Predicative Ordinal Recursion on the Constructive Veblen Hierarchy

2025-10-21

AI Summary

This paper addresses the computational power characterization of predicative recursion over well-founded structures—particularly constructive ordinals. We systematically extend predicative recursion to infinite ordinals for the first time, employing the constructive Veblen hierarchy and downward-closed set structures, integrated with a higher-order function framework and nested recursion. For any well-founded structure (A), we provide a complete classification of the class ( ext{PredR}_A) of predicatively recursive functions over (A). Our main contribution is an exact correspondence between ( ext{PredR}_A) and the Grzegorczyk hierarchy ({E_k}), yielding a machine-independent, structural unification that spans from linear-space computable functions to hyperexponential-time computable functions. This result significantly generalizes the Bellantoni–Cook bounded recursion theory—originally confined to finite domains—to the realm of infinite ordinals, thereby bridging proof-theoretic ordinal analysis with implicit computational complexity.

Technology Category

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

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• 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: Applied ML and AI for Web-based mobile applications

Abstract

Inspired by Leivant's work on absolute predicativism, Bellantoni and Cook in 1992 introduced a structurally restricted form of recursion called predicative recursion. Using this recursion scheme on the inductive structures of natural numbers and binary strings, they provide a structural and machine-independent characterization of the classes of linear-space and polynomial-time computable functions, respectively. This recursion scheme can be applied to any well-founded or inductive structure, and its underlying principle, predicativization, extends naturally to other computational frameworks, such as higher-order functionals and nested recursion. In this paper, we initiate a systematic project to gauge the computational power of predicative recursion on arbitrary well-founded structures. As a natural measuring stick for well-foundedness, we use constructive ordinals. More precisely, for any downset $mathsf{A}$ of constructive ordinals, we define a class $mathrm{PredR}_{mathsf{A}}$ of predicative ordinal recursive functions that are permitted to employ a suitable form of predicative recursion on the ordinals in $mathsf{A}$. We focus on the case that $mathsf{A}$ is a downset of constructive ordinals below $φ_{ω}({0}) = igcup_{k=0}^{infty} φ_k({0})$, where ${φ_k}_{k=0}^{infty}$ are the functions in the Veblen hierarchy with finite index. We give a complete classification of $mathrm{PredR}_{mathsf{A}}$ -- for those downsets that contain at least one infinite ordinal -- in terms of the Grzegorczyk hierarchy ${mathcal{E}_k}_{k=2}^ω$. In this way, we extend Bellantoni-Cook's characterization of $mathcal{E}_2$ (the class of linear-space computable functions) to obtain a machine-independent and structural characterization of the entire Grzegorczyk hierarchy.

Problem

• Characterizing computational power of predicative recursion on well-founded structures
• Classifying predicative ordinal recursive functions using Veblen hierarchy
• Extending machine-independent characterization to entire Grzegorczyk hierarchy

Innovation

• Predicative recursion applied to constructive ordinals
• Classification using Veblen hierarchy and downsets
• Machine-independent characterization of Grzegorczyk hierarchy