P versus NP
Résumé
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 related function problem in $FNP$ for the same input, even though it has other several solutions. This event is not necessarily true when the solution does not exist for the inputs in these function problems in $FEXP-complete$. We also show there is no possible reduction between these problems, because there is not any computable function in logarithmic space that matches when overlap the solutions for the same input. Indeed, we are not trying to prove that $EXP$ is in $NP$, because this would lead us to a relativizing proof. In this way, if $FP = FNP$, then we might have the chance of resolve the solutions of the inputs in some of these function problems in $FEXP-complete$ by a polynomial time algorithm and this would only happen when the solutions exist, but this is not possible by the time hierarchy theorem. Therefore, by reductio ad absurdum, $P \neq NP$.
Domaines
Complexité [cs.CC]
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