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arXiv · 2505.20009

Proof Compression via Subatomic Logic and Guarded Substitutions

Victoria Barrett·Alessio Guglielmi·Benjamin Ralph·Lutz Straßburger
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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.

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Victoria Barrett, Alessio Guglielmi, Benjamin Ralph, Lutz Straßburger. 2025-05-26. Proof Compression via Subatomic Logic and Guarded Substitutions. https://arxiv.org/abs/2505.20009

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