A short cut to parallelization theorems

The third list-homomorphism theorem states that if a function is both foldr and foldl , it has a divide-and-conquer parallel implementation as well. In this paper, we develop a theory for obtaining such parallelization theorems. The key is a new proof of the third list-homomorphism theorem based on shortcut deforestation. The proof implies that there exists a divide-and-conquer parallel program of the form of h (x 'merge' y) = h1 x - h2 y, where h is the subject of parallelization, merge is the operation of integrating independent substructures, h1 and h2 are computations applied to substructures, possibly in parallel, and - merges the results calculated for substructures, if (i) h can be specified by two certain forms of iterative programs, and (ii) merge can be implemented by a function of a certain polymorphic type. Therefore, when requirement (ii) is fulfilled, h has a divide-and-conquer implementation if h has two certain forms of implementations. We show that our approach is applicable to structure-consuming operations by catamorphisms (folds), structure-generating operations by anamorphisms (unfolds), and their generalizations called hylomorphisms.