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Diagonalization of Polynomial-Time Deterministic Turing Machines Via Nondeterministic Turing Machine. (arXiv:2110.06211v20 [cs.CC] UPDATED)

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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 over 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 working within time
$O(n^k)$ for all $k\in\mathbb{N}_1$. By this, we further show that
$L_d\in\mathcal{NP}$ . That is, we present a proof that $\mathcal{P}$ and
$\mathcal{NP}$ differ.

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