Abstract
For various polynomial-time reducibilities r, this paper asks 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 is not equal to NP, this paper shows that for k-truth-table reductions, k > 1, equivalence and reducibility to sparse sets provably diier. Though Gavalda and 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, this paper shows that extending their result to the 2-truth-table case would provide a proof that P is not equal to NP. Additionally, this paper studies the relative power of diierent notions of reducibility, and proves that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of Ko.