Andrej Bauer: Instance reducibility and 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 set-indexed infima distribute over 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 anti-monotone, and the ¬¬-dense degrees coincide with those that are reducible to 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 ¬¬-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 mix uniform and non-uniform reductions, and by generally widening the scope of Weihrauch reducibility.