On Sets Polynomially Enumerable by Iteration

Abstract

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 over 1, si -self-reducible are polynomially enumerable by iteration, and that all recursive sets that are ≤p over 1, si -self-reducible are bi-enumerable. We also show that when the ≤p over 1, si-self-reduction is via a function whose inverse is computable in polynomial time, then the above results not only hold, but also 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