On low for speed oracles

TL;DR

This paper explores the properties and structure of 'low for speed' oracles in computational complexity, building on prior work to answer open questions and demonstrate their rich theoretical landscape.

Contribution

It advances understanding of low for speed oracles by answering open questions and revealing their complex structure in computational theory.

Findings

Existence of non-computable low for speed oracles confirmed

Construction of a computably enumerable low for speed oracle

Rich structural properties of the class of low for speed oracles

Abstract

Relativizing computations of Turing machines to an oracle is a central concept in the theory of computation, both in complexity theory and in computability theory(!). Inspired by lowness notions from computability theory, Allender introduced the concept of "low for speed" oracles. An oracle A is low for speed if relativizing to A has essentially no effect on computational complexity, meaning that if a decidable language can be decided in time $f(n)$ with access to oracle A, then it can be decided in time poly(f(n)) without any oracle. The existence of non-computable such A's was later proven by Bayer and Slaman, who even constructed a computably enumerable one, and exhibited a number of properties of these oracles as well as interesting connections with computability theory. In this paper, we pursue this line of research, answering the questions left by Bayer and Slaman and give further evidence that the structure of the class of low for speed oracles is a very rich one.