119-418: The P versus NP problem is a major unsolved problem in theoretical computer science . Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm that solves the task and runs in polynomial time (as opposed to, say, exponential time ) exists, meaning the task completion time is bounded above by a polynomial function on
238-526: A differential geometer , he learned about the conjecture that any Riemannian manifold is isometric to a submanifold of Euclidean space . Nash's results proving the conjecture are now known as the Nash embedding theorems , the second of which Mikhael Gromov has called "one of the main achievements of mathematics of the twentieth century". Nash's first embedding theorem was found in 1953. He found that any Riemannian manifold can be isometrically embedded in
357-509: A Euclidean space by a continuously differentiable mapping. Nash's construction allows the codimension of the embedding to be very small, with the effect that in many cases it is logically impossible that a highly-differentiable isometric embedding exists. (Based on Nash's techniques, Nicolaas Kuiper soon found even smaller codimensions, with the improved result often known as the Nash–Kuiper theorem .) As such, Nash's embeddings are limited to
476-590: A body of results paving the way for a systematic understanding of elliptic and parabolic partial differential equations . Their De Giorgi–Nash theorem on the smoothness of solutions of such equations resolved Hilbert's nineteenth problem on regularity in the calculus of variations , which had been a well-known open problem for almost sixty years. In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated for schizophrenia . After 1970, his condition slowly improved, allowing him to return to academic work by
595-532: A ceremony in Oslo. In 1951, the Massachusetts Institute of Technology (MIT) hired Nash as a C. L. E. Moore instructor in the mathematics faculty. About a year later, Nash began a relationship with Eleanor Stier, a nurse he met while admitted as a patient. They had a son, John David Stier, but Nash left Stier when she told him of her pregnancy. The film based on Nash's life, A Beautiful Mind ,
714-561: A classical (non-quantum) computer? Can the discrete logarithm be computed in polynomial time on a classical (non-quantum) computer? Can the shortest vector of a lattice be computed in polynomial time on a classical or quantum computer? Can the graph isomorphism problem be solved in polynomial time on a classical computer? Is graph canonization polynomial time equivalent to the graph isomorphism problem? Can leaf powers and k -leaf powers be recognized in polynomial time? Can parity games be solved in polynomial time? Can
833-530: A consultant. Not long after breaking up with Stier, Nash met Alicia Lardé Lopez-Harrison , a naturalized U.S. citizen from El Salvador . Lardé was graduated from MIT , having majored in physics. They married in February 1957. Although Nash was an atheist , the ceremony was performed in an Episcopal church . In 1958, Nash was appointed to a tenured position at MIT, and his first signs of mental illness soon became evident. He resigned his position at MIT in
952-576: A deterministic fixed gap sequence? Can 3SUM be solved in strongly sub-quadratic time, that is, in time O ( n ) for some ϵ>0 ? Can the edit distance between two strings of length n be computed in strongly sub-quadratic time? (This is only possible if the strong exponential time hypothesis is false.) Can X + Y sorting be done in o ( n log n ) time? What is the fastest algorithm for matrix multiplication ? Can all-pairs shortest paths be computed in strongly sub-cubic time, that is, in time O ( V ) for some ϵ>0 ? Can
1071-418: A finite number of possible grids. In this case the problem is in P, as the answer can be found by table lookup.) The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" (and independently by Leonid Levin in 1973). Although the P versus NP problem was formally defined in 1971, there were previous inklings of
1190-576: A full benefit of the George Westinghouse Scholarship, initially majoring in chemical engineering . He switched to a chemistry major and eventually, at the advice of his teacher John Lighton Synge , to mathematics. After graduating in 1948, with both a B.S. and M.S. in mathematics, Nash accepted a fellowship to Princeton University , where he pursued further graduate studies in mathematics and sciences. Nash's adviser and former Carnegie professor Richard Duffin wrote
1309-596: A geometrical language, the work is almost entirely to do with the mathematical analysis of partial differential equations . After proving his two isometric embedding theorems , Nash turned to research dealing directly with partial differential equations, where he discovered and proved the De Giorgi–Nash theorem, thereby resolving one form of Hilbert's nineteenth problem . In 2011, the National Security Agency declassified letters written by Nash in
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#17327805841751428-532: A graduate student, Nash found a new result in the mathematical field of real algebraic geometry . He announced his theorem in a contributed paper at the International Congress of Mathematicians in 1950, although he had not yet worked out the details of its proof. Nash's theorem was finalized by October 1951, when Nash submitted his work to the Annals of Mathematics . It had been well-known since
1547-606: A human of more conventional circumstances and return to mathematical research. In these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le problème de Cauchy pour les équations différentielles d'un fluide général"; the idea that Prof. Heisuke Hironaka called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data". But after my return to
1666-610: A letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius." Nash was also accepted at Harvard University . However, the chairman of the mathematics department at Princeton, Solomon Lefschetz , offered him the John S. Kennedy fellowship, convincing Nash that Princeton valued him more. Further, he considered Princeton more favorably because of its proximity to his family in Bluefield. At Princeton, he began work on his equilibrium theory, later known as
1785-491: A limited nesting depth of Kleene stars ? Separating words problem : How many states are needed in a deterministic finite automaton that behaves differently on two given strings of length n {\displaystyle n} ? What is the Turing completeness status of all unique elementary cellular automata ? The problem to determine all positive integers n {\displaystyle n} such that
1904-409: A machine. After all, one would simply have to choose the natural number n so large that when the machine does not deliver a result, it makes no sense to think more about the problem. Similarly, Stephen Cook (assuming not only a proof, but a practically efficient algorithm) says: ... it would transform mathematics by allowing a computer to find a formal proof of any theorem which has a proof of
2023-482: A mechanical method that could solve any problem would revolutionize mathematics: If there really were a machine with φ( n ) ∼ k ⋅ n (or even ∼ k ⋅ n ), this would have consequences of the greatest importance. Namely, it would obviously mean that in spite of the undecidability of the Entscheidungsproblem , the mental work of a mathematician concerning Yes-or-No questions could be completely replaced by
2142-715: A messenger or having a special function of some kind, of having supporters and opponents and hidden schemers, along with a feeling of being persecuted and searching for signs representing divine revelation. During his psychotic phase, Nash also referred to himself in the third person as "Johann von Nassau". Nash suggested his delusional thinking was related to his unhappiness, his desire to be recognized, and his characteristic way of thinking, saying, "I wouldn't have had good scientific ideas if I had thought more normally." He also said, "If I felt completely pressureless I don't think I would have gone in this pattern". Nash reported that he started hearing voices in 1964, then later engaged in
2261-426: A process of consciously rejecting them. He only renounced his "dream-like delusional hypotheses" after a prolonged period of involuntary commitment in mental hospitals—"enforced rationality". Upon doing so, he was temporarily able to return to productive work as a mathematician. By the late 1960s, he relapsed. Eventually, he "intellectually rejected" his "delusionally influenced" and "politically oriented" thinking as
2380-488: A proof either way would have profound implications for mathematics, cryptography , algorithm research, artificial intelligence , game theory , multimedia processing, philosophy , economics and many other fields. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute , each of which carries a US$ 1,000,000 prize for the first correct solution. Consider
2499-580: A proof if a "reasonable" size proof exists, would essentially end this struggle. Donald Knuth has stated that he has come to believe that P = NP, but is reserved about the impact of a possible proof: [...] if you imagine a number M that's finite but incredibly large—like say the number 10↑↑↑↑3 discussed in my paper on "coping with finiteness"—then there's a humongous number of possible algorithms that do n bitwise or addition or shift operations on n given bits, and it's really hard to believe that all of those algorithms fail. My main point, however,
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#17327805841752618-466: A reasonable length, since formal proofs can easily be recognized in polynomial time. Example problems may well include all of the CMI prize problems . Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries to prove. A method guaranteed to find
2737-429: A recipient of the prestigious Fields Medal in 1958. Although the medal committee's reasoning is not fully known, and was not purely based on questions of mathematical merit, archival research has shown that Nash placed third in the committee's vote for the medal, after the two mathematicians ( Klaus Roth and René Thom ) who were awarded the medal that year. Although Nash's mental illness first began to manifest in
2856-487: A restatement of the problem in The P Versus NP Problem as "Does P = NP?" According to polls, most computer scientists believe that P ≠ NP. A key reason for this belief is that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than 3,000 important known NP-complete problems (see List of NP-complete problems ). These algorithms were sought long before
2975-571: A solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice. In this case the initial proof would be mainly of interest to theoreticians, but the knowledge that polynomial time solutions are possible would surely spur research into better (and possibly practical) methods to achieve them. A solution showing P = NP could upend
3094-480: A solution to the problem of partitioning tri-partite graphs into triangles, which could then be used to find solutions for the special case of SAT known as 3-SAT, which then provides a solution for general Boolean satisfiability. So a polynomial-time solution to Sudoku leads, by a series of mechanical transformations, to a polynomial time solution of satisfiability, which in turn can be used to solve any other NP-problem in polynomial time. Using transformations like this,
3213-427: A speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required. When one substitutes "linear time on a multitape Turing machine" for "polynomial time" in the definitions of P and NP, one obtains
3332-457: A vast class of seemingly unrelated problems are all reducible to one another, and are in a sense "the same problem". Although it is unknown whether P = NP, problems outside of P are known. Just as the class P is defined in terms of polynomial running time, the class EXPTIME is the set of all decision problems that have exponential running time. In other words, any problem in EXPTIME
3451-523: A waste of effort. In 1995, he said that he did not realize his full potential due to nearly 30 years of mental illness. Nash wrote in 1994: I spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release. And it did happen that when I had been long enough hospitalized that I would finally renounce my delusional hypotheses and revert to thinking of myself as
3570-459: A well-known conjecture in the field of elliptic partial differential equations . In 1938, Charles Morrey had proved a fundamental elliptic regularity result for functions of two independent variables, but analogous results for functions of more than two variables had proved elusive. After extensive discussions with Nirenberg and Lars Hörmander , Nash was able to extend Morrey's results, not only to functions of more than two variables, but also to
3689-591: Is a list of notable unsolved problems in computer science . A problem in computer science is considered unsolved when no solution is known or when experts in the field disagree about proposed solutions. Computational complexity [ edit ] Main article: Computational complexity theory P versus NP problem What is the relationship between BQP and NP ? NC = P problem NP = co-NP problem P = BPP problem P = PSPACE problem L = NL problem PH = PSPACE problem L = P problem L = RL problem Unique games conjecture Is
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3808-422: Is at least one input for which that algorithm will not produce the right answer; it will either produce the wrong answer, finish without giving a conclusive answer, or otherwise run forever without producing any answer at all. It is also possible to consider questions other than decision problems. One such class, consisting of counting problems, is called #P : whereas an NP problem asks "Are there any solutions?",
3927-538: Is different from Wikidata Dynamic lists All articles with unsourced statements Articles with unsourced statements from September 2024 John Forbes Nash Jr. John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015), known and published as John Nash , was an American mathematician who made fundamental contributions to game theory , real algebraic geometry , differential geometry , and partial differential equations . Nash and fellow game theorists John Harsanyi and Reinhard Selten were awarded
4046-464: Is known. From the definition alone it is unintuitive that NP-complete problems exist; however, a trivial NP-complete problem can be formulated as follows: given a Turing machine M guaranteed to halt in polynomial time, does a polynomial-size input that M will accept exist? It is in NP because (given an input) it is simple to check whether M accepts the input by simulating M ; it is NP-complete because
4165-528: Is now known as the De Giorgi–Nash theorem or the De Giorgi–Nash–Moser theory (which is distinct from the Nash–Moser theorem ). De Giorgi and Moser's methods became particularly influential over the next several years, through their developments in the works of Olga Ladyzhenskaya , James Serrin , and Neil Trudinger , among others. Their work, based primarily on the judicious choice of test functions in
4284-572: Is overconfident to believe P ≠ NP and that researchers should also explore proofs of P = NP. For example, in 2002 these statements were made: The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. [...] The resolution of Fermat's Last Theorem also shows that very simple questions may be settled only by very deep theories. Being attached to
4403-523: Is reducible to in polynomial time and whose solution is still verifiable in polynomial time. That is, any NP problem can be transformed into any NP-complete problem. Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP. NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time. For instance,
4522-443: Is solvable by a deterministic Turing machine in O (2) time, where p ( n ) is a polynomial function of n . A decision problem is EXPTIME-complete if it is in EXPTIME, and every problem in EXPTIME has a polynomial-time many-one reduction to it. A number of problems are known to be EXPTIME-complete. Because it can be shown that P ≠ EXPTIME, these problems are outside P, and so require more than polynomial time. In fact, by
4641-455: Is straightforward to verify "yes" instances of this generalized Sudoku problem given a candidate solution. However, it is not known whether there is a polynomial-time algorithm that can correctly answer "yes" or "no" to all instances of this problem. Therefore, generalized Sudoku is in NP (quickly verifiable), but may or may not be in P (quickly solvable). (It is necessary to consider a generalized version of Sudoku, as any fixed size Sudoku has only
4760-429: Is studied in computational complexity theory , the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are time (how many steps it takes to solve a problem) and space (how much memory it takes to solve a problem). In such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that
4879-462: Is that I don't believe that the equality P = NP will turn out to be helpful even if it is proved, because such a proof will almost surely be nonconstructive. A proof of P ≠ NP would lack the practical computational benefits of a proof that P = NP, but would represent a great advance in computational complexity theory and guide future research. It would demonstrate that many common problems cannot be solved efficiently, so that
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4998-465: Is the simplex algorithm in linear programming , which works surprisingly well in practice; despite having exponential worst-case time complexity , it runs on par with the best known polynomial-time algorithms. Finally, there are types of computations which do not conform to the Turing machine model on which P and NP are defined, such as quantum computation and randomized algorithms . Cook provides
5117-576: Is the Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turing machines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof by reduction provided a simpler way to show that many other problems are also NP-complete, including the game Sudoku discussed earlier. In this case, the proof shows that a solution of Sudoku in polynomial time could also be used to complete Latin squares in polynomial time. This in turn gives
5236-537: Is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k . No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If
5355-607: Is the number of vertices in H . On the other hand, even if a problem is shown to be NP-complete, and even if P ≠ NP, there may still be effective approaches to the problem in practice. There are algorithms for many NP-complete problems, such as the knapsack problem , the traveling salesman problem , and the Boolean satisfiability problem , that can solve to optimality many real-world instances in reasonable time. The empirical average-case complexity (time vs. problem size) of such algorithms can be surprisingly low. An example
5474-509: Is very easy to tell whether solutions exist, but thought to be very hard to tell how many. Many of these problems are #P-complete , and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems. In 1975, Richard E. Ladner showed that if P ≠ NP, then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem ,
5593-512: The Aanderaa–Karp–Rosenberg conjecture true? Černý Conjecture : If a deterministic finite automaton with n {\displaystyle n} states has a synchronizing word , must it have one of length at most ( n − 1 ) 2 {\displaystyle (n-1)^{2}} ? Generalized star-height problem : Can all regular languages be expressed using generalized regular expressions with
5712-589: The Abel Prize in Norway. The driver of the taxicab they were riding in from Newark Airport lost control of the cab and struck a guardrail. Both passengers were ejected and killed. At the time of his death, Nash was a longtime resident of New Jersey. He was survived by two sons, John Charles Martin Nash, who lived with his parents at the time of their death, and elder child John Stier. Following his death, obituaries appeared in scientific and popular media throughout
5831-503: The Boolean satisfiability problem is NP-complete by the Cook–Levin theorem , so any instance of any problem in NP can be transformed mechanically into a Boolean satisfiability problem in polynomial time. The Boolean satisfiability problem is one of many NP-complete problems. If any NP-complete problem is in P, then it would follow that P = NP. However, many important problems are NP-complete, and no fast algorithm for any of them
5950-760: The Golden Globe Award for Best Actor – Motion Picture Drama at the 59th Golden Globe Awards and the BAFTA Award for Best Actor at the 55th British Academy Film Awards . Crowe was nominated for the Academy Award for Best Actor at the 74th Academy Awards ; Denzel Washington won for his performance in Training Day . Four of Nash's game-theoretic papers (Nash 1950a , 1950b , 1951 , 1953 ) and three of his pure mathematics papers (Nash 1952b , 1956 , 1958 ) were collected in
6069-558: The Nash equilibrium , a crucial concept in non-cooperative games. A version of his thesis was published a year later in the Annals of Mathematics . In the early 1950s, Nash carried out research on a number of related concepts in game theory, including the theory of cooperative games . For his work, Nash was one of the recipients of the Nobel Memorial Prize in Economic Sciences in 1994. In 1949, while still
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#17327805841756188-401: The Nash equilibrium . Nash did not publish extensively, although many of his papers are considered landmarks in their fields. As a graduate student at Princeton, he made foundational contributions to game theory and real algebraic geometry . As a postdoctoral fellow at MIT , Nash turned to differential geometry . Although the results of Nash's work on differential geometry are phrased in
6307-608: The Nobel Memorial Prize in Economic Sciences (along with John Harsanyi and Reinhard Selten ) for his game theory work as a Princeton graduate student. In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he was the John Nash and that his new work had value. They formed part of the nucleus of a group that contacted the Bank of Sweden 's Nobel award committee and were able to vouch for Nash's mental health and ability to receive
6426-553: The Riemann hypothesis , the lecture was incomprehensible. Colleagues in the audience immediately realized that something was wrong. In April 1959, Nash was admitted to McLean Hospital for one month. Based on his paranoid, persecutory delusions , hallucinations , and increasing asociality , he was diagnosed with schizophrenia . In 1961, Nash was admitted to the New Jersey State Hospital at Trenton . Over
6545-513: The Schwartz–Zippel lemma for polynomial identity testing be derandomized ? Does linear programming admit a strongly polynomial -time algorithm? (This is problem #9 in Smale's list of problems.) How many queries are required for envy-free cake-cutting ? What is the algorithmic complexity of the minimum spanning tree problem ? Equivalently, what is the decision tree complexity of
6664-601: The University of Naples Federico II in 2003, an honorary doctorate in economics from the University of Antwerp in 2007, an honorary doctorate of science from the City University of Hong Kong in 2011, and was keynote speaker at a conference on game theory. Nash also received honorary doctorates from two West Virginia colleges: the University of Charleston in 2003 and West Virginia University Tech in 2006. He
6783-449: The calculus of variations solve elliptic partial differential equations, Hilbert's nineteenth problem (on the smoothness of these minimizers), conjectured almost sixty years prior, was directly amenable to the De Giorgi–Nash theory. Nash received instant recognition for his work, with Peter Lax describing it as a "stroke of genius". Nash would later speculate that had it not been for De Giorgi's simultaneous discovery, he would have been
6902-408: The calculus of variations . Nash found the construction of smoothly differentiable isometric embeddings to be unexpectedly difficult. However, after around a year and a half of intensive work, his efforts succeeded, thereby proving the second Nash embedding theorem. The ideas involved in proving this second theorem are largely separate from those used in proving the first. The fundamental aspect of
7021-2350: The coding theory are also the unsolved problems in mathematics . References [ edit ] ^ Fellows, Michael R. ; Rosamond, Frances A. ; Rotics, Udi; Szeider, Stefan (2009), "Clique-width is NP-complete" (PDF) , SIAM Journal on Discrete Mathematics , 23 (2): 909–939, doi : 10.1137/070687256 , MR 2519936 , S2CID 18055798 , archived from the original (PDF) on 2019-02-27 . ^ Demaine, Erik D. ; O'Rourke, Joseph (2007), "24 Geodesics: Lyusternik–Schnirelmann", Geometric folding algorithms: Linkages, origami, polyhedra , Cambridge: Cambridge University Press, pp. 372–375, doi : 10.1017/CBO9780511735172 , ISBN 978-0-521-71522-5 , MR 2354878 . ^ Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2006), "Simultaneous graph embeddings with fixed edges" (PDF) , Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers (PDF) , Lecture Notes in Computer Science, vol. 4271, Berlin: Springer, pp. 325–335, doi : 10.1007/11917496_29 , ISBN 978-3-540-48381-6 , MR 2290741 . External links [ edit ] Open problems around exact algorithms by Gerhard J. Woeginger , Discrete Applied Mathematics 156 (2008) 397–405. The RTA list of open problems – open problems in rewriting . The TLCA List of Open Problems – open problems in area typed lambda calculus . v t e Well-known unsolved problems by discipline Astronomy Biology Chemistry Computer science Economics Fair division Geoscience Information theory Mathematics Medicine Neuroscience Physics Statistics Retrieved from " https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_computer_science&oldid=1257436060 " Categories : Conjectures Lists of unsolved problems Unsolved problems in computer science Hidden categories: Articles with short description Short description
7140-399: The discrete logarithm problem , and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete. The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic . An important unsolved problem in complexity theory is whether
7259-410: The exponential time hypothesis true? Is the strong exponential time hypothesis (SETH) true? Do one-way functions exist? Is public-key cryptography possible? Log-rank conjecture Polynomial versus nondeterministic-polynomial time for specific algorithmic problems [ edit ] Main article: NP-intermediate Can integer factorization be done in polynomial time on
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#17327805841757378-420: The rotation distance between two binary trees be computed in polynomial time? Can graphs of bounded clique-width be recognized in polynomial time? Can one find a simple closed quasigeodesic on a convex polyhedron in polynomial time? Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time? Can the square-root sum problem be solved in polynomial time in
7497-460: The time hierarchy theorem , they cannot be solved in significantly less than exponential time. Examples include finding a perfect strategy for chess positions on an N × N board and similar problems for other board games. The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer and Rabin proved in 1974 that every algorithm that decides
7616-549: The travelling salesman problem . Efficient solutions to these problems would have enormous implications for logistics. Many other important problems, such as some problems in protein structure prediction , are also NP-complete; making these problems efficiently solvable could considerably advance life sciences and biotechnology. These changes could be insignificant compared to the revolution that efficiently solving NP-complete problems would cause in mathematics itself. Gödel, in his early thoughts on computational complexity, noted that
7735-423: The weak formulation of partial differential equations, is in strong contrast to Nash's work, which is based on analysis of the heat kernel . Nash's approach to the De Giorgi–Nash theory was later revisited by Eugene Fabes and Daniel Stroock , initiating the re-derivation and extension of the results originally obtained from De Giorgi and Moser's techniques. From the fact that minimizers to many functionals in
7854-426: The 1930s that every closed smooth manifold is diffeomorphic to the zero set of some collection of smooth functions on Euclidean space . In his work, Nash proved that those smooth functions can be taken to be polynomials . This was widely regarded as a surprising result, since the class of smooth functions and smooth manifolds is usually far more flexible than the class of polynomials. Nash's proof introduced
7973-621: The 1950s, Nash discovered and proved the Nash embedding theorems by solving a system of nonlinear partial differential equations arising in Riemannian geometry . This work, also introducing a preliminary form of the Nash–Moser theorem , was later recognized by the American Mathematical Society with the Leroy P. Steele Prize for Seminal Contribution to Research . Ennio De Giorgi and Nash found, with separate methods,
8092-415: The 1950s, in which he had proposed a new encryption –decryption machine. The letters show that Nash had anticipated many concepts of modern cryptography , which are based on computational hardness . Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games . The thesis, written under the supervision of doctoral advisor Albert W. Tucker , contained the definition and properties of
8211-593: The 1994 Nobel Prize in Economics . In 2015, he and Louis Nirenberg were awarded the Abel Prize for their contributions to the field of partial differential equations. As a graduate student in the Princeton University Department of Mathematics , Nash introduced a number of concepts (including Nash equilibrium and the Nash bargaining solution ) which are now considered central to game theory and its applications in various sciences. In
8330-504: The 2019 answers became 99% believed P ≠ NP. These polls do not imply whether P = NP, Gasarch himself stated: "This does not bring us any closer to solving P=?NP or to knowing when it will be solved, but it attempts to be an objective report on the subjective opinion of this era." To attack the P ;= NP question, the concept of NP-completeness is very useful. NP-complete problems are problems that any other NP problem
8449-657: The MST problem? The optimal algorithm to compute MSTs is known , but it relies on decision trees, so its complexity is unknown. Gilbert–Pollack conjecture : Is the Steiner ratio of the Euclidean plane equal to 2 / 3 {\displaystyle 2/{\sqrt {3}}} ? Programming language theory [ edit ] Main article: Programming language theory POPLmark Barendregt–Geuvers–Klop conjecture Other problems [ edit ] Is
8568-518: The P versus NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time. If P ≠ NP, which is widely believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. The problem has been called the most important open problem in computer science . Aside from being an important problem in computational theory ,
8687-507: The Turing machine model? Other algorithmic problems [ edit ] The dynamic optimality conjecture : Do splay trees have a bounded competitive ratio? Can a depth-first search tree be constructed in NC ? Can the fast Fourier transform be computed in o ( n log n ) time? What is the fastest algorithm for multiplication of two n -digit numbers? What is the lowest possible average-case time complexity of Shellsort with
8806-514: The attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place. List of unsolved problems in computer science List of unsolved computational problems This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources . This article
8925-526: The award. Nash's later work involved ventures in advanced game theory, including partial agency, which show that, as in his early career, he preferred to select his own path and problems. Between 1945 and 1996, he published 23 scientific papers. Nash has suggested hypotheses on mental illness. He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on strike " from an economic point of view. He advanced views in evolutionary psychology about
9044-585: The basic logic of analysis and differential geometry. Judging from the classical perspective, what Nash has achieved in his papers is as impossible as the story of his life ... [H]is work on isometric immersions ... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored. While spending time at the Courant Institute in New York City, Louis Nirenberg informed Nash of
9163-418: The classes DLIN and NLIN . It is known that DLIN ≠ NLIN. One of the reasons the problem attracts so much attention is the consequences of the possible answers. Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well. A proof that P = NP could have stunning practical consequences if the proof leads to efficient methods for solving some of
9282-511: The computer is deterministic (given the computer's present state and any inputs, there is only one possible action that the computer might take) and sequential (it performs actions one after the other). In this theory, the class P consists of all decision problems (defined below ) solvable on a deterministic sequential machine in a duration polynomial in the size of the input; the class NP consists of all decision problems whose positive solutions are verifiable in polynomial time given
9401-400: The concatenation of n {\displaystyle n} and n 2 {\displaystyle n^{2}} in base b {\displaystyle b} uses at most k {\displaystyle k} distinct characters for b {\displaystyle b} and k {\displaystyle k} fixed and many other problems in
9520-406: The concept of NP-completeness was even defined ( Karp's 21 NP-complete problems , among the first found, were all well-known existing problems at the time they were shown to be NP-complete). Furthermore, the result P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NP and P = PH . It is also intuitively argued that
9639-464: The concepts now known as Nash function and Nash manifold , which have since been widely studied in real algebraic geometry. Nash's theorem itself was famously applied by Michael Artin and Barry Mazur to the study of dynamical systems , by combining Nash's polynomial approximation together with Bézout's theorem . During his postdoctoral position at MIT , Nash was eager to find high-profile mathematical problems to study. From Warren Ambrose ,
9758-580: The context of parabolic partial differential equations . In his work, as in Morrey's, uniform control over the continuity of the solutions to such equations is achieved, without assuming any level of differentiability on the coefficients of the equation. The Nash inequality was a particular result found in the course of his work (the proof of which Nash attributed to Elias Stein ), which has been found useful in other contexts. Soon after, Nash learned from Paul Garabedian , recently returned from Italy, that
9877-404: The corresponding #P problem asks "How many solutions are there?". Clearly, a #P problem must be at least as hard as the corresponding NP problem, since a count of solutions immediately tells if at least one solution exists, if the count is greater than zero. Surprisingly, some #P problems that are believed to be difficult correspond to easy (for example linear-time) P problems. For these problems, it
9996-443: The dream-like delusional hypotheses in the later 60s I became a person of delusionally influenced thinking but of relatively moderate behavior and thus tended to avoid hospitalization and the direct attention of psychiatrists. Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with
10115-414: The existence of problems that are hard to solve but whose solutions are easy to verify matches real-world experience. If P = NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in "creative leaps", no fundamental gap between solving a problem and recognizing the solution once it's found. On the other hand, some researchers believe that it
10234-586: The field of cryptography , which relies on certain problems being difficult. A constructive and efficient solution to an NP-complete problem such as 3-SAT would break most existing cryptosystems including: These would need modification or replacement with information-theoretically secure solutions that do not assume P ≠ NP. There are also enormous benefits that would follow from rendering tractable many currently mathematically intractable problems. For instance, many problems in operations research are NP-complete, such as types of integer programming and
10353-422: The following yes/no problem: given an incomplete Sudoku grid of size n 2 × n 2 {\displaystyle n^{2}\times n^{2}} , is there at least one legal solution where every row, column, and n × n {\displaystyle n\times n} square contains the integers 1 through n 2 {\displaystyle n^{2}} ? It
10472-532: The form of paranoia , his wife later described his behavior as erratic. Nash thought that all men who wore red ties were part of a communist conspiracy against him. He mailed letters to embassies in Washington, D.C., declaring that they were establishing a government. Nash's psychological issues crossed into his professional life when he gave an American Mathematical Society lecture at Columbia University in early 1959. Originally intended to present proof of
10591-546: The graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to László Babai , runs in quasi-polynomial time . The integer factorization problem
10710-464: The ideas of Nash's proof were applied for various constructions of turbulent solutions of the Euler equations in fluid mechanics . In the 1970s, Mikhael Gromov developed Nash's ideas into the general framework of convex integration , which has been (among other uses) applied by Stefan Müller and Vladimír Šverák to construct counterexamples to generalized forms of Hilbert's nineteenth problem in
10829-418: The important problems in NP. The potential consequences, both positive and negative, arise since various NP-complete problems are fundamental in many fields. It is also very possible that a proof would not lead to practical algorithms for NP-complete problems. The formulation of the problem does not require that the bounding polynomial be small or even specifically known. A non-constructive proof might show
10948-471: The mid-1980s. Nash's life was the subject of Sylvia Nasar 's 1998 biographical book A Beautiful Mind , and his struggles with his illness and his recovery became the basis for a film of the same name directed by Ron Howard , in which Nash was portrayed by Russell Crowe . John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Virginia . His father and namesake, John Forbes Nash Sr.,
11067-415: The middle of the night. He is referred to in a novel set at Princeton, The Mind-Body Problem , 1983, by Rebecca Goldstein . Sylvia Nasar 's biography of Nash, A Beautiful Mind , was published in 1998. A film by the same name was released in 2001, directed by Ron Howard with Russell Crowe playing Nash; it won four Academy Awards , including Best Picture . For his performance as Nash, Crowe won
11186-404: The next nine years, he spent intervals of time in psychiatric hospitals , where he received both antipsychotic medications and insulin shock therapy . Although he sometimes took prescribed medication, Nash later wrote that he did so only under pressure. According to Nash, the film A Beautiful Mind inaccurately implied he was taking atypical antipsychotics . He attributed the depiction to
11305-608: The potential benefits of apparently nonstandard behaviors or roles. Nash criticized Keynesian ideas of monetary economics which allowed for a central bank to implement monetary policies . He proposed a standard of "Ideal Money" pegged to an "industrial consumption price index " which was more stable than "bad money." He noted that his thinking on money and the function of monetary authority paralleled that of economist Friedrich Hayek . Nash received an honorary degree, Doctor of Science and Technology, from Carnegie Mellon University in 1999, an honorary degree in economics from
11424-404: The problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP = co-NP). The most efficient known algorithm for integer factorization is the general number field sieve , which takes expected time to factor an n -bit integer. The best known quantum algorithm for this problem, Shor's algorithm , runs in polynomial time, although this does not indicate where
11543-445: The problem lies with respect to non- quantum complexity classes . All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as Cobham's thesis . It is a common assumption in complexity theory; but there are caveats. First, it can be false in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, rendering it impractical. For example,
11662-585: The problem of deciding whether a graph G contains H as a minor , where H is fixed, can be solved in a running time of O ( n ), where n is the number of vertices in G . However, the big O notation hides a constant that depends superexponentially on H . The constant is greater than 2 ↑ ↑ ( 2 ↑ ↑ ( 2 ↑ ↑ ( h / 2 ) ) ) {\displaystyle 2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow (h/2)))} (using Knuth's up-arrow notation ), and where h
11781-477: The problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA , speculating that cracking a sufficiently complex code would require time exponential in the length of the key. If proved (and Nash was suitably skeptical), this would imply what is now called P ≠ NP, since a proposed key can be verified in polynomial time. Another mention of
11900-469: The proof is an implicit function theorem for isometric embeddings. The usual formulations of the implicit function theorem are inapplicable, for technical reasons related to the loss of regularity phenomena. Nash's resolution of this issue, given by deforming an isometric embedding by an ordinary differential equation along which extra regularity is continually injected, is regarded as a fundamentally novel technique in mathematical analysis . Nash's paper
12019-423: The rejection of politically oriented thinking as essentially a hopeless waste of intellectual effort. So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists. In 1978, Nash was awarded the John von Neumann Theory Prize for his discovery of non-cooperative equilibria, now called Nash Equilibria. He won the Leroy P. Steele Prize in 1999. In 1994, he received
12138-417: The resulting implicit function theorem is known as the Nash–Moser theorem . It has been extended and generalized by a number of other authors, among them Gromov, Richard Hamilton , Lars Hörmander , Jacob Schwartz , and Eduard Zehnder . Nash himself analyzed the problem in the context of analytic functions . Schwartz later commented that Nash's ideas were "not just novel, but very mysterious," and that it
12257-538: The right information, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine. Clearly, P ⊆ NP. Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes: Since 2002, William Gasarch has conducted three polls of researchers concerning this and related questions. Confidence that P ≠ NP has been increasing – in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002. When restricted to experts,
12376-491: The screenwriter who was worried about the film encouraging people with mental illness to stop taking their medication. Nash did not take any medication after 1970, nor was he committed to a hospital ever again. Nash recovered gradually. Encouraged by his then former wife, Lardé, Nash lived at home and spent his time in the Princeton mathematics department where his eccentricities were accepted even when his mental condition
12495-402: The setting of low differentiability. For this reason, Nash's result is somewhat outside the mainstream in the field of differential geometry , where high differentiability is significant in much of the usual analysis. However, the logic of Nash's work has been found to be useful in many other contexts in mathematical analysis . Starting with work of Camillo De Lellis and László Székelyhidi,
12614-425: The size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is " P " or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be verified in polynomial time is "NP" , standing for "nondeterministic polynomial time". An answer to
12733-593: The spring of 1959. His son, John Charles Martin Nash, was born a few months later. The child was not named for a year because Alicia felt that Nash should have a say in choosing the name. Due to the stress of dealing with his illness, Nash and Lardé divorced in 1963. After his final hospital discharge in 1970, Nash lived in Lardé's house as a boarder . This stability seemed to help him, and he learned how to consciously discard his paranoid delusions . Princeton allowed him to audit classes. He continued to work on mathematics and
12852-422: The then-unknown Ennio De Giorgi had found nearly identical results for elliptic partial differential equations. De Giorgi and Nash's methods had little to do with one another, although Nash's were somewhat more powerful in applying to both elliptic and parabolic equations. A few years later, inspired by De Giorgi's method, Jürgen Moser found a different approach to the same results, and the resulting body of work
12971-438: The truth of Presburger statements of length n has a runtime of at least 2 2 c n {\displaystyle 2^{2^{cn}}} for some constant c . Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems , such as the halting problem . They cannot be completely solved by any algorithm, in the sense that for any particular algorithm there
13090-407: The underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann . Gödel asked whether theorem-proving (now known to be co-NP-complete ) could be solved in quadratic or linear time , and pointed out one of the most important consequences—that if so, then the discovery of mathematical proofs could be automated. The relation between the complexity classes P and NP
13209-457: The verifier for any particular instance of a problem in NP can be encoded as a polynomial-time machine M that takes the solution to be verified as input. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists. The first natural problem proven to be NP-complete was the Boolean satisfiability problem, also known as SAT. As noted above, this
13328-444: The world. In addition to their obituary for Nash, The New York Times published an article containing quotes from Nash that had been assembled from media and other published sources. The quotes consisted of Nash's reflections on his life and achievements. At Princeton in the 1970s, Nash became known as "The Phantom of Fine Hall" (Princeton's mathematics center), a shadowy figure who would scribble arcane equations on blackboards in
13447-659: Was a prolific guest speaker at a number of events, such as the Warwick Economics Summit in 2005, at the University of Warwick . Nash was elected to the American Philosophical Society in 2006 and became a fellow of the American Mathematical Society in 2012. On May 19, 2015, a few days before his death, Nash, along with Louis Nirenberg , was awarded the 2015 Abel Prize by King Harald V of Norway at
13566-878: Was an electrical engineer for the Appalachian Electric Power Company . His mother, Margaret Virginia (née Martin) Nash, had been a schoolteacher before she was married. He was baptized in the Episcopal Church . He had a younger sister, Martha (born November 16, 1930). Nash attended kindergarten and public school, and he learned from books provided by his parents and grandparents. Nash's parents pursued opportunities to supplement their son's education, and arranged for him to take advanced mathematics courses at nearby Bluefield College (now Bluefield University ) during his final year of high school. He attended Carnegie Institute of Technology (which later became Carnegie Mellon University) through
13685-551: Was awarded the Leroy P. Steele Prize for Seminal Contribution to Research in 1999, where his "most original idea" in the resolution of the loss of regularity issue was cited as "one of the great achievements in mathematical analysis in this century". According to Gromov: You must be a novice in analysis or a genius like Nash to believe anything like that can be ever true and/or to have a single nontrivial application. Due to Jürgen Moser 's extension of Nash's ideas for application to other problems (notably in celestial mechanics ),
13804-491: Was criticized during the run-up to the 2002 Oscars for omitting this aspect of his life. He was said to have abandoned her based on her social status, which he thought to have been beneath his. In Santa Monica, California , in 1954, while in his twenties, Nash was arrested for indecent exposure in a sting operation targeting gay men. Although the charges were dropped, he was stripped of his top-secret security clearance and fired from RAND Corporation , where he had worked as
13923-539: Was eventually allowed to teach again. In the 1990s, Lardé and Nash resumed their relationship, remarrying in 2001. John Charles Martin Nash earned a PhD in mathematics from Rutgers University and was diagnosed with schizophrenia as an adult. On May 23, 2015, Nash and his wife died in a car accident on the New Jersey Turnpike in Monroe Township, New Jersey while returning home from receiving
14042-469: Was poor. Lardé credits his recovery to maintaining "a quiet life" with social support . Nash dated the start of what he termed "mental disturbances" to the early months of 1959, when his wife was pregnant. He described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic ' ". For Nash, this included seeing himself as
14161-416: Was very hard to "get to the bottom of it." According to Gromov: Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to do it. ... what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical' – it is something that brings about a dramatic alteration of our understanding of
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