R_{1-tt}^{SN}(NP) Distinguishes Robust Many-One and Turing Completeness

TL;DR

This paper constructs a relativized world where a natural complexity class has Turing-complete sets but no many-one complete sets, highlighting a separation between these notions of completeness.

Contribution

It demonstrates that in a relativized setting, certain complexity classes can have Turing-complete sets without having many-one complete sets, and explores related completeness notions.

Findings

Existence of a relativized world with Turing-complete but no many-one complete sets.

The class has 2-truth-table complete sets but lacks 1-truth-table complete sets.

Multiple equivalent forms of the class related to ordered and parallel access to NP and NP∩coNP.

Abstract

Do complexity classes have many-one complete sets if and only if they have Turing-complete sets? We prove that there is a relativized world in which a relatively natural complexity class-namely a downward closure of NP, \rsnnp - has Turing-complete sets but has no many-one complete sets. In fact, we show that in the same relativized world this class has 2-truth-table complete sets but lacks 1-truth-table complete sets. As part of the groundwork for our result, we prove that \rsnnp has many equivalent forms having to do with ordered and parallel access to $\np$ and $\npinterconp$.