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, $ {{\rm R}_{1\mbox{-}{tt}}^{\cal SN}({\rm 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 $ {{\rm R}_{1\mbox{-}{tt}}^{\cal SN}({\rm NP})} $ has many equivalent forms having to do with ordered and parallel access to NP and NP ∩ coNP.
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