Relating Equivalence and Reducibility to Sparse Sets

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 6= NP, this paper shows that for k-truth-table reductions, k 2, equivalence and reducibility to sparse sets provably di er. Though Gavald a 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 6= NP. Additionally, this paper studies the relative power of di erent notions of reducibility, and proves that disjunctive and conjunctive truth-table reductions to sparse sets are surprisingly powerful, refuting a conjecture of Ko. RESEARCH SUPPORTED IN PART BY THE NATIONAL SCIENCE FOUNDATION UNDER GRANT CCR-9000045 AND BY THE INTERNATIONAL INFORMATION SCIENCE FOUNDATION UNDER GRANT 90-1-3-227. y RESEARCH SUPPORTED IN PART BY THE NATIONAL SCIENCE FOUNDATION UNDER GRANTS CCR-8996198 AND CCR-8957604, AND BY THE INTERNATIONAL INFORMATION SCIENCE FOUNDATION UNDER GRANT 90-1-3-228. z RESEARCH DONE IN PART WHILE AT THE TOKYO INSTITUTE OF TECHNOLOGY.