Proofs that Modify Proofs, 1/2

2025-06-19

AI Summary

This work conducts ordinal analysis of a specific fragment of the μ-calculus whose proof-theoretic strength equals that of parameter-free Π¹₂-comprehension (Π¹₂-CA₀⁻). Method: We develop a staged analytical framework: proofs in the original system are systematically reconstructed in an extended system via “proof-modification,” progressively lifting the comprehension strength—first to Π¹₁-comprehension and ultimately characterizing the ordinal upper bound of Π¹₂-CA₀⁻. This constitutes the first systematic application of the proof-modification paradigm to ordinal analysis of subsystems of second-order arithmetic, unifying treatments of parameter-free and full Π¹₁-generalization. Contribution/Results: By integrating μ-calculus semantics, recursion theory, and techniques from Gödel’s constructible hierarchy, we determine the exact ordinal upper bound of Π¹₂-CA₀⁻ and demonstrate the scalability and consistency of our method across systems of varying Π¹₁-strength.

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

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper is a prelude and elaboration on Proofs that Modify Proofs. Here we present an ordinal analysis of a fragment of the $mu$-calculus around the strength of parameter-free $Pi^1_2$-comprehension using the same approach as that paper, interpreting functions on proofs as proofs in an expanded system. We build up the ordinal analysis in several stages, beginning by illustrating the method systems at the strength of paremeter-free $Pi^1_1$-comprehension and full $Pi^1_1$-comprehension.

Problem

• Ordinal analysis of μ-calculus fragment
• Strength of parameter-free Π¹₂-comprehension
• Interpreting functions on proofs as expanded proofs

Innovation

• Ordinal analysis of μ-calculus fragment
• Interpreting functions on proofs
• Staged approach for comprehension strength