P = NP Concepts

The Golden Ticket: P, NP, and the Search for the Impossible

by Lance Fortnow · 30 Mar 2013 · 236pp · 50,763 words

a procedure that finds golden tickets, such as the solutions to all the other Millennium Prize Problems. Prove P = NP and get $6 million for solving the six unsolved Millennium Problems. Showing either P = NP or P ≠ NP won’t be easy. If you want $6 million, you’ll have a better chance playing the lottery

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some problems, like factoring large numbers, that remain out of reach for current machines. Quantum mechanics can also give us unbreakable secrets whether or not P = NP. What about the future? The great challenges of computing still lie ahead of us. How do we deal with computers that have to work together

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algorithm that solves NP problems. Let’s jump to the year 2026 and explore this beautiful world, almost surely a fantasy world, the world of P = NP. First let’s see how this world developed. The Urbana Algorithm In 2016, a Czech mathematician, Milena Pavel, emailed around a theoretical algorithm for solving

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solutions can also be found quickly. Conversely, if there is even one problem in NP where no algorithm can find a quick solution, then P ≠ NP. Determining whether P = NP is the most important question in computer science and perhaps in all of mathematics. Many have tried hard to find good algorithms for problems

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of NP. If you have an efficient algorithm for solving satisfiability, then all the problems whose solutions we can efficiently check have efficient algorithms, and P = NP. You just need one efficient algorithm for satisfiability and the Clay Institute’s million dollars is yours. Satisfiability is itself an NP problem since we

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expression true, making satisfiability the hardest of all problems in NP. Determining whether or not satisfiability has efficient algorithms is the same as determining whether P = NP. Steve Cook presented his paper at the STOC conference on May 4, 1971, in what was then the Stouffer’s Somerset Inn in Shaker Heights

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hardest problems in NP. Like satisfiability, if we had some quick algorithm to solve clique, then every problem in NP would have efficient algorithms and P = NP. Cook showed how to reduce problems like clique to satisfiability. Karp showed how to reduce satisfiability to clique. The very different-looking problems of satisfiability

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path, map coloring, and max-cut. Solve any of these problems efficiently and you have an efficient algorithm for all of them and have shown P = NP. If P ≠ NP, then none of these twenty-one problems (including satisfiability) has a quick algorithm. Karp did not invent these twenty-one problems, nor were these

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such an embarrassment, and in fact I’m willing to give a prize of one live turkey to the first person who proves that P = NP.” So prove P = NP and you will win a million dollars and a turkey. Beyond Karp After Karp’s paper, an industry in computer science arose showing the

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talks about the satisfiability problem and formulates the P versus NP question in different terminology. He suggests that if we lived in a world where P = NP, “the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by a machine. … Now it seems to me, however,

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Annals of the History of Computing 6, no. 4 (October 1984). Chapter 6 DEALING WITH HARDNESS IN CHAPTER 2 WE SAW THE BEAUTIFUL WORLD where P = NP, and life was good. We could optimize everything and learn anything. We had machines that could do just about anything we could imagine them to

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immediately after Cook, Karp, and Levin brought the problem and its importance to the world, computer scientists and mathematicians tried to formally prove that P = NP or P ≠ NP. All the usual techniques failed to work. By the end of the 1970s the general consensus was that “substantially new proof techniques will probably be

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versus NP problem, claiming to show P ≠ NP, or P = NP, or that it is impossible to determine whether or not P = NP, or that the P versus NP question doesn’t even make sense. Every year dozens of these manuscripts are emailed

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message. Hellman could use his private key on “tzljcnnfekktis” to recover the original message, “Attack at noon.” Is public-key cryptography even possible? Not if P = NP. If P = NP, there would be an efficient algorithm to recover the correct private key from the public key. Even by 1976 most computer scientists believed that

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could find the prime factors of numbers that are millions of digits long. P = NP would break the RSA protocol. P = NP breaks all public-key protocols, for if P = NP, you can always recover the private key from the public key. If P = NP, it is impossible to send secret messages to someone you have never communicated

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messages, all encryptions are equally likely, and it is mathematically impossible to learn anything about the message given the pad, no matter whether or not P = NP. So why don’t we all use one-time pads instead of the more complicated and possibly breakable schemes based on factoring numbers? You have

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created by some trusted organization (the U.S. government?) with some method of providing the same pad securely to Facebook and other companies as well. P = NP makes most things much easier, but it makes cryptography a bit more painful. Facebook could also use quantum mechanics to both create and transmit a

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off computation time for better pseudorandom generators is a tricky choice to make. Pseudorandom number generators work well if we have suitably complex problems. If P = NP, every efficiently computable process can be inverted in some sense. It would be hard, if not impossible, to create computationally random coin tosses. Rock-paper

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Grover algorithm is the best one can do for NP-complete problems using a quantum computer, so it is unlikely that a quantum version of P = NP could be true. Even if physicists managed to build quantum computers, our hardest problems would remain beyond their reach. That doesn’t mean quantum

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to the light bulbs we read by. How do we make the best use of this ultraconnected world? Whether or not it turns out that P = NP or P ≠ NP, thinking about P and NP shapes how we address these challenges. Parallel Computing In 1965, Gordon Moore noticed that the number of basic components

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random noise or redundant data. Finding the crucial useful part of a data collection is difficult, and then we need to interpret that information. If P = NP, we would have algorithms that could work their way through the data, working through the important parts and using tools based on the principle of

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Nearly everything in this chapter, except the section on Occam’s razor, is a figment of my imagination meant to illustrate the unlikely world of P = NP. Chapter 3 On Milgram’s experiment, see Stanley Milgram, “The Small World Problem,” Psychology Today 2, no. 1 (1967): 60–67. The Bacon number calculation

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, 35–36, 76–77 efficiency: of algorithms, 36; defining, 76–78 efficient computation. See P Einstein, Albert, 21 ellipsoid algorithm, 69–70 employment, effect of P = NP on, 27 Enigma machine, 124–26, 125 entertainment, automated creation of, 25 Entscheidungsproblem, 49 Erdős, Paul, 32, 32n Euler, Leonhard, 38–39 Eulerian paths, 39

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–83, 84, 167 Kolmogorov complexity, 83 Komsomol, 85 language, sentences in, 75, 75–76 language translation, 18, 23 Large Hadron Collider, 158 law enforcement, and P = NP, 25–26 laws of motion, 20–21 Levin, Leonid, 6, 79, 83–85 LHC Computing Grid, 158 Liapunov, Alexy, 79 liar’s paradox, 110–13

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, 16–19 map coloring: explanation of problem, 42–43, 43; heuristic for, 92–96, 93, 94, 95, 96; as NP, 46 market equilibrium, 49 marketing, P = NP and, 27 Martian rule, 86–87 Marxism, and probability theory, 81 Massachusetts Institute of Technology, 85 matchmaking, 33–36, 34, 35, 117–18 mathematics, NP

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–50 math puzzles, usefulness of, 4–5 Matsuoka, Yoky, 5–6, 165 max-cut problem, 45, 46 McCulloch, Warren, 75 McLean, Malcolm, 160 medicine, and P = NP, 14–15 Merkle, Roger, 127 messenger RNA (mRNA), 47–48 Meyer, Albert, 85 microprocessors: capabilities of, 90–92; parallel, 155, 156–57; speed of, 156

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, 80 Perelman, Grigori, 7, 12 personalized recommendations, 23, 25 physics, NP problems in, 48, 48 Pippenger, Nicholas, 157 Pitts, Walter, 75 P = NC, 157–58 P = NP: big data and, 159; cryptography and, 129–30; imagined possibilities of, 12–19, 23–27; implications of, ix, 6, 9, 10, 46; importance of question

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, 104 Poe, Edgar Allan, 124 Poincaré conjecture, 7 poker protocol, 137 polyalphabetic cipher, 124 polytope, 69–70, 70 prime numbers, 67–69, 129 privacy, and P = NP, 26–27 private-key cryptography, 26 probability theory, Kolmogorov and, 81–82, 167 products, in computations, 138 programs: contradictions in, 112; for hand control, 5

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–6 protein folding, 47–48 protein threading, 48 pseudorandomness, 140 public-key cryptography: factoring in, 140–41; P = NP and, 26, 127; randomness in, 136–37 public randomness, 136 P versus NP: circuit size in, 116; clique circuit computation and, 117; Eastern history of

Hands-On Machine Learning With Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems

by Aurelien Geron · 14 Aug 2019

can be reduced in polynomial time. An NP-Complete problem is both NP and NP-Hard. A major open mathematical question is whether or not P = NP. If P ≠ NP (which seems likely), then no polynomial algorithm will ever be found for any NP-Complete problem (except perhaps on a quantum computer). 3 log2

Hands-On Machine Learning With Scikit-Learn and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems

by Aurélien Géron · 13 Mar 2017 · 1,331pp · 163,200 words

can be reduced in polynomial time. An NP-Complete problem is both NP and NP-Hard. A major open mathematical question is whether or not P = NP. If P ≠ NP (which seems likely), then no polynomial algorithm will ever be found for any NP-Complete problem (except perhaps on a quantum computer). 3 log2

Natural language processing with Python

by Steven Bird, Ewan Klein and Edward Loper · 15 Dec 2009 · 504pp · 89,238 words

_pairs = nltk.bigrams(brown_news_tagged) >>> list(nltk.FreqDist(a[1] for (a, b) in word_tag_pairs if b[1] == 'N')) ['DET', 'ADJ', 'N', 'P', 'NP', 'NUM', 'V', 'PRO', 'CNJ', '.', ',', 'VG', 'VN', ...] This confirms our assertion that nouns occur after determiners and adjectives, including numeral adjectives (tagged as NUM). Verbs Verbs

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phrase: I shot an elephant in my pajamas. First we need to define a simple grammar: >>> ... ... ... ... ... ... ... ... ... groucho_grammar = nltk.parse_cfg(""" S -> NP VP PP -> P NP NP -> Det N | Det N PP | 'I' VP -> V NP | VP PP Det -> 'an' | 'my' N -> 'elephant' | 'pajamas' V -> 'shot' P -> 'in' """) 8.1 Some

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[SEM=?pp] VP[SEM=(?v + ?ap)] -> IV[SEM=?v] AP[SEM=?ap] NP[SEM=(?det + ?n)] -> Det[SEM=?det] N[SEM=?n] PP[SEM=(?p + ?np)] -> P[SEM=?p] NP[SEM=?np] AP[SEM=?pp] -> A[SEM=?a] PP[SEM=?pp] NP[SEM='Country="greece"'] -> 'Greece' NP[SEM='Country="china"'] -> 'China' Det[SEM='SELECT

Python for Finance

by Yuxing Yan · 24 Apr 2014 · 408pp · 85,118 words

as np, datetime ticker='AAPL' path='http://www.google.com/finance/getprices?q=ttt&i=60&p=1d&f=d,o,h,l,c ,v' p=np.array(pd.read_csv(path.replace('ttt',ticker),skiprows=7,header=No ne)) date=[] for i in arange(0,len(p)): if p[i][0

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their conversion is vital. Assume that we have four prices and we choose the first and last three prices as follows: >>>import numpy as np >>>p=np.array([1,1.1,0.9,1.05]) [ 185 ] Statistical Analysis of Time Series It is important how these prices are sorted. If the first

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quotes_historical_yahoo begdate=(2013,10,1) enddate=(2013,10,30) ticker='IBM' #WMT data= quotes_historical_yahoo(ticker, begdate, enddate,asobject=True, adjusted=True) p=np.array(data.aclose) dollar_vol=np.array(data.volume*p) ret=np.array((p[1:] - p[:-1])/p[1:]) illiq=mean(np.divide(abs(ret

The Simpsons and Their Mathematical Secrets

by Simon Singh · 29 Oct 2013 · 262pp · 65,959 words

arrangement of Euler’s equation is visible over Homer’s left shoulder. This equation also appears in “MoneyBART.” Finally, in the same image, the relationship P = NP can be seen over Homer’s right shoulder. Although the majority of viewers would not have noticed these three letters, let alone given them a

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second thought, P = NP represents a statement about one of the most important unsolved problems in theoretical computer science. P = NP is a statement concerning two types of mathematical problems. P stands for polynomial and NP for nondeterministic polynomial

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, this would jeopardize the security of everything from personal online purchases to high-level international political and military communications. The problem is often summarized as “P = NP or P ≠ NP?”, which asks the question: Will apparently difficult problems (NP) one day be shown to be just as easy as simple problems (P), or not

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? Finding the solution to the mystery of P = NP or P ≠ NP? is on the mathematicians’ most wanted list, and there is even a prize on its head. The Clay Mathematics Institute, established in Cambridge, Massachusetts

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, listed this puzzle as one of its seven Millennium Prize Problems in 2000, offering a $1 million reward for a definitive answer to the question P = NP or P ≠ NP? David S. Cohen, who explored P-type and NP-type problems while studying for his master’s degree in computer science at the University

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of California, Berkeley, has a hunch that NP-type problems are indeed much easier than we currently think, which is why P = NP appears behind Homer in his 3-D universe. However, Cohen holds a minority view. When William Gasarch, a computer scientist at the University of Maryland

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, polled one hundred researchers in 2002, only 9 percent thought that P = NP, while 61 percent favored P ≠ NP. He repeated the poll in 2010, and this time 81 percent favored P ≠ NP. Of course, truth in mathematics is not decided by a popularity contest, but if the majority turn

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out to be right, then Cohen’s positioning of P = NP in the landscape of “Homer3” will look somewhat incongruous. This, however, should not prove to be an issue in the short term, as half of

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be the case that I made a wise decision.” Although he has neither invented a radical new computing technology nor cracked the mystery of whether P = NP or P ≠ NP, Cohen still feels that he might have made an indirect contribution to research: “I really would have preferred to live my whole life as

Artificial Intelligence: A Modern Approach

by Stuart Russell and Peter Norvig · 14 Jul 2019 · 2,466pp · 668,761 words

P = NP look at a subclass of NP called the NP-complete problems. The word “complete” is used here in the sense of “most extreme” and thus

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propositional logic, is there an assignment of truth values to the proposition symbols of the sentence that makes it true? Unless a miracle occurs and P = NP, there can be no algorithm that solves all satisfiability problems in polynomial time. However, AI is more interested in whether there are algorithms that perform

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that PSPACE-hard problems are worse than NP-complete problems, although it could turn out that NP = PSPACE, just as it could turn out that P = NP. A.2Vectors, Matrices, and Linear Algebra Mathematicians define a vector as a member of a vector space, but we will use a more concrete definition

Algorithms to Live By: The Computer Science of Human Decisions

by Brian Christian and Tom Griffiths · 4 Apr 2016 · 523pp · 143,139 words

to check—but finding a route less than 1,000 miles, or proving that it’s impossible, is another feat entirely. The question of whether P = NP (i.e., whether it’s possible to jump efficiently to the solutions of NP problems) is the greatest unsolved mystery in computer science. The main

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are certain problems with a special status: if one of them can be solved efficiently, then any problem in NP can be solved efficiently and P = NP (Cook, “The Complexity of Theorem-Proving Procedures”). These are known as “NP-hard” problems. In the absence of an answer to whether

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P = NP, problems in NP cannot be solved efficiently, which is why we refer to them as “intractable.” (In “A Terminological Proposal,” Donald Knuth suggested this as

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an appropriate label for NP-hard problems, in addition to offering a live turkey to anybody who could prove P = NP.) The intractable scheduling problems that Eugene Lawler encountered in chapter 5 fall into this category. An NP-hard problem that is itself in NP is

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2002 survey of one hundred leading theoretical computer scientists, sixty-one thought P ≠ NP and only nine thought P = NP (Gasarch, “The P =? NP Poll”). While proving P = NP could be done by exhibiting a polynomial-time algorithm for an NP-complete problem, proving P ≠ NP requires making complex arguments about the limits of polynomial-time algorithms, and there

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six major open problems in mathematics for whose solutions the Clay Mathematics Institute will award a “Millennium Prize” of $1 million. The question of whether P = NP, which we saw in chapter 8, is also a Millennium Prize problem.) “Michael, this is Vaughan”: Rabin tells this story in Shasha and Lazere, Out

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, A. J. M., and P. Coles. “Bayesian Inductive Inference and the Anthropic Cosmological Principle.” Comments on Astrophysics. 17 (1993): 23–47. Gasarch, William I. “The P =? NP Poll.” SIGACT News 33, no. 2 (2002): 34–47. Gauthier, David P. Morals by Agreement. New York: Oxford University Press, 1985. Geman, Stuart, Elie Bienenstock

Elliptic Tales: Curves, Counting, and Number Theory

by Avner Ash and Robert Gross · 12 Mar 2012

verify that in the example we used before, where p = 7, the group has 8 elements.) We will be most concerned with the number ap = p − Np . Why is the formula for ap different for singular curves than for nonsingular 146 CHAPTER 9 Table 9.4. Counting points on singular cubics Type

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p, we worked out Np in chapter 9. The results, taken from table 9.4, are that Np = p if E has additive reduction at p, Np = p − 1 if E has split multiplicative reduction at p, and Np = p + 1 if E has nonsplit multiplicative reduction at p. Next we review

The Language Instinct: How the Mind Creates Language

by Steven Pinker · 1 Jan 1994 · 661pp · 187,613 words

, and an optional prepositional phrase.” VP VNP(PP) “A verb phrase can consist of a verb, a noun phrase, and an optional prepositional phrase.” PP P NP “A prepositional phrase can consist of a preposition and a noun phrase.” N boy, girl, dog, cat, ice cream, candy, hotdogs “The nouns in the