On $NP$ versus ${\rm co}NP$ and Frege Systems

We prove in this paper that there is a language $Ld$ accepted by some nondeterministic Turing machines but not by any ${\rm co}\mathcal{NP}$-machines (defined later). We further show that $Ld$ is in $\mathcal{NP}$, thus proving that $\mathcal{NP}\neq{\rm co}\mathcal{NP}$. The techniques used in this paper are lazy-diagonalization and the novel new technique developed in author's recent work \cite{Lin21}. As a by-product, we reach the important result \cite{Lin21} that $\mathcal{P}\neq\mathcal{NP}$ once again, which is clear from the above outcome and the well-known fact that $\mathcal{P}={\rm co}\mathcal{P}$. Next, we show that the complexity class ${\rm co}\mathcal{NP}$ has intermediate languages, i.e., there are language $L_{inter}\in{\rm co}\mathcal{NP}$ which is not in $\mathcal{P}$ and not ${\rm co}\mathcal{NP}$-complete. We also summarize other direct consequences in the area of proof complexity implied by our main outcome. Lastly, we show a lower bounds result for Frege proof systems, i.e., no Frege proof systems can be polynomial bounded.