What mathematical theorem, the first major theorem to be solved using a computer, was proven in 1976 by Kenneth Appel and Wolfgang Haken with a set of 1936 maps?
"he volunteered in mathematics enrichment programs in Dover and in southern Maine public schools. He believed ""that students should be afforded the opportunity to study mathematics at the level of their ability, even if it is well above their grade level."" Kenneth Appel is known for his work in topology, the branch of mathematics that explores certain properties of geometric figures. His biggest accomplishment was proving the four color theorem in 1976 with Wolfgang Haken. The New York Times wrote in 1976: Now the four-color conjecture has been proved by two University of Illinois mathematicians, Kenneth Appel and Wolfgang Haken."
"a smaller counter-example.) Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the"
"in over 400 pages of microfiche, which had to be checked by hand with the assistance of Haken's daughter Dorothea Blostein . Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the University of Illinois used a postmark stating ""Four colors suffice."" At the same time the unusual nature of the proof—it was the first major theorem to be proved with extensive computer assistance—and the complexity of the human-verifiable portion aroused considerable controversy . In the early 1980s, rumors spread of a flaw in the Appel–Haken proof. Ulrich Schmidt at"
"into the 1950s and 1960s, and later in some specialized applications. The principle of the modern computer was first described by computer scientist Alan Turing, who set out the idea in his seminal 1936 paper, ""On Computable Numbers"". Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was"
"colorings of the graphs embedded on surfaces with arbitrary genus. Tait's reformulation generated a new class of problems, the ""factorization problems"", particularly studied by Petersen and Kőnig. The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, ""extremal graph theory"". The four color problem remained unsolved for more than a century. In 1969 Heinrich Heesch published a method for solving the problem using computers. A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of ""discharging"""
"appeared in 1989 and explained Schmidt's discovery and several further errors found by others . Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only O(""n"") time, where ""n"" is the number of vertices. In 1996, Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas created a quadratic-time algorithm, improving on a quartic-time algorithm based on Appel and Haken’s proof This new proof is similar to Appel and Haken's but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations. Both the unavoidability and reducibility parts"
"formula. An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained the conjecture. In other words, Ringel, Youngs and others had to construct extreme examples for every genus g = 1,2,3... If g = 12s + k, the genera fall into 12 cases according as k = 0,1,2,3,4,5,6,7,8,9,10,11."
"developed by Heesch. The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney. Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist"
"Turing independently demonstrated that Hilbert's ""Entscheidungsproblem"" (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, when put together, are able to produce any computation. The work of Gödel showed that the notion of computation is essentially unique. In 1941 Konrad Zuse completed the Z3 (computer), the first working Turing-complete machine; this was the first digital computer in the modern sense. Computability theory characterizes problems as having, or not having, computational solutions. The first result of computability theory"
"is the Tarski–Seidenberg theorem, which has been implemented in computers by using the cylindrical algebraic decomposition. Entscheidungsproblem In mathematics and computer science, the (, German for ""decision problem"") is a challenge posed by David Hilbert in 1928. The problem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers ""Yes"" or ""No"" according to whether the statement is ""universally valid"", i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid"
"the face of logic in the twentieth century"". Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's ""Entscheidungsproblem"" in 1936 and 1937, respectively. The ""Entscheidungsproblem"" asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution. Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It"
"algorithm is a computable function. Although initially skeptical, by 1946 Gödel argued in favor of this thesis: With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided. Church (1936a, 1936b) and Turing (1936), inspired by techniques used by Gödel (1931) to prove his incompleteness theorems, independently demonstrated that the Entscheidungsproblem is not effectively decidable. This result showed that there is no algorithmic procedure that can correctly decide whether arbitrary mathematical propositions are true or false. Many problems in mathematics have been shown to be undecidable after these initial examples"
"as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory. Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states"
"machines—were proved about their formal computational capabilities. In the 1960s, cellular automata were studied as a particular type of dynamical system and the connection with the mathematical field of symbolic dynamics was established for the first time. In 1969, Gustav A. Hedlund compiled many results following this point of view in what is still considered as a seminal paper for the mathematical study of cellular automata. The most fundamental result is the characterization in the Curtis–Hedlund–Lyndon theorem of the set of global rules of cellular automata as the set of continuous endomorphisms of shift spaces. In 1969, German computer pioneer"
"although he recognized this gave a proof of the loop theorem, he did not write up a detailed proof. The essential ingredient of this proof is the concept of Haken hierarchy. Proofs were later written up, by Klaus Johannson, Marc Lackenby, and Iain Aitchison with Hyam Rubinstein. Loop theorem In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if there is a map"
"in pure mathematics. In 1951, they started development work on Meg, the Mark 1's successor, which would include a floating point unit. In 1936, mathematician Alan Turing published a definition of a theoretical ""universal computing machine"", a computer which held its program on tape, along with the data being worked on. Turing proved that such a machine was capable of solving any conceivable mathematical problem for which an algorithm could be written. During the 1940s, Turing and others such as Konrad Zuse developed the idea of using the computer's own memory to hold both the program and data, instead of"
"throughout its history. When access to digital computers became possible in the middle fifties, a few scientists instinctively recognized that a machine that could manipulate numbers could also manipulate symbols and that the manipulation of symbols could well be the essence of human thought. This was a new approach to creating thinking machines. In 1955, Allen Newell and (future Nobel Laureate) Herbert A. Simon created the ""Logic Theorist"" (with help from J. C. Shaw). The program would eventually prove 38 of the first 52 theorems in Russell and Whitehead's ""Principia Mathematica"", and find new and more elegant proofs for some."
"consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term ""the continuum"" for the real numbers. Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain . It became the first on David Hilbert's list of important open questions that was presented at the International Congress of Mathematicians in the year 1900 in Paris. Axiomatic set theory was at that point not yet formulated. Kurt Gödel proved in 1940 that the negation of the continuum"
"an important result, the so-called speed-up theorem. The field began to flourish in 1971 when the Stephen Cook and Leonid Levin proved the existence of practically relevant problems that are NP-complete. In 1972, Richard Karp took this idea a leap forward with his landmark paper, ""Reducibility Among Combinatorial Problems"", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its computational intractability, are NP-complete. In the 1980s, much work was done on the average difficulty of solving NP-complete problems—both exactly and approximately. At that time, computational complexity theory was at its height, and it was"
"early 20th century, David Hilbert led a program to axiomatize all of mathematics with precise axioms and precise logical rules of deduction which could be performed by a machine. Soon, it became clear that a small set of deduction rules are enough to produce the consequences of any set of axioms. These rules were proved by Kurt Gödel in 1930 to be enough to produce every theorem. The actual notion of computation was isolated soon after, starting with Gödel's incompleteness theorem. This theorem showed that axiom systems were limited when reasoning about the computation which deduces their theorems. Church and"
"coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done. The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park with the guidance of Alan Turing and his seminal work, On Computable Numbers. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool"
"RWTH Aachen examined Appel and Haken's proof for his master's thesis . He had checked about 40% of the unavoidability portion and found a significant error in the discharging procedure . In 1986, Appel and Haken were asked by the editor of ""Mathematical Intelligencer"" to write an article addressing the rumors of flaws in their proof. They responded that the rumors were due to a ""misinterpretation of [Schmidt's] results"" and obliged with a detailed article . Their magnum opus, ""Every Planar Map is Four-Colorable"", a book claiming a complete and detailed proof (with a microfiche supplement of over 400 pages),"
"A century later, in 1936, mathematician Alan Turing published his description of what became known as a Turing machine, a theoretical concept intended to explore the limits of mechanical computation. Turing was not imagining a physical machine, but a person he called a ""computer"", who acted according to the instructions provided by a tape on which symbols could be read and written sequentially as the tape moved under a tape head. Turing proved that if an algorithm can be written to solve a mathematical problem, then a Turing machine can execute that algorithm. Konrad Zuse's Z3 was the world's first"
"in 1926. Her doctoral thesis, ""Die Frage der endlich vielen Schritte in der Theorie der Polynomideale"" (in English ""The Question of Finitely Many Steps in Polynomial Ideal Theory""), published in ""Mathematische Annalen"", is the foundational paper for computer algebra. It first established the existence of algorithms (including complexity bounds) for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings. Hermann's algorithm for primary decomposition is still in contemporary use. From 1925 to 1927, Hermann worked as assistant for Leonard Nelson. Together with Minna Specht, she posthumously published Nelson's work ""System der philosophischen Ethik und"
"of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems. 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 is widely conjectured that the answer is no. 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"
"were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post. Kleene (1943) introduced the concepts of relative computability, foreshadowed by Turing (1939), and the arithmetical hierarchy. Kleene later generalized recursion theory to higher-order functionals. Kleene and Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems. These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language. The systems of propositional logic and first-order logic are the most widely"
"one of the most famous machines of its period. In 1960, Rosenblatt and colleagues were able to show that the perceptron could in finite training cycles learn any task that its parameters could embody. The perceptron convergence theorem was proved for single-layer neural nets. During this period, neural net research was a major approach to the brain-machine issue that had been taken by a significant number of individuals. Reports by the New York Times and statements by Rosenblatt claimed that neural nets would soon be able to see images, beat humans at chess, and reproduce. At the same time, new"
"Numbers, with an Application to the Entscheidungsproblem"" (1936). In this paper, Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines. The ""Entscheidungsproblem"" (decision problem) was originally posed by German mathematician David Hilbert in 1928. Turing proved that his ""universal computing machine"" would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the ""decision problem"" by first showing that the halting"
"correctly settles the problem for all given domino sets. In 1966, Wang's student Robert Berger solved the domino problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem (the problem of testing whether a Turing machine eventually halts) then implies the undecidability of Wang's tiling problem. Combining Berger's undecidability result with Wang's observation shows that there must exist a finite set of"
"twenty papers spanning over 500 pages from 1983 to 2004. Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 independently by Joseph Kruskal and S. Tarkowski. A minor of an undirected graph ""G"" is any graph that may be obtained from ""G"" by a sequence of zero or more contractions of edges of ""G"" and deletions of edges and vertices"
"solution ""f"" of the Beltrami equation for which ""f"" is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞. A similar result is true with C replaced by the unit disk D. Their proof used the Beurling transform, a singular integral operator. Measurable Riemann mapping theorem In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of"
"field or of an elliptic curve with complex multiplication). This is more standard, as the ""D"" are then fundamental discriminants. This result was first conjectured by Gauss in Section 303 of his ""Disquisitiones Arithmeticae"". It was essentially proven by Kurt Heegner in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967, which had many commonalities to Heegner's work, but sufficiently many differences that Stark considers the proofs to be different. Heegner ""died before anyone really understood what he had done"". Stark formally filled in the gap"
"to the inception of modern computing theory in the papers of von Neumann and Kleene. Harel also writes that the more generic name was proposed by H.D. Mills as ""The Structure Theorem"" in the early 1970s. This version of the theorem replaces all the original program's control flow with a single global codice_1 loop that simulates a program counter going over all possible labels (flowchart boxes) in the original non-structured program. Harel traced the origin of this folk theorem to two papers marking the beginning of computing. One is the 1946 description of the von Neumann architecture, which explains how"