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arXiv · 2401.12385

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

Patrick Baillot·Ugo Dal Lago·Cynthia Kop·Deivid Vale
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Abstract

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 $\mathtt{BFF}_2$.

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Patrick Baillot, Ugo Dal Lago, Cynthia Kop, Deivid Vale. 2024-01-22. A Characterization of Basic Feasible Functionals Through Higher-Order Rewriting and Tuple Interpretations. https://doi.org/10.46298/lmcs-21(4%3A19)2025

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