Algebraic effects & handlers have become a standard approach for side-effects in functional programming. Their modular composition with other effects and clean separation of syntax and semantics make them attractive to a wide audience. However, not all effects can be classified as algebraic; some need a more sophisticated handling. In particular, effects that have or create a delimited scope need special care, as their continuation consists of two parts-in and out of the scope-and their modular composition introduces additional complexity. These effects are called scoped and have gained attention by their growing applicability and adoption in popular libraries. While calculi have been designed with algebraic effects & handlers built in to facilitate their use, a calculus that supports scoped effects & handlers in a similar manner does not yet exist. This work fills this gap: we present $\lambda_{\mathit{sc}}$, a calculus with native support for both algebraic and scoped effects & handlers. It addresses the need for polymorphic handlers and explicit clauses for forwarding unknown scoped operations to other handlers. Our calculus is based on Eff, an existing calculus for algebraic effects, extended with Koka-style row polymorphism, and consists of a formal grammar, operational semantics, a (type-safe) type-and-effect system and type inference. We demonstrate $\lambda_{\mathit{sc}}$ on a range of examples.

Algebraic effects & handlers are a modular approach for modeling side-effects in functional programming. Their syntax is defined in terms of a signature of effectful operations, encoded as a functor, that are plugged into the free monad; their denotational semantics is defined by fold-style handlers that only interpret their part of the syntax and forward the rest. However, not all effects are algebraic: some need to access an internal computation. For example, scoped effects distinguish between a computation in scope and out of scope; parallel effects parallellize over a computation, latent effects defer a computation. Separate definitions have been proposed for these higher-order effects and their corresponding handlers, often leading to expedient and complex monad definitions. In this work we propose a generic framework for higher-order effects, generalizing algebraic effects & handlers: a generic free monad with higher-order effect signatures and a corresponding interpreter. Specializing this higher-order syntax leads to various definitions of previously defined (scoped, parallel, latent) and novel (writer, bracketing) effects. Furthermore, we formally show our framework theoretically correct, also putting different effect instances on formal footing; a significant contribution for parallel, latent, writer and bracketing effects.

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.

The Flemish Government has imposed a new and challenging set of learning objectives for secondary education. An important part comprises Computational Thinking for 3rd and 4th year students, and Computer Science for students in their 5th and 6th year. Yet, most teachers, who have mostly not been trained to teach these subjects, do not feel confident in programming-related topics. With eTeacher, we take away this burden from the teacher and allow students to work independently and at their own pace. In particular, we present a platform with an integrated course text for Computational Thinking. eTeacher is available on every device, gives automatic code-feedback both content-wise and on programming style and provides exercises of different types and levels. Teachers can monitor their students individually and consult statistics. The platform was extensively evaluated in a pilot at 20 schools.

Abstract Automatic differentiation (AD) is a range of algorithms to compute the numeric value of a function’s (partial) derivative, where the function is typically given as a computer program or abstract syntax tree. AD has become immensely popular as part of many learning algorithms, notably for neural networks. This paper uses Prolog to systematically derive gradient-based forward- and reverse-mode AD variants from a simple executable specification: evaluation of the symbolic derivative. Along the way we demonstrate that several Prolog features (DCGs, co-routines) contribute to the succinct formulation of the algorithm. We also discuss two applications in probabilistic programming that are enabled by our Prolog algorithms. The first is parameter learning for the Sum-Product Loop Language and the second consists of both parameter learning and variational inference for probabilistic logic programming.

Abstract Algebraic effects offer a versatile framework that covers a wide variety of effects. However, the family of operations that delimit scopes are not algebraic and are usually modelled as handlers, thus preventing them from being used freely in conjunction with algebraic operations. Although proposals for scoped operations exist, they are either ad-hoc and unprincipled, or too inconvenient for practical programming. This paper provides the best of both worlds: a theoretically-founded model of scoped effects that is convenient for implementation and reasoning. Our new model is based on an adjunction between a locally finitely presentable category and a category of functorial algebras. Using comparison functors between adjunctions, we show that our new model, an existing indexed model, and a third approach that simulates scoped operations in terms of algebraic ones have equal expressivity for handling scoped operations. We consider our new model to be the sweet spot between ease of implementation and structuredness. Additionally, our approach automatically induces fusion laws of handlers of scoped effects, which are useful for reasoning and optimisation.