Proof Compression via Subatomic Logic and Guarded Substitutions

2025-05-26

AI Summary

This work addresses the inefficiency of proof compression and the information loss or redundancy caused by traditional linearization in propositional classical logic. Methodologically, it introduces a novel proof representation framework based on subatomic logic and controlled explicit substitution, featuring a “guarded substitution” mechanism—substitution is applied only to protected occurrences of free variables—enabling superposition-state modeling of derivations and achieving p-simulation of substitution-based Frege systems without the cut rule. Key contributions include: (i) the first strictly linear, non-erasing, and non-redundant derivation system; (ii) elimination of unit introduction/elimination overhead; (iii) polynomial-length proof compression; and (iv) preservation of proof strength without relying on the cut rule. The framework establishes a new paradigm for proof complexity analysis and efficient proof verification.

Technology Category

Knowledge Representation and Reasoning: Logic ProgrammingConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSecurity and Privacy: Data transparency and provenanceSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Subatomic logic is a recent innovation in structural proof theory where atoms are no longer the smallest entity in a logical formula, but are instead treated as binary connectives. As a consequence, we can give a subatomic proof system for propositional classical logic such that all derivations are strictly linear: no inference step deletes or adds information, even units. In this paper, we introduce a powerful new proof compression mechanism that we call guarded substitutions, a variant of explicit substitutions, which substitute only guarded occurrences of a free variable, instead of all free occurrences. This allows us to construct ''superpositions'' of derivations, which simultaneously represent multiple subderivations. We show that a subatomic proof system with guarded substitution can p-simulate a Frege system with substitution, and moreover, the cut-rule is not required to do so.

Problem

• Develop subatomic logic for linear derivations in classical logic
• Introduce guarded substitutions to enable proof superposition
• Achieve p-simulation of Frege systems without cut-rule

Innovation

• Subatomic logic treats atoms as binary connectives
• Guarded substitutions enable superposition of derivations
• Subatomic system p-simulates Frege without cut-rule