A Solution of the P versus NP Problem based on specific property of clique function

Abstract

Circuit lower bounds are important since it is believed that a super-polynomial circuit lower bound for a problem in NP implies that P!=NP. Razborov has proved superpolynomial lower bounds for monotone circuits by using method of approximation. By extending this approach, researchers have proved exponential lower bounds for the monotone network complexity of several different functions. But until now, no one could prove a non-linear lower bound for the non-monotone complexity of any Boolean function in NP. While we show that in this paper by replacement of each Not gates into constant 1 equivalently in standard circuit for clique problem, it can be proved that non-monotone network has the same or higher lower bound compared to the monotone one for computing the clique function. This indicates that the non-monotone network complexity of the clique function is super-polynomial which implies that P!=NP.