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 \naive" 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

中文翻译：

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证明压缩和 NP 与 PSPACE II：附录

在 [3] 中，我们通过高级证明理论方法证明了猜想 NP = PSPACE，该方法将 Hudelmaier 的最小逻辑 (HSC) [5] 的无割连续演算与相应的最小 Prawitz 式自然推导 (ND) 中的水平压缩相结合 [ 6]。在本附录中，我们展示了如何在不参考 HSC 的情况下证明较弱的结果 NP = coNP。潜在的想法（由于第二作者）是省略完整的最小逻辑并仅压缩对给定非哈密顿图中哈密顿圈存在的\朴素的普通树状 ND 反驳，因为 NP 完全中的哈密顿图问题. 因此，松散地说，NP = coNP 的证明可以通过 HSC-elimination 从我们的 NP = PSPACE [3] 证明中获得。 [3] L. Gordeev, EH Haeusler, Proof Compression and NP Versus PSPACE II, Bulletin of逻辑部分（49）（3）：213-230 (2020) http://dx.doi.org/10.18788/0138-0680.2020.16 [1907.03858] [5] J. Hudelmaier，直觉命题逻辑的 O (n log n)-空间决策程序，J . 逻辑计算。(3): 1-13 (1993) [6] D. Prawitz, Natural deduction: a proof-theoretical study。阿尔姆奎斯特和维克塞尔，1965