We study the pm-degrees and pT-degrees of immune and bi-immune sets. We demonstrate the existence of incomparable pT-immune degrees in deterministic time classes.

We investigate the relationship between two different notions of reducibility among prediction (learning) problems within the distribution-free learning model of Valiant (PAC learning model). The notions of reducibility we consider are the analogues for prediction problems of the many-one reducibility and of the Turing reducibility. The former is the notion of prediction preserving reducibility developed by Pitt and Warmuth, and its generalization. Concerning these two notions of reducibility, we show that there exist a pair of prediction problems A and B, whose membership problems are polynomial time solvable, such that A is reducible to B with respect to the Turing reducibility, but not with respect to the prediction preserving reducibility. We show this result by making use of the notion of a class of polynomially sparse variants of a concept representation class. We first show that any class A of polynomially sparse variants of another class B is reducible to B with respect to the Turing reducibility'. We then prove the existence of a prediction problem R and a class R of polynomially sparse variants of R, such that R does not reduce to R with respect to the prediction preserving reducibility.