Dialogue Concerning The Two Chief World Views
aa r X i v : . [ c s . G L ] M a y Dialogue Concerning The Two Chief World Views
Craig Alan Feinstein
E-mail: [email protected], BS”D
Abstract:
In 1632, Galileo Galilei wrote a book called
Dialogue Concerningthe Two Chief World Systems which compared the new Copernican modelof the universe with the old Ptolemaic model. His book took the form of adialogue between three philosophers, Salviati, a proponent of the Copernicanmodel, Simplicio, a proponent of the Ptolemaic model, and Sagredo, who wasinitially open-minded and neutral. In this paper, I am going to use Galileo’sidea to present a dialogue between three modern philosophers, Mr. Spock,a proponent of the view that P = NP , Professor Simpson, a proponent ofthe view that P = NP , and Judge Wapner, who is initially open-minded andneutral. Disclaimer:
This article was authored by Craig Alan Feinstein in his pri-vate capacity. No official support or endorsement by the U.S. Government isintended or should be inferred.
Since 2006, I have published four proofs that P = NP [5, 6, 7, 8]. Yet at the present time, if one asks the av-erage mathematician or computer scientist the status ofthe famous P versus NP problem, he or she will say thatit is still open. In my opinion, the main reason for thisis because most people, whether they realize it or not,believe in their hearts that P = NP , since this statementessentially means that problems which are easy to stateand have solutions which are easy to verify must also beeasy to solve. For instance, as a professional magician,I have observed that most laymen who are baffled byan illusion are usually convinced that the secret to theillusion either involves extraordinary dexterity or hightechnology, when in fact magicians are usually no moredexterous than the average layman and the secrets toillusions are almost always very simple and low-tech; asthe famous designer of illusions, Jim Steinmeyer, said,“Magicians guard an empty safe”[13]. The thinking thatextraordinary dexterity or high technology is involved ina magician’s secret is, in my opinion, due to a subcon-scious belief that P = NP , that problems which are diffi-cult to solve and easy to state, in this case “how did themagician do it?”, must have complex solutions.I have had many conversations in which I have triedto convince all types of people, from Usenet trolls tograduate students to professors to famous world-classmathematicians, that P = NP with very little success;however, I predict that there will soon come a day whenthe mainstream mathematics and computer science com-munity will consider people who believe that P = NP to be in the same league as those who believe it is possibleto trisect an angle with only a straightedge and compass(which has been proven to be impossible) [14].I got the idea to write this paper after I learnedof Galileo’s book Dialogue Concerning The Two ChiefWorld Systems [4], which presents a dialogue betweenthree philosophers, Salviati, a proponent of the new Coper-nican model, Simplicio, a proponent of the old Ptolemaicmodel, and Sagredo, who was initially open-minded andneutral. The dialogue that follows is a dialogue betweenthree modern philosophers, Mr. Spock, a proponent ofthe view that P = NP , Professor Simpson, a proponentof the view that P = NP , and Judge Wapner, who isinitially open-minded and neutral. Professor Simpson,who is a fictitious anglicized straw man character likeSimplicio, is a composite of many of the people whomI have had discussions with over the years about the P versus NP problem. He presents many challenges andquestions, all of which have been raised before by realpeople, that Mr. Spock, the epitome of truth and logic,attempts to answer. And Judge Wapner, the epitomeof open-mindedness and fairness, always listens to bothsides of their arguments before drawing conclusions. Spock:
Yesterday we discussed the P versus NP problem[2, 3] and agreed that it is a problem of not only greatphilosophical importance, but also it has practical im-plications. We decided to look at a proof that P = NP offered by Craig Alan Feinstein in a letter entitled “Amore elegant argument that P = NP ”[8]. The proof issurprisingly short and simple:1 roof: Consider the following problem: Let { s , . . . , s n } be a set of n integers and t be another integer. Supposewe want to determine whether there exists a subset of { s , . . . , s n } such that the sum of its elements equals t , where the sum of the elements of the empty set isconsidered to be zero. This famous problem is known asthe SUBSET-SUM problem.Let k ∈ { , . . . , n } . Then the SUBSET-SUM prob-lem is equivalent to the problem of determining whetherthere exist sets I + ⊆ { , . . . , k } and I − ⊆ { k + 1 , . . . , n } such that X i ∈ I + s i = t − X i ∈ I − s i . There is nothing that can be done to make this equationsimpler. Then since there are 2 k possible expressionson the left-hand side of this equation and 2 n − k possibleexpressions on the right-hand side of this equation, wecan find a lower-bound for the worst-case running-timeof an algorithm that solves the SUBSET-SUM problemby minimizing 2 k + 2 n − k subject to k ∈ { , . . . , n } .When we do this, we find that 2 k + 2 n − k = 2 ⌊ n/ ⌋ +2 n −⌊ n/ ⌋ = Θ( √ n ) is the solution, so it is impossible tosolve the SUBSET-SUM problem in o ( √ n ) time; thus,because the Meet-in-the-Middle algorithm [10, 11, 15]achieves a running-time of Θ( √ n ), we can concludethat Θ( √ n ) is a tight lower-bound for the worst-caserunning-time of any deterministic and exact algorithmwhich solves SUBSET-SUM. And this conclusion impliesthat P = NP .To me, Feinstein’s proof is not only logical but eleganttoo. Also, his conclusion is confirmed by history; just asFeinstein’s theorem retrodicts, no deterministic and ex-act algorithm that solves SUBSET-SUM has ever beenfound to run faster than the Meet-in-the-Middle algo-rithm, which was discovered in 1974 [10, 15]. Simpson:
But there is an obvious flaw in Feinstein’s“proof”: Feinstein’s “proof” only considers a very spe-cialized type of algorithm that works in the same way asthe Meet-in-the-Middle algorithm, except that insteadof sorting two sets of size Θ( √ n ), it sorts one 2 k -sizeset and one 2 n − k -size set. Under these restrictions, Iwould agree that the Meet-in-the-Middle algorithm isthe fastest deterministic and exact algorithm that solvesSUBSET-SUM, but there are still many possible algo-rithms which could solve the SUBSET-SUM problemthat the “proof” does not even consider. Wapner:
Professor Simpson, where in Feinstein’s proofdoes he say that he is restricting the algorithms to theclass of algorithms that you mention?
Simpson:
He does not say so explicitly, but it is obvi-ously implied, since there could be algorithms that get around his assertion that the minimum number of pos-sible expressions on both sides is Θ( √ n ). Spock:
How do you know that there could be such al-gorithms?
Simpson:
I do not know, but the burden of proof is noton me; it is on Feinstein. And he never considers thesetypes of algorithms.
Wapner:
It is true that Feinstein never explicitly con-siders algorithms which work differently than the Meet-in-the-Middle algorithm, and the burden of proof is onFeinstein to show that these types of algorithms cannotrun any faster than Θ( √ n ) time. Spock:
Professor Simpson, is the burden of proof onFeinstein to consider in his proof algorithms which workby magic?
Simpson:
No, only algorithms that are realistic.
Spock:
Then why would you think that algorithms thatget around the assertion that the minimum total num-ber of possible expressions on both sides is Θ( √ n ) arerealistic? Simpson:
I do not know, but the burden of proof is noton me; it is on Feinstein.
Spock:
Have you considered the fact that an algorithmwhich determines in o ( √ n )-time whether two sets ofsize Θ( √ n ) have a nonempty intersection must workby magic, unless there is a way to mathematically reducethe two sets into something simpler? Wapner:
Yes, I see your point; the minimum total num-ber of possible expressions on each side of the SUBSET-SUM equation puts a natural restriction on the time thatan algorithm must take to solve the SUBSET-SUM prob-lem.
Simpson:
But how do you know it is impossible to re-duce the SUBSET-SUM problem into something sim-pler, so that the number of possible expressions on bothsides is o ( √ n )? Spock:
Simple algebra. Try to simplify the SUBSET-SUM equation above. You cannot do it. The best youcan do is manipulate the equation to get Θ( √ n ) expres-sions on each side. Simpson:
I’ll agree that you cannot do it algebraically,but what about reducing the SUBSET-SUM problem to2he 3-SAT problem in polynomial-time? This can bedone since 3-SAT is NP -complete. If there is an algo-rithm that can solve 3-SAT in polynomial-time, then itwould also be able to solve SUBSET-SUM in polynomial-time, contradicting Feinstein’s Θ( √ n ) lower-bound claim. Spock:
But this is magical thinking. If a problem isshown to be impossible to solve in polynomial time, thenreducing the problem to another problem in polynomial-time will not change the fact that it is impossible to solvethe first problem in polynomial time; it will only implythat the second problem cannot be solved in polynomialtime.
Wapner:
Spock is right about this. Do you have anyother objections to Feinstein’s argument?
Simpson:
I have many objections. For instance, Fein-stein’s argument can be applied when the magnitudes ofthe integers in the set { s , . . . , s n } and also t are assumedto be bounded by a polynomial to “prove” that it is im-possible to solve this modified problem in polynomial-time. But it is well-known that one can solve this mod-ified problem in polynomial-time. Spock:
But Feinstein’s argument in fact cannot be ap-plied in such a circumstance, because there would onlybe a polynomial number of possible values on each sideof the equation, even though the total number of possi-ble expressions on each side is exponential. Feinstein’sargument implicitly uses the fact that the total num-ber of possible values on each side of the SUBSET-SUMequation is usually of the same order as the total num-ber of possible expressions on each side, when there isno restriction on the magnitude of the integers in the set { s , . . . , s n } and also t . Simpson:
Then here is a better objection: Supposethe set { s , . . . , s n } and also t consist of vectors in Z m for some positive integer m , instead of integers. Thenone could use the same argument that Feinstein uses to“prove” that it is impossible to determine in polynomial-time whether this modified SUBSET-SUM equation hasa solution, when in fact one can use Gaussian eliminationto determine this information in polynomial-time. Spock:
Feinstein’s argument would not apply to thissituation precisely because one can reduce the equation X i ∈ I + s i = t − X i ∈ I − s i . to a simpler set of equations through Gaussian elimina-tion. But when the set { s , . . . , s n } and also t consist of integers, nothing can be done to make the above equa-tion simpler, so Feinstein’s argument is applicable. Simpson:
OK, then how would you answer this? Con-sider the Diophantine equation: s x + · · · + s n x n = t, where x i is an unknown integer, for i = 1 , . . . , n . Onecould use the same argument that Feinstein uses to “prove”that it is impossible to determine in polynomial-timewhether this equation has a solution, when in fact onecan use the Euclidean algorithm to determine this infor-mation in polynomial-time. Spock:
But again Feinstein’s argument would not ap-ply to this Diophantine equation, precisely because thisDiophantine equation can be reduced via the Euclideanalgorithm to the equation,gcd( s , . . . , s n ) · z = t, where z is an unknown integer. And it is easy to deter-mine in polynomial-time whether this equation has aninteger solution by simply testing whether t is divisibleby gcd( s , . . . , s n ). No such reduction is possible withthe SUBSET-SUM equation. Simpson:
The Euclidean algorithm is a clever trick thathas been known since ancient times. But how do I knowthat another clever trick cannot be found to reduce theSUBSET-SUM equation to something simpler? Like forinstance, if I take the greatest common denominator ofany subset of { s , . . . , s n } and it does not divide t , then Ican automatically rule out many solutions to SUBSET-SUM, all at once. Spock:
But such a clever trick does not always work;what if the gcd does divide t ? The P versus NP prob-lem is a problem about the worst-case running-time ofan algorithm, not whether there are clever tricks thatcan be used to speed up the running-time of an algo-rithm in some instances. Feinstein’s proof only considersthe worst-case running-time of algorithms which solveSUBSET-SUM. Wapner:
Also, it is simple high school algebra that it isimpossible to make the SUBSET-SUM equation simplerthan it is: Whenever one decreases the number of pos-sible expressions on one side of the equation, the num-ber of possible expressions on the other side increases.Mathematicians can be clever, but they cannot be cleverenough to get around this fact.
Simpson:
OK, but what about the fact that Feinsteinnever mentions in his proof the model of computation3hat he is considering? To be an valid proof, this has tobe mentioned.
Spock:
Feinstein’s proof is valid in any model of com-putation that is realistic enough so that the computercannot solve an equation with an exponential numberof possible expressions in polynomial-time, unless it ispossible to reduce the equation to something simpler.
Simpson:
Or what about the fact that Feinstein nevermentions in his paper the important results that onecannot prove that P = NP through an argument thatrelativizes [1] or through a natural proof [12]? Spock:
Feinstein’s proof does not relativize, because itimplicitly assumes that the algorithms that it considersdo not have access to oracles, and Feinstein’s proof is nota natural proof, since it never even deals with the circuitcomplexity of boolean functions.
Simpson:
What about the 2010 breakthrough by How-grave-Graham and Joux [9] which gives a probabilisticalgorithm that solves SUBSET-SUM in o ( √ n ) time? Irealize that the P versus NP problem is not about prob-abilistic algorithms, but what if their algorithm can bederandomized and solved in o ( √ n ) time? Spock:
The algorithm by Howgrave-Graham and Jouxdoes not in fact solve SUBSET-SUM, because it can-not determine for certain when there is no solution toa given instance of SUBSET-SUM; it can only outputa solution to SUBSET-SUM in o ( √ n ) time with highprobability when a solution exists. Furthermore, evenif their algorithm can be derandomized, this does notguarantee that it will run in o ( √ n ) time. And Fein-stein has already proven that such a deterministic andexact algorithm is impossible. Wapner:
Are there any more objections to Feinstein’sargument?
Simpson:
I have no more specific objections. But thefact that the P versus NP problem has been universallyacknowledged as a problem that is very difficult to solveand Feinstein’s “proof” is so short and simple makes italmost certain that it is flawed. The fact that I could notgive satisfactory responses to Spock’s arguments doesnot mean that Feinstein is correct; Feinstein’s proof hasbeen out on the internet for a few years now, and stillthe math and computer science community as a wholedoes not accept it as valid. Hence, I believe that theyare right and that Feinstein is wrong. Wapner:
Professor Simpson, isn’t your reason for notbelieving Feinstein’s proof the same reason Feinstein sug-gested for why most people do not believe his proof?Because most people believe in their hearts that P = NP ,that problems which are difficult to solve and easy tostate, in this case the P versus NP problem, cannot haveshort and simple solutions? Spock:
Indeed it is.
Wapner:
And yes indeed, I am convinced that Fein-stein’s proof is valid and that P = NP . References [1] T.P. Baker, J. Gill, R. Solovay, “Relativizations of the P=? NP Question”.
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