🤖 AI Summary
Digital circuits lack a fully compositional theoretical foundation.
Method: This paper establishes the first complete compositional semantic framework for sequential digital circuits. It introduces a non-delay-guarded feedback elimination reduction, defines a novel family of equations enabling equivalent transformation of circuits into canonical form, and unifies denotational, operational, and algebraic semantics—grounded in symmetric trace categories, stream function semantics, structured operational semantics, and observational equivalence theory.
Contribution/Results: It provides the first mutual consistency verification and soundness-and-completeness proofs across all three semantic styles; establishes a rigorous mathematical foundation for black-box, freely composable circuit design; and enables precise behavioral characterization, equivalence reasoning, and the development of formal verification and synthesis tools.
📝 Abstract
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