Reductions to Sets of Low Information Content

In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very exible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces to a sparse set, then P = NP. We also show that any set A that reduces to some sparse set (via various types of reductions) in fact reduces by the same type of reduction to a sparse set that is simple relative to A. We give a complete characterization of the sets of low instance complexity in terms of reductions to tally sets; it follows that if P 6 = NP, then no set of low instance complexity can be complete for NP with respect to disjunctive reductions or conjunctive reductions.