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an interesting observation about Natural Proofs
Posted:
Nov 9, 2011 11:58 AM


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 RazborovRudich 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 polynomialtime. 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:
SUBSETSUM (where the target integer is zero) is equivalent to the problem of determining whether the set of subsetsums of {s1,...,sk} and the set of subsetsums of {s_{k+1},...,s_n} intersect nontrivially. There are 2^k members of the set of subsetsums of {s1,...,sk} and 2^{nk} members of the set of subsetsums 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 SUBSETSUM problem is about is 2^k+2^{nk}, for k in {1,...,n}, we can get a lower bound for the worstcase runningtime of an algorithm that solves SUBSETSUM by minimizing 2^k+2^{nk}, 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 SUBSETSUM.
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.



