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$P\neq NP$
arXiv - CS - Computational Complexity Pub Date : 2020-03-22 , DOI: arxiv-2003.09791
Tianrong Lin

The whole discussions are divided into two parts, one is for $|\Sigma|\geq 2$ (general case), another is for $|\Sigma|=1$ (special case). The main contribution of the present paper is that a series of results are obtained. Specifically, we prove in general case that: (1) Let $L_1\in NP-P$ and $L_2\in P$, then the complexity of problem on reducibility from $L_1$ to $L_2$ is $\Omega(m^{n^k})$, $m\geq 2$ is a constant, $n=|\omega|$ and $\omega\in\Sigma^*$ the input; (2) There exists no polynomial-time algorithm for {\it SAT}; (3) An immediate corollary of (1) and (2) is that $P\neq NP$, which also can be deduced from (6), see Remark \ref{rk_7.1}; (4) Let $L_1\in coNP-coP$ and $L_2\in coP$, then the complexity of problem on reducibility from $L_1$ to $L_2$ is $\Omega(m^{n^k})$, $m\geq 2$ is a constant, where $n=|\omega|$ and $\omega\in\Sigma^*$ the input; (5) There exists no polynomial-time algorithm for {\it TAUT}; (6) An immediate corollary of (4) and (5) is that $coP\neq coNP$; We next study the problem in special case. It is shown that: (1) the complexity of problem of reducibility from $L_1\in NP-P$ (resp.~ $L_1\in coNP-coP$) to $L_2\in P$ (resp.~$L_2\in coP$) is less than $n^k+k$ where $k\in\mathbb{N}$ and $n=|\omega|$ is the size of input $\omega$; (2) an immediate corollary is that $P=NP$ and $coP=coNP$ in the special case. However, the title of the paper will only reflect the general case.
更新日期:2020-04-23

 

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