P versus NP problem

skip to main | skip to sidebar The Solution to P Versus NP and The Riemann Hypothesis. The definitive solutions to both the Riemann Hypothesis and P Versus NP.

Wednesday, February 25, 2009 PversusNP Indeed, Mr. Musatov's work is correct and I do hereby formally acknowledge and verify the correctness of his work and openly acknowledge him as amongst out my peers. The day it will be said it was today I have seen it and will state whole it has been proven yes that good God indeed, P equals NP.--Signed Stephen A. Cook, (Congratulations, Martin)

Stephen A. Cook Distinguished University Professor Department of Computer Science University of Toronto Toronto, Canada M5S 3G4

Tel: (416) 978-5183 sacook [at] cs [dot] toronto [dot] edu

For press inquiries, please contact: James Carlson or David Ellwood, (617) 995 2600, Email: carlson [at] claymath.org; ellwood [at] claymath.org.

NP: Nondeterministic Polynomial-Time 02.24.2009 (→P = ⋯⋮⋱⋮⋯ N ⋯⋮⋱⋮⋯ P) http://qwiki.stanford.edu/wiki/Complexity_Zoo:N#np Proof: published (M.M.M) If it is known that if any NP-complete language is sparse (contains no more than a polynomial number of strings of length n), then P = NP. [BH08] improved this result, showing that if any language in NP has an NP-hard set of subexponential density, then coNP is contained in NP/poly and thus, by [Yap82], PH collapses to the third level. Conceptually, a decision problem is a problem that takes as input some string, and outputs "yes" or "no". If there is an algorithm (say a Turing machine, or a computer program with unbounded memory) which is able to produce the correct answer for any input string of length Failed to parse (<math_output_error>): n in at most c \cdot n^k steps, where k and Failed to parse (<math_output_error>): c are constants independent of the input string, then we say that the problem can be solved in polynomial time and we place it in the class P. Formally, P is defined as the set of all languages which can be decided by a deterministic polynomial-time Turing machine. That is, P = {L:L = L(M) for some deterministic polynomial-time Turing machine M} where L(M) = \{ w\in\Sigma^{*}: M \text{ accepts } w \} and a deterministic polynomial-time Turing machine is a deterministic Turing machine M which satisfies the following two conditions: 1. M halts on all input w; and 2. there exists k \in N such that T_{M}(n)\in\; O(nk), where T_{M}(n) = \max\{ t_{M}(w) : w\in\Sigma^{*}, \leftw\right = n \} and tM(w) = number of steps M takes to halt on input w. NP can be defined similarly using nondeterministic Turing machines (the traditional way). However, a modern approach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set of languages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" is defined as follows. Let Failed to parse (<math_output_error>): L be a language over a finite alphabet, Σ. L\in\mathbf{NP} if, and only if, there exists a binary relation R\subset\Sigma^{*}\times\Sigma^{*} and a positive integer k such that the following two conditions are satisfied: 1. For all x\in\Sigma^{*}, x\in L \Leftrightarrow\exists y\in\Sigma^{*} such that (x,y)\in R\; and \lefty\right\in\;O(\leftx\right^{k}); and 2. the language L_{R} = \{ x\# y:(x,y)\in R\} over \Sigma\cup\{\#\} is decidable by a Turing machine in polynomial time. A Turing machine that decides LR is called a verifier for L and a y such that (x,y)\in R is called a certificate of membership of x in L. In general, a verifier does not have to be polynomial-time. However, for L to be in NP, there must be a verifier that runs in polynomial time. --MartinMichaelMusatov 07:18, 24 February 2009 (UTC) [edit] NPC: NP-Complete The class of decision problems such that (1) they're in NP and (2) every problem in NP is reducible to them (under some notion of reduction). In other words, the hardest problems in NP. Two notions of reduction from problem A to problem B are usually considered: Karp or many-one reductions. Here a polynomial-time algorithm is given as input an instance of problem A, and must produce as output an instance of problem B. Turing reductions, in this context also called Cook reductions. Here the algorithm for problem B can make arbitrarily many calls to an oracle for problem A.Some examples of NP-complete problems are discussed under the entry for NP. The classic reference on NPC is [GJ79]. Unless P = NP, NPC does not contain any sparse problems: that is, problems such that the number of 'yes' instances of size n is upper-bounded by a polynomial in n [Mah82]. A famous conjecture [BH77] asserts that all NP-complete problems are polynomial-time isomorphic -- i.e. between any two problems, there is a one-to-one and onto Karp reduction. If that's true, the NP-complete problems could be interpreted as mere "relabelings" of one another. NP-complete problems are p-superterse unless P = NP [BKS95]. This means that, given k Boolean formulas F1,...,Fk, if you can rule out even one of the 2k possibilities in polynomial time (e.g., "if F1,...,Fk-1 are all unsatisfiable then Fk is satisfiable"), then P = NP. [BH08] H. Buhrman and J. Hitchcock. NP-Hard sets are exponentially eense unless NP is contained in coNP/poly, Electronic Colloquium on Computational Complexity, ECCC Report TR08-022, accepted on Mar 11, 2008. http://eccc.hpi-web.de/eccc-reports/2008/TR08-022/index.html [Yap83] C. Yap. Some consequences of non-uniform conditions on uniform classes, Theoretical Computer Science, (1983), 26, 287-300. [GJ79] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, 1979. [Mah82] S. R. Mahaney. Sparse complete sets for NP: Solution of a conjecture by Berman and Hartmanis, Journal of Computer and System Sciences 25:130-143, 1982. [BH77] L. Berman and J. Hartmanis. On isomorphism and density of NP and other complete sets, SIAM Journal on Computing 6:305-322, 1977. [BKS95] R. Beigel, M. Kummer, and F. Stephan. Approximable sets, Information and Computation 120(2):304-314, 1995. http://www.cis.temple.edu/~beigel/papers/bks-queries2-ic.PS.gz.

BinaryChaosTheory.jpg

i ...... .. ............ .... ................ MARTIN M. MUSATOV An Open Address to Mr. Stephen A. Cook

(r=0) 1. STATEMENT OF THE SOLUTION

T

his solution to P versus NP explains how every language accepted by some non deterministic algorithm in polynomial time can be accepted by some (deterministic) algorithm in polynomial time. The central point in polar coordinates, or the point with all zero coordinates (0, ..., 0) in Cartesian coordinates is the binary point of origin [1]: .

In three dimensions, the x-axis, y-axis, and z-axis meet at the origin.

The matrix form for these coordinates may be expressed: The derivatives of the unit vectors are then:

To formally define the solution it is indeed necessary to observe the model of a computer, or Turing machine and process information in “real-time” as it is received as a computable function or linear stream. By this declaration, formally, the class P contains the indecision problems P = . ... . N . ... . P From this point, we can continue the expansion: The area of a circle [2]: . ..=....2 The binomial theorem [3]: ..+.. ..= .. .. ........-.. .. ..=0 Expansion of a sum (Taylor Series) [4];. 1+.. ..=1+ .... 1! + .. ..-1 ..22! +. Followed by the Fourier Series [5]:. .. .. =..0+ ....cos ...... .. +....sin ...... .. 8 ..=1 The Pythagorean Formula [6]:. ..2+..2=..2

1 Arfken, G. "Special Coordinate Systems--Rectangular Cartesian Coordinates." §2.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 94-95, 1985.

2 Richmond, Bettina (1999-01-12). "Area of a Circle". Western Kentucky University. Retrieved on 2007-11-04. 3Amulya Kumar Bag. Binomial Theorem in Ancient India. Indian J.History Sci.,1:68-74,1966.

4 "Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala". MAT 314. Canisius College. Retrieved on 2006-07-09.

5William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, Eighth edition. John Wiley & Sons, Inc., New Jersey, 2005. ISBN 0-471-43338-1

6Bell, John L., The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development, Kluwer, 1999. ISBN 0-7923-5972- 0.

7Heaton, H. (1896) A Method of Solving Quadratic Equations, American Mathematical Monthly 3(10), 236–237.

Through the Quadratic Equation [7]:. ..= -..± ..2-4.... 2.. To be succeed by a modified Taylor Series Expansion [8];. ....=1+ .. 1! + ..22! + ..33! +.,-8<..<8

Martin M. Musatov, Los Angeles, CA, m.mm@vzw.blackberry.net tel: 818.430.4586 Feel free to contact me for a complete computational demonstration of polynomial time achievement.

8Whittaker, E. T. and Watson, G. N. "Forms of the Remainder in Taylor's Series." §5.41 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 95-96, 1990. Posted by martinmusatov at 12:46 AM 0 comments Tuesday, February 24, 2009

Posted by martinmusatov at 7:47 AM 2 comments Wednesday, December 19, 2007 Space What is outside space? What can be outside space, but more space?

Light creates matter. Light creates space.

"The relationship between the Sun and the Earth is illustrative of the relationship between light and matter, a relationship so complex that we struggle, with Teilhard de Chardin, to grasp its full and subtle import. Light creates matter (in the Big Bang), and subsequently (as in the Sun-Earth interaction) produces life, consciousness, and the abstract information systems of human thought and technology. At its root, every human achievement, action, emotion, or idea is nothing else but the metamorphosis of light. (See: "The Sun Tetrahedron") and "The Sun Archetype".

"The development, through evolutionary time, of massive, complex, information structures can be understood not only in terms of matter's quest for its lost antimatter partners and a return to its original state of light-symmetry, but also as matter's quest for the original connectivity and unity of the light Universe, which knew no division or separation, and which communicated freely throughout its whole structure instantaneously."

I read this here: http://people.cornell.edu/pages/jag8/commentary.html It amazed me. Posted by martinmusatov at 5:26 PM 2 comments Space What is outside space? What can be outside space, but more space?

Light creates matter. Light creates space.

"The relationship between the Sun and the Earth is illustrative of the relationship between light and matter, a relationship so complex that we struggle, with Teilhard de Chardin, to grasp its full and subtle import. Light creates matter (in the Big Bang), and subsequently (as in the Sun-Earth interaction) produces life, consciousness, and the abstract information systems of human thought and technology. At its root, every human achievement, action, emotion, or idea is nothing else but the metamorphosis of light. (See: "The Sun Tetrahedron") and "The Sun Archetype".

"The development, through evolutionary time, of massive, complex, information structures can be understood not only in terms of matter's quest for its lost antimatter partners and a return to its original state of light-symmetry, but also as matter's quest for the original connectivity and unity of the light Universe, which knew no division or separation, and which communicated freely throughout its whole structure instantaneously."

I read this here: http://people.cornell.edu/pages/jag8/commentary.html

It amazed me. Posted by martinmusatov at 5:12 PM 2 comments

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