Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although many researchers appreciate and know how to apply AD, it remains a challenge to truly understand the underlying processes. From an algebraic point of view, however, AD appears surprisingly natural: it originates from the differentiation laws. In this work we use Algebra of Programming techniques to reason about different AD variants, leveraging Haskell to illustrate our observations. Our findings stem from three fundamental algebraic abstractions: (1) the notion of semimodule, (2) Nagata's construction of the ‘idealization of a module’, and (3) Kronecker's delta function, that together allow us to write a single-line abstract definition of AD. From this single-line definition, and by instantiating our algebraic structures in various ways, we derive different AD variants, that have the same extensional behaviour, but different intensional properties, mainly in terms of (asymptotic) computational complexity. We show the different variants equivalent by means of Kronecker isomorphisms, a further elaboration of our Haskell infrastructure which guarantees correctness by construction. With this framework in place, this paper seeks to make AD variants more comprehensible, taking an algebraic perspective on the matter.

Abstract Delimited control is a powerful mechanism for programming language extension which has been recently proposed for Prolog (and implemented in SWI-Prolog). By manipulating the control flow of a program from inside the language, it enables the implementation of powerful features, such as tabling, without modifying the internals of the Prolog engine. However, its current formulation is inadequate: it does not capture Prolog’s unique non-deterministic nature which allows multiple ways to satisfy a goal. This paper fully embraces Prolog’s non-determinism with a novel interface for disjunctive delimited control, which gives the programmer not only control over the sequential (conjunctive) control flow, but also over the non-deterministic control flow. We provide a meta-interpreter that conservatively extends Prolog with delimited control and show that it enables a range of typical Prolog features and extensions, now at the library level: findall, cut, branch-and-bound optimisation, probabilistic programming, …

This work presents PλωNK, a functional probabilistic network programming language that extends Probabilistic NetKAT (PNK). Like PNK, it enables probabilistic modelling of network behaviour, by providing probabilistic choice and infinite iteration (to simulate looping network packets). Yet, unlike PNK, it also offers abstraction and higher-order functions to make programming much more convenient. The formalisation of PλωNK is challenging for two reasons: Firstly, network programming induces multiple side effects (in particular, parallelism and probabilistic choice) which need to be carefully controlled in a functional setting. Our system uses an explicit syntax for thunks and sequencing which makes the interplay of these effects explicit. Secondly, measure theory, the standard domain for formalisations of (continuous) probablistic languages, does not admit higher-order functions. We address this by leveraging ω-Quasi Borel Spaces (ωQBSes), a recent advancement in the domain theory of probabilistic programming languages. We believe that our work is not only useful for bringing abstraction to PNK, but that—as part of our contribution—we have developed the meta-theory for a probabilistic language that combines advanced features like higher-order functions, iteration and parallelism, which may inform similar meta-theoretic efforts.

Abstract Tabling is a powerful resolution mechanism for logic programs that captures their least fixed point semantics more faithfully than plain Prolog. In many tabling applications, we are not interested in the set of all answers to a goal, but only require an aggregation of those answers. Several works have studied efficient techniques, such as lattice-based answer subsumption and mode-directed tabling, to do so for various forms of aggregation. While much attention has been paid to expressivity and efficient implementation of the different approaches, soundness has not been considered. This paper shows that the different implementations indeed fail to produce least fixed points for some programs. As a remedy, we provide a formal framework that generalises the existing approaches and we establish a soundness criterion that explains for which programs the approach is sound.