This paper describes a technique for assembling both data types and functions from isolated individual components. We also explore how the same technology can be used to combine free monads and, as a result, structure Haskell’s monolithic IO monad. 1

We consider the natural combinations of algebraic computational effects such as side-effects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.

Abstract. During the last two decades, monads have become an indispensable tool for structuring functional programs with computational effects. In this setting, the mathematical notion of a monad is extended with operations that allow programmers to manipulate these effects. When several effects are involved, monad transformers can be used to build up the required monad one effect at a time. Although this seems to be modularity nirvana, there is a catch: in addition to the construction of a monad, the effect-manipulating operations need to be lifted to the resulting monad. The traditional approach for lifting operations is nonmodular and ad-hoc. We solve this problem with a principled technique for lifting operations that makes monad transformers truly modular. 1

Modularity in programming language semantics derives from abstracting over the structure of underlying denotations, yielding semantic descriptions that are more abstract and reusable. One such semantic framework is Liang’s modular monadic semantics in which the underlying semantic structure is encapsulated with a monad. Such abstraction can be at odds with program verification, however, because program specifications require access to the (deliberately) hidden semantic representation. The techniques for reasoning about modular monadic definitions of imperative programs introduced here overcome this barrier. And, just like program definitions in modular monadic semantics, our program specifications and proofs are representation-independent and hold for whole classes of monads, thereby yielding proofs of great generality.

As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages to facilitate their modeling. Most of the existing probabilistic languages, however, focus only on discrete distributions, and there has been little effort to develop probabilistic languages whose expressive power is beyond discrete distributions. This dissertation presents a probabilistic language, called PTP (ProbabilisTic Programming), which supports all kinds of probability distributions.

A fundamental question, in modelling computational effects, is how to give a unified semantic account of modularity, i.e., a mathematical theory that supports the various combinations one naturally makes of computational effects such as exceptions, side-effects, interactive input/output, nondeterminism, and, particularly

The incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a preexisting monad. A complication of this approach is that the operations associated to the pre-existing monad need to be lifted to the new monad. In a companion paper by Jaskelioff, the lifting problem has been addressed in the setting of system F ω. Here, we recast and extend those results in a category-theoretic setting. We abstract and generalize from monads to monoids (in a monoidal category), and from monad transformers to monoid transformers. The generalization brings more simplicity and clarity, and opens the way for lifting of operations with applicability beyond monads. Key words: Monad, Monoid, Monoidal Category

Functional programming is well suited for equational reasoning on programs. In this paper, we are trying to use this capability for program comprehension purposes. Specifically, in a program understanding process, higher-order operators can work like abstract schemes in which we can fit formal specifications calculated from the source code. Such specifications are calculated by a transformational process which we call reverse program calculation that operates on both notations: pointwise and pointfree. Once a specification matches an abstract schema, a new refactoring phase leading to a clearer source code takes place. At the same time, an unambiguous behavioural understanding is reached because we give a mathematical description of the abstract schemes. To provide a more complete and realistic perspective of the approach, we use recursive operators that can handle side effects.

A fundamental question, in modelling computational effects, is how to give a unified semantic account of modularity, i.e., a mathematical theory that supports the various combinations one naturally makes of computational effects such as exceptions, side-effects, interactive input/output, nondeterminism, and, particularly for this workshop, continuations [2, 3, 5]. We have begun to give such an account over recent years for all of these effects other than continuations [8], describing the sum and the tensor, or commutative combination, of effects, starting from Eugenio Moggi's proposal to use monads to give semantics for each individual effect [15]. That has yielded the most commonly used combinations of the various effects. Here, we extend our account to include continuations. We consider three distinct ways in which continuations combine with the other effects: sum, tensor, and by applying the continuations monad transformer C(-); we analyse each of these in the following three Detections. We did not...

Despite their invaluable contribution to the programming language community, monads as a foundation for the study of effects have three problems: they make it difficult to combine effects; they enforce sequentialization of computations by the syntax; they prohibit effect-free evaluations from invoking e#ectful computations. Building on the judgmental formulation and the possible worlds interpretation of modal logic, we propose a logical analysis of effects based upon the view monads are not identified with effects. Our analysis leads to a language called # # which distinguishes between control e#ects and world e#ects, enforces sequentialization of computations only by the semantics, and logically explains the invocation of computations from evaluations. # # also serves as a unified framework for studying Haskell and ML, which have traditionally been studied separately.