P versus NP

Frank Vega (1)

1 Datys

There are some function problems in $FEXP$ that are also in $EXP-complete$, which have a corresponding function problem in $FNP$, such that each of these function problems in $FEXP$ 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 $FEXP$. We discuss there is no possible reduction between these function problems, because there is not any computable function in logarithmic space that matches when the solutions overlap 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 could have the chance of resolve the solutions of the inputs in any of these function problems in $FEXP$ which belongs to $EXP-complete$ by a polynomial time algorithm and this might happen when the solutions exist, but this is not possible by the time hierarchy theorem. Therefore by reductio ad absurdum, $P \neq NP$.

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

Domaines: Complexité [cs.CC]

Frank Vega. P versus NP. 2014. ⟨hal-00984866v4⟩