Quantified propositional calculi and narrow implicit proofs

2026-06-09

AI Summary

This work investigates the effective compression of propositional proof length while preserving expressive power and inference efficiency, focusing on *narrow implicit proofs*—a class of implicit proof systems in which each line is restricted to polynomial size and the entire proof is succinctly encoded by a Boolean circuit. By employing cut-elimination constructions and formalizing the analysis through circuit encodings within quantified propositional proof systems \( G_i \), the paper establishes that for every \( i \geq 1 \), the quantified propositional system \( G_{i+1} \) is equivalent to narrow implicit \( G_i \), and further shows that \( G_1 \) is equivalent to implicit resolution. These results uncover a deep connection between the quantified propositional calculus hierarchy and implicit proof systems, providing a solid theoretical foundation for efficient proof representation.

Abstract

In the implicit version of a propositional proof system Q, we work with Q-proofs that are not written down directly, but are succinctly encoded by circuits. Thus implicit Q-proofs are potentially exponentially shorter than usual Q-proofs. We study narrow implicit proofs, a restricted version of this notion, in which lines in the encoded proof can only have polynomial size. We use a cut-elimination construction to show that G_{i+1} is equivalent to narrow implicit G_i, for i >= 1, where G_i is the extension of Frege allowing reasoning with Sigma^q_i quantified propositional formulas. We show that G_1 is equivalent to implicit resolution.

Problem

• quantified propositional calculus
• implicit proofs
• narrow proofs
• Frege systems
• proof complexity

Innovation

• narrow implicit proofs
• quantified propositional calculus
• cut-elimination
• implicit resolution
• proof complexity