Quasipolynomial Normalisation in Deep Inference via Atomic Flows and Threshold Formulae

TL;DR

This paper presents a direct quasipolynomial-time cut-elimination method for classical propositional logic in deep inference, utilizing atomic flows to simplify and faithfully represent the process.

Contribution

It introduces a new direct proof of quasipolynomial cut-elimination in deep inference using atomic flows, improving upon previous indirect methods.

Findings

Quasipolynomial cut-elimination is achievable directly in deep inference.

Atomic flows effectively represent the cut-elimination process.

The method simplifies the proof of Jeřábek's result.

Abstract

Je\v{r}\'abek showed that cuts in classical propositional logic proofs in deep inference can be eliminated in quasipolynomial time. The proof is indirect and it relies on a result of Atserias, Galesi and Pudl\'ak about monotone sequent calculus and a correspondence between that system and cut-free deep-inference proofs. In this paper we give a direct proof of Je\v{r}\'abek's result: we give a quasipolynomial-time cut-elimination procedure for classical propositional logic in deep inference. The main new ingredient is the use of a computational trace of deep-inference proofs called atomic flows, which are both very simple (they only trace structural rules and forget logical rules) and strong enough to faithfully represent the cut-elimination procedure.