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On Computable Numbers, With An Application To The Entscheidungsproblem
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 ...
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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 ...
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Rice's Theorem
In computability theory, Rice's theorem states that all non-trivial semantic properties of programs are undecidable problem, undecidable. A ''semantic'' property is one about the program's behavior (for instance, "does the program halting problem, terminate for all inputs?"), unlike a syntactic property (for instance, "does the program contain an if-then-else statement?"). A ''non-trivial'' property is one which is neither true for every program, nor false for every program. The theorem generalizes the undecidability of the halting problem. It has far-reaching implications on the feasibility of static program analysis, static analysis of programs. It implies that it is impossible, for example, to implement a tool that checks whether any given program is correctness (computer science), correct, or even executes without error (it is possible to implement a tool that always overestimates or always underestimates, so in practice one has to decide what is less of a problem). The theorem ...
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Universal Machine
In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a human in the process of computing a real number to a machine which is only capable of a finite number of conditions ; which will be called "-configurations". He then described the operation of such machine, as described below, and argued: Turing introduced the idea of such a machine in 1936–1937. Introduction Martin Davis makes a persuasive argument that Turing's conception of what is now known as "the stored-program computer", of placing the "action table"—the instructions for the machine—in the same "memory" as the input data, strongly influenced John von Neumann's conception of the first Ameri ...
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Martin Davis (mathematician)
Martin David Davis (March 8, 1928 – January 1, 2023) was an American mathematician and computer scientist who contributed to the fields of computability theory and mathematical logic. His work on Hilbert's tenth problem led to the MRDP theorem. He also advanced the Post–Turing model and co-developed the Davis–Putnam–Logemann–Loveland (DPLL) algorithm, which is foundational for Boolean satisfiability solvers. Davis won the Leroy P. Steele Prize, the Chauvenet Prize (with Reuben Hersh), and the Lester R. Ford Award. He was a fellow of the American Academy of Arts and Sciences and a fellow of the American Mathematical Society. Early life and education Davis's parents were Jewish immigrants to the United States from Łódź, Poland, and married after they met again in New York City. Davis was born in New York City on March 8, 1928. He grew up in the Bronx, where his parents encouraged him to obtain a full education.. He graduated from the prestigious Bronx High ...
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Universal Turing Machine
In computer science, a universal Turing machine (UTM) is a Turing machine capable of computing any computable sequence, as described by Alan Turing in his seminal paper "On Computable Numbers, with an Application to the Entscheidungsproblem". Common sense might say that a universal machine is impossible, but Turing proves that it is possible. He suggested that we may compare a human in the process of computing a real number to a machine which is only capable of a finite number of conditions ; which will be called "-configurations". He then described the operation of such machine, as described below, and argued: Turing introduced the idea of such a machine in 1936–1937. Introduction Martin Davis makes a persuasive argument that Turing's conception of what is now known as "the stored-program computer", of placing the "action table"—the instructions for the machine—in the same "memory" as the input data, strongly influenced John von Neumann's conception of the first Amer ...
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Formal System
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Concepts A formal system has the following: * Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules). * Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal language. A formal system is said to be recursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules are decidable ...
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Post–Turing Machine
A Post machine or Post–Turing machineRajendra Kumar, ''Theory of Automata'', Tata McGraw-Hill Education, 2010, p. 343. is a "program formulation" of a type of Turing machine, comprising a variant of Emil Post's Turing-equivalent model of computation. Post's model and Turing's model, though very similar to one another, were developed independently. Turing's paper was received for publication in May 1936, followed by Post's in October. A Post–Turing machine uses a binary alphabet, an infinite sequence of binary storage locations, and a primitive programming language with instructions for bi-directional movement among the storage locations and alteration of their contents one at a time. The names "Post–Turing program" and "Post–Turing machine" were used by Martin Davis in 1973–1974 (Davis 1973, p. 69ff). Later in 1980, Davis used the name "Turing–Post program" (Davis, in Steen p. 241). 1936: Post model In his 1936 paper "Finite Combinatory Processes&mdas ...
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Exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product (mathematics), product of multiplying bases: b^n = \underbrace_.In particular, b^1=b. The exponent is usually shown as a superscript to the right of the base as or in computer code as b^n. This binary operation is often read as " to the power "; it may also be referred to as " raised to the th power", "the th power of ", or, most briefly, " to the ". The above definition of b^n immediately implies several properties, in particular the multiplication rule:There are three common notations for multiplication: x\times y is most commonly used for explicit numbers and at a very elementary level; xy is most common when variable (mathematics), variables are used; x\cdot y is used for emphasizing that one ta ...
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James R
James may refer to: People * James (given name) * James (surname) * James (musician), aka Faruq Mahfuz Anam James, (born 1964), Bollywood musician * James, brother of Jesus * King James (other), various kings named James * Prince James (other) * Saint James (other) Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Film and television * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada-language film * "James", a television Adventure Time (season 5)#ep42, ...
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Ernest Nagel
Ernest Nagel (; ; November 16, 1901 – September 20, 1985) was an American philosopher of science. Suppes, Patrick (1999)Biographical memoir of Ernest Nagel In '' American National Biograph''y (Vol. 16, pp. 216-218). New York: Oxford University Press. uthor eprint, archived/ref> Along with Rudolf Carnap, Hans Reichenbach, and Carl Hempel, he is sometimes seen as one of the major figures of the logical positivist movement. His 1961 book '' The Structure of Science'' is considered a foundational work in the logic of scientific explanation. Life and career Nagel was born in Nové Mesto nad Váhom (now in Slovakia, then Vágújhely and part of the Austro-Hungarian Empire) to Jewish parents. His mother, Frida Weiss, was from the nearby town of Vrbové (or Verbo). He emigrated to the United States at the age of 10 and became a U.S. citizen in 1919. He received a BSc from the City College of New York (CCNY) in 1923, and earned his PhD from Columbia University in 1931, with a di ...
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On Formally Undecidable Propositions Of Principia Mathematica And Related Systems
"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November 17, 1930, it was originally published in German in the 1931 volume of '' Monatshefte für Mathematik und Physik.'' Several English translations have appeared in print, and the paper has been included in two collections of classic mathematical logic papers. The paper contains Gödel's incompleteness theorems, now fundamental results in logic that have many implications for consistency proofs in mathematics. The paper is also known for introducing new techniques that Gödel invented to prove the incompleteness theorems. Outline and key results The main results established are Gödel's first and second incompleteness theorems, which have had an enormous impact on the field of mathematical logic. These appear as theorems VI and XI, respect ...
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Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \rightarrow q \equiv \neg p \vee q :#p \rightarrow q \equiv \neg q \rightarrow \neg p :#p \vee q \equiv \neg p \rightarrow q :#p \wedge q \equiv \neg (p \rightarrow \neg q) :#\neg (p \rightarrow q) \equiv p \wedge \neg q :#(p \righta ...
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