The Categorical Multi-Combinator Machine: CMCM

Implementations of functional programming languages can take a number of different forms; the first implementations used the SECD machine of Landin [Lan], which is a generalisation of the abstract machine underlying implementations of imperative programming languages like Pascal. Making the implementation lazy can be done with some cost in efficiency (details can be found in Henderson’s book [Hen]), but a more natural way of proceeding is provided by an implementation which uses rewriting of source-level expressions. Such implementations were pioneered by Wadsworth’s work on graphreduction for the λ-calculus [Wad], but only achieved reasonable performance after Turner’s observation that the translation of programs into a variable-free, or combinator, form allowed for more efficient execution. Turner’s work on the graph reduction of combinator code (together with that of Abdali [Abd]) was followed up by a number of researchers, including Hughes ([Hug]), Johnsson ([Joh]) and Lins ([Lins1) each of whom developed different systems of combinators, and algorithms for translating source programs into combinator form. Each of these implementations can be characterised as interpretive, as the reduction and transformation of the combinator graph can be seen as interpretation of the graph, and the obvious question raised by this is that of whether a true compiler for lazy functional programs can be produced. The first attempt to do this was the G-machine (‘G’ for graph) of Augustsson and Johnsson [Joh] in which many of the manipulations of the graph are transformed into abstract machine instructions, which can be rendered into machine code. Another approach is given by the Three Instruction Machine, or TIM [FaiWr], which comes full circle in using closures to represent the expressions under evaluation.