  Digital circuits, despite having been studied for nearly a century and used
at scale for about half that time, have until recently evaded a fully
compositional theoretical understanding, in which arbitrary circuits may be
freely composed together without consulting their internals. Recent work
remedied this theoretical shortcoming by showing how digital circuits can be
presented compositionally as morphisms in a freely generated symmetric traced
category. However, this was done informally; in this paper we refine and expand
the previous work in several ways, culminating in the presentation of three
sound and complete semantics for digital circuits: denotational, operational
and algebraic. For the denotational semantics, we establish a correspondence
between stream functions with certain properties and circuits constructed
syntactically. For the operational semantics, we present the reductions
required to model how a circuit processes a value, including the addition of a
new reduction for eliminating non-delay-guarded feedback; this leads to an
adequate notion of observational equivalence for digital circuits. Finally, we
define a new family of equations for translating circuits into bisimilar
circuits of a 'normal form', leading to a complete algebraic semantics for
sequential circuits
