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P versus NP problem

The P versus NP problem is a major unsolved problem in computer science. [1]

133 relations: Advanced Encryption Standard, AKS primality test, Alexander Razborov, Algorithm, American Mathematical Society, Anil Nerode, Artificial intelligence, Average-case complexity, Avi Wigderson, Aviezri Fraenkel, Bonnie Berger, Boolean satisfiability problem, Certificate (complexity), Clay Mathematics Institute, Co-NP, Cobham's thesis, Complexity class, Composite number, Computational complexity theory, Computer programming, Constructive proof, Cook–Levin theorem, Cornell University, David Eppstein, Decision problem, Descriptive complexity theory, Deterministic automaton, Discrete logarithm, Donald Knuth, Economics, Elementary (TV series), Entscheidungsproblem, Eugene M. Luks, EXPTIME, F. Thomson Leighton, Fermat's Last Theorem, First-order logic, Fixed-point combinator, Game complexity, Game theory, General number field sieve, Gerhard J. Woeginger, Graph (mathematics), Graph isomorphism, Graph isomorphism problem, Halting problem, Hash function, HP Labs, Independence (mathematical logic), Information-theoretic security, ..., InformIT, Integer, Integer factorization, Integer programming, Introduction to Algorithms, IP (complexity), John Markoff, John von Neumann, Journal of the Operational Research Society, Karp's 21 NP-complete problems, Knapsack problem, Kurt Gödel, Lance Fortnow, László Babai, Leonid Levin, Linear programming, List of NP-complete problems, List of unsolved problems in computer science, List of unsolved problems in mathematics, Massachusetts Institute of Technology, Michael O. Rabin, Millennium Prize Problems, MIT Press, Moshe Vardi, Natural proof, Neil Immerman, Non-deterministic Turing machine, NP (complexity), NP-completeness, NP-hardness, One-way function, Operations research, Oracle machine, P (complexity), Palo Alto, California, Peano axioms, PH (complexity), Philosophy, Polymath Project, Polynomial, Polynomial hierarchy, Presburger arithmetic, Princeton University, Protein structure prediction, PSPACE, Public-key cryptography, Quantum algorithm, Quantum computing, Randomized algorithm, Reduction (complexity), Rice University, Richard J. Lipton, Robert M. Solovay, RSA (cryptosystem), Russell Impagliazzo, Scott Aaronson, Second-order logic, Shor's algorithm, SIAM Journal on Computing, Signature (logic), Simplex algorithm, Stephen Cook, Steven Rudich, String (computer science), Subset, Subset sum problem, Symmetric-key algorithm, Symposium on Theoretical Aspects of Computer Science, The New Yorker, Theoretical computer science, Theory of computation, Time complexity, Time hierarchy theorem, Total order, Travelling Salesman (2012 film), Travelling salesman problem, Triple DES, Turing machine, Undecidable problem, Unique games conjecture, Universal quantification, W. H. Freeman and Company, Zermelo–Fraenkel set theory. Expand index (83 more) »

Advanced Encryption Standard

The Advanced Encryption Standard (AES), also known as Rijndael (its original name), is a specification for the encryption of electronic data established by the U.S. National Institute of Standards and Technology (NIST) in 2001.

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AKS primality test

The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled "PRIMES is in P".

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Alexander Razborov

Aleksandr Aleksandrovich Razborov (Алекса́ндр Алекса́ндрович Разбо́ров; born February 16, 1963), sometimes known as Sasha Razborov, is a Soviet and Russian mathematician and computational theorist.

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Algorithm

In mathematics and computer science, an algorithm is a self-contained step-by-step set of operations to be performed.

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American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

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Anil Nerode

Anil Nerode (born 1932) is an American mathematician.

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Artificial intelligence

Artificial intelligence (AI) is the intelligence exhibited by machines or software.

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Average-case complexity

In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the algorithm, averaged over all possible inputs.

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Avi Wigderson

Avi Wigderson (אבי ויגדרזון; born 9 September 1956) is an Israeli mathematician and computer scientist.

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Aviezri Fraenkel

Aviezri Siegmund Fraenkel (אביעזרי פרנקל) (born June 7, 1929) is an Israeli mathematician who has made notable contributions to combinatorial game theory.

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Bonnie Berger

Bonnie Anne Berger is an American mathematician and computer scientist, who works as a professor of applied mathematics and computer science at the Massachusetts Institute of Technology.

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Boolean satisfiability problem

In computer science, the Boolean Satisfiability Problem (sometimes called Propositional Satisfiability Problem and abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula.

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Certificate (complexity)

In computational complexity theory, a certificate (also called a witness) is a string that certifies the answer to a computation, or certifies the membership of some string in a language.

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Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Providence, Rhode Island.

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Co-NP

In computational complexity theory, co-NP is a complexity class.

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Cobham's thesis

Cobham's thesis, also known as Cobham–Edmonds thesis (named after Alan Cobham and Jack Edmonds), asserts that computational problems can be feasibly computed on some computational device only if they can be computed in polynomial time; that is, if they lie in the complexity class P. Formally, to say that a problem can be solved in polynomial time is to say that there exists an algorithm that, given an n-bit instance of the problem as input, can produce a solution in time O(nc), where c is a constant that depends on the problem but not the particular instance of the problem.

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Complexity class

In computational complexity theory, a complexity class is a set of problems of related resource-based complexity.

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Composite number

A composite number is a positive integer that has at least one positive divisor other than one or the number itself.

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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.

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Computer programming

Computer programming (often shortened to programming) is a process that leads from an original formulation of a computing problem to executable computer programs.

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Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object.

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Cook–Levin theorem

In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete.

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Cornell University

Cornell University is an American private Ivy League and federal land-grant research university located in Ithaca, New York.

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David Eppstein

David Arthur Eppstein (born 1963) is an American computer scientist and mathematician.

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Decision problem

In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters.

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Descriptive complexity theory

Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them.

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Deterministic automaton

In computer science, a deterministic automaton is a concept of automata theory in which the outcome of a transition from one state to another is determined by the input.

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Discrete logarithm

In mathematics, a discrete logarithm is an integer k solving the equation, where b and g are elements of a finite group.

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Donald Knuth

Donald Ervin Knuth (born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University.

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Economics

Economics is the social science that seeks to describe the factors which determine the production, distribution and consumption of goods and services.

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Elementary (TV series)

Elementary is an American crime drama series that presents a contemporary update of Sir Arthur Conan Doyle's character Sherlock Holmes.

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Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem (German for 'decision problem') is a challenge posed by David Hilbert in 1928.

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Eugene M. Luks

Eugene Michael Luks (born circa 1940) is an American mathematician and computer scientist, a professor emeritus of computer and information science at the University of Oregon.

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EXPTIME

In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. In terms of DTIME, We know and also, by the time hierarchy theorem and the space hierarchy theorem, that so at least one of the first three inclusions and at least one of the last three inclusions must be proper, but it is not known which ones are.

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F. Thomson Leighton

Frank Thomson "Tom" Leighton is a professor of Applied Mathematics at the Massachusetts Institute of Technology and the CEO of Akamai Technologies.

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Fermat's Last Theorem

In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn.

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First-order logic

First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science.

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Fixed-point combinator

In computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function y that satisfies the equation, It is so named because, by setting x.

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Game complexity

Combinatorial game theory has several ways of measuring game complexity.

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Game theory

Game theory is the study of strategic decision-making.

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General number field sieve

In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 100 digits.

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Gerhard J. Woeginger

Gerhard J. Woeginger is an Austrian mathematician and computer scientist who works in the Netherlands as a professor at the Eindhoven University of Technology (TU/e), where he chairs the combinatorial optimization group in the department of mathematics and computer science.

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Graph (mathematics)

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links.

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Graph isomorphism

In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is generally called "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.

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Graph isomorphism problem

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic.

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Halting problem

In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running or continue to run forever.

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Hash function

A hash function is any function that can be used to map data of arbitrary size to data of fixed size.

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HP Labs

HP Labs (or HP Laboratories) is the exploratory and advanced research group for Hewlett-Packard.

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Independence (mathematical logic)

In mathematical logic, independence refers to the unprovability of a sentence from other sentences.

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Information-theoretic security

A cryptosystem is information-theoretically secure if its security derives purely from information theory.

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InformIT

InformIT, a subsidiary of Pearson Education, is an online book vendor and an electronic publisher of technology and education content.

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Integer

An integer (from the Latin ''integer'' meaning "whole")Integer 's first, literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

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Integer factorization

In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers.

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Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.

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Introduction to Algorithms

Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein.

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IP (complexity)

In computational complexity theory, the class IP (which stands for Interactive Polynomial time) is the class of problems solvable by an interactive proof system.

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John Markoff

John Markoff (born October 24, 1949) is a journalist best known for his work at The New York Times, and a book and series of articles about the 1990s pursuit and capture of hacker Kevin Mitnick.

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John von Neumann

John von Neumann (Hungarian: Neumann János,; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, polymath, and polyglot.

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Journal of the Operational Research Society

The Journal of the Operational Research Society is a peer-reviewed academic journal covering operations research.

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Karp's 21 NP-complete problems

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.

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Knapsack problem

The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a mass and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

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Kurt Gödel

Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.

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Lance Fortnow

Lance Jeremy Fortnow (born August 15, 1963) is a computer scientist known for major results in computational complexity and interactive proof systems.

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László Babai

László (Laci) Babai (born July 20, 1950 in Budapest) from Babai's web site, retrieved 2010-07-30.

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Leonid Levin

Leonid Anatolievich Levin (Леони́д Анато́льевич Ле́вин; Леоні́д Анато́лійович Ле́він; born November 2, 1948) is a Soviet-American computer scientist.

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Linear programming

Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

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List of NP-complete problems

This is a list of some of the more commonly known problems that are NP-complete when expressed as decision problems.

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List of unsolved problems in computer science

This article is a list of unsolved problems in computer science.

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List of unsolved problems in mathematics

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still elude solution.

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Massachusetts Institute of Technology

The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts.

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Michael O. Rabin

Michael Oser Rabin (מִיכָאֵל עוזר רַבִּין, born September 1, 1931), is an Israeli computer scientist and a recipient of the Turing Award.

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Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.

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MIT Press

The MIT Press is a university press affiliated with the Massachusetts Institute of Technology (MIT) in Cambridge, Massachusetts (United States).

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Moshe Vardi

Moshe Ya'akov Vardi (משה יעקב ורדי) is an Israeli computer scientist.

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Natural proof

In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one.

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Neil Immerman

Neil Immerman (24 November 1953, Manhasset, New York) is an American theoretical computer scientist, a professor of computer science at the University of Massachusetts Amherst.

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Non-deterministic Turing machine

In theoretical computer science, a Turing machine is a theoretical machine that is used in thought experiments to examine the abilities and limitations of computers.

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NP (complexity)

In computational complexity theory, NP is one of the most fundamental complexity classes.

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NP-completeness

In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard.

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NP-hardness

NP-hardness (''n''on-deterministic ''p''olynomial-time hard), in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP".

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One-way function

In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input.

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Operations research

Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions.

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Oracle machine

In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems.

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P (complexity)

In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is one of the most fundamental complexity classes.

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Palo Alto, California

Palo Alto (Spanish: palo: literally "stick", colloquial: "tree" and alto: "tall"; meaning: "tall tree") is a charter city located in the northwest corner of Santa Clara County, California, in the San Francisco Bay Area of the United States.

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Peano axioms

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.

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PH (complexity)

In computational complexity theory, the complexity class PH is the union of all complexity classes in the polynomial hierarchy: PH was first defined by Larry Stockmeyer.

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Philosophy

Philosophy is the study of the general and fundamental nature of reality, existence, knowledge, values, reason, mind, and language.

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Polymath Project

The Polymath Project is a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

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Polynomial hierarchy

In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes P, NP and co-NP to oracle machines.

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Presburger arithmetic

Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929.

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Princeton University

Princeton University is a private Ivy League research university in Princeton, New Jersey, United States.

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Protein structure prediction

Protein structure prediction is the prediction of the three-dimensional structure of a protein from its amino acid sequence — that is, the prediction of its folding and its secondary, tertiary, and quaternary structure from its primary structure.

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PSPACE

In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space.

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Public-key cryptography

Public-key cryptography, also known as asymmetric cryptography, is a class of cryptographic protocols based on algorithms that require two separate keys, one of which is secret (or private) and one of which is public.

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Quantum algorithm

In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation.

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Quantum computing

Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data.

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Randomized algorithm

A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic.

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Reduction (complexity)

In computability theory and computational complexity theory, a reduction is an algorithm for transforming one problem into another problem.

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Rice University

William Marsh Rice University, commonly referred to as Rice University or Rice, is a private research university located on a campus in Houston, Texas, United States.

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Richard J. Lipton

Richard Jay "Dick" Lipton (born September 6, 1946) is an American computer scientist who has worked in computer science theory, cryptography, and DNA computing.

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Robert M. Solovay

Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.

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RSA (cryptosystem)

RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission.

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Russell Impagliazzo

Russell Impagliazzo is a professor of computer science at the University of California, San Diego.

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Scott Aaronson

Scott Joel Aaronson (born May 21, 1981) is a theoretical computer scientist and faculty member in the Electrical Engineering and Computer Science department at the Massachusetts Institute of Technology.

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Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic.

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Shor's algorithm

Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization formulated in 1994.

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SIAM Journal on Computing

The SIAM Journal on Computing (SICOMP) is a scientific journal focusing on the mathematical and formal aspects of computer science.

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Signature (logic)

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language.

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Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.

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Stephen Cook

Stephen Arthur Cook, (born December 14, 1939) is a renowned American-Canadian computer scientist and mathematician who has made major contributions to the fields of complexity theory and proof complexity.

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Steven Rudich

Steven Rudich (born October 4, 1961) is a professor in the Carnegie Mellon School of Computer Science.

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String (computer science)

In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.

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Subset

In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

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Subset sum problem

In computer science, the subset sum problem is one of the important problems in complexity theory and cryptography.

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Symmetric-key algorithm

Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext.

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Symposium on Theoretical Aspects of Computer Science

Symposium on Theoretical Aspects of Computer Science (STACS) is an academic conference in the field of computer science.

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The New Yorker

The New Yorker is an American magazine of reportage, commentary, criticism, essays, fiction, satire, cartoons, and poetry.

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Theoretical computer science

Theoretical computer science is a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation.

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Theory of computation

In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm.

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Time complexity

In computer science, the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input.

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Time hierarchy theorem

In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines.

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Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X which is transitive, antisymmetric, and total.

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Travelling Salesman (2012 film)

Travelling Salesman is a 2012 intellectual thriller film about four mathematicians solving the P versus NP problem, one of the most challenging mathematical problems in history.

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Travelling salesman problem

The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.

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Triple DES

In cryptography, Triple DES (3DES) is the common name for the Triple Data Encryption Algorithm (TDEA or Triple DEA) symmetric-key block cipher, which applies the Data Encryption Standard (DES) cipher algorithm three times to each data block.

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Turing machine

A Turing machine is an abstract "machine" that manipulates symbols on a strip of tape according to a table of rules; to be more exact, it is a mathematical model that defines such a device.

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Undecidable problem

In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is known to be impossible to construct a single algorithm that always leads to a correct yes-or-no answer.

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Unique games conjecture

In computational complexity theory, the Unique Games Conjecture is a conjecture made by Subhash Khot in 2002.

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Universal quantification

In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".

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W. H. Freeman and Company

W.

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Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets free of paradoxes such as Russell's paradox.

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References

[1] https://en.wikipedia.org/wiki/P_versus_NP_problem

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