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Digital circuits, despite having been studied for nearly a century and usedat scale for about half that time, have until recently evaded a fullycompositional theoretical in which arbitrary circuits may be freely composedtogether without consulting their internals. Recent work remedied thistheoretical shortcoming by showing how digital circuits can be presentedcompositionally as morphisms in a freely generated symmetric traced category.However, this was done informally; in this paper we refine and expand theprevious work in several ways, culminating in the presentation of three soundand complete semantics for digital circuits: denotational, operational andalgebraic. For the denotational semantics, we establish a correspondencebetween stream functions with certain properties and circuits constructedsyntactically. For the operational semantics, we present the reductionsrequired to model how a circuit processes a value, including the addition of anew reduction for eliminating non-delay-guarded feedback; this leads to anadequate notion of observational equivalence for digital circuits. Finally, wedefine a new family of equations for translating circuits into bisimilarcircuits of a 'normal form', leading to a complete algebraic semantics forsequential circuits.

A Complete Theory of Sequential Digital Circuits: Denotational, Operational and Algebraic Semantics