a categorical theory of digital circuits based on monoidal categories and graph rewriting. The main goal of this paper is conceptual: to fill a foundational gap in reasoning about digital circuits, which is currently almost exclusively semantic (simulations). The level of abstraction we target is circuits with discrete signal levels, discrete time, and explicit delays, which is appropriate for modelling a range of components such as boolean gates or transistors working in saturation mode. We start with an algebraic signature consisting of the basic electronic components of a given class of circuits and extend it gradually (and in a free way) with further algebraic structure (representing circuit combinations, delays, and feedback), while quotienting it with a notion of equivalence corresponding to input-output observability. Using well-known results about the correspondence between free monoidal categories and graph-like structures we can develop, in a principled way, a graph rewriting system which is shown to be useful in reasoning about such circuits. We illustrate the power of our system by reasoning equationally about a challenging class of circuits: combinational circuits with feedback.