ON SYMMETRICAL DIFFERENCES OF NP-HARD SETS WITH WEAKLY-P-SELECTIVE SETS

The symmetric differences of NP - hard sets with weakly -P - selective sets are investigated in this paper. We show that if there exist a weakly-P-selective set A and a NP- less-than-or-equal-to m(P) -hard set H such that H - A is-an-element-of P(btt) (Sparse) and A - H is-an-element-of P(m) (Sparse) then P = NP So no NP- less-than-or-equal-to m(P) -hard set has sparse symmetric difference with any weakly -P - selective set unless P = NP. In addition we show there exists a P-selective set which has exponentially dense symmetric difference with every set in P(btt)(Sparse).