On reductions of NP sets to sparse sets

M. Ogiwara and O. Watanabe (1990) showed that if SAT is bounded truth-table reducible to a sparse set, then P=NP. In the present work, the authors simplify their proof, strengthen the result, and use it to obtain several new results. Among the new results are the following: applications of the main theorem to log-truth-table and log-Turing reductions of NP sets to sparse sets; generalizations of the main theorem which yield results similar to the main result at arbitrary levels of the polynomial hierarchy; and the construction of an oracle relative to which P not=NP but there are NP-complete sets which are f(n)-tt-reducible to a tally set, for any f(n) in Omega (log n).<
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