# Proof Compression and NP Versus PSPACE II: Addendum

In [3] we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC) [5] with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction (ND) [6]. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea (due to the second author) is to omit full minimal logic and compress only " normal tree-like ND refutations of the existence of Hamiltonian cycles in given non-Hamiltonian graphs, since the Hamiltonian graph problem in NP-complete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE [3]. [3] L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE II, Bulletin of the Section of Logic (49) (3): 213-230 (2020) http://dx.doi.org/10.18788/0138-0680.2020.16 [1907.03858] [5] J. Hudelmaier, An O (n log n)-space decision procedure for intuitionistic propositional logic, J. Logic Computat. (3): 1-13 (1993) [6] D. Prawitz, Natural deduction: a proof-theoretical study. Almqvist Wiksell, 1965

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