For a while now, I have long since wondered if it were possible to prove once and for all if we are living in a vast simulation.
On the surface it seems impossible to prove whether we are inside some kind of matrix-type simulation, however I believe that I have come up with a hypothesis that may prove once and for all if we are or not . .
Before we come to that, let us examine the components that lead up to the proposed theory.
P Versus NP
For several decades there has been a raging debate in mathematics and computer science whether P = NP or not.
P stands for polynomial, which refers to the amount of time it takes to solve a problem, and NP stands for nondeterministic polynomial time.
A classic example of an NP problem is the travelling salesman problem (TSP).
The Travelling Salesman Problem describes a salesman who must travel between N cities. The order in which he does so is something he does not care about, as long as he visits each once during his trip, and finishes where he was at first.
Each city is connected to other close by cities, or nodes, by air, road or rail. Each of those links between the cities has one or more cost attached to it.
So the problem is what is the most optimum route for the salesman to take; whereby he travels for the least amount of time, for the smallest amount of money.
In mathematics we use algorithms as a means of a shortcut to finding answers.
If we were to set the most powerful computer on earth the task of coming up with an algorithm that could be applied to any group of cities in any country, and still come up with the correct answer, it would take that machine several times the age of the universe to work it out.
In other words the problem can not be solved in polynomial time, therefore it is called an NP problem.
It is worth mentioning that most mathematicians working in the field believe that P = NP, in other words there is no such thing as NP problems, and it is just a matter of discovery.
A Case For NP
The French scientist Pierre Laplace opined that if we were to know every position and velocity of every particle in the universe, along with the complete laws of physics, we would be able to predict the future.
Whilst that statement might have been true in a universe governed by Newtonian physics, alas the same cannot be said for one where quantum mechanics is apparent.
Quantum uncertainty states that we can never know the position and velocity of a particle before we actually measure it. It goes on to state that the measurement of one state, destroys the other.
So for example if we observe a particle moving through a vacuum, by measuring its speed we affect its location, and vice versa.
So in a rather oblique way, the uncertainty principle can be said to invoke NP mathematics, as there is no way for a computer to come up with an algorithm that can determine at once speed and velocity of a given particle.
Various experiments we have done with entangled photons have proved once and for all that the quantum uncertainty principle is real, and not just a matter of us getting better observational equipment.
So perhaps then this is the biggest clue that NP is real and in fact P is not equal to NP (P ≠ NP)
A Case For P
In 1972 mathematician Richard Karp solved 21 NP problems, and thus relegated them from NP to NP Hard, which meant that an answer to these set of problems, could be checked in P time.
This offers the question then; if NP can be reduced to NP-Hard, then surely NP-Hard can be one day boiled down to P? In other words, P = NP.
If that is the case, then all NP problems will one day be reduced to P and as far as mathematics is concerned, there will be no mysteries left in the universe.
P and NP As A Function In Simulation Theory
So all that being said; let us now look at how P = NP or indeed P ≠ NP has any bearing on whether we're living in a simulation or not.
My hypothesis states that if P does indeed equal NP, then we cannot rule out the assertion that we may indeed be living in a simulation.
However if P ≠ NP, and NP problems truly exist regardless of what computational heights a computer may achieve, then the chances of us living in a sim are vanishingly small.
The reason being that it is difficult to conceive a computer, alien or otherwise, creating a universe in which there are problems which it itself cannot solve.
This would be the equivalent of a locksmith creating a lock that he couldn't make a key for and open.
So, let's hear it for P ≠ NP, because in my humble opinion, that would prove once and for all that we are not living in a simulation.
Whether we'll prove it is another question, and in a wonderful logical loop, the proving of such, may turn out to be NP in itself!
WHAT DO YOU THINK OF MY LOGICAL REASONING? EVEN IF I AM CORRECT, WILL WE EVER BE ABLE TO PROVE THAT P ≠ NP, OR ARE WE DOOMED TO SEARCH FOR THE ANSWER FOR ALL OF HUMAN HISTORY? AS EVER, LET ME KNOW BELOW!
Extra Reading
Karp's 21 NP-complete problems
NP-completeness
MIT News P = NP?
P versus NP problem
Travelling salesman problem