On Sets Reducible to Sparse Sets∗

The main results of this paper may be summarized as follows: (1) For every k > 0, there exist a sparse set S and a set L such that L≤
^P
(k+1)-tt
S but there is no sparse set S′ such that L≤
^P
k-tt
S′. Thus, the class of sets that are bounded truth-table reducible to sparse sets can be decomposed into a properly infinite hierarchy based on bounding the number of (nonadaptive) queries that are allowed. (2) There exist a sparse set S and a set L such that L ≤
^P
tt
S′, but there is no integer k such that for some sparse set S′, L≤
^P
k-tt
S′. Thus the class of sets that are bounded truth-table reducible to sparse sets is properly included in the class of sets that are truth-table reducible to sparse sets. (3) The class of sets that are bounded truth-table reducible to tally sets is equal to the class of sets that axe many-one reducible to tally sets. (4) The class of sets having polynomial size circuits is equal to the class of sets that are truth-table reducible to tally sets.