We prove in this paper that there is a language $L_s$ accepted by some nondeterministic Turing machine that runs within time $O(n^k)$ for any positive integer $k\in\mathbb{N}_1$ but not by any ${\rm co}\mathcal{NP}$ machines. Then we further show that $L_s$ is in $\mathcal{NP}$, thus proving a groundbreaking result that $$\mathcal{NP}\neq{\rm co}\mathcal{NP}. $$ The main techniques used in this paper are simulation and the novel new techniques developed in the author's recent work. Our main result has profound implications, such as $\mathcal{P}\neq\mathcal{NP}$, etc. Further, if there exists some oracle $A$ such that $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$, we then explore what mystery lies behind it and show that if $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$ and under some rational assumptions, then the set of all ${\rm co}\mathcal{NP}^A$ machines is not enumerable, thus showing that the simulation techniques are not applicable for the first half of the whole step to separate $\mathcal{NP}^A$ from ${\rm co}\mathcal{NP}^A$. Finally, a lower bounds result for Frege proof systems is presented (i.e., no Frege proof systems can be polynomially bounded).

翻译：本文证明存在一种语言 $L_s$，该语言可由某个非确定性图灵机在 $O(n^k)$ 时间内接受，其中 $k$ 为任意正整数 $k\in\mathbb{N}_1$，但无法被任何 ${\rm co}\mathcal{NP}$ 机接受。我们进一步证明 $L_s$ 属于 $\mathcal{NP}$，从而得出一个突破性结果：$$\mathcal{NP}\neq{\rm co}\mathcal{NP}. $$ 本文采用的主要技术是模拟方法以及作者近期工作中提出的新颖技术。我们的主要结果具有深远的影响，例如可推导出 $\mathcal{P}\neq\mathcal{NP}$ 等结论。进一步地，若存在预言机 $A$ 使得 $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$，我们将探究其背后的奥秘，并证明在 $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$ 及某些合理假设下，所有 ${\rm co}\mathcal{NP}^A$ 机的集合是不可枚举的，从而说明模拟技术不适用于分离 $\mathcal{NP}^A$ 与 ${\rm co}\mathcal{NP}^A$ 整个步骤的前半部分。最后，本文给出了弗雷格证明系统的下界结果（即不存在多项式有界的弗雷格证明系统）。