The categorical notion of monad, used by Moggi to structure denotational descriptions, has proved to be a powerful tool for structuring combinator libraries. Moreover, the monadic programming style provides a convenient syntax for many kinds of computation, so that each library defines a new sublanguage. Recently, several workers have proposed a generalization of monads, called variously “arrows ” or Freyd-categories. The extra generality promises to increase the power, expressiveness and efficiency of the embedded approach, but does not mesh as well with the native abstraction and application. Definitions are typically given in a point-free style, which is useful for proving general properties, but can be awkward for programming specific instances. In this paper we define a simple extension to the functional language Haskell that makes these new notions of computation more convenient to use. Our language is similar to the monadic style, and has similar reasoning properties. Moreover, it is extensible, in the sense that new combining forms can be defined as expressions in the host language. 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.

In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in call-by-value programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κ-categories. In this paper, we investigate environments in call-by-value languages by analysing a finegrain variant of Moggi’s computational λ-calculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κ-categories.

We investigate continuations in the context of idealized call-by-value programming languages. On the semantic side, we analyze the categorical structures that arise from continuation models of call-by-value languages. On the syntactic side, we study the call-by-value continuation-passing transformation as a translation between equational theories. Among the novelties are an unusually simple axiomatization of control operators and a strengthened completeness result with a proof based on a delaying transform.

Abstract. We introduce the arrow calculus, a metalanguage for manipulating Hughes’s arrows with close relations both to Moggi’s metalanguage for monads and to Paterson’s arrow notation. Arrows are classically defined by extending lambda calculus with three constructs satisfying nine (somewhat idiosyncratic) laws. In contrast, the arrow calculus adds four constructs satisfying five laws. Two of the constructs are arrow abstraction and application (satisfying beta and eta laws) and two correspond to unit and bind for monads (satisfying left unit, right unit, and associativity laws). The five laws were previously known to be sound; we show that they are also complete, and hence that the five laws may replace the nine. We give a translation from classic arrows into the arrow calculus to complement Paterson’s desugaring and show that the two translations form an equational correspondence in the sense of Sabry and Felleisen. We are also the first to publish formal type rules (which are unusual in that they require two contexts), which greatly aided our understanding of arrows. The first fruit of our new calculus is to reveal some redundancies in the classic formulation: the nine classic arrow laws can be reduced to eight, and the three additional classic arrow laws for arrows with apply can be reduced to two. The calculus has also been used to clarify the relationship between idioms, arrows and monads and as the inspiration for a categorical semantics of arrows. 1

We introduce the arrow calculus, a metalanguage for manipulating Hughes’s arrows with close relations both to Moggi’s metalanguage for monads and to Paterson’s arrow notation. Arrows are classically defined by extending lambda calculus with three constructs satisfying nine (somewhat idiosyncratic) laws; in contrast, the arrow calculus adds four constructs satisfying five laws (which fit two well-known patterns). The five laws were previously known to be sound; we show that they are also complete, and hence that the five laws may replace the nine. 1

This paper gives a formal basis for the closure conversion phase of functional programming languages with imperative features, using a graphical semantics for the language. We present normal forms of graphs, one corresponding to procedural languages, and one corresponding to object-oriented languages. Using closure conversion, we can prove normalization results for both normal forms. Thus, we obtain sound algorithms for compiling the language into either procedural or object-oriented code. We discuss efficiency issues of the translation and suggest some improvements on the algorithm.