There is no recursive axiomatization for feasible functionals of type 2

Abstract

The author shows a class of type-two feasible functionals, C/sub 2/, that satisfies Cook's conditions, (1990) and cannot be expressed as the lambda closure of type-one poly-time functions and any recursively enumerable set of type-two feasible functionals. Further, no class of total type-two functionals containing this class is representable as the lambda closure of a recursively enumerable set of type-two total computable functionals and type-one poly-time functions. The definition of C/sub 2/ provides a clear computational procedure for functionals of C/sub 2/. Using functionals of class C/sub 2/ a more general notion of polynomial-time reducibility between two arbitrary type-one functions can be introduced.<>