NP!= P. By Liu Ran. Table of Contents. The P vs. NP problem is a major unsolved problem in computer

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1 NP!= P By Lu Ra Table of Cotets. Itroduce 2. Strategy 3. Prelmary theorem 4. Proof 5. Expla 6. Cocluso. Itroduce The P vs. NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer ca quckly solve every problem whose soluto ca be quckly verfed by a computer also. It was troduced 97 by Stephe Cook hs semal paper "The complexty of theorem provg procedures" ad s cosdered by may to be the most mportat ope problem computer scece. The formal term used above meas the exstece of a algorthm for the task that rus polyomal tme quckly. The geeral class of questos for whch some algorthm ca provde a aswer polyomal tme s called "class P" or just "P". For some questos, there s o kow

2 way to quckly fd a aswer, but f oe s provded wth formato showg what the aswer s, t may be possble to quckly verfy the aswer. The class of questos for whch a aswer ca be verfed polyomal tme s called NP. NP-complete problems are a set of problems to each of whch ay other NP-problem ca be reduced polyomal tme, ad whose soluto may be verfed polyomal tme stll. That s, ay NP problem ca be trasformed to ay of the NP-complete problems. Iformally, a NP-complete problem s at least as "tough" as ay other problem NP. NP-hard problems are those at least as hard as NP-complete problems,.e., all NP-problems ca be reduced to them polyomal tme. NP-hard problems eed ot be NP,.e., they eed ot have solutos verfable polyomal tme. I formato theory, etropy s a measure of the ucertaty a radom varable. I ths cotext, the term usually refers to the Shao etropy, whch quatfes the expected value of the formato cotaed a message. Etropy s measured bts, ats, or bas typcally. Shao etropy s the average upredctablty a radom varable, whch s equvalet to ts formato cotet.

3 Claude Elwood Shao (Aprl 30, 96 February 24, 200) was a Amerca mathematca, electroc egeer, ad cryptographer kow as "The father of formato theory". Shao s famous for havg fouded formato theory wth a ladmark paper that he publshed 948. The Shao etropy, a measure of ucertaty (see further below) ad deoted by H(x), s defed by Shao as H(x)=E[I( x )]=E[ log(2,/p( x )) ]= - å p( x )log(2, p( x )) (,2,..) Where p( x ) s the probablty mass fucto of outcome x. Specally, f there are N outcomes, ad p( x )=p( x 2 )= =p( x )=/N. H(x) = -å p( x )log(2, p( x )) = -å(/ N)log(2,(/ N)) = å (/ N)log(2, N) = log(2,n).

4 2. Strategy Ay NP problem ca reduce to logc crcut, we ca prove the etropy of P problem, H(P) = 0, whle the etropy of NP problem, H(NP) > 0. Moreover, whe put from crease to +, the delta etropy of P problem, D H(P) = H(+) H() = 0, whle the delta etropy of NP problem, D H(NP) = H(+) H() =. If NP problem ca resolve to P problem polyomal tme, we ca prove D H(NP) = H(+) H() ¹. It s cotradctory wth D H(NP) = H(+) H() =.

5 3. Prelmary theorem (2.) Polyomal detcal theorem: k k- f(x) = a x + a x a x+ a, a ¹ 0 ; k k- 0 k k k- g(x) = b x + b x b x+ b, b ¹ 0; k k- 0 k f(x) = g(x) Û (2.2) Bomal theorem: a = b, a = b,..., a = b, a = b, a ¹ 0 k k k k k Bomal theorem calls also Newto bomal theorem. Newto brought forth t year. ( a+ b) = C 0 a + C a b C a b C b,>0,< C expresses combatoal umber of takg freely elemets from elemets, C =!/ (( - )!!) (2.3) The etropy of P problem s zero,.e. H(P) = 0. Because every step of P class problem s determstc, ts happe probablty s always. Base o the defto of P problem, P problem ca va polyomal steps to get determstc outcome. Defe T() = O( k ), we ca express the happe probablty as p = T ( ) Õ p, p = Þ p = Þ there s oly outcome for P problem Þ etropy of P problem H(P) = - log(2,) = 0. (2.4) The etropy of NP problem s above zero,.e. H(NP) > 0. Because every step of NP class problem s o-determstc, ts happe

6 probablty s always <. Base o the defto of NP problem, NP problem ca oly va polyomal steps to verfy outcome s aswer or ot. Every step has more tha choce to calculate. Defe T () = O( k ),we ca express the happe probablty of oe outcome as p( x ) = T '( ) Õ p, p < Þ p( x ) <. Because H(NP) = - å p( x )log(2, p( x )) = å p( x )log(2,/ p( x )) > å p( x )log(2,) =0. Base o the defto of NP problem, NP problem s easy to verfy polyomal tme. Every outcome provded s determstc, but why H(NP) > 0? Because every outcome provded s determstc. Every step s determstc. But the fal outcome s o-determstc to be the correct aswer. It s oly oe of may outcomes. Deote the happe probablty of oe outcome as p( x ), the we ca express the happe probablty as p = p( x ) T ( ) Õ p =p( x ) = p( x )<. Specally, whe the happe probablty of every outcome s detcal. If there are N outcomes, p( x ) = /N. It s s determstc to verfy oe outcome va polyomal steps, but happe probablty of the outcome beg correct aswer s /N. (2.5) If a NP problem ca reduce to P problems, every P problem must be oe of may outcomes. If there s oly oe outcome for NP problem, the happe probablty of oe outcome s p( x )=, the happe probablty s p =

7 p( x ) T ( ) Õ p =p( x ) = p( x )=. The H(NP) = 0, t s cotradctory wth (2.4) H(NP) > 0 Þ There are may outcome for NP problem. Otherwse, t s a P problem. To expla (2.5) clearly, I draw a chart below. A NP problem ca reduce to may parallel P problems. Because every step of P problem s determstc ad (2.4) H(NP) > 0 Þthe oly o-determstc step s the outcome, Þ NP problem must have may outcomes ad every P problem s outcome s oly oe of all outcomes. Þ A NP problem ca reduce to may parallel P problems. From (2.5.) ad (2.5.2) Þ A NP problem ca reduce to may parallel P problems, every P problem must be oe of may outcomes.

8 4. Proof Ay NP problem ca be trasformed to ay of the NP-complete problems. The frst NP-complete problem s the logc crcut. That s, f the logc crcut equal to P problem, NP = P s prove; f ot equal to, NP P s prove. Let s cosderate a logc crcut lke below. puts va k gates, the output. Every put ca be value 0 or, suppose puts ca geerate N outcomes. Express as G() = N.

9 Iput crease to +, because put s value 0 or, ew out ca be express as G(+) = ( N + = 0) + ( N + = ) = 2N. It meas that puts from to geerate N outcomes, ad put (+) s value 0; puts from to geerate N outcomes, ad put + s value. The total outcomes are 2N. The logc crcut s etropy s log(2,n) whe puts; the logc crcut s etropy s log(2,2n) whe + puts. The delta etropy V H = H(+) H(N) = log(2,2n) log(2,n )= (3.). (3.2) Suppose NP = P, t meas that (3.2.) Ay NP problem ca reduce to oe P problem polyomal tme. I.e. NP = P ; (3.2.2) Ay NP problem ca reduce to polyomal parallel P problem polyomal tme. I.e. T ( ) =å P, T() = O( k NP( ) ( ) ); (3.2.3) Ay NP problem ca reduce to expoetal parallel P problem polyomal tme. I.e. T ( ) p( ) =å P, T() = O( NP( ) ( ) k ); (3.2.4) Ay NP problem ca reduce to more tha expoetal parallel P problem polyomal tme. I.e. T ( ) p( ) =å P, T() > O( NP( ) ( ) k ). For (3.2.), f NP = P, from prelmary theorem (2.3) H(P) = 0 ÞV H =H( P (+)) H( P ())= 0 0 = 0. It s cotradctory wth (3.);

10 For (3.2.2), T ( ) =å P, T() = O( k NP( ) ( ) ), because prelmary theorem (2.3) H(P) = 0 ad (2.4) H(NP) > 0 Þevery P () s oe of may outcomes, whch clude formato quatty ad reduce determacy. Ad because (3.) Þ V H =H( P (+)) H( P ()) = log(2, T(+) - log(2, T()= Þ log(2, T(+) - log(2, T() = log(2, T(+)/T()) = Þ T(+)/T() = 2, deote T() k ' k '- = ak ' x ak '-x ax a0 ak ' , ¹ 0 Þ T ( ) a ( ) a ( )... a ( ) a + = = 2T() = k ' ( k '-) k ' k '- 0 k ' ( k '-) 2( ak ' ak '-... a a ), because of (2.) Polyomal detcal k ' k ' theorem ad (2.2) Bomal theorem Þ 2a a k ' ( k '-) cotradctory wth a k ' ¹ 0, T() = - k ' = Þ a ' 0 k ' a + a a + a, a ¹ 0. k ' k ' 0 k ' k =, t s For (3.2.3), T ( ) p( ) =å P, T() = O( k NP( ) ( ) ), because prelmary theorem (2.3) H(P) = 0 ad (2.4) H(NP) > 0 Þevery P () s oe of may outcomes, whch clude formato quatty ad reduce p( ) determacy Þ NP() s complexty >= T(). = O( k )). >= p( ) O( k ) Þ NP() s complexty s expoetal, t s cotradctory wth NP = P. For (3.2.4), T ( ) p( ) =å P, T() > O( k NP( ) ( ) ), because prelmary theorem (2.3) H(P) = 0 ad (2.4) H(NP) > 0 Þevery P () s oe of may outcomes, whch clude formato quatty ad reduce p( ) p( ) determacy Þ NP() s complexty >= T(). > O( k ). >= O( k )

11 Þ NP() s complexty s more tha expoetal, t s cotradctory wth NP = P. 5. Expla To expla my proof clearly, I draw a flow chart to deote that computer hadles NP problem process. å deotes parallel relatoshp ad Õ deotes seral relatoshp below fgures. More detaled flow chart s below. Ay NP problem must reduce to P problem ad every P problem s oe of may outcomes. Ay P problem must reduce to basc structo. But f NP=P, t volates etropy theorem. Ay NP problem ca t reduce to polyomal P problem. Whe NP reduces to P problem, the delta etropy s always zero. Whe NP reduces to polyomal P problem, the delta etropy does ot equal to. Whe NP reduces to expoetal P problem or more complex, the

12 complexty has become cotradctory wth defto of P problem. All scearos are cotradctory, so NP = P s wrog, 6. Cocluso I essece, P problem s a determstc problem, whch ca reduce to basc structos polyomal tme. NP problem s a o-determstc problem, whch ca t reduce to P problem polyomal tme. If NP = P, t meas that determstc problem equals to o-determstc problem, whch volates formato etropy prcple. So ay o-determstc problem s ot easy to calculate.

13 Refereces [] Ihara, Shusuke (993). Iformato theory for cotuous systems. World Scetfc. p. 2. ISBN [2] I ths cotext, a 'message' meas a specfc realzato of the radom varable. [3] Brllou, Léo (2004). Scece & Iformato Theory. Dover Publcatos. p ISBN [4] Shao, Claude E. (July/October 948). "A Mathematcal Theory of Commucato". Bell System Techcal Joural27 (3): [5] Gose, Fracos & Olla, Stefao (2008). Etropy methods for the Boltzma equato: lectures from a specal semester at the Cetre Émle Borel, Isttut H. Pocaré, Pars, 200. Sprger. p. 4. ISBN [6] A b Scheer, B: Appled Cryptography, Secod edto, page 234. Joh Wley ad Sos. [7] A b Shao, Claude E.: Predcto ad etropy of prted Eglsh, The Bell System Techcal Joural, 30:50 64, Jauary 95.