Instance reducibility and Weihrauch degrees

TL;DR

This paper introduces a new notion of reducibility called instance reducibility, which classifies principles in reverse constructive mathematics and extends Weihrauch degrees with a richer lattice structure and additional computational features.

Contribution

It defines instance reducibility, shows it forms a complete lattice, and connects it to Weihrauch degrees, enhancing their structure with computable infima, suprema, and logical operations.

Findings

Instance degrees form a complete lattice (a frame).

Extended degrees include computable infima, suprema, and implications.

$ eg eg$-dense degrees correspond to Weihrauch degrees.

Abstract

We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility.