# P≠ NP

The whole discussion is divided into two parts: one is for |Σ|≥ 2 (general case), and another is for |Σ|=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∈ NP-P, for any language L∈ P, the lower bound on reducibility from AL to L is Ω(m^n) where m≥ 2 is a constant, n=|ω| and ω∈Σ^* the input; (2) There exists no polynomial-time algorithm for SAT; (3) An immediate corollary of (1) and (2) is that P≠ NP, which also can be deduced from (6); (4) There exists a language coAL∈ coNP-coP, for any language L∈ coP, the lower bound on reducibility from coAL to L is Ω(m^n) where m≥ 2 is a constant, n=|ω| and ω∈Σ^* the input; (5) There exists no polynomial-time algorithm for TAUT; (6) An immediate corollary of (4) and (5) is that coP≠ coNP; We next study the problem in special case. It is shown that: (1) there exists k∈ℕ and a reducibility φ from an arbitrary language L_1∈ NP-P (resp. L_1∈ coNP-coP) to an another arbitrary language L_2∈ P (resp. L_2∈ coP) such that T_φ(n)≤ n^k+k where n=|ω| and ω∈Σ^n is the input; (2) an immediate corollary is that P=NP and coP=coNP in the special case.

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