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 not=N...

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Published in[1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference pp. 220 - 229
Main Authors Allender, E., Hemachandra, L.A., Ogiwara, M., Watanabe, O.
Format Conference Proceeding
LanguageEnglish
Published IEEE Comput. Soc. Press 1991
Subjects
Circuits
Complexity theory
Computer science
Decoding
Information science
Polynomials
Robustness
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Summary: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 not=NP, the authors show that for k-truth-table reductions, k>or=2, equivalence and reducibility to sparse sets provably differ. Though R. Gavalda and 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 not=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).< >
ISBN:9780818622557
0818622555
DOI:10.1109/SCT.1991.160264