On the computational complexity of type 2 functionals

Abstract

Cook has proposed three necessary conditions that a class C of type 2 functionals must satisfy in order to be the intuitively correct class of polynomial time computable functionals. We consider a strengthening of Cook's conditions, by replacing the notion of closure under functional substitution with the notion of uniform closure, introduced by Seth, and we show that there is a unique maximal class C _max of type 2 functionals that satisfies this stronger version of Cook's conditions. We give an Oracle Turing Machine characterization of C _max. We show C _max to be different from Cook's and Kapron's BFF and Seth's C ₂. We also show that when the input functions are restricted to range over PTIME, BFF and C _max still differ. However, if the input functions are restricted to range over the real numbers in the sense of Ko and Friedman, then BFF and C _max coincide. Work by Seth suggests that Cook's conditions might not be sufficient, and Seth has proposed to add a new condition. We give a Turing Machine characterization of a class C _S that is bigger than BFF and satisfies both Cook's and Seth's conditions. We propose a new condition, in our opinion more natural than Seth's, and show that only BFF satisfies this new condition together with Cook's.