A compositional theory of digital circuits

A theory is compositional if complex components can be constructed out of simpler ones on the basis of their interfaces, without inspecting their internals. Digital circuits, despite being studied for nearly a century and used at scale for about half that time, have until recently evaded a fully compositional theoretical understanding. The sticking point has been the need to avoid feedback loops that bypass memory elements, the so called 'combinational feedback' problem. This requires examining the internal structure of a circuit, defeating compositionality. Recent work remedied this theoretical shortcoming by showing how digital circuits can be presented compositionally as morphisms in a freely generated Cartesian traced (or dataflow) category. The focus was to support a better syntactical understanding of digital circuits, culminating in the formulation of novel operational semantics for digital circuits. In this paper we shift the focus onto the denotational theory of such circuits, interpreting them as functions on streams with to certain properties. These ensure that the model is fully abstract, i.e. the equational theory and the semantic model are in perfect agreement. To support this result we introduce two key equations: the first can reduce circuits with combinational feedback to circuits without combinational feedback via finite unfoldings of the loop, and the second can translate between open circuits with the same behaviour syntactically by reducing the problem to checking a finite number of closed circuits. The most important consequence of this new semantics is that we can now give a recipe that ensures a circuit always produces observable output, thus using the denotational model to inform and improve the operational semantics.