On the Power of Positive Turing Reductions

TL;DR

This paper explores the computational power of positive Turing reductions, revealing that for the class DP, positive Turing reductions are as powerful as Turing reductions, contrasting earlier results for NP.

Contribution

It demonstrates that positive Turing reductions to DP are equivalent to Turing reductions, extending the understanding of reducibility in the boolean hierarchy.

Findings

Positive Turing reductions to DP capture all sets Turing reducible to NP.

Positive Turing and Turing reducibility to DP are equivalent.

P(NP[1]) can replace DP in this context.

Abstract

In the early 1980s, Selman's seminal work on positive Turing reductions showed that positive Turing reduction to NP yields no greater computational power than NP itself. Thus, positive Turing and Turing reducibility to NP differ sharply unless the polynomial hierarchy collapses. We show that the situation is quite different for DP, the next level of the boolean hierarchy. In particular, positive Turing reduction to DP already yields all (and only) sets Turing reducibility to NP. Thus, positive Turing and Turing reducibility to DP yield the same class. Additionally, we show that an even weaker class, P(NP[1]), can be substituted for DP in this context.