Parallel Haskell: the Vectorisation Monad

It has long been known that some of the most common uses of for and while-loops in imperative programs can easily be expressed using the standard higher-order functions fold and map. With this correspondence as a starting point, we derive parallel implementations of various iterative constructs, each having a better complexity than their sequential counterparts, and explore the use of monads to guarantee the soundness of the parallel implementation. As an aid to the presentation of the material, we use the proposed syntax for parallel Haskell 27] ((gure 1) as a vehicle in which imperative functional programs will be expressed. Surprisingly, incorporating imperative features into a purely functional language has become an active area of research within the functional programming community 30, 24, 36, 20]. One of the techniques gaining widespread acceptance as a model for imperative functional programming is monads 38, 37, 26]. Typically monads are used to guarantee single threaded-ness, enabling side eeects to be incorporated into a purely functional language without losing referential transparency. We take a diierent approach. First for-loops are translated into a monadic framework. Next, by ensuring that the monad satisses the programming identities usually associated with the successful vectorisation of imperative constructs, all well-typed for-loops in which the body is expressed using the vectorisation monad can be parallelised. The technique is not just restricted to a data-parallel environment. It could be implemented by a divide and conquer technique on a multi-processor platform, or by a parallel implementation of graph reduction such as that ooered by the GRIP 12] machine. 1 Background \ `Should declarative languages be mixed with imperative languages?', clearly has the answer that they should, because at the moment we don't know how to do everything in pure declarative languages."|C. Strachey, 1966 22]. The work presented in this paper uses a model of parallel computation based upon the Bird Meertens Formalism 6]. To make things simple, all parallelism is expressed in terms of a O(1) complexity map function, and O(log N) complexity fold and scan functions which both require an associative operator. Unlike our earlier work, we use the BMF as is|all the operations presented here can be interpreted as though they were being expressed on lists. This stance contradicts our earlier paper 15] in which we said that lists were unsuitable for data-parallel evaluation in a non-strict language. We still believe this to be true, but we have tried to …