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 fn-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 NATI...

• Type: Conference
• Year: 1991
• Where: COCO
• Authors: Eric Allender, Lane A. Hemachandra, Mitsunori Ogiwara, Osamu Watanabe
• Keywords: Algorithms, COCO 1991, Foundation Under Grant, Science Foundation, Sparse Sets