Relating Equivalence and Reducibility to Sparse Sets

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 $P \ne NP$, this paper shows that for k-truth-table reductions, $k \geq 2$, equivalence and reducibility to sparse sets provably differ. Though Gavaldà and Watanabe have shown that, for any polynomial-time computable unbounded function $f( \cdot )$, 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 $P\ne NP$. Additionally, this paper studies the relative power of different notions of reducibility, and proves that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of Ko.