Modular SOS (MSOS) is a variant of conventional Structural Operational Semantics (SOS). Using MSOS, the transition rules for each construct of a programming language can be given incrementally, once and for all, and do not need reformulation when further constructs are added to the language. MSOS thus provides an exceptionally high degree of modularity in language descriptions, removing a shortcoming of the original SOS framework. After sketching the background and reviewing the main features of SOS, the paper explains the crucial differences between SOS and MSOS, and illustrates how MSOS descriptions are written. It also discusses standard notions of semantic equivalence based on MSOS. Appendix A shows how the illustrative MSOS rules given in the paper would be formulated in conventional SOS.

We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the a-calculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to obtain the logic, which is a classical firstorder multi-sorted logic with higher-order value and computation types, as in Levy’s call-by-push-value, a principle of induction over computations, a free algebra principle, and predicate fixed points. This logic embraces Moggi’s computational λ-calculus, and also, via definable modalities, Hennessy-Milner logic, and evaluation logic, though Hoare logic presents difficulties. 1

Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, side-effects, input/output and continuations. We present generalisations of both constructs, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, side-effects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering monoidal parameterisation, we extend the range of effects to cover separated side-effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our categorical definitions — with and without monoidal parameterisation — and act as prototypical languages with parameterised effects.

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We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi's side-effects monad transformer (an application is the combination of side-effects with nondeterminism). And we give a theory...

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Monads are commonplace programming devices that are used to uniformly structure computations with effects such as state, exceptions, and I/O. This paper further develops the monadic programming paradigm by investigating the extent to which monadic computations can be optimised by using generalisations of short cut fusion to eliminate monadic structures whose sole purpose is to “glue together ” monadic program components. We make several contributions. First, we show that every inductive type has an associated build combinator and an associated short cut fusion rule. Second, we introduce the notion of an inductive monad to describe those monads that give rise to inductive types, and we give examples of such monads which are widely used in functional programming. Third, we generalise the standard augment combinators and cata/augment fusion rules for algebraic data types to types induced by inductive monads. This allows us to give the first cata/augment rules for some common data types, such as rose trees. Fourth, we demonstrate the practical applicability of our generalisations by providing Haskell implementations for all concepts and examples in the paper. Finally, we offer deep theoretical insights by showing that the augment combinators are monadic in nature, and thus that our cata/build and cata/augment rules are arguably the best generally applicable fusion rules obtainable.

Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.

We develop a model of concurrent imperative programming with threads. We focus on a small imperative language with cooperative threads which execute without interruption until they terminate or explicitly yield control. We define and study a trace-based denotational semantics for this language; this semantics is fully abstract but mathematically elementary. We also give an equational theory for the computational effects that underlie the language, including thread spawning. We then analyze threads in terms of the free algebra monad for this theory. 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.