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| | On computable numbers, with an application to the Entscheidungsproblem - A. M. Turing, 1936 |
 | | It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps. | | | The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. | | | Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. |
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http://www.abelard.org/turpap2/tp2-ie.asp
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| | On computable numbers, with an application to the Entscheidungsproblem - A. M. Turing, 1936 |
| | The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. | | | It would be true if we could enumerate the computable sequences by finite means, but the problem of enumerating computable sequences is equivalent to the problem of finding out whether a given number is the D.N of a circle-free machine, and we have no general process for doing this in a finite number of steps. | | | Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. |
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http://www.abelard.org/turpap2/tp2-ie.asp
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| | constructive |
| | I think that in fact the computable numbers satisfy the axioms for real numbers, and so the collection of all computable numbers is a model of the reals. | | | I think that in fact the computable >numbers satisfy the axioms for real numbers, and so the collection of all >computable numbers is a model of the reals. | | | It's consistent with constructive mathematics that every real number is recursive (computable), and every function on the reals is recursive. |
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http://www.math.niu.edu/~rusin/known-math/00_incoming/constructive
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| | Recursion Theory |
| | (It is clear, incidentally, that any real number which happens to be rational is, on this definition, straightforwardly computable, but not every computable real need be rational.) Intuitively, a real number is computable if it can be approximated to an arbitrary degree of accuracy by an algorithmic method. | | | A real number is said to be computable when there is a computable sequence of rationals which converges effectively to it. | | | It is interesting that the set of all computable reals, like the set of rationals, is of course only denumerably infinite, while the set of all reals is uncountably infinite. |
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http://www.mulhauser.net/research/tutorials/computability/recursion.html
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| | nrich.maths.org::Mathematics Enrichment::NRICH |
| | I'm not sure what a computable number is, but maybe it means one in which there is an algorithm (or computer program) with which you can calculate the nth decimal place. | | | Yes, the definition of a computable number is one for which there is a program that computes the nth place of the number. | | | If this is true then I think we can say (using a bit of set theory) that most real numbers are not computable. |
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http://www.nrich.maths.org.uk/askedNRICH/edited/1172.html
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| | natural theology > development > 2 model > 3 a transfinite network |
| | Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. | | | The part of the Cantor universe that is represented by the natural numbers, and so is computable, is an infinitesimally small fraction of the whole. | | | "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. |
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http://www.naturaltheology.net/Development/Dev02_Model/model03TransNet.html
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| | natural theology > development > 2 model > 3 a transfinite network |
| | Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. | | | The maximum number is aleph(0), ie one different instruction corresponding to each different natural number. | | | The number of steps required to solve some computational problems, however, grows exponentially with the size of the input. |
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http://www.naturaltheology.net/Development/Dev02_Model/model03TransNet.html
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| | Alan Turing (1912-1954) |
| | However he was able to produce a diagonalisation argument not unlike that of Cantor when proving the uncountability of the real numbers to show that there was an uncountable number of numbers which are not computable. | | | He defined a computable number as a real number whose decimal expansion could be produced by a Turing machine starting with a blank tape. | | | He found the answer to be in the negative by devising a concept called a computable number and mecanical processes (later to be called Turing Machines). |
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http://www.amt.edu.au/turingb.html
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| | Computable number - Wikipedia, the free encyclopedia |
| | The computable numbers form a real closed field and can be used in the place of real numbers for many, but by no means all, mathematical purposes. | | | The computable complex numbers form an algebraically closed field, and for many purposes is large enough already without requiring the noncomputable construction of the real and complex numbers. | | | An example of a definable, non-computable real number is Chaitin's constant, Ω. |
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http://en.wikipedia.org/wiki/Computable_number
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| | Good Candidates for Computable Reals |
| | For example Bishop85 talks of a real number as a regular sequence of rationals A sequence is regular if the difference between the n th and m th elements in the sequence is not greater than (1/m+1/n). | | | All the operations which one would expect to be computable over the computable reals (which is probably the same as the constructive functions over constructive reals) must also be computable over the chosen representatives. | | | For a representation of the computable reals to work properly its not sufficient for the representations of the computable reals to be computable. |
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http://www.rbjones.com/rbjpub/cs/cs009.htm
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| | On Computable Numbers, with an Application to the Entscheidungsproblem [Decision Problem] [encyclopedia] |
| | We have described a number which is not computable - this seems to be a paradox since we appear to have described in finite terms, a number which cannot be described in finite terms. | | | And a computable real number is one for which there is a computer program or algorithm for calculating the digits one by one. | | | To compute the Nth digit of this number, you get the Nth computer program and then you start it running until it puts out an Nth digit, and at that point you change it. |
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http://kosmoi.com/Computer/Science/Entscheidungsproblem
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| | Computable number - Wikipedia, the free encyclopedia |
| | In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. | | | There are however many real numbers which are not computable: the set of all computable numbers is countable (because the set of algorithms is) while the set of real numbers is not (see Cantor's diagonal argument). | | | The computable numbers form a real closed field and can be used in the place of real numbers for many, but by no means all, mathematical purposes. |
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http://en.wikipedia.org/wiki/Computable_number
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| | PlanetMath: computable number |
| | This is version 7 of computable number, born on 2003-04-08, modified 2004-09-28. | | | A complex number is called computable if its real and imaginary parts are computable. | | | There are however many real numbers which are not computable: the set of all computable numbers is countable (because the set of algorithms is) while the set of real numbers is |
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http://planetmath.org/encyclopedia/ComputableNumber.html
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| | PlanetMath: computable number |
| | There are however many real numbers which are not computable: the set of all computable numbers is countable (because the set of algorithms is) while the set of real numbers is | | | This is version 7 of computable number, born on 2003-04-08, modified 2004-09-28. | | | Every computable number is definable, but not vice versa. |
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http://planetmath.org/encyclopedia/ComputableNumber.html
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| | Talk:Computable number - Wikipedia, the free encyclopedia |
| | For computable numbers, to even start the diagonal argument, you have to order the algorithms, and there is no algorithm which can do that, so the result is not a computable number. | | | The set of numbers described by this definition is not even closed under addition (plus, you would get a different set of numbers in different bases). | | | One thing I still don't understand (and which is not sufficiently clear in definable number) is the following: So one can prove the existence of an enumeration of the set of definable numbers, say by lexicographically ordering the corresponding formulas. |
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http://en.wikipedia.org/wiki/Talk:Computable_number
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| | Recursive set - Term Explanation on IndexSuche.Com |
| | In recursion_theory (otherwise called the theory of computability) a set ''S'' of integers, or natural numbers, or literal_string s, or tuples of any of the above, is | | | or Computable or Decidable if there is an algorithm that, when given a number or literal string or tuple (as the case may be) returns a correct yes-or-no answer to the question of whether the input number, string, or tuple is a member of the set ''S''. | | | A copy of the license is included in the section entitled |
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http://www.indexsuche.com/Recursive_set.html
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| | Is a deterministic universe logically consistent with a probabilistic Quantum Theory? |
| | We take Church’s Thesis to essentially state that a number-theoretic function is effectively computable if, and only if, it is recursive, and a partial number-theoretic function is effectively computable if, and only if, it is partial recursive (3, p227). | | | We note too, that a number-theoretic function is Turing-computable if, and only if, it is partial recursive (3, p233, Corollary 5.13 and p237, Corollary 5.15). | | | Classically (3, p120, 121, 214), a partial function f of n arguments is called partial recursive if, and only if, f can be obtained from the initial functions (zero function), projection functions, and successor function (of classical recursive function theory) by means of substitution, recursion and the classical, unrestricted, μ-operator. |
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http://alixcomsi.com/CTG_08_Quantum_consistency1.htm
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| | Rober Rosen - Effective Processes, Computation and Complexity |
| | As he says, "Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable." In other words, whereas the set of real numbers is so large as to be uncountably infinite, the set of computable numbers is countably (enumerably) infinite. | | | Turing begins his paper: "The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. | | | [2] Turing, A. "On computable numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, ser. |
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http://www.panmere.com/rosen/effprocess1.htm
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| | Abstracts of Christopher K. Mccord |
| | The Lefschetz number L(f), on the other hand, is readily computable, but does not give a lower bound for the number of fixed points. | | | The Lefschetz number L(f), on the other hand, is readily computable, but usually does not estimate the number of fixed points. | | | By considering the Nielsen and Lefschetz coincidence numbers, we show that N(f) >= L(f)$ for all self-maps on compact infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable group of finite index). |
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http://math.uc.edu/~meyer/abstract.htm
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| | Who Can Name the Bigger Number? |
| | Some mathematicians thought that ‘computable’ coincided with a technical notion called ‘primitive recursive.’ But in 1928 Wilhelm Ackermann disproved them by constructing a sequence of numbers that’s clearly computable, yet grows too quickly to be primitive recursive. | | | Ackermann number is bigger than X. The inverse grows as slowly as Ackermann’s original sequence grows quickly; for all practical purposes, the inverse is at most 4. | | | The famous mathematician G. Hardy quipped that Skewes’ was "the largest number which has ever served any definite purpose in mathematics.") What’s more, Ackermann’s briskly-rising cavalcade performs an occasional cameo in computer science. |
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http://www.scottaaronson.com/writings/bignumbers.html
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| | solution |
| | Even if fn(x) determines a 'real number', which function it is is only determinable from its ordinal place amongst all computable functions, not from its ordinal place amongst the real-number functions, with the result that, if the latter is 'n', then fn(x) is not a calculable function of n. | | | If we were to define 'real numbers' not in terms of Platonic limits, but merely convergent sequences of rationals, as the Intuitionists have done, then we would be identifying 'real numbers' with certain functions, since sequences are functions from the natural numbers. | | | And that leads to the representation of every real number as not just the would-be limit of a sequence of finite decimals, but also a limit which is actually reached, so that a real number is identical with a certain infinite decimal ([26], pp189, 191). |
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http://www.arts.uwa.edu.au/philoswww/Staff/solution.html
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| | Luciano Floridi |
| | Alan Turing's contributions to computer science are so outstanding that two of his seminal papers, "On Computable Numbers with an application to the Entscheidungsproblem" and "Computing Machinery and Intelligence", have provided the foundations for the development of the theory of computability, recursion functions and artificial intelligence. | | | CTT implies that we shall never be able to provide a formalism F that both captures the former notion and is more powerful than a Turing Machine, where "more powerful" means that all TM-computable functions are F-computable but not vice versa. | | | CTT remains a "working hypothesis", still falsifiable if it is possible to prove that there is a class of functions that are effectively computable in the sense of {1,2,3,4} but are not TM-computable. |
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http://www.wolfson.ox.ac.uk/~floridi/ctt.htm
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| | natural theology > synopsis > 27 Alan Turing |
| | Turing concentrated on 'computable numbers' which he described as 'the real numbers whose expression as a decimal is calculable by finite means'. | | | Further, Turing noted that the class of computable numbers is denumerable (ie its cardinal number is aleph(0), whereas the class of real numbers is non denumerable (its cardinal number is aleph(>0). | | | The essence of the decision problem could be captured by dealing with the computable numbers, and then extended to other fields of mathematics. |
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http://naturaltheology.net/Synopsis/s27Turing.html
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| | On computable numbers, with an application to the Entscheidungsproblem - A. M. Turing, 1936 |
| | The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. | | | Although the class of computable numbers is so great, and in many ways similar to the class of real numbers, it is nevertheless enumerable. | | | It may be thought that arguments which prove that the real numbers are not enumerable[5] would also prove that the computable numbers and sequences cannot be enumerable. |
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http://www.cs.umass.edu/~immerman/cs601/turingReference.html
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| | CSE 3323, Assignment 1, Solution |
| | From the Turing "Entscheidungsproblem" paper, "a number is computable if its decimal can be written down by a machine." And "although the class of computable numbers is so great,... | | | I was shocked at the number of spelling and grammatical errors, showing that students failed to proof-read their work. | | | The presence of signal corresponded to a 1 and absence corresponded to a 0, hence a binary number could be formed. |
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http://www.csse.monash.edu.au/courseware/cse3323/CSE3323-2000/cse3323-assign-solution.html
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| | Function (mathematics) - Enpsychlopedia |
| | The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. | | | A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. | | | This argument shows that there are functions from integers to integers that are not computable. |
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http://www.grohol.com/wiki/Function_(mathematics)
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| | Computability and Complexity - Introduction |
| | But Turing says that these computable numbers constitute a small part of the set of all numbers and for the most part the great percentage of numbers are not computable. | | | So a number is computable if there is such a TM that will stop after printing n number of 1s on an input tape of zeroes. | | | The first abstract model of computation was defined by Alan Turing in 1936. |
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http://hypatia.math.uri.edu/~kulenm/mth381pr/comput/computab.html
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| | Function (mathematics) - Wikipedia, the free encyclopedia |
| | The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. | | | A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. | | | This argument shows that there are functions from integers to integers that are not computable. |
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http://en.wikipedia.org/wiki/Function_(mathematics)
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| | Function (mathematics) - Encyclopedia.WorldSearch |
| | The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. | | | A very common type of function occurs when the argument and the function value are both numbers, the functional relationship is expressed by a formula, and the value of the function is obtained by direct substitution of the argument into the formula. | | | This argument shows that there are functions from integers to integers that are not computable. |
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http://encyclopedia.worldsearch.com/function_(mathematics).htm
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