No value restriction is needed for algebraic effects and handlers

We present a straightforward, sound Hindley-Milner polymorphic type system for algebraic effects and handlers in a call-by-value calculus, which allows type variable generalisation of arbitrary computations, not just values. This result is surprising. On the one hand, the soundness of unrestricted call-by-value Hindley-Milner polymorphism is known to fail in the presence of computational effects such as reference cells and continuations. On the other hand, many programming examples can be recast to use effect handlers instead of these effects. Analysing the expressive power of effect handlers with respect to state effects, we claim handlers cannot express reference cells, and show they can simulate dynamically scoped state

Functional Programming for Modular Bayesian Inference

We present an architectural design of a library for Bayesian modelling and inference in modern functional programming languages. The novel aspect of our approach are modular implementations of existing state-of- the-art inference algorithms. Our design relies on three inherently functional features: higher-order functions, inductive data-types, and support for either type-classes or an expressive module system. We provide a perfor- mant Haskell implementation of this architecture, demonstrating that high-level and modular probabilistic programming can be added as a library in sufficiently expressive languages. We review the core abstractions in this architecture: inference representations, inference transformations, and inference representation transformers. We then implement concrete instances of these abstractions, counterparts to particle filters and Metropolis-Hastings samplers, which form the basic building blocks of our library. By composing these building blocks we obtain state-of-the-art inference algorithms: Resample-Move Sequential Monte Carlo, Particle Marginal Metropolis-Hastings, and Sequential Monte Carlo Squared. We evaluate our implementation against existing probabilistic programming systems and find it is already com- petitively performant, although we conjecture that existing functional programming optimisation techniques could reduce the overhead associated with the abstractions we use. We show that our modular design enables deterministic testing of inherently stochastic Monte Carlo algorithms. Finally, we demonstrate using OCaml that an expressive module system can also implement our design

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming

Partially-Static Data as Free Extension of Algebras

Partially-static data structures are a well-known technique for improving binding times. However, they are often defined in an ad-hoc manner, without a unifying framework to ensure full use of the equations associated with each operation. We present a foundational view of partially-static data structures as free extensions of algebras for suitable equational theories, i.e. the coproduct of an algebra and a free algebra in the category of algebras and their homomorphisms. By precalculating these free extensions, we construct a high-level library of partially-static data representations for common algebraic structures. We demonstrate our library with common use-cases from the literature: string and list manipulation, linear algebra, and numerical simplification.Supported by the European Research Council grant ‘events causality and symmetry Ð the next- generation semantics’; the Engineering and Physical Sciences Research Council grant EP/N007387/1 ‘Quantum computation as a programming language’, and a Balliol College Oxford Career Development Fellowshi

On the Expressive Power of User-Defined Effects: Effect Handlers, Monadic Reflection, Delimited Control

We compare the expressive power of three programming abstractions for user-defined computational effects: Bauer and Pretnar's effect handlers, Filinski's monadic reflection, and delimited control without answer-type-modification. This comparison allows a precise discussion about the relative expressiveness of each programming abstraction. It also demonstrates the sensitivity of the relative expressiveness of user-defined effects to seemingly orthogonal language features. We present three calculi, one per abstraction, extending Levy's call-by-push-value. For each calculus, we present syntax, operational semantics, a natural type-and-effect system, and, for effect handlers and monadic reflection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: safety, termination, and, where applicable, soundness and adequacy. Using Felleisen's notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro-expressed while preserving typeability either by monadic reflection or by delimited control. We supplement our development with a mechanised Abella formalisation