Ricerc@Sapienza

The Euler's Sieve refines the Sieve of Eratosthenes to compute prime numbers, by crossing off each non prime number just once. Euler's Sieve is considered hard to be faithfully and efficiently coded as a purely functional stream based program. We propose three Haskell programs implementing the Euler's Sieve, all based on the idea of generating just once each composite to be crossed off. Their faithfulness with respect to the Euler's Sieve is up to costly stream unions imposed by the sequential nature of streams.

In this paper, we propose and explore a new approach to abstract machines and optimal reduction via streams, infinite sequences of elements. We first define a sequential abstract machine capable of performing directed virtual reduction (DVR) and then we extend it to its parallel version, whose equivalence is explained through the properties of DVR itself. The result is a formal definition of the λ-calculus interpreter called Parallel Environment for Lambda Calculus Reduction (PELCR), a software for λ-calculus reduction based on the Geometry of Interaction.