A Characterization of Basic Feasible Functionals Through Higher-Order Rewriting and Tuple Interpretations

The class of type-two basic feasible functionals (BFF2) is the analogue of FP (polynomial time functions) for type-2 functionals, that is, functionals that can take (first-order) functions as arguments. BFF2 can be defined through Oracle Turing Machines with running time bounded by second-order polynomials. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. We address the problem of characterizing BFF2 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. In this paper, we consider a recently introduced notion of cost–size tuple interpretations for higher-order term rewriting and see second order rewriting as ways of computing type-2 functionals. We then prove that the class of functionals represented by higher-order terms admitting polynomially bounded cost–size interpretations exactly corresponds to BFF2.