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Topic: On Computable Numbers

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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.
		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.
		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).
	en.wikipedia.org /wiki/Computable_number (1350 words)

	Wikipedia:WikiProject Numbers - Wikipedia, the free encyclopedia
		It is a prime number, a Mersenne prime, a Fermat prime, a permutable prime, a palindromic prime, a composite number, a highly composite number, an abundant number, a surreal number, and an amicable number with 0.
		It is the Xth prime number, the previous being Number N - 2x, with which it comprises a twin prime, the next being [[Number N + 2x]], with which it comprises a twin prime.
		User:Egil's proposal for naming the articles Number N (number), and making the spelled out names of the numbers be redirects, (e.g, Four hundred and ninety-six redirects to 496 (number)), received the most votes, and with the initiative of User:Dysprosia, the articles were moved accordingly.
	en.wikipedia.org /wiki/Wikipedia:WikiProject_Numbers (3828 words)

	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
		Computable numbers were introduced by Alan Turing in 1936.
		This is version 7 of computable number, born on 2003-04-08, modified 2004-09-28.
	planetmath.org /encyclopedia/ComputableNumber.html (285 words)

	On computable numbers, with an application to the Entscheidungsproblem - A. M. Turing, 1936 (Site not responding. Last check: 2007-11-06)
		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.
		The numbers of letters “A” and “C” are to agree with the numbers chosen in §5, so that, in particular, “0” is replaced by “DC”, “1” by “DCC”, and the blanks by “D”.
	www.cs.umass.edu /~immerman/cs601/turingReference.html (6790 words)

	Search Results for numbers
		The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
		In addition to his important work in the number theory of transcendental numbers (that is, numbers that are not the solution of an algebraic equation with rational coefficients) Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable.
		Each number from 0 to 9 was represented by the position of a large and small peg in a square array, and numbers with 2, 3 or larger numbers of digits were represented by placing 2, 3 or a larger number of squares in a horizontal row.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=numbers&CONTEXT=1 (15603 words)

	natural theology > synopsis > 27 Alan Turing (Site not responding. Last check: 2007-11-06)
		Turing concentrated on 'computable numbers' which he described as 'the real numbers whose expression as a decimal is calculable by finite means'.
		The essence of the decision problem could be captured by dealing with the computable numbers, and then extended to other fields of mathematics.
		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).
	naturaltheology.net /Synopsis/s27Turing.html (1169 words)

	MathAction and Axiom RealNumbers
		In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line.
		Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type.
		The computable numbers form a real closed field and can be used in the place of real numbers for some, but by no means all, mathematical purposes.
	www.axiom-developer.org /zope/mathaction/RealNumbers (767 words)

	Rober Rosen - Effective Processes, Computation and Complexity
		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.
		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.
		Placing the majority of real numbers out of the realm of Turing-computability may seem like a devastating blow to a commonsense notion of calculability, but the reality is that there is a priori no good reason to suppose that there should exist a finite recursive function for calculating any arbitrary real number.
	www.panmere.com /rosen/effprocess1.htm (3198 words)

	Home Business Forums (Site not responding. Last check: 2007-11-06)
		A real number a is said to be computable if it can be approximated by some algorithm (or Turing machine), in the following sense: given any integer n ≥ 1, the algorithm produces an integer k such that:
		The computable numbers include all specific real numbers which appear in practice, including all algebraic numbers, e, $\pi, et cetera.$
		It has been hypothesized that most of analysis could be reconstructed using computable numbers, although as of yet no one has taken the time to undertake this project.
	homebusinessforums.com /index.php?title=Computable_number (1527 words)

	Alan Turing (Site not responding. Last check: 2007-11-06)
		For the set of all Turing machines is countable, and hence so is the class of computable numbers, which is in one-to-one correspondence with a subset of that set.
		The number h whose nth digit is 1 if the nth TM [in some ordering] halts when started on a blank tape and is 0 if it doesn't halt.
		In real computer programmes, you write the formula however it drops out, but then, of course, you are dealing not with real numbers but with a computer approximation to real numbers, and numbers like 1/3 don't [usually] even have a representation.
	www.arrakis.es /~carlisg/lectura_5.htm (1474 words)

	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.
		Therefore there are a countably infinite number of computer programs obviously (in the same way that there are a countably infinite number of decimals which have a finite number of places, under any base).
		Yes, the definition of a computable number is one for which there is a program that computes the nth place of the number.
	www.nrich.maths.org.uk /askedNRICH/edited/1172.html (1260 words)

	Technical Report # 36 (Site not responding. Last check: 2007-11-06)
		Some irrational numbers are "random" in a sense which implies that no algorithm can compute their decimal expansions to an arbitrarily high degree of accuracy.
		This feature of (most) irrational numbers has been claimed to be at the heart of the deterministic, but chaotic behavior exhibited by many non-linear dynamical systems.
		In this paper, a number of now-classical chaotic systems are shown to remain chaotic when their domains are restricted to the computable real numbers, providing counter-examples to the above claim.
	www.cogs.indiana.edu /Publications/techreps1991/36.html (110 words)

	[No title]
		I believe that there is a constructive function of a real number, which is constructively continuous, such that the location of the maximum is not 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.
		What we're relying upon is the fact that every machine which does compute a real number will eventually manage to compute an epsilon_n/2 approximation of its number, on the slies of time we give it, and thus the associated interval will get included in the list and cover the associated real number.
	www.math.niu.edu /~rusin/known-math/00_incoming/constructive (1191 words)

	Glenn's Presentation Outline
		In 1899, Giuseppe Peano axiomatized the arithmetic of cardinal numbers.
		We agree to associate with the formula the unique number that is the product of the first ten primes in order of magnitude, each prime being raised to a power equal to the Gödel number of the corresponding elementary sign.
		In a similar fashion, a unique number, the product of as many primes as there are signs (each prime being raised to a power equal to the Gödel number of the corresponding sign), can be assigned to every finite sequence of elementary signs and, in particular, to every formula.
	www.physics.ucla.edu /~chester/CES/november/glenn.html (1210 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		They perform, or at least are expected to perform, computations on sets like the set of real numbers, the open subsets or the compact subsets of real numbers, or the set of continuous functions from the real unit interval to the real numbers.
		Scientific computation is the domain of computation which is based mainly on the equations of physics.
		Computer science is oriented by the digital nature of machines and by its discrete foundations given by Turing machines.
	www.cs.columbia.edu /cacnet/digest/cac09.05 (2017 words)

	Citations: with an application to the Entscheidung problems - Turing, real (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		He described the recursive real numbers as a subset of the real numbers obtained by imposing restrictions on the definition of a real number.
		The main issue in the present work is to study the analogous problems in the case of p adic numbers and verify which ones carry over and which ones fail and the reason for the failures.
		In fact, in terms of computational complexity, the fact that quintic equations may not have radical expressions for their roots is largely irrelevant; it simply rules out one mode of expression.
	citeseer.ist.psu.edu /context/272810/0 (1145 words)

	natural theology > notes > 24 february 2002 (Site not responding. Last check: 2007-11-06)
		'The "computable" numbers maybe 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 as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth.
		I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another.
	www.naturaltheology.net /Notes/Notes02/notesM02D24.html (624 words)

	First edition of On Computable Numbers by Alan Turing offered by The Manhattan Rare Book Company
		On Computable Numbers, with an Application to the Entscheidungsproblem, In Proceedings of The London Mathematical Society, Series 2, vol 42, Part 3 and 4, pp.
		Turing interpreted this to mean a computing machine and set out to design one capable of resolving all mathematical problems, but in the process he proved in his seminal paper “On Computable Numbers, with an Application to the Entscheidungsproblem [‘Halting Problem']” (1936) that no such universal mathematical solver could ever exist.
		His work characterized the abstract essence of any computing device so well that it was in effect a challenge to actually build one… Turing's work was an inspiration to pursue something of which most of the world had not even conceived: a universal computing machine” (Britannica).
	www.theworldsgreatbooks.com /turing1.htm (492 words)

	PlanetMath: computable real function
		A real function is computable if it is both sequentially computable and effectively uniformly continuous.
		It is not hard to generalize these definitions to functions of more than one variable or functions only defined on a subset of
		This is version 3 of computable real function, born on 2004-09-29, modified 2004-09-29.
	planetmath.org /encyclopedia/EffectiveUniformContinuity.html (147 words)

	Untitled Document
		10 I give some arguments with the intention of showing that the computable numbers include all numbers which could naturally be regarded as computable.
		p.234, and, in fact, any computable sequence is capable of being described in terms of such a table.
		§2, and corresponding to any machine of this type a computing machine can be constructed to compute the same sequence, that is to say the sequence computed by the computer.
	www.crumpled.com /cp/classics/turing1.html (6068 words)

	On Task-Specific Initialization
		Some instruction numbers, in particular the small ones, are computable by very short programs, others are not.
		In general, programs consisting of many instructions that are not so easily computable, given the initial arithmetic instructions (Section A.2.1), tend to be less probable.
		Similarly, as the number of frozen programs grows, those with higher addresses in general become harder to access, that is, the address computation may require longer subprograms.
	www.idsia.ch /~juergen/oopsweb/node31.html (233 words)

	CSE 3323, Assignment 1, Solution
		The presence of signal corresponded to a 1 and absence corresponded to a 0, hence a binary number could be formed.
		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.
	www.csse.monash.edu.au /courseware/cse3323/CSE3323-2000/cse3323-assign-solution.html (833 words)

	Turing’s Golden (Site not responding. Last check: 2007-11-06)
		Then he suggests substituting a computer for A (so that, perversely and curiously, the computer’s assignment briefly appears to be that of simulating a man simulating a woman (Moody 1993)).
		It is, for example, child’s play to program a computer to simulate human like mistakes and slowness in mathematical questions but extraordinarily hard to display the talents humans characteristically deploy in ordinary conversation over a range of topics (it certainly hasn’t been done).
		While computers can simulate or exceed the performance of human experts in narrowly defined areas, they are laughably far from simulating the core human competencies most called for in the Turing test.
	www.hfac.uh.edu /phil/leiber/Turing'sGolden.htm (11273 words)

	Computing machinery and intelligence
		This paper is often said to mark the beginning of the cognitivist revolution in psychology by arguing that computing machines that think are possible, thus defending the appropriateness of computational models of intelligence and, by extension, other cognitive processes.
		The paper also recommends the controversial Turing test, according to which a computing machine that can simulate a thinking, speaking human so well that a human judge cannot detect the simulation should be deemed to possess genuine intelligence.
		The 1936(?) paper on computability is the ultimate foundation of the entire "computational metaphor" that sets cognitive science apart from the more general "cognitive psychology".
	www.cogsci.umn.edu /OLD/calendar/past_events/millennium/files/1112224936.html (465 words)

	Non-Computability of Thought (Site not responding. Last check: 2007-11-06)
		As you must know, there are real numbers that are not computable by computers: After all, there are at most a countably infinite set of computers, and a countably infinite set of computer programs, whereas there is an uncountably infinite number of real numbers.
		Suppose the two computers run at clock speeds that are independent of each other (they are not synchronized).
		The most trivial of these are R itself (both computers generate the strings of alternating ones and zeros 010101010101...
	linas.org /plants/RCS/noncompute.html,v (584 words)

	Theoretische Informatik I - Computable Analysis
		Computable analysis supplies exact definitions for these and many other similar questions and tries to solve them.
		Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis.
		It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics.
	www.informatik.fernuni-hagen.de /import/thi1/klaus.weihrauch/book.html (181 words)

	Making Sense of Information II
		Hence, investigators of other phenomena ought not to presume that time reversibility is necessarily applicable in, or should be taken as a model for, their own inquiries.
		Perhaps also important in that such numbers can be generated in other ways and so may play a role in "reality".
		Hence, investigators of other phenomena ought not to presume that either arithmetic (and, by inference, formal mathematics generally) or "Turing-computability" is necessarily applicable in, or should be taken as a model for, their own inquiries.
	serendip.brynmawr.edu /local/scisoc/information/grobstein15july04.html (788 words)

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