The class of type-two basic feasible functionals ($\mathtt{BFF}_2$) is the analogue of $\mathtt{FP}$ (polynomial time functions) for type-2 functionals, that is, functionals that can take (first-order) functions as arguments. $\mathtt{BFF}_2$ 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 $\mathtt{BFF}_2$ 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 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 $\mathtt{BFF}_2$.

@article{paperbot3351,
  title = {A Characterization of Basic Feasible Functionals Through Higher-Order Rewriting and Tuple Interpretations},
  author = {Patrick Baillot and Ugo Dal Lago and Cynthia Kop and Deivid Vale},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 4},
  year = {2025},
  doi = {10.46298/lmcs-21(4:19)2025}
}