P versus NP

Frank Vega (1)

1 Datys

There are some function problems in $FEXP-complete$, which has a corresponding function problem in $FNP$, such that each of these function problems in $FEXP-complete$ could be solved by some solution that has the corresponding function problem in $FNP$ for the same input, when the solution for this input in the $FEXP-complete$ function problem exists. This event is not necessarily true when the solution does not exist for the inputs in these function problems in $FEXP-complete$. In the first case, when the solution exists in some input of one of these function problem in $FEXP-complete$, the related function problem in $FNP$ could have several solutions for the same input and only one of them coincide with the solution of the $FEXP-complete$ function problem. In this way, if $FP = FNP$, then we might have the possibility of resolve the solutions of the inputs in some of these function problems in $FEXP-complete$ by a polynomial time algorithm and this would happen only when the solutions exist, but this is not possible by the time hierarchy theorem. Therefore, $P \neq NP$.

Mots clés: Complexity classes; Turing Machines; NP; EXP

Domaines: Complexité [cs.CC]

Frank Vega. P versus NP. 2014. ⟨hal-00984866v2⟩