Stashing And Parallelization Pentagons

Abstract

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".