Non-deterministic computation and the Jayne-Rogers Theorem

Abstract

We provide a simple proof of a computable analogue to the Jayne-Rogers Theorem from descriptive set theory. The difficulty of the proof is delegated to a simulation result pertaining to non-deterministic type-2 machines. Thus, we demonstrate that developments in computational models can have applications in fields thought to be far removed from it.

Universality, optimality, and randomness deficiency

“…We thank the anonymous referee for helpful comments improving the clarity of this work and for pointing out the connection between Theorems 5.15 and 5.16 and the work in [23].…”

“…2 set A ⊆ 2 ω is a special case of the fact that a function between two computable metric spaces is effectively ∆ 0 2 -measurable if and only if it Weihrauchreduces to C N [23]. Similarly, one can prove Theorem 5.16 by building a Σ 0 2 set A ⊆ 2 ω such that for no ∆ 0 2 set B is A ∩ MLR = B ∩ MLR and appealing to the results of [23].…”

“…Similarly, one can prove Theorem 5.16 by building a Σ 0 2 set A ⊆ 2 ω such that for no ∆ 0 2 set B is A ∩ MLR = B ∩ MLR and appealing to the results of [23]. This is the approach taken by Davie, Fouché, and Pauly [9].…”

A comparison of concepts from computable analysis and effective descriptive set theory

“…While these spaces do not carry a meaningful topology, they can nevertheless be studied as represented spaces. This was done implicitly in [17], and more explicitly in [1,32,26] and [25,27].…”

“…• The study of Weihrauch reducibility often draws on concepts from descriptive set theory via results that identify various classes of measurable functions as lower cones for Weihrauch reducibility [1,2,26]. The Weihrauch lattice is used as the setting for a metamathematical investigation of the computable content of mathematical theorems [3,9,21].…”