On the difference between polynomial-time many-one and truth-table reducibilities on distributional problems

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P 6 = NP. As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT using a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT using a less redundant distribution unless P = NP. We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of DistNP , this relation is proper unless P = NP . (2) C contains DistP , but it is not contained in AveP unless DistNP ⊆ AveZPP . (3) C has a ≤m-complete set. (4) C has a ≤ptt-complete set that is not ≤ p m-complete unless P = NP . This shows that under the assumption that P 6 = NP , the two completeness notions differ on some non-trivial subclass of DistNP .