Effective randomness, strong reductions and Demuth's theorem

TL;DR

This paper explores generalizations of Demuth's Theorem, demonstrating its validity for Schnorr and computable randomness, and analyzing the degrees of reals random under computable measures.

Contribution

It extends Demuth's Theorem to Schnorr and computable randomness and clarifies limitations when replacing Turing equivalence with weaker notions.

Findings

Demuth's Theorem holds for Schnorr and computable randomness.

The theorem cannot be strengthened to wtt-equivalence.

Results on Turing and tt-degrees of reals random under computable measures.

Abstract

We study generalizations of Demuth's Theorem, which states that the image of a Martin-L\"of random real under a tt-reduction is either computable or Turing equivalent to a Martin-L\"of random real. We show that Demuth's Theorem holds for Schnorr randomness and computable randomness (answering a question of Franklin), but that it cannot be strengthened by replacing the Turing equivalence in the statement of the theorem with wtt-equivalence. We also provide some additional results about the Turing and tt-degrees of reals that are random with respect to some computable measure.