Computer-Simulation Model Theory (P= NP is not provable) (1906.09873v1)

Abstract: The simulation hypothesis says that all the materials and events in the reality (including the universe, our body, our thinking, walking and etc) are computations, and the reality is a computer simulation program like a video game. All works we do (talking, reasoning, seeing and etc) are computations performed by the universe-computer which runs the simulation program. Inspired by the view of the simulation hypothesis (but independent of this hypothesis), we propose a new method of logical reasoning named "Computer-Simulation Model Theory", CSMT. Computer-Simulation Model Theory is an extension of Mathematical Model Theory where instead of mathematical-structures, computer-simulations are replaced, and the activity of reasoning and computing of the reasoner is also simulated in the model. (CSMT) argues that: For a formula $\phi$, construct a computer simulation model $S$, such that 1- $\phi$ does not hold in $S$, and 2- the reasoner $I$ $($human being, the one who lives inside the reality$)$ cannot distinguish $S$ from the reality $(R)$, then $I$ cannot prove $\phi$ in reality. Although $\mathrm{CSMT}$ is inspired by the simulation hypothesis, but this reasoning method is independent of the acceptance of this hypothesis. As we argue in this part, one may do not accept the simulation hypothesis, but knows $\mathrm{CSMT}$ a valid reasoning method. As an application of Computer-Simulation Model Theory, we study the famous problem P vs NP. We let $\phi \equiv\mathrm{ [P= NP]} $ and construct a computer simulation model $E$ such that $\mathrm{P= NP}$ does not hold in $E$.