Robin Kaarsgaard

Chris Heunen, Robin Kaarsgaard, Louis Lemonnier

Controlled commands -- computations whose execution depends on a separate input -- play a central role in reversible Boolean circuits and quantum circuits. However, existing formalisms typically treat control only implicitly, entangled with other aspects of computation. From a semantic perspective, control is most naturally expressed in semisimple rig categories, which -- unlike standard circuit models such as props -- support both parallel and conditional composition. We present a construction that freely adjoins an explicit syntactic notion of control to a circuit theory specified as a suitable prop, subject to eight universally quantified equations. Our main result is that these equations are sound and complete for the intended semantics of control: the resulting theory satisfies a universal property, identifying it exactly as the circuit subtheory of the free semisimple rig completion. The proof combines coherence for rig categories with a new method based on induction over Gray codes. We illustrate the usefulness of the framework by showing that it simplifies several existing sound and complete axiomatisations of quantum circuits, isolating a small and conceptually clean set of generators and equations. In addition, the same equations yield a sound and complete axiomatisation of the multiply controlled Toffoli gate set, that is universal for reversible Boolean circuits.

Martin Elsman, Fritz Henglein, Robin Kaarsgaard et al.

We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars, vectors, matrices or tensors represented as multi-dimensional arrays. We show that both symbolic and automatic differentiation can be performed using a differential calculus for generating linear functions representing Fréchet derivatives based on rules for primitive, constant, linear and bilinear functions as well as their sequential and parallel composition. Linear functions are represented in a combinatory domain-specific language. Finally, we provide a calculus for symbolically computing the adjoint of a derivative without using matrices, which are too inefficient to use on high-dimensional spaces. The resulting symbolic representation of a derivative retains the data-parallel operations from the input program. The combination of combinatory differentiation and computing formal adjoints turns out to be behaviorally equivalent to reverse-mode automatic differentiation. In particular, it provides opportunities for optimizations where matrices are too inefficient to represent linear functions.

Joachim Tilsted Kristensen, Robin Kaarsgaard, Michael Kirkedal Thomsen

Algorithms are ways of mapping problems to solutions. An algorithm is invertible precisely when this mapping is injective, such that the initial problem can be uniquely inferred from its solution. While invertible algorithms can be described in general-purpose languages, no guarantees are generally made by such languages as regards invertibility, so ensuring invertibility requires additional (and often non-trivial) proof. On the other hand, while reversible programming languages guarantee that their programs are invertible by restricting the permissible operations to those which are locally invertible, writing programs in the reversible style can be cumbersome, and may differ significantly from conventional implementations even when the implemented algorithm is, in fact, invertible. In this paper we introduce Jeopardy, a functional programming language that guarantees program invertibility without imposing local reversibility. In particular, Jeopardy allows the limited use of uninvertible -- and even nondeterministic! -- operations, provided that they are used in a way that can be statically determined to be invertible. To this end, we outline an \emph{implicitly available arguments analysis} and three further approaches that can give a partial static guarantee to the (generally difficult) problem of guaranteeing invertibility.