Yet another argument in favour of NP=CoNP

Abstract

This article shows yet another proof of NP = CoNP. In a previous article we proved that NP = PSPACE and from it we can conclude that NP = CoNP immediatly. The former proof shows how to obtain polynomial and, polynomial in time ckeckable Dag-like proofs for all purely implicational Minimal logic tautologies. From the fact that Minimal implicational logic is PSPACE-complete we get the proof that NP = PSPACE. This first proof of NP = CoNP uses Hudelmaier linear upper-bound on the height of Sequente Calculus minimal implicational logic proofs. In an addendum to the proof of NP = PSPACE we observe that we do not need to use Hudelmaier upper-bound, since any proof of non-hamiltonicity for any graph is linear upper-bounded. By the CoNP-completeness of nonhamiltonicity we obtain NP = CoNP as a corollary of the first proof. In this article we show a third proof of CoNP = NP, also providing polynomial size and polynomial verifiable certificates that are Dags. They are generated from normal Natural Deduction proofs, linear height upper-bounded too, by removing redundancy, i.e., repeated parts. The existence of repeated parts is consequence of the redundancy theorem for family of superpolynomial proofs in the purely implicational Minimal logic. Its mandatory to read at least two previous articles to get the details of the proof presented here. The article that proves the redundabcy theorem and the article that shows how to remove the repeated parts of a normal Natural Deduction proof to have a polynomial Dag certificate for minimal implicational logic tautologies.