On the relative proof complexity of deep inference via atomic flows

TL;DR

This paper investigates the proof complexity of deep inference systems for propositional logic using atomic flows, showing polynomial simulations and limitations of certain proof systems.

Contribution

It introduces atomic flows as a tool to analyze proof size and demonstrates polynomial simulations and non-simulation results for deep inference systems.

Findings

Polynomial simulation of Resolution variants.

KS cannot be polynomially simulated by bounded-depth Frege.

Polynomial-size proofs of propositional pigeonhole principle in KS.

Abstract

We consider the proof complexity of the minimal complete fragment, KS, of standard deep inference systems for propositional logic. To examine the size of proofs we employ atomic flows, diagrams that trace structural changes through a proof but ignore logical information. As results we obtain a polynomial simulation of versions of Resolution, along with some extensions. We also show that these systems, as well as bounded-depth Frege systems, cannot polynomially simulate KS, by giving polynomial-size proofs of certain variants of the propositional pigeonhole principle in KS.