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**1 - 1**of**1**### On the Power of Positive Turing Reductions

"... 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, positiveTuring and Turing reducibility to NP di er sharply unless the polynomial hierarchy collapses. We show ..."

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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, positiveTuring and Turing reducibility to NP di er sharply unless the polynomial hierarchy collapses. We show that the situation is quite di erent 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 toDP yield the same class. Additionally, we show that an even weaker class, P NP[1] , can be substituted for DP in this context. Category: F.1 Key Words: computational complexity, NP, positive Turing reductions