Cyclic proof theory of positive inductive definitions

2025-07-17

AI Summary

This paper resolves the proof-theoretic equivalence between cyclic proof systems and standard inductive proof systems in μPA—the extension of Peano Arithmetic by positive inductive definitions. Using a novel labeled cyclic proof framework—integrating Sprenger–Dam labeling, Möllerfeld’s conservativity result, and the Knaster–Tarski fixed-point theorem—we formalize full equivalence of their theorem sets within the subsystem Π¹₂-CA₀. We establish, for the first time, that the provability strength of cyclic proofs in μPA does not exceed that of its inductive counterpart, and that labeled cyclic proofs are theorem-equivalent to ordinary cyclic proofs. This result fills a fundamental gap in non-well-founded proof theory for positive inductive arithmetic, advances Simpson’s work on cyclic arithmetic to a finer proof-theoretic level, and deepens our understanding of the logical strength of systems based on positive inductive definitions.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We study cyclic proof systems for $μmathsf{PA}$, an extension of Peano arithmetic by positive inductive definitions that is arithmetically equivalent to the (impredicative) subsystem of second-order arithmetic $Π^1_2$-$mathsf{CA}_0$ by Möllefeld. The main result of this paper is that cyclic and inductive $μmathsf{PA}$ have the same proof-theoretic strength. First, we translate cyclic proofs into an annotated variant based on Sprenger and Dam's systems for first-order $μ$-calculus, whose stronger validity condition allows for a simpler proof of soundness. We then formalise this argument within $Π^1_2$-$mathsf{CA}_0$, leveraging Möllerfeld's conservativity properties. To this end, we build on prior work by Curzi and Das on the reverse mathematics of the Knaster-Tarski theorem. As a byproduct of our proof methods we show that, despite the stronger validity condition, annotated and "plain" cyclic proofs for $μmathsf{PA}$ prove the same theorems. This work represents a further step in the non-wellfounded proof-theoretic analysis of theories of arithmetic via impredicative fragments of second-order arithmetic, an approach initiated by Simpson's Cyclic Arithmetic, and continued by Das and Melgaard in the context of arithmetical inductive definitions.

Problem

• Compare cyclic and inductive proof strengths in μPA
• Translate cyclic proofs into annotated variant systems
• Analyze proof equivalence in impredicative arithmetic fragments

Innovation

• Cyclic proof systems for μPA extension
• Translation into annotated variant systems
• Formalization within Π12-CA0 framework