```Date: Nov 10, 2011 11:29 AM
Author: Craig Feinstein
Subject: Re: an interesting observation about Natural Proofs

On Nov 9, 11:58 am, Craig Feinstein <cafei...@msn.com> wrote:> Unless it is an axiom that P!=NP, any proof of P!=NP has to be> essentially be of the form:>> f solves SAT -> f in C> f in C -> f cannot be computed in polynomial time.> Therefore, f solves SAT -> f cannot be computed in polynomial time.>> The Razborov-Rudich paper on "Natural Proofs" essentially shows that> if (1) C is a property that is common (Largeness) and (2) it is easy> to check whether a function f has property C (Constructivity or Low> Complexity), then the above proof cannot be valid unless "pigs can> fly".>> Suppose C is an uncommon property, so (1) is broken. In the most> extreme case, C=the class of functions which solve SAT. Then the above> proof would be of the form:>> f solves SAT -> f solves SAT> f solves SAT -> f cannot be computed in polynomial time.> Therefore, f solves SAT -> f cannot be computed in polynomial time.>> So this proof would be of the form "P!=NP because P!=NP".>> Now, suppose it is difficult to check whether a function f has> property C, so (2) is broken. In the most extreme case, C = the class> of functions f that cannot be computed in polynomial-time. Then the> above proof would be of the form:>> f solves SAT -> f cannot be computed in polynomial time> f cannot be computed in polynomial time -> f cannot be computed in> polynomial time.> Therefore, f solves SAT -> f cannot be computed in polynomial time.>> So this proof would be of the form "P!=NP because P!=NP".>> So we see that breaking one of the two conditions for a natural proof,> largeness or constructivity, leads one in the direction of a proof of> the flavor "P!=NP because P!=NP". And the two conditions, largeness> and constructivity, are "natural" because they lead a proof away from> a proof of the flavor "P!=NP because P!=NP". From this observation, we> can conclude that a proof that P!=NP cannot involve deep mathematics,> since deep mathematics is "natural", at least "natural" to> mathematicians. For example, here is one such proof:>> SUBSET-SUM (where the target integer is zero) is equivalent to the> problem of determining whether the set of subset-sums of {s1,...,sk}> and the set of subset-sums of {-s_{k+1},...,-s_n} intersect> nontrivially. There are 2^k members of the set of subset-sums of> {s1,...,sk} and 2^{n-k} members of the set of subset-sums of {-s_{k> +1},...,-s_n}, and it is impossible to simplify the problem further.>> Hence, since the combined size of the sets that the SUBSET-SUM problem> is about is 2^k+2^{n-k}, for k in {1,...,n}, we can get a lower bound> for the worst-case running-time of an algorithm that solves SUBSET-SUM> by minimizing 2^k+2^{n-k}, s.t. k in {1,...,n}. We get 2^{n/2}, so> Omega(2^{n/2}) is an exponential lower bound for algorithms which> solve SUBSET-SUM.>> This is an example of a "supernatural proof". It is not natural,> because in this case, C=the set of functions that cannot be computed> in o(2^{n/2}) time, so it violates the constructivity property above.> And it also does not involve deep mathematics. This supernatural proof> is almost of the form "P!=NP because P!=NP", but not quite or else it> would be a circular argument.>> I believe that the reason why the P vs NP problem has been considered> so difficult is because of human psychology. Most people either want> there to be some deep reason why P!=NP (some almost magic secret using> some deep mathematics like the proof of FLT had) or they refuse to> believe that P!=NP. They don't want to accept that there is no deep> reason why P!=NP. It essentially just is.Not everything I said in this post is technically correct. But I thinkthe idea is correct.
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