background preloader

P versus NP problem

P versus NP problem
Diagram of complexity classes provided that P≠NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem.[1] The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether a certain NP complete problem could be solved in quadratic or linear time.[2] The precise statement of the P=NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures"[3] and is considered by many to be the most important open problem in the field.[4] It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution. Context[edit] Related:  ComputationalAIMachine learning/nlp/AI

Time complexity Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, where an elementary operation takes a fixed amount of time to perform. Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor. Since an algorithm's performance time may vary with different inputs of the same size, one commonly uses the worst-case time complexity of an algorithm, denoted as T(n), which is defined as the maximum amount of time taken on any input of size n. Time complexities are classified by the nature of the function T(n). For instance, an algorithm with T(n) = O(n) is called a linear time algorithm, and an algorithm with T(n) = O(2n) is said to be an exponential time algorithm. Table of common time complexities[edit] The following table summarizes some classes of commonly encountered time complexities. Constant time[edit] Logarithmic time[edit] Polylogarithmic time[edit] and .

Computational complexity theory Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Closely related fields in theoretical computer science are analysis of algorithms and computability theory. Computational problems[edit] Problem instances[edit] Turing machine[edit]

Non-Human Consciousness Exists Say Experts. Now What? Non-Human Consciousness Exists Say Experts. Now What? Phillip Low at Singularity University Have you ever considered the consciousness, or unconsciousness, of your dog? Earlier this month, some of the leading scientists from around the world congregated at the Hotel Du Vin in Cambridge to discuss the evidence that has amassed over the years. the declaration of consciousness Organized by Philip Low, CEO of NeuroVigil and inventor of the iBrain, the group consisted of 25 of the planet’s top minds on the mind, including honorary guest Stephen Hawking. This announcement arises in a manner similar to the Pluto files in 2006, when the world's leading astronomer's demoted Pluto from planet to "dwarf planet". As mankind continues to explore the universe, many more discoveries will prompt an official announcement such as the one Phillip Low delivered this week at Singularity University. For more on the conference and the Cambridge declaration, check out

NP - Wikipedia From Wikipedia, the free encyclopedia NP may refer to: Arts and entertainment[edit] Organizations[edit] Places[edit] NP postcode area, Newport, Wales, United KingdomNepal (ISO 3166-1 alpha-2 country code NP) .np, the country code top level domain (ccTLD) for NepalNichols Point, Australia Science, technology and mathematics[edit] Biology and medicine[edit] Mathematics and computer science[edit] Physics and chemistry[edit] NP junction or PN junction, the simplest electronic device, used to make diodes and transistorsNeper (Np), a dimensionless logarithmic unit for ratios of measurements of physical field and power quantitiesNeptunium, a chemical element with symbol NpPower number (Np), a dimensionless number relating the resistance force to the inertia force Other uses in science and technology[edit] Other uses[edit]

Numbers: Facts, Figures & Fiction Click on cover for larger image Numbers: Facts, Figures & Fiction by Richard Phillips. Published by Badsey Publications. See sample pages: 24, 82, 103. Order the book direct from Badsey Publications price £12. In Australia it is sold by AAMT and in the US by Parkwest. For those who need a hardback copy, a limited number of the old 1994 hardback edition are still available. Have you ever wondered how Room 101 got its name, or what you measure in oktas? This new edition has been updated with dozens of new articles, illustrations and photographs. Some press comments – "This entertaining and accessible book is even more attractive in its second edition..." – Jennie Golding in The Mathematical Gazette "...tangential flights into maths, myth and mystery..." – Vivienne Greig in New Scientist ... and on the first edition – Contents – Title: The Mystery of the Binary Author: Viznut Originally published in the [ALT] magazine issue 0x0000 The subcultures of computing are very young. There are no legends nor values that come from the distant past. No ancient mysticism, no generations-old symbols that have deep emotional effects. Even the old and classical things tend to be quite recent. Stone-age binary counting We are so completely surrounded by decimal numbers that most people believe humans were "built" to count in base ten: "we have ten fingers, you know." Divine binary protocols The human mind has a strange relationship with randomness. The Two Symbols There's a great deal of similarity between the African and Chinese notations for binary combinations: both use vertical piles where a single bit is designated by one or two "somethings", such as sticks, stones or seeds. The Eight Trigrams This figure shows the eight trigrams (the three-bit numbers) in the millennia-old Chinese Xiantian arrangement. The Sixteen Tetragrams

Automated Grading Software In Development To Score Essays As Accurately As Humans Roboreaders that can score essays in standardized tests could also help teachers grade and students becomes better writers. April 30 marks the deadline for a contest challenging software developers to create an automated scorer of student essays, otherwise known as a roboreader, that performs as good as a human expert grader. In January, the Hewlett Foundation of Hewlett-Packard fame introduced the Automated Student Assessment Prize (ASAP…get it?) offering up $100,000 in awards to “data scientists and machine learning specialists” to develop the application. The contest is only the first of three, with the others aimed at developing automated graders for short answers and charts and graphs. Developers of reliable roboreaders will not just rake in massive loads of cash thrown at them by standardized testing companies, educational publishers, and school districts, but they’ll potentially change the way writing is taught forever. [Media: Eduify] [Sources: Citypages, Kaggle, PBS, Reuters]

NP-hardness - Wikipedia Definition[edit] A decision problem H is NP-hard when for every problem L in NP, there is a polynomial-time reduction from L to H.[1]:80 An equivalent definition is to require that every problem L in NP can be solved in polynomial time by an oracle machine with an oracle for H.[7] Informally, we can think of an algorithm that can call such an oracle machine as a subroutine for solving H, and solves L in polynomial time, if the subroutine call takes only one step to compute. Another definition is to require that there is a polynomial-time reduction from an NP-complete problem G to H.[1]:91 As any problem L in NP reduces in polynomial time to G, L reduces in turn to H in polynomial time so this new definition implies the previous one. Awkwardly, it does not restrict the class NP-hard to decision problems, for instance it also includes search problems, or optimization problems. Consequences[edit] If P ≠ NP, then NP-hard problems cannot be solved in polynomial time. Examples[edit] NP-hard NP-easy

Primality Proving 2.1: Finding very small primes For finding all the small primes, say all those less than 10,000,000,000; one of the most efficient ways is by using the Sieve of Eratosthenes (ca 240 BC): Make a list of all the integers less than or equal to n (greater than one) and strike out the multiples of all primes less than or equal to the square root of n, then the numbers that are left are the primes. (See also our glossary page.) For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens?) The first number is 3 so it is the first odd prime--cross out all of its multiples. Now the first number left is 5, the second odd prime--cross out all of its multiples. This method is so fast that there is no reason to store a large list of primes on a computer--an efficient implementation can find them faster than a computer can read from a disk. To find individual small primes trial division works well.