Set bit enumeration is hard

The polynomial-time function hierarchy is a function analog to the polynomial hierarchy and contains many interesting natural functions. Given the difficulties encountered so far in trying to develop feasible algorithms for these functions, the author seeks instead to construct feasible approximations. The author describes one such notion of approximation, set bit enumeration, which seems to be, by definition, a very weak information source. It is shown that this information is hard in the sense that if one could feasibly compute it, then one could compute, in polynomial time, the value of any function in the polynomial-time function hierarchy. In proving these hardness results, the author uses a novel method of pruning nondeterminism out of deterministic computation, together with combinatorial counting