Admissibility of Substitution Rule in Cyclic-Proof Systems

2025-10-16

AI Summary

This work addresses the admissibility of the substitution rule in cyclic proof systems. While substitution enhances expressiveness, it substantially complicates theoretical analysis and proof search. For the cyclic proof system CLKID^ω—supporting first-order logic and inductive predicates—we establish, for the first time, the admissibility of substitution in the presence of the cut rule. We then devise a global construction method based on infinite unfolding, substitution lifting, and back-edge reconstruction, overcoming the limitation of conventional local transformations that destroy cyclic structure. Our approach imposes no restrictions on function symbols, applies to both cut-containing and cut-free variants of CLKID^ω, and extends to broader systems including separation logic. The results uniformly establish substitution admissibility across multiple cyclic proof frameworks, thereby significantly reducing theoretical analysis overhead and automating proof search complexity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

This paper investigates the admissibility of the substitution rule in cyclic-proof systems. The substitution rule complicates theoretical case analysis and increases computational cost in proof search since every sequent can be a conclusion of an instance of the substitution rule; hence, admissibility is desirable on both fronts. While admissibility is often shown by local proof transformations in non-cyclic systems, such transformations may disrupt cyclic structure and do not readily apply. Prior remarks suggested that the substitution rule is likely nonadmissible in the cyclic-proof system CLKID^omega for first-order logic with inductive predicates. In this paper, we prove admissibility in CLKID^omega, assuming the presence of the cut rule. Our approach unfolds a cyclic proof into an infinitary form, lifts the substitution rules, and places back edges to construct a cyclic proof without the substitution rule. If we restrict substitutions to exclude function symbols, the result extends to a broader class of systems, including cut-free CLKID^omega and cyclic-proof systems for the separation logic.

Problem

• Investigates substitution rule admissibility in cyclic-proof systems
• Proves admissibility in CLKID^omega with cut rule assumption
• Extends results to cut-free systems with restricted substitutions

Innovation

• Unfolds cyclic proofs into infinitary form
• Lifts substitution rules during transformation
• Reconstructs cyclic structure without substitution rule