Relating equivalence and reducibility to sparse sets

For various polynomial-time reducibilities r, the authors ask whether being r-reducible to a sparse set is a broader notion than being r-equivalent to a sparse set. Although distinguishing equivalence and reducibility to sparse sets, for many-one or 1-truth-table reductions, would imply that P ≠ NP, the authors show that for k-truth-table reductions, k ≥ 2, equivalence and reducibility to sparse sets provably differ. Though R. Gavaldaand D. Watanabe have shown that, for any polynomial-time computable unbounded function f(·), some sets f(n)-truth-table reducible to sparse sets are not even Turing equivalent to sparse sets, the authors show that extending their result to the 2-truth-table case would provide a proof that P ≠ NP. Additionally, the authors study the relative power of different notions of reducibility and show that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of K. Ko (1989).