Some desirable conditions for feasible functionals of type 2

We consider functionals of type 2 as transformers between functions of type 1. An intuitively feasible functional must preserve the complexity of the input function in some broad sense. We show that the well quasi-order functional, which has been proposed by S.A. Cook (1990) as being intuitively feasible, fails to preserve the class of Kalmar elementary functions. For the basic feasible functionals (BFF), we show that there are arbitrarily large complexity classes of type 1 functions, under the classical definition of a complexity class, which contain polynomial-time functions and are closed under composition but are not preserved by the BFF. However, for a more natural definition of a complexity class of type 1 functions, BFF is shown to preserve all such complexity classes. BFF is the largest known class with this property. We prove BFF to be the largest class of type 2 functionals which satisfies Cook's conditions and the Ritchie-Cobham property, and preserves all classes of type 1 computable functions that contain polynomial-time functions and are closed under composition and limited recursion on notation. These results give some evidence that basic feasible functionals may be the right notion of type 2 feasibility.<
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