Solved: Homer Simpson's Mysterious Equation: P = NP.

Today I solved the Mysterious Homer Simpson Equation.

The Homer Simpson mystery is described in an article by Larry Hardesty at physorg.com:

“In the 1995 Halloween episode of The Simpsons, Homer Simpson finds a portal to the mysterious Third Dimension behind a bookcase, and desperate to escape his in-laws, he plunges through. He finds himself wandering across a dark surface etched with green gridlines and strewn with geometric shapes, above which hover strange equations. One of these is the deceptively simple assertion that P = NP. [emphasis original]

“In fact, in a 2002 poll, 61 mathematicians and computer scientists said that they thought P probably didn’t equal NP, to only nine who thought it did — and of those nine, several told the pollster that they took the position just to be contrary. But so far, no one’s been able to decisively answer the question one way or the other. Frequently called the most important outstanding question in theoretical computer science, the equivalency of P and NP is one of the seven problems that the Clay Mathematics Institute will give you a million dollars for proving — or disproving.”

The article pursues a proposed computing aspect of the equation, and says,

“A mathematical expression that involves N’s and N²s and N’s raised to other powers is called a polynomial, and that’s what the “P” in “P = NP” stands for. P is the set of problems whose solution times are proportional to polynomials involving N's.”

But the proposed solution is actually saying that N is fully contained within P, i.e. a subset of P (wholly owned subsidiary?). That makes the solution trivial on its face, and tautological on inspection. Because if the intersection of N and P is less than P, it cannot equal P; the equation is only valid if N = P. This makes the equation an identity or definition of N = P, merely a tautology without overall meaning. So the solution proposed by the article is less than philosophically robust, and without any dramatic meaning that can be derived from it.

The real answer to Homer’s Mystery Equation comes directly from George Boole (note 1), who mathematically codified human reason in his Boolean algebra. Boole gave us the following definition (note 2):

x² = x

This is the mathematical definition of a universal set. Here’s why:

Factoring, x ( 1 - x) = 0 ;

If x is the set of “x things”, then 1-x is the set of ALL “non-x things”; the union of these sets is the universal set.

So for P=NP, the universal set occurs when N=P: P=PP, or P=P².

Now the relationship N=P has meaning, because as Boole shows, this equation has only two solutions, 1 and 0, and the impact of this is that all three of the First Principles of Rational thought are confirmed mathematically. Tautology / identity has been produced; Non-contradiction is shown (either one or zero but not both); and Excluded Middle is shown (there is no intermediated solution between 0 and 1).

Thus Homer’s Mystery Equation, P=NP, is the universal axiom equation for rational thought and logic, when P=N, and when P does not equal N, the equation is either trivial (N < P) or nonviable (N > P).

What did you do with your day today?


Note 1: A History of Mathematics, 1968; Carl B. Boyer; pages 577 - 581.

Note 2: An Investigation of The Laws of Thought, 1853; George Boole; Ch II, page 22.