A complexity theory for finite type functionals is called for in many areas of Computer Science and Mathematics; for example for dealing with complexity issues in programming languages that allow passing functions as parameters or to define and study the complexity of real functions or to give a version of Gödel's Dialectica interpretation to constructive theories.

Cook was the first one to study the notion of feasible computable functional. He and Kapron have defined and studied a class of functionals called BFF; it is generally agreed that every functional in BFF is intuitively feasible, but it is open to decide whether BFF provides the right formalization for the intuitive notion of feasible computable functional. Cook has also stated some necessary, but probably not sufficient as some work by Seth suggests, conditions that a class C of functionals must satisfy in order to be the intuitively correct class of feasible functionals. We show that there is a unique maximal class CM of type two functionals satisfying Cook's conditions, and we give an Oracle Turing Machine characterization of it. Even if CM does not seem to provide the right notion of polynomial time computable functional our result and characterization of CM is of interest; we conjecture there is a maximal class M satisfying Cook's conditions and an additional condition on the preservation of complexity classes of functions suggested by Seth, and that M=BFF. We hope the methods used in the paper may prove useful in settling this conjecture.

In addition we also consider the question of type two feasibility on some restricted domains: in feasible mathematics only polynomial time computable functions are considered; we show that even on such a restricted domain CM is different from BFF and we don't see any reason to reject functionals in CM - BFF as not intuitively feasible in this setting. On the other hand we show that if the only input functions allowed are real numbers as defined by Ker-I-Ko and Friedman then CM = BFF ="Polynomial time computable real function" as defined by by Ker-I-Ko and Friedman, so at least in the case of real analysis it seems that a good definition of polynomial time computable functional has been reached.