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Parallel reduction is a major component of parallel programming and widely used for summarization and aggregation. It is not well understood, however, what sorts of nontrivial summarizations 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 parallelization 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 by 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 jumps, prefix-sum patterns, and even tree manipulations.

Lambda calculus with algebraic simplification for reduction parallelization by equational reasoning