Some observations on NP real numbers and P-selective sets

Abstract

We show that each standard left cut of a real number is a p-selective set. Classification results about NP real numbers and recursively enumerable real numbers follow from similar results about p-selective and semirecursive sets. In particular, it is proved that no left cut can be NP-hard unless the polynomial hierarchy collapses to ϵ 2 P . This result is surprising because it shows that the McLaughlin-Martin construction of a ⩽ tt -complete r.e. semirecursive set fails at the polynomial time complexity level.