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What is Simulation of Polynomial-Time Nondeterministic Turing Machines via Universal Nondeterministic Turing Machines about?

We prove in this paper that there is a language $L_s$ accepted by some nondeterministic Turing machines but not by any ${\rm co}\mathcal{NP}$ machines. Then we further show that $L_s$ is in $\mathcal{NP}$, thus proving 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. 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 ${\rm co}\mathcal{NP}^A$. Our main result has profound implications, such as $\mathcal{P}\neq\mathcal{NP}$ and a lower bounds result for Frege proof systems (i.e., no Frege proof systems can be polynomially bounded), etc.

Author

Lin, Tianrong

Published

2024

Language

EN