Coarse Reducibility and Algorithmic Randomness

TL;DR

This paper explores how noncomputable information can be recovered from coarse descriptions of random sets, revealing connections between randomness, triviality, and coarse reducibility in computability theory.

Contribution

It establishes new results linking randomness notions to coarse reducibility and degree structures, including the non-surjectivity of certain embeddings and minimal pairs in coarse degrees.

Findings

If A is 1-random and B is reducible from all coarse descriptions of A, then B is K-trivial.

For weakly 2-random A, B is computable if reducible from all coarse descriptions.

Mutually weakly 3-random sets have coarse degrees forming a minimal pair.

Abstract

A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if A is a 1-random set which Turing-reduces to 0', then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Phi such that if D is a coarse description of X, then Phi^D is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.