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：本文证明存在一个语言 $L_s$，它可被某个在时间 $O(n^k)$ 内运行的非确定性图灵机接受（其中 $k\in\mathbb{N}_1$ 为任意正整数），但无法被任何 ${\rm co}\mathcal{NP}$ 机器接受。进一步地，我们证明 $L_s \in \mathcal{NP}$，从而得出突破性结论：$$\mathcal{NP}\neq{\rm co}\mathcal{NP}. $$ 本文采用的主要技术包括模拟方法以及作者近期工作中提出的新型技术。我们的主要结果具有深远影响，例如 $\mathcal{P}\neq\mathcal{NP}$ 等。此外，若存在预言机 $A$ 使得 $\mathcal{P}^A \ne \mathcal{NP}^A = {\rm co}\mathcal{NP}^A$，我们将探究其背后奥秘，并证明：若 $\mathcal{P}^A \ne \mathcal{NP}^A = {\rm co}\mathcal{NP}^A$ 且满足若干合理假设，则所有 ${\rm co}\mathcal{NP}^A$ 机器构成的集合不可枚举，从而表明在分离 $\mathcal{NP}^A$ 与 ${\rm co}\mathcal{NP}^A$ 的完整步骤中，模拟技术不适用于前半部分。最后，本文给出 Frege 证明系统的下界结果（即不存在多项式有界的 Frege 证明系统）。