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

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 runnin...

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Bibliographic Details
Published in:Logical methods in computer science Vol. 21, Issue 4
Main Authors: Baillot, Patrick, Lago, Ugo Dal, Kop, Cynthia, Vale, Deivid
Format: Journal Article
Language:English
Published: Logical Methods in Computer Science Association 05.11.2025
Subjects:
Computer Science
Basic Feasible Functions Higher-Order Term Rewriting Tuple Interpretation Implicit Computational Complexity
Higher-Order Term Rewriting
Tuple Interpretation
Basic Feasible Functions
Implicit Computational Complexity
ISSN:1860-5974, 1860-5974
Online Access:Get full text
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Summary: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 $\mathtt{BFF}_2$.
ISSN:1860-5974
1860-5974
DOI:10.46298/lmcs-21(4:19)2025