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Topic: Church Turing thesis

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	Church–Turing thesis - Wikipedia, the free encyclopedia
		However, the thesis is a definition and not a theorem, and hence cannot be proved true.
		The success of the Church–Turing thesis prompted supertheses that extend the thesis, including the conjecture that there is a polynomial transformation from the representation of computable functions in one formalization to their representation in another, and the conjecture that every model of computation can be step-by-step simulated by a Turing machine.
		The universe is not a Turing machine (ie, the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer.
	en.wikipedia.org /wiki/Church-Turing_thesis (1418 words)

	Alonzo Church - Wikipedia, the free encyclopedia
		He and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative "mechanical processes for computation" had equivalent computational abilities.
		Church remained a professor of mathematics at Princeton until 1967, when he moved to California.
		Church's lambda calculus influenced the design of the Lisp family of computer languages as well as functional programming languages in general.
	en.wikipedia.org /wiki/Alonzo_Church (262 words)

	Church-Turing thesis - Encyclopedia, History and Biography (Site not responding. Last check: 2007-10-07)
		In computability theory the Church-Turing thesis, Church's thesis, Church's conjecture or Turing's thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers.
		The thesis claims that any calculation which is possible, can be performed by an algorithm running on a computer, provided that sufficient time and storage space are available.
		However, the thesis is a definition and not a theorem, and hence cannot be proven true.
	www.arikah.com /encyclopedia/Church-Turing_thesis (1452 words)

	The Church-Turing Thesis
		Turing introduced this thesis in the course of arguing that the Entscheidungsproblem, or decision problem, for the predicate calculus - posed by Hilbert (Hilbert and Ackermann 1928) -- is unsolvable.
		Turing introduces his machines with the intention of providing an idealised description of a certain human activity, the tedious one of numerical computation, which until the advent of automatic computing machines was the occupation of many thousands of people in business, government, and research establishments.
		Thus when Turing maintains that every number or function that "would naturally be regarded as computable" can be calculated by a Turing machine he is asserting not thesis M but a thesis concerning the extent of the effectively calculable numbers and functions.
	www.science.uva.nl /~seop/archives/win2002/entries/church-turing (4911 words)

	Church-Turing thesis : The Church-Turing thesis (Site not responding. Last check: 2007-10-07)
		The thesis, which is now generally assumed to be true, is also known as Church's thesis or Church's conjecture (named after Alonzo Church) and Turing's thesis (named after Alan Turing).
		However, the thesis doesn't have the status of a theorem and cannot be proven; it is conceivable but unlikely that it could be disproven by exhibiting a method which is universally accepted as being an effective algorithm but which cannot be performed on a Turing machine.
		The universe isn't a Turing machine (ie, the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a hypercomputer.
	www.city-search.org /th/the-church-turing-thesis.html (1117 words)

	The Church-Turing Thesis
		Turing introduced this thesis in the course of arguing that the Entscheidungsproblem, or decision problem, for the predicate calculus - posed by Hilbert (Hilbert and Ackermann 1928) - is unsolvable.
		Other writers maintain thesis M (or some equivalent or near equivalent) on the spurious ground that the various and prima facie very different attempts - by Turing, Church, Post, Markov, and others - to characterise in precise terms the informal notion of an effective procedure have turned out to be equivalent to one another.
		Turing introduces his machines with the intention of providing an idealised description of a certain human activity, the tedious one of numerical computation, which until the advent of automatic computing machines was the occupation of many thousands of people in commerce, government, and research establishments.
	setis.library.usyd.edu.au /stanford/archives/win1998/entries/church-turing (4902 words)

	The Church – Turing Thesis (Site not responding. Last check: 2007-10-07)
		The strategies taken have been to show that either there is a computable function that cannot be modeled by a Turing Machine, or to show that the Church – Turing Thesis leads to a contradiction when it is subjected to mathematical proof techniques.
		There also is no way that the Church – Turing Thesis may be proven mathematically to be true because it is virtually impossible to show that all computable problems in existence can be modeled by a Turing Machine.
		But, this thesis is very similar to theories that have been proposed in the natural sciences which in essence are also not provable in the sense of mathematics.
	ucsu.colorado.edu /~stephanb/projects/CSCI5444.htm (534 words)

	Turing's Thesis (Site not responding. Last check: 2007-10-07)
		Turing's thesis has withstood the test of time and many examinations.
		They are equivalent to a kind of Turing machine with separate read- and write- heads that move in one direction only, with the tape in between.
		In order to show a certain function is computable by a Turing machine, it suffices to give a verbal description of a computational process, even without the use of a formal model.
	www.cs.hmc.edu /claremont/keller/webBook/ch06/sec06.html (1816 words)

	The Paradigms and Paradoxes of Intelligence, Part 2: The Church-Turing Thesis
		Turing, along with mathematician and philosopher Alonzo Church, advanced, imdependently, an assertion that has become known as the Church-Turing thesis: If a problem that can be presented to a Turing Machine is not solvable by one, then it is also not solvable by human thought.
		Although the existence of Turing's unsolvable problems is a mathematical certainty, the Church-Turing thesis is not a mathematical proposition at all.
		Accepting this thesis means that there are questions for which answers can be shown to exist but which can never be found (and to date, no human has ever solved one of Turing's unsolvable problems, or at least has never proven that he or she has).
	www.kurzweilai.net /articles/art0256.html (2020 words)

	The Church-Turing Thesis
		One of Turing's achievements in his paper of 1936 was to present a formally exact predicate with which the informal predicate ‘can be calculated by means of an effective method’ may be replaced.
		The further proposition, very different from Turing's own thesis, that a Turing machine can compute whatever can be computed by any machine working on finite data in accordance with a finite program of instructions, is sometimes also referred to as (a version of) the Church-Turing thesis or Church's thesis.
		Copeland, B.J. ‘Turing's O-machines, Penrose, Searle, and the Brain’.
	plato.stanford.edu /entries/church-turing (4930 words)

	Michael Nielsen: Interesting problems: The Church-Turing-Deutsch Principle (Site not responding. Last check: 2007-10-07)
		So Church and Turing mathematically defined a class of functions — called, unsurprisingly, the “computable functions” — that they claimed corresponded to the class of functions that a human mathematician would reasonably be able to compute.
		Turing makes a rather more convincing case, talking at some length about the sorts of algorithmic processes that he could imagine a mathematician performing with pencil and paper, arguing that each such process could be simulated on a Turing machine.
		Turing’s arguments were a creditable foundation for the Church-Turing thesis, but they do not rule out the possibility of finding somewhere in Nature a process that cannot be simulated on a Turing machine.
	www.qinfo.org /people/nielsen/blog/archive/000071.html (4351 words)

	Ray Kurzweil Response to Searle Review (Site not responding. Last check: 2007-10-07)
		It is remarkable that Searle refers to the Church-Turing Thesis as a "mathematical result." He must be confusing the Church-Turing Thesis (CTT) with Church and Turing theorems.
		CTT is not a mathematical theorem at all, but rather a philosophical conjecture which relates to a proposed relationship between what a human brain can do and what a Turing machine can do.
		This thesis states that all problems that a human being can solve can be reduced to a set of algorithms, supporting the idea that machine intelligence and human intelligence are essentially equivalent).
	www.kurzweiltech.com /Searle/searle_response_10.htm (427 words)

	Church-Turing thesis
		We then came to see that other mathematically precise analyses were demonstrably equivalent to Turing's (and Church's), and some employed the notion of a machine, for ex.
		But any attempt to have the thesis claim that procedures beyond algorithmic ones are Turing computable\register machine computable\etc. seems to reflect a confusion that can be cleared up by reflecting on this history of the emergence of the provably equivalent analyses of the intuitive informal notion of an algorithmic procedure in terms of idealized machines.
		In either event, we need no reference to his machines, or any machines on the lefthand side of the C-T thesis, and whatever the notion is used to fix the scope of the lefthand side, the resulting scope must be arguably equivalent to that of "algorithmic proceudre".
	philo.at /phlo/199803/msg00061.html (312 words)

	CSC 110 Topic 29
		Turing defined his machines in a simple way because this made them more believable as models of mechanical processes (and it also made them conceptually easier to work with in mathematical proofs).
		One such definition, proposed by mathematician Alonzo Church in 1936 (the same year that Turing invented the TM), is based on recursive functions instead of Turing machines.
		The important thing is that the binary serial number that encodes a Turing machine is unique in that the same serial number cannot encode two different TMs (since the rules of a TM can be reconstructed exactly from its serial number).
	www.augustana.ab.ca /~hackw/csc110/topic/top29.html (1100 words)

	BletchleyPark.net
		Alonzo Church's λ-calculus paper and Alan Turing's "Machines" paper in 1936 connected the informal notion of an algorithm or computable function to a precise definition.
		We have the (1) machines paradigm from Turing, the (2) recursive functions from Church and (3) languages that emerged after Church and Turing.
		The Church-Turing thesis, although is not a proof of algorithm, it remains true in the philosophical sense, since it hasn't been disapproved.
	www.bletchleypark.net /computation/church_turing.html (260 words)

	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.
	www.wolfson.ox.ac.uk /~floridi/ctt.htm (5326 words)

	The Church-Turing Thesis: Breaking the Myth \| Lambda the Ultimate
		It seems to shoot at fish in a barrel (the so-called "strong" Church-Turing thesis, of which there are actually several, including the rather ill-formed proposition which is debunked in the paper), and while the SCTT as described is indeed ill-formed, this is hardly a remarkable or notable result.
		In either case, the fact that the CTT disclaims IO/interactivity (and Turing's so-called "A" machine was formulated explicitly to avoid it, by requiring inputs to be pre-loaded onto the tape before execution) is well-known.
		Turing himself fashioned this limitation to exclude the possibility of what he called O-machines (a Turing machine communiticating with a hyper-Turing "oracle"--such an aggregation would certainly be hyper-Turing in power).
	lambda-the-ultimate.org /node/view/1038 (9306 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		Historically, Church and Turing hit on their respective theses (Church's thesis, Turing's thesis) independently, using different but equivalent formalisms.
		The Church-Turing thesis can't be proved mathematically because it asserts the equality of two things, only one of which has a precise definition.
		Nevertheless, to those who believe that the old informal notion of computability has some definite meaning, the Church-Turing thesis is not just an arbitrary definition, but an assumption which might be false, and could conceivably be refuted one day, if someone comes up with a way of computing a Turing-uncomputable function.
	www.math.niu.edu /~rusin/known-math/99/church (612 words)

	Church-Turing thesis (Site not responding. Last check: 2007-10-07)
		Any computer program in any of the conventional programming languages can be translated into a Turing machine, and any Turing machine can be translated into most programming languages, so the thesis is equivalent to saying that the conventional programming languages are sufficient to express any algorithm.
		The notion of "effective method" is intuitively clear but is not formally defined since it is not exactly clear what a "simple and precise instruction" is, and what exactly the "required intelligence to execute these instructions" is. (See for example effective results in number theory for cases well beyond the Euclidean algorithm.)
		However, the thesis does not have the status of a theorem and cannot be proven; it is conceivable but unlikely that it could be disproven by exhibiting a method which is universally accepted as being an effective algorithm but which cannot be performed on a Turing machine.
	www.knowallabout.com /c/ch/church_turing_thesis.html (963 words)

	Church-Turing thesis
		Imagine a "Turing Test" in which the interrogators must be convinced that the participant is a normally sighted individual.
		In short, Turing suggested that a machine that could behave in a manner indistinguishable from a human could be considered to be "thinking." For many researchers, the goal is simply to pass the Turing Test.
		The Church Awakens "The AIDS pandemic is the greatest humanitarian crisis," Casey said.
	www.stargeek.com /item/165629.html (2896 words)

	Church Turing Thesis
		One of Turing's achievements in his paper of 1936 [OnComputableNumbers] was to present a formally exact predicate with which the informal predicate 'can be calculated by means of an effective method' may be replaced.
		Turing's thesis says that all ModelsOfComputation are no stronger than the computational power of a TuringMachine.
		Turing also introduced another kind of machine, the OMachine, which is not deterministic this is wrong, an oracle must be deterministic, but it can answer any question, even ones that are not computable by a TM.
	c2.com /cgi/wiki?ChurchTuringThesis (1846 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The Church-Turing Thesis: Algorithm = Computable by some TM That is, everything intuitively computable (using the intuitive notion of an algorithm) can be computed by a TM (and vice versa).
		Evidence in support of the Church-Turing thesis: - it has not been disproved yet, - a few other formalizations of algorithm were proposed, but all of them turn out to be equivalent to the notion of Turing machine, - Turing's argument how TM models a human computer is convincing.
		Remark: It is plausible that someone may come up with a different notion of algorithm (e.g., using yet to be discovered laws of physics) and show that certain problems unsolvable by Turing machines are solvable in the new model.
	www.cs.sfu.ca /~kabanets/cmpt308/lectures/3.txt (281 words)

	[No title]
		The result is also known as Church's thesis or Church's conjecture and Turing's thesis.
		The thesis might be rephrased as saying that the notion of effective or mechanical method in logic and mathematics is captured by Turing machines.
		Stephen Kleene (Church 1932, 1936a, 1941, Kleene 1935) and recursive functions by
	en-cyclopedia.com /wiki/Church-Turing_thesis (1055 words)

	Tony Santolupo (Site not responding. Last check: 2007-10-07)
		The Church-Turing thesis is the brainchild of Alan Turing and Alonzo Church.
		Both Turing and Church reached the hypothesis independently and in different forms.
		By realizing that the neuronic level of the brain functions according to the same cause-and-effect paradigm as the rest of reality we could conclude that all faculties of the mind are ‘mechanical processes’.
	community.middlebury.edu /~schar/Courses/fs023.F02/paper2/tony.htm (2446 words)

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