On sets polynomially enumerable by iteration

Sets whose members are enumerated by some Turing machine are called recursively enumerable. We define a set to be polynomially enumerable by iteration if its members are efficiently enumerated by iterated application of some Turing machine. We prove that many complex sets—including all exponential-time complete sets, all NP-complete sets yet obtained by direct construction, and the complements of all such sets—are polynomially enumerable by iteration. These results follow from more general results. In fact, we show that all recursively enumerable sets that are ^p_1si-self-reducible are polynomially enumerable by iterations, and that all recursive sets that are ^p_1si-self-reducible are bi-enumerable. We also show that when the ^p_1si-self-reduction is via a function whose inverse is computable in polynomial time, then the above results hold with the polynomial enumeration given by a function whose inverse is computable in polynomial time. In the final section of the paper we show that no NP-complete set can be iteratively enumerated in lexicographically increasing order unless the polynomial time hierarchy collapses to NP. We also show that the sets that are monotonically bi-enumerable are “essentially” the same as the sets in parity polynomial time.