$P\neq NP$

The whole discussion is divided into two parts: one is for $|\Sigma|\geq 2$ (general case), and another is for $|\Sigma|=1$ (special case). The main contribution of the paper is that a series of results are obtained. Specifically, we prove in general case that : (1) There exists a language $AL\in NP-P$, for any language $L\in P$, the lower bound on reducibility from $AL$ to $L$ is $\Omega(m^n)$ where $m\geq 2$ 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); (4) There exists a language $coAL\in coNP-coP$, for any language $L\in coP$, the lower bound on reducibility from $coAL$ to $L$ is $\Omega(m^n)$ where $m\geq 2$ is a constant, $n=|\omega|$ 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) there exists $k\in\mathbb{N}$ and a reducibility $\varphi$ from an arbitrary language $L_1\in NP-P$ (resp.~$L_1\in coNP-coP$) to an another arbitrary language $L_2\in P$ (resp.~$L_2\in coP$) such that $T_{\varphi}(n)\leq n^k+k$ where $n=|\omega|$ and $\omega\in\Sigma^n$ is the input; (2) an immediate corollary is that $P=NP$ and $coP=coNP$ in the special case.