Many-one reductions and the category of multivalued functions

TL;DR

This paper develops a category-theoretic framework for multi-valued functions, revealing that many-one degrees form a distributive lattice and highlighting differences from single-valued functions in computational reducibility.

Contribution

It introduces a general category-theoretic approach to multi-valued functions, establishing their degree structures and identifying open questions in the field.

Findings

Many-one degrees form a distributive lattice under general conditions

Some classical reduction results extend to multi-valued functions, others do not

Category-theoretic properties influence reducibility and degree structures

Abstract

Multi-valued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the moniker search problems leading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees). A more general understanding is possible, if the category-theoretic properties of multi-valued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multi-valued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch). Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multi-valued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multi-valued functions.