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Entscheidungsproblem
In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid, i.e., valid in every Structure (mathematical logic), structure. Such an algorithm was proven to be impossible by Alonzo Church and Alan Turing in 1936. Completeness theorem By Gödel's completeness theorem, the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the ' can also be viewed as asking for an algorithm to decide whether a given statement is provable using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the ' is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Turing Machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. The machine operates on an infinite memory tape divided into discrete mathematics, discrete cells, each of which can hold a single symbol drawn from a finite set of symbols called the Alphabet (formal languages), alphabet of the machine. It has a "head" that, at any point in the machine's operation, is positioned over one of these cells, and a "state" selected from a finite set of states. At each step of its operation, the head reads the symbol in its cell. Then, based on the symbol and the machine's own present state, the machine writes a symbol into the same cell, and moves the head one step to the left or the right, or halts the computation. The choice of which replacement symbol to write, which direction to move the head, and whet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Turing's Proof
Turing's proof is a proof by Alan Turing, first published in November 1936 with the title "On Computable Numbers, with an Application to the ". It was the second proof (after Church's theorem) of the negation of Hilbert's ; that is, the conjecture that some purely mathematical yes–no questions can never be answered by computation; more technically, that some decision problems are " undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K Principia_Mathematica.html" ;"title="'Principia Mathematica">'Principia Mathematica''. Turing followed this proof with two others. The second and third both rely on the first. All rely on his development of typewriter-like "Turing machine, computing m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Church–Turing Thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a wiktionary:thesis, thesis about the nature of computable functions. It states that a function (mathematics), function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable'' to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formal system, formalize the notion of computability: * In 1933, Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Halting Problem
In computability theory (computer science), 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. The halting problem is ''Undecidable problem, undecidable'', meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically Definable set, definable but not Computable function, computable. A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program that might determine whether programs halt, that a "pathological" program exists for which makes an incorrect determination. Specifically, is the program that, when called with some input, passes its own s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general-purpose computer. Turing is widely considered to be the father of theoretical computer science. Born in London, Turing was raised in southern England. He graduated from University of Cambridge, King's College, Cambridge, and in 1938, earned a doctorate degree from Princeton University. During World War II, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre that produced Ultra (cryptography), Ultra intelligence. He led Hut 8, the section responsible for German naval cryptanalysis. Turing devised techniques for speeding the breaking of Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Moses Schönfinkel
Moses Ilyich Schönfinkel (; 29 September 1888 – ) was a logician and mathematician, known for the invention of combinatory logic. Life Moses Schönfinkel was born on in Ekaterinoslav, Russian Empire (now Dnipro, Ukraine). He was born to a Jewish family. His father was Ilya Girshevich Schönfinkel, a merchant of first guild, who was in а grocery store trade, and his mother, Maria “Masha” Gertsovna Schönfinkel (née Lurie) came from a prominent Lurie family. Moses had siblings named Deborah, Natan, Israel and Grigoriy. Schönfinkel attended the Novorossiysk University of Odessa, studying mathematics under Samuil Osipovich Shatunovskii (1859–1929), who worked in geometry and the foundations of mathematics. From 1914 to 1924, Schönfinkel was a member of David Hilbert's group at the University of Göttingen in Germany. On 7 December 1920 he delivered a talk entitled ''Elemente der Logik'' ("Elements of Logic") to the group where he outlined the concept of combinatory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Max Newman
Maxwell Herman Alexander Newman, FRS (7 February 1897 – 22 February 1984), generally known as Max Newman, was a British mathematician and codebreaker. His work in World War II led to the construction of Colossus, the world's first operational, programmable electronic computer, and he established the Royal Society Computing Machine Laboratory at the University of Manchester, which produced the world's first working, stored-program electronic computer in 1948, the Manchester Baby. Early life and education Newman was born Maxwell Herman Alexander Neumann in Chelsea, London, England, to a Jewish family, on 7 February 1897. His father was Herman Alexander Neumann, originally from the German city of Bromberg (now in Poland), who had emigrated with his family to London at the age of 15.William Newman, "Max Newman – Mathematician, Codebreaker and Computer Pioneer", pp. 176–188 in Herman worked as a secretary in a company, and married Sarah Ann Pike, an Irish schoolteacher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, the Church–Turing thesis, proving the unsolvability of the ''Entscheidungsproblem'' ("decision problem"), the Frege–Church ontology, and the Church–Rosser theorem. Alongside his doctoral student Alan Turing, Church is considered one of the founders of computer science. Life Alonzo Church was born on June 14, 1903, in Washington, D.C., where his father, Samuel Robbins Church, was a justice of the peace and the judge of the Municipal Court for the District of Columbia. He was the grandson of Alonzo Webster Church (1829–1909), United States Senate Librarian from 1881 to 1901, and great-grandson of Alonzo Church, a professor of Mathematics and Astronomy and 6th President of the University of Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wilhelm Ackermann
Wilhelm Friedrich Ackermann (; ; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic and the Ackermann function, an important example in the theory of computation. Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis ''Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit'', which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought. From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (''Academy of Sciences'') in Göttingen, and was an honorary professor at the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Computable Function
Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precise definition of the concept of algorithm, every formal definition of computability must refer to a specific model of computation. Many such models of computation have been proposed, the major ones being Turing machines, register machines, lambda calculus and general recursive functions. Although these four are of a very different nature, they provide exactly the same class of computable functions, and, for every model of computation that has ever been proposed, the computable functions for such a model are computable for the above four models of computation. The Church–Turing thesis is the unprovable assertion that every notion of computability that can be imagined can compute only functions that are computable in the above sense. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Paul Bernays
Paul Isaac Bernays ( ; ; 17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator of David Hilbert. Biography Bernays was born into a distinguished German-Jewish family of scholars and businessmen. His great-grandfather, Isaac ben Jacob Bernays, served as chief rabbi of Hamburg from 1821 to 1849. Bernays spent his childhood in Berlin, and attended the Köllnische Gymnasium, 1895–1907. At the University of Berlin, he studied mathematics under Issai Schur, Edmund Landau, Ferdinand Georg Frobenius, and Friedrich Schottky; philosophy under Alois Riehl, Carl Stumpf and Ernst Cassirer; and physics under Max Planck. At the University of Göttingen, he studied mathematics under David Hilbert, Edmund Landau, Hermann Weyl, and Felix Klein; physics under Voigt and Max Born; and philosophy under Leonard Nelson. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |