Simulating Polynomial-Time Nondeterministic Turing Machines via Nondeterministic Turing Machines
For complexity theory, this claims to resolve major open problems (P vs NP, NP vs coNP), but the result is almost certainly incorrect given the long-standing nature of these problems and lack of peer review.
The paper claims to prove NP ≠ coNP by constructing a language accepted by a nondeterministic Turing machine in O(n^k) time but not by any coNP machine, with implications including P ≠ NP. It also discusses oracle separations and lower bounds for Frege proof systems.
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).