# On the Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems

## Abstract

In this paper we separate many-one reducibility from truth- table reducibility for distributional problems in Dist *NP* under the hy- pothesis that *P* ≠ *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 Dist *NP* , this relation is proper unless *P* = *NP*. (2) *C* contains Dist *P* , but it is not contained in Ave *P* unless Dist *NP* ≠ Ave *ZPP*. (3) *C* has a ≤^{p} _{m}-complete set. (4) *C* has a ≤^{p} _{tt}-complete set that is not ≤^{p} _{m}-complete unless *P* = *NP*. This shows that under the assumption that *P* ≠ *NP* , the two complete- ness notions differ on some non-trivial subclass of Dist *NP*.

## Keywords

Dominance Condition Distributional Problem Partial Assignment Standard Distribution Equivalent Formula## Preview

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