On Basic Feasible Functionals and the Interpretation Method

The class of basic feasible functionals (BFF) is the analog of FP (polynomial time functions) for type-two functionals, that is, functionals that can take (first-order) functions as arguments. BFF can be defined by means of oracle Turing machines of time bounded by a second-order polynomial. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. We address the problem of characterizing the class BFF by higher-order term rewriting. Various kinds of interpretations for first-order term rewriting have been introduced in the literature for proving termination and characterizing (first-order) complexity classes. Here we consider a recently introduced notion of cost-size interpretations for higher-order term rewriting and see definitions as ways of computing functionals. We then prove that the class of functionals represented by higher-order terms admitting a certain kind of cost-size interpretation is exactly BFF.