Sequentialization for full N-Graphs via sub-N-Graphs

TL;DR

This paper extends the sequentialization theorem to full propositional classical N-Graphs, introducing a method to find split nodes even with global discharging rules, advancing proof theory for complex logical systems.

Contribution

It provides a sequentization method for full propositional classical N-Graphs, including the handling of global discharging rules, which was not previously addressed.

Findings

Established a sequentization procedure for full propositional classical N-Graphs.

Demonstrated how to identify split nodes in proofs with global discharging.

Extended the sequentialization theorem to more complex N-Graph systems.

Abstract

Since proof-nets for MLL- were introduced by Girard (1987), several studies have appeared dealing with its soundness proof. Bellin & Van de Wiele (1995) produced an elegant proof based on properties of subnets (empires and kingdoms) and Robinson (2003) proposed a straightforward generalization of this presentation for proof-nets from sequent calculus for classical logic. In 2014 it was presented an extension of these studies to obtain a proof of the sequentialization theorem for the fragment of N-Graphs with conjunction, disjunction and negation connectives, via the notion of sub-N-Graphs. N-Graphs is a symmetric natural deduction calculus with multiple conclusions that adopts Danos-Regnier's criterion and has defocussing switchable links. In this paper, we present a sequentization for full propositional classical N-Graphs, showing how to find a split node in the middle of the proof even with a global rule for discharging hypothesis.