  In this paper we demonstrate that the class of basic feasible functionals has
recursion theoretic properties which naturally generalize the corresponding
properties of the class of feasible functions. We also improve the Kapron -
Cook result on mashine representation of basic feasible functionals. Our proofs
are based on essential applications of logic. We introduce a weak fragment of
second order arithmetic with second order variables ranging over functions from
N into N which suitably characterizes basic feasible functionals, and show that
it is a useful tool for investigating the properties of basic feasible
functionals. In particular, we provide an example how one can extract feasible
"programs" from mathematical proofs which use non-feasible functionals (like
second order polynomials).
