Conjunctive and Disjunctive Reducibilities to Sparse and Tally Sets Revisited

We continue the investigation of polynomial-time reducibilities along the directions of [8]. We show that there is a set which is f(n)-conjunctive (similarly f(n)-disjunctive) deterministic polynomial time reducible to a tally set but is not g(n)-conjunctive (respectively g(n)-disjunctive) deterministic polynomial time reducible to any sparse set. We also show that there is a set which is f(n)-conjunctive deterministic polynomial time reducible to a tally set but is not g(n)-turing non-deterministic polynomial time reducible to any tally set. Several interesting corollaries about the closure of sparse and tally sets under such reducibilities follow from these results.