Computing Downwards Accumulations on Trees Quickly

Downwards accumulations on binary trees are essentially functions which pass information down a tree, from the root towards the leaves. Under certain conditions, a downwards accumulation is bothèecient' (computable in a functional style in parallel time proportional to the depth of the tree) and`manipulable' (enjoying a number of distributivity properties useful in program construction). In this paper, we show that these conditions do in fact yield a stronger conclusion: the accumulation can be computed in parallel time proportional to the logarithm of the depth of the tree, on a Crew Pram machine.