On Different Reducibility Notions for Function Classes

This paper continues research of Toda (The Complexity of Finding Medians, 31 st Symposium on Foundations of Computer Science (1990), pp. 778{787) on problems complete for function classes like FP # P and Mid P under Krentel's metric reductions. We rst show that metric reductions wipe out the diierence between Mid P and other related classes of functions which are probably diierent from Mid P. In order to obtain a more detailed classiication of naturally arising functional problems we then deene and examine a stricter notion of reducibility and show that a number of problems, among them those proved by Toda to be hard for Mid P under metric reductions, are complete for diierent classes of median functions related to Mid P under our stricter reducibil-ity. Finally, we use these results to exhibit new natural complete sets for the well studied classes of sets PP, PP NP , and P PP .