Diagonalization of Polynomial-Time Deterministic Turing Machines Via Nondeterministic Turing Machine

The diagonalization technique was invented by Georg Cantor to show that there are more real numbers than algebraic numbers and is very important in {\em theoretical computer science}. In this work, we enumerate all polynomial-time deterministic Turing machines and diagonalize against all of them by a universal nondeterministic Turing machine. As a result, we obtain that there is a language $L_d$ not accepted by any polynomial-time deterministic Turing machines but accepted by a nondeterministic Turing machine running within time $O(n^k)$ for any $k\in\mathbb{N}_1$. By this, we further show that $L_d\in\mathcal{NP}$ . That is, we present a proof that $\mathcal{P}$ and $\mathcal{NP}$ differ. Meanwhile, we also show that there exists a language $L_s$ in $\mathcal{P}$ but the machine accepting it runs within time $O(n^k)$ for all $k\in\mathbb{N}_1$ and any deterministic $n^k+k$ time-bounded Turing machine for a fixed $k\in\mathbb{N}_1$ can not accept it. Lastly, we show that if $\mathcal{P}^O=\mathcal{NP}^O$ and on some rational assumptions then the set $P^O$ of all polynomial-time deterministic oracle Turing machines with $O$ is not enumerable.