Completion of Choice

TL;DR

This paper explores the completion of choice problems within the Weihrauch lattice, revealing how completion relates to total reducibility and characterizing the complexity of various choice problems.

Contribution

It introduces the concept of completion for choice problems in the Weihrauch lattice, linking it to total Weihrauch reducibility and classifying the completeness of problems based on space properties.

Findings

Choice problems related to compact spaces are complete.

Choice problems for unbounded spaces are typically not complete.

Completion helps analyze independence of problems from premises.

Abstract

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal role in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete.