Sunday, April 16, 2017
It has been obvious for some time, in the field of logic, that there is an analogy with electro magnetism with the battery - the battery consisting of an ampere driven through a volt against a resistance of an ohm. In the area of logic, the equivalence are a Gödel of propositions, through a Turing busy beaver, against a Nash of resistance. However certain details have not been blocked through in the area of P != NP.
The phrasing of the prize for this is wrong, because the inequality means that one side will demand that the other side solves problems which do not need to be solved. Fortunately there is a solution in Paul J. Cohen, and his breakdown of forcing.
Let me take an example, which is well known to anyone who studies the problem - it is even featured on the Wikipedia page. I speak of the subset sum problem. As phrased, the problem can only be P != NP, because of all of the calculations that the prime hunter has to do. Their is no prize, because it is obvious to anyone who looks, and summary answers can be discarded.
But if one does not take a greedy algorithm for it, one can easily see that Cohen's forcing example provides a simple answer, though rather hard to compute. 1st we must answer the question “What is the smallest answer that can be given by the sum to the alternate party?” One can see obviously that this is a classic Nash set up, with the equality answer to be the question. The answer to this is ”yes/no” only in terms that the sum must answer whether or not he has found an answer - not which answers, because he does not need to answer this - his counterpart does.
With this answer given, the next question is how to do this on an arbitrary computer? 1 would think that it would be by addition, but with a computer this is incorrect. Because before condition can be done, the computer must copy and then check the numbers - so that condition can be supplied. The computer must check the validity of the answers before actually computing them. Thus we may take “validity” as a test. And if we have a test, then by recursion 1 can test the number - and assume that the tested number is undecidable. We then recurse upwards to the actual question and by sorting the numbers on a binary search tree, have an answer. It is the checker that needs to fill in all of the details at that point.
Fortunately, on page 117 of “Set Theory and the Continuum Hypothesis” The 1st finite forcing will do the trick, because it only asks if there is the existence of the set.
This can be done for any problem, because Cohen proved that while it could not be done everywhere it can be done anywhere.
There is also the problem in Cohen of the 3 different approaches - but this too is a Gödelian proof. The 3 approaches are idealist, realist, and does not apply. If you then think of this in terms of P != NP, one will see that the idealist point of view is a flat and infinitely small theory of points, similar to Euclid's geometry - or any that might be of interest to a idealist mathematician. The realist point of view is a physical point of view, though the physics may not exist in our universe. This does not compare point of view dismisses the entire project - which is to say a real computer is like this, even if it pretends not to be.
With the substitution of a parsimonious algorithm for a greedy algorithm, and the substitution of a quality for inequality, certain answers open themselves up to examination - though every individual case must be forced through Cohen.
Stirling S. Newberry