A minimal set low for speed

TL;DR

This paper constructs a set of minimal Turing degree that is low-for-speed, answering an open question about the existence of such sets using a novel combination of forcing and approximation techniques.

Contribution

It introduces a new method combining forcing and approximation to construct a minimal degree set that is low-for-speed, resolving a previously open problem.

Findings

Constructed a minimal degree set that is low-for-speed.

Demonstrated the compatibility of minimal degree and low-for-speed properties.

Developed a novel forcing and approximation method for this construction.

Abstract

An oracle $A$ is low-for-speed if it is unable to speed up the computation of a set which is already computable: if a decidable language can be decided in time $t(n)$ using $A$ as an oracle, then it can be decided without an oracle in time $p(t(n))$ for some polynomial $p$. The existence of a set which is low-for-speed was first shown by Bayer and Slaman who constructed a non-computable computably enumerable set which is low-for-speed. In this paper we answer a question previously raised by Bienvenu and Downey, who asked whether there is a minimal degree which is low-for-speed. The standard method of constructing a set of minimal degree via forcing is incompatible with making the set low-for-speed; but we are able to use an interesting new combination of forcing and full approximation to construct a set which is both of minimal degree and low-for-speed.