Dan R. Ghica

Dan R. Ghica, George Kaye, David Sprunger

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 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.

Mario Alvarez-Picallo, Dan R. Ghica, David Sprunger et al.

We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.

Dan R. Ghica, Todd Waugh Ambridge

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.