The relationship between the complexity classes P and NP is an unsolved question in the field of theoretical computer science. In this paper, we look at the link between the P - NP question and the "Deterministic" versus "Non Deterministic" nature of a problem, and more specifically at the temporal nature of the complexity within the NP class of problems. Let us remind that the NP class is called the class of "Non Deterministic Polynomial" languages. Using the meta argument that results in Mathematics should be "time independent" as they are reproducible, the paper shows that the P!=NP assertion is impossible to prove in the a-temporal framework of Mathematics. A similar argument based on randomness shows that the P=NP assertion is also impossible to prove, so that the P - NP problem turns out to be "unprovable" in Mathematics. This is not an undecidability theorem, as undecidability points to the paradoxical nature of a proposition. In fact, this paper highlights the time dependence of the complexity for any NP problem, linked to some pseudo-randomness in its heart.
|Published - 4 Apr 2009