# R_{1−tt}^{SN}(NP) distinguishes robust many-one and Turing completeness

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## 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, R _{1−tt} ^{ SN } (NP)—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 R _{1−tt} ^{ SN } (NP) has many equivalent forms having to do with ordered and parallel access to NP and NP ∩ coNP.

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