On the Computational Content of Moduli of Regularity and their Logical Strength

2026-03-04

AI Summary

This study investigates the computational significance of the existence of modulus of regularity and its logical strength in the frameworks of reverse mathematics and Weihrauch reducibility. By integrating techniques from reverse mathematics, Weihrauch computability analysis, and uniformity principles from nonstandard analysis, the work establishes a direct link between the modulus of regularity and algorithmic computability: its existence enables the effective construction of algorithms for computing zeros of continuous real-valued functions on compact metric spaces, and further yields minimal-norm zeros in uniformly convex Banach spaces. The paper precisely characterizes the degree of the law of excluded middle required for these results and demonstrates that replacing compactness with mere metric boundedness fails to preserve classical quantitative bounds. Thus, it affirms the pivotal role of the modulus of regularity in generating effective computational procedures across diverse mathematical structures.

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 incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We continue the investigation into the computational status of the existence of moduli of regularity (and their use for rates of convergence) in the sense of Kohlenbach, Lopez and Nicolae (2019), carried out w.r.t. classical reverse mathematics and Weihrauch degrees in a previous paper and determine the amount of LEM involved. We also show that the existence of a modulus of regularity always yields an algorithm for the computation of a zero in the case of continuous real-valued functions F on a compact metric space K (in F equipped with a modulus of uniform continuity and K given in standard representation) whenever such a zero exists. If K is a compact subset of a uniformly convex Banach space X and the zero set of F is convex one can compute even the zero of minimal norm. A modulus of regularity can also be used to compute the left-most infinite path of an infinite 0/1-tree. We also show that there is no proof-theoretically tame nonstandard uniformity principle which would make it possible to replace in the regularity assumption compactness by metric boundedness and still guarantee classically correct bounds.

Problem

• modulus of regularity
• computational content
• logical strength
• reverse mathematics
• Weihrauch degrees

Innovation

• modulus of regularity
• computational content
• Weihrauch degrees
• reverse mathematics
• algorithmic zero finding