Abstract
Abstract Parallel reduction is a major component of parallel programming and widely used for summarisation and aggregation. It is not well understood, however, what sorts of non-trivial summarisations can be implemented as parallel reductions. This paper develops a calculus named λAS, a simply typed lambda calculus with algebraic simplification. This calculus provides a foundation for studying a parallelisation of complex reductions by equational reasoning. Its key feature is δ abstraction. A δ abstraction is observationally equivalent to the standard λ abstraction, but its body is simplified before the arrival of its arguments using algebraic properties such as associativity and commutativity. In addition, the type system of λAS guarantees that simplifications due to δ abstractions do not lead to serious overheads. The usefulness of λAS is demonstrated on examples of developing complex parallel reductions, including those containing more than one reduction operator, loops with conditional jumps, prefix sum patterns and even tree manipulations.
Highlights
Functional programming is commonly regarded as a promising approach in parallel programming
Functional programming makes parallel programming easy and intuitive. Another benefit of using functional programs in parallel programming is equational reasoning, which helps certify the correctness of parallel implementations
Several parallel implementations of reduction-related computations are developed by introducing δ abstractions, and their correctness is checked by equational reasoning
Summary
Functional programming is commonly regarded as a promising approach in parallel programming. In the following recursive Fibonacci function: fib n = if n ≤ 1 1 else fib (n − 1) + fib (n − 2). It is syntactically clear that the two recursive calls, fib (n − 1) and fib (n − 2), can be simultaneously evaluated. For this reason, functional programming makes parallel programming easy and intuitive. Consider the following parallel implementation of the fib function in Haskell: fib n = if n ≤ 1 1 else par x (pseq y (x + y)) where x = fib (n − 1) y = fib (n − 2)
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