Diagonalization of PolynomialTime Turing Machines Via Nondeterministic Turing Machine
Abstract
The diagonalization technique was invented by Cantor to show that there are more real numbers than algebraic numbers, and is very important in computer science. In this work, we enumerate all polynomialtime deterministic Turing machines and diagonalize over all of them by an universal nondeterministic Turing machine. As a result, we obtain that there is a language $L_d$ not accepted by any polynomialtime deterministic Turing machines but accepted by a nondeterministic Turing machine working within $O(n^k)$ for any $k\in\mathbb{N}_1$, i.e. $L_d\in \mathcal{NP}$ . That is, we present a proof that $\mathcal{P}$ and $\mathcal{NP}$ differ. Key words: Diagonalization, PolynomialTime deterministic Turing machine, Universal nondeterministic Turing machine
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06211
 Bibcode:
 2021arXiv211006211L
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory;
 68Q15;
 68Q17;
 F.1.3
 EPrint:
 V10 further made some proofs more accurate, feedbacks are welcome. arXiv admin note: text overlap with arXiv:2110.05942, arXiv:2106.11886