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

2024-01-22

AI Summary

This paper characterizes the class BFF₂—type-2 functions computable by second-order polynomial-time oracle Turing machines. To this end, it introduces the first exact algebraic characterization of BFF₂ via higher-order term rewriting: by defining a cost-size interpretation, it establishes that higher-order rewrite systems induced by polynomially bounded interpretations precisely capture all and only the functions in BFF₂. The approach models type-2 function semantics as higher-order term reduction and integrates second-order polynomial bounding analysis to yield a purely rewriting-based characterization of BFF₂. The main contribution is the first tight correspondence between higher-order rewriting and BFF₂, providing a novel algebraic framework for complexity analysis, termination verification, and feasibility certification of higher-order programs. This result lays a theoretical foundation for reasoning about computational resources in higher-order functional computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $mathtt{BFF}_2$.

Problem

• Characterize type-two basic feasible functionals (BFF₂) via higher-order rewriting
• Link BFF₂ to polynomially bounded cost-size interpretations
• Establish equivalence between BFF₂ and second-order term rewriting

Innovation

• Higher-order term rewriting for BFF_2
• Cost-size interpretations for complexity
• Second-order polynomial time bounds