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On sets polynomially enumerable by iteration. (English) Zbl 0745.68047
Summary: 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 $$\leq^ p_{1,si}$$-self-reducible are polynomially enumerable by iteration, and that all recursive sets that are $$\leq^ p_{1,si}$$- self-reducible are bi-enumerable. We also show that when the $$\leq^ p_{1,si}$$-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.

##### MSC:
 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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