We show that hard sets S for NP must have exponential density, i.e. |S =n | ges 2n epsi for some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n 1-isin queries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n 1-isin queries.
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Buhrman et al. (2008) studied this question.
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