Polymonadic Programming

Monads are a popular tool for the working functional programmer to structure effectful computations. This paper presents polymonads, a generalization of monads. Polymonads give the familiar monadic bind the more general type forall a,b. L a -> (a -> M b) -> N b, to compose computations with three different kinds of effects, rather than just one. Polymonads subsume monads and parameterized monads, and can express other constructions, including precise type-and-effect systems and information flow tracking; more generally, polymonads correspond to Tate's productoid semantic model. We show how to equip a core language (called lambda-PM) with syntactic support for programming with polymonads. Type inference and elaboration in lambda-PM allows programmers to write polymonadic code directly in an ML-like syntax--our algorithms compute principal types and produce elaborated programs wherein the binds appear explicitly. Furthermore, we prove that the elaboration is coherent: no matter which (type-correct) binds are chosen, the elaborated program's semantics will be the same. Pleasingly, the inferred types are easy to read: the polymonad laws justify (sometimes dramatic) simplifications, but with no effect on a type's generality.Comment: In Proceedings MSFP 2014, arXiv:1406.153

Polymonad programming in Haskell

Polymonads were recently introduced by Hicks et al. as a unified approach to programming with different notions of monads. Their work was mainly focussed on foundational aspects of the approach. In this article, we show how to incorporate the notion of polymonads into Haskell, which is the first time this has been done in a full-scale language. In particular, we show how polymonads can be represented in Haskell, give a justification of the representation through proofs in Agda, and provide a plugin for the Glasgow Haskell Compiler (GHC) that enables their use in practice. Finally, we demonstrate the utility of our system by means of examples concerned with session types and the parameterized effect monad. This work provides a common representation of a number of existing approaches to generalized monads in Haskell

Polymonad programming in Haskell

Polymonads were recently introduced by Hicks et al. as a unified approach to programming with different notions of monads. Their work was mainly focussed on foundational aspects of the approach. In this article, we show how to incorporate the notion of polymonads into Haskell, which is the first time this has been done in a full-scale language. In particular, we show how polymonads can be represented in Haskell, give a justification of the representation through proofs in Agda, and provide a plugin for the Glasgow Haskell Compiler (GHC) that enables their use in practice. Finally, we demonstrate the utility of our system by means of examples concerned with session types and the parameterized effect monad. This work provides a common representation of a number of existing approaches to generalized monads in Haskell

Differentially Private Bayesian Programming

We present PrivInfer, an expressive framework for writing and verifying differentially private Bayesian machine learning algorithms. Programs in PrivInfer are written in a rich functional probabilistic programming language with constructs for performing Bayesian inference. Then, differential privacy of programs is established using a relational refinement type system, in which refinements on probability types are indexed by a metric on distributions. Our framework leverages recent developments in Bayesian inference, probabilistic programming languages, and in relational refinement types. We demonstrate the expressiveness of PrivInfer by verifying privacy for several examples of private Bayesian inference

Unifying graded and parameterised monads

Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model of effects by replacing the single type constructor of a monad with an indexed family of constructors. Most notably, graded monads (indexed by a monoid) model effect systems and parameterised monads (indexed by pairs of pre- and post-conditions) model program logics. This paper studies the relationship between these two generalisations of monads via a third generalisation. This third generalisation, which we call category-graded monads, arises by generalising a view of monads as a particular special case of lax functors. A category-graded monad provides a family of functors T f indexed by morphisms f of some other category. This allows certain compositions of effects to be ruled out (in the style of a program logic) as well as an abstract description of effects (in the style of an effect system). Using this as a basis, we show how graded and parameterised monads can be unified, studying their similarities and differences along the way