https://en.wikipedia.org/wiki/Minimax
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One iteration of the middle-square method, showing a 6-digit seed, which is then squared, and the resulting value has its middle 6 digits as the output value (and also as the next seed for the sequence). “}}
https://en.wikipedia.org/wiki/Middle-square_method
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One iteration of the middle-square method, showing a 6-digit seed, which is then squared, and the resulting value has its middle 6 digits as the output value (and also as the next seed for the sequence). “}}
https://en.wikipedia.org/wiki/Entscheidungsproblem
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By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers[2] showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of “effectively calculable” is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis. “}}
https://en.wikipedia.org/wiki/Universal_Turing_machine
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Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet. In that sense it behaves like a computer with a fixed program. However, we can encode the action table of any Turing machine in a string. Thus we can construct a Turing machine that expects on its tape a string describing an action table followed by a string describing the input tape, and computes the tape that the encoded Turing machine would have computed. Turing described such a construction in complete detail in his 1936 paper: “It is possible to invent a single machine which can be used to compute any computable sequence. If this machine U is supplied with a tape on the beginning of which is written the S.D [“standard description” of an action table] of some computing machine M, then U will compute the same sequence as M.”[3] Stored-program computer 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 American discrete-symbol (as opposed to analog) computer—the EDVAC. Davis quotes Time magazine to this effect, that “everyone who taps at a keyboard… is working on an incarnation of a Turing machine,” and that “John von Neumann [built] on the work of Alan Turing” (Davis 2000:193 quoting Time magazine of 29 March 1999). “}}
https://en.wikipedia.org/wiki/Kurt_G%C3%B6del
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Incompleteness theorem Kurt Gödel’s achievement in modern logic is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement. — John von Neumann[17] … In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this using a process known as Gödel numbering…..Gödel suffered periods of mental instability and illness. Following the assassination of his close friend Moritz Schlick,[35] Gödel developed an obsessive fear of being poisoned, and would eat only food prepared by his wife Adele. Adele was hospitalized beginning in late 1977, and in her absence Gödel refused to eat;[36] he weighed 29 kilograms (65 lb) when he died of “malnutrition and inanition caused by personality disturbance” in Princeton Hospital on January 14, 1978[37] He was buried in Princeton Cemetery. Adele died in 1981.[38]
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