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 $L_s$，它可由某个在 $O(n^k)$ 时间内运行的非确定性图灵机接受，其中 $k$ 为任意正整数 $k\in\mathbb{N}_1$，但该语言不能被任何 ${\rm co}\mathcal{NP}$ 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以及作者近期工作中提出的新颖方法。我们的主要结果具有深远影响，例如可推导出 $\mathcal{P}\neq\mathcal{NP}$ 等结论。此外，若存在某个谕示 $A$ 使得 $\mathcal{P}^A\ne\mathcal{NP}^A={\rm co}\mathcal{NP}^A$，我们探究其背后隐含的奥秘，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，从而表明模拟技术不适用于分离 $\mathcal{NP}^A$ 与 ${\rm co}\mathcal{NP}^A$ 全过程的前半阶段。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）。