Towards a Theory of Parallel Algorithms on Concrete Data Structures

Building on Kahn and Plotkin's theory of concrete data structures and sequential functions, Berry and Curien de ned an intensional model of sequential algorithms between concrete data structures. In this paper we report on an attempt to develop a similar intensional model of concurrent computation. We present a notion of parallel algorithm between concrete data structures, together with suitable application and currying operations. We de ne an intensional strictness ordering on parallel algorithms, with respect to which application is well behaved (at rst order types). We de ne the input-output function computed by a parallel algorithm, and we show that every parallel algorithm computes a continuous function. Thus, a parallel algorithm may be viewed as a continuous function together with a parallel computation strategy. In contrast, a Berry-Curien sequential algorithm may be viewed as a sequential function together with a sequential computation strategy. The intensional strictness ordering on parallel algorithms corresponds to the pointwise ordering on the functions they compute, in the same sense that the set inclusion ordering used by Berry and Curien on sequential algorithms corresponds to the stable ordering on the functions they compute. We believe that the ideas and results presented here constitute a rst step towards a fuller understanding of the intensional semantics of parallelism, even though the model presented here is not yet general enough to provide a satisfactory account of higher order algorithms, and lacks a notion of composition for algorithms. We present some ideas for overcoming these de ciencies, and some directions for further research.