Consequences of the Existence of Sparse Sets Hard for NP under a Subclass of Truth-Table Reductions

The consequences of the existence of sparse hard sets for NP have been investigated for nearly 20 years. Many surprising results have been found, e.g. the result of Ogihara and Watanabe that P = NP is implied by the existence of sparse pbtt -hard sets for NP. It is clear that the existence of sparse hard sets for a weak reduction (e.g. pm) implies stronger results than the existence of sparse hard sets for strong reductions (e.g. pT ). It is astonishing that by now we have no result which separates ptt and pT -reductions, although our intuition says that pT -reductions are stronger than ptt -reductions. This paper shows weakened truth-table reductions for which we obtain an improvement to the best known pT -reduction result. We show that the existence of sparse pk CNF tt -hard ( pk DNF tt -hard, resp.) sets for NP implies PH P2 .