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1. The Church-Turing Thesis (Stanford Encyclopedia Of Philosophy)
There are various equivalent formulations of the churchturing thesis. A common one is that every effective computation can be carried out by a Turing
http://plato.stanford.edu/entries/church-turing/
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##### The Church-Turing Thesis
First published Wed Jan 8, 1997; substantive revision Mon Aug 19, 2002 There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind.
##### The Thesis and its History
The Church-Turing thesis concerns the notion of an effective or mechanical
• M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols); M will, if carried out without error, produce the desired result in a finite number of steps; M can (in practice or in principle) be carried out by a human being unaided by any machinery save paper and pencil; M demands no insight or ingenuity on the part of the human being carrying it out.
• 2. ChurchÃ¢ÂÂTuring Thesis - Wikipedia, The Free Encyclopedia
They would not however invalidate the original or Physical churchturing thesis, since a quantum computer can always be simulated by a Turing machine.
http://en.wikipedia.org/wiki/Church-Turing_thesis
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##### ChurchÃ¢ÂÂTuring thesis
(Redirected from Church-Turing thesis Jump to: navigation search In computability theory the ChurchÃ¢ÂÂTuring thesis (also known as Church's thesis Church's conjecture and Turing's thesis ) is a combined hypothesis about the nature of effectively calculable (computable) functions by recursion (Church's Thesis), by mechanical device equivalent to a Turing machine (Turing's Thesis) or by use of Church's ÃÂ»-calculus
Church's Thesis: " Every effectively calculable function (effectively decidable predicate) is general recursive " (Kleene 1952:300) Turing's Thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX. " (Kleene 1952:376)
The three computational processes (recursion, ÃÂ»-calculus, and Turing machine) were shown to be equivalent by Alonzo Church Stephen Kleene and J.B. Rosser

3. Church-Turing Thesis -- From Wolfram MathWorld
The churchturing thesis (formerly commonly known simply as Church s thesis) says that any real-world computation can be translated into an equivalent
http://mathworld.wolfram.com/Church-TuringThesis.html
 Search Site Algebra Applied Mathematics Calculus and Analysis ... Rowland, Todd Church-Turing Thesis The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata combinators register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing. There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an

4. The Turing-Church Thesis
The term churchturing thesis seems to have been first introduced by Kleene, with a small flourish of bias in favour of Church
http://www.alanturing.net/turing_archive/pages/Reference Articles/The Turing-Chu
AlanTuring.net Reference Articles
##### By Jack Copeland
There are various equivalent formulations of the Turing-Church thesis (which is also known as Turing's thesis, Church's thesis, and the Church-Turing thesis). One formulation of the thesis is that every effective computation can be carried out by a Turing machine.
##### Effective Methods
The Turing-Church thesis concerns the notion of an effective or mechanical method in logic and mathematics. 'Effective' and its synonym 'mechanical' are terms of art in these disciplines: they do not carry their everyday meaning. A method, or procedure, M, for achieving some desired result is called 'effective' or 'mechanical' just in case
• M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols);
• M will, if carried out without error, always produce the desired result in a finite number of steps;
• 5. The Church-Turing Thesis
The churchturing thesis. Turing proposed the following hypothesis. Every function which would naturally be regarded as computable can be computed by the
http://alumni.imsa.edu/~matth/quant/299/paper/node6.html
Next: Complexity Classes Up: The Classical Computer Previous: Turing Machines Contents
##### The Church-Turing Thesis
Turing proposed the following hypothesis: Every 'function which would naturally be regarded as computable' can be computed by the universal Turing machine. it should be noted that there is ambiguity as to what, precisely, a function which would naturally be regarded as computable means. Due to this ambiguity, this statement is not subject to rigorous proof. There is strong evidence for this hypothesis; many diverse models of computation have been shown to compute the same set of functions as a Turing machine, as yet there have been no counterexamples to the thesis. This thesis gives us insight into the ``power'' of computing machines. If a computing device can solve all the problems a Turing machine can solve, then it is as powerful as a Turing machine.
Matthew Hayward 2005-02-17

6. The Church-Turing Thesis: Breaking The Myth | Lambda The Ultimate
churchturing thesis Whenever there is an effective method (algorithm) for obtaining the values of a mathematical function, the function can be computed by
http://lambda-the-ultimate.org/node/1038
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##### The Church-Turing Thesis: Breaking the Myth
This paper seeks to explode the myth that Turing Machines (TM) are the universal model for all computation. Church-Turing Thesis: Whenever there is an effective method (algorithm) for obtaining the values of a mathematical function, the function can be computed by a TM. [...] The Church-Turing thesis has since been reinterpreted to imply that Turing Machines model all computations, rather than just functions. This claim, which we call the Strong Church-Turing Thesis , is part of the mainstream theory of computation. In particular, it can be found in today's popular undergraduate theory textbooks: Strong Church-Turing Thesis: A TM can do (compute) anything that a computer can do. It is a myth that the original Church-Turing thesis is equivalent to this interpretation of it; Turing himself would have denied it. [...] In fact, the Strong Church-Turing Thesis is incorrect - the function-based behaviour of algorithms does not capture all forms of computation. For example, as explained in [Weg97], interactive protocols are not algorithmic. [...] The reasons for the widespread acceptance of what we call the "Turing Thesis Myth" can be traced to the establishment of computer science as a separate discipline in the 1960's. To understand these reasons better, we can identify three distinct claims that make up the myth, and examine them individually:

 7. The Church-Turing Thesis: Consensus And Opposition Research Workshop of the Israel Science Foundation Professor Martin Davis, New York University The churchturing thesis Consensus and Oppositionhttp://www.vanleer.org.il/eng/videoShow.asp?id=318

8. Computationalism And The Church-Turing Thesis
The churchturing thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent
http://www.umsl.edu/~piccininig/Computationalism and the Church-Turing Thesis 12
 Computationalism, the Church-Turing Thesis, and the Church-Turing Fallacy Gualtiero Piccinini Department of Philosophy University of Missouri St. Louis 599 Lucas Hall (MC 73) One University Blvd. St. Louis , MO 63121-4499 piccininig@umsl.edu Abstract The Church-Turing Thesis (CTT) is often employed in arguments for computationalism. I scrutinize the most prominent of such arguments in light of recent work on CTT and argue that they are unsound. Although CTT does nothing to support computationalism, it is not irrelevant to it. By understanding correctly the relationship between CTT and computationalism, we deepen our appreciation of computationalism as an empirical hypothesis. Computationalism, or the Computational Theory of Mind, is the view that mental capacities are explained by inner computations. In the case of human beings, computationalists typically assume that inner computations are realized by neural processes; I will borrow a term from current neuroscience and refer to them as neural computations. Typically, computationalists also maintain that neural computations are Turing-computable, that is, computable by Turing Machines (TMs).

9. Skeltoac Â» Church-Turing Thesis
Tag Archives churchturing thesis. Carry the one. December 6, 2007 Â 215 am. IÂve been nibbling through Douglas R. HofstadterÂs Pulitzer Prize-winning
http://skeltoac.com/tag/church-turing-thesis/
##### Carry the one
By Andy Skelton Posted in Creamy Filling Unvisible Wobble Also tagged algorithms Godel Escher Bach Hofstadter mathematics ... Comments (4)
• ##### Andy Skelton?
Yup! I'm known as "skeltoac" in many places. If you need to email me, use that three-syllable word followed by the "at" symbol and this domain: gmail.com.
##### Cyphergories

10. Church-Turing Thesis - Simple English Wikipedia, The Free Encyclopedia
The churchturing thesis (also known as Church s thesis, Church s conjecture and Turing s thesis) is a statement about computers.
http://simple.wikipedia.org/wiki/Church-Turing_thesis
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##### From Simple English Wikipedia - the free encyclopedia that anyone can change
Jump to: navigation search The Church-Turing thesis (also known as Church's thesis Church's conjecture and Turing's thesis ) is a statement about computers . It states that every computer program (every algorithm ) can be remade to run on a simple type of computer known as a Turing machine This short article can be made longer. You can help Wikipedia by adding to it Retrieved from " http://simple.wikipedia.org/wiki/Church-Turing_thesis Category Computer science Views Personal tools Getting around Search Toolbox In other languages

11. Computational Complexity: The Efficient Church-Turing Thesis
The churchturing thesis roughly states that everything computable is computable by a Turing machine. I strongly believe the church-turing thesis and have
http://weblog.fortnow.com/2006/12/efficient-church-turing-thesis.html
##### Computational Complexity
About Computational complexity and other fun stuff in math and computer science as viewed by Bill Gasarch. Blog created and written until March 2007 by Lance Fortnow. My Links Bill's Home Page Lance's Home Page Weblog Home Weblog Archives and Search ... Favorite Theorems Recent Posts Shifting Time On Paper Titles My International Day The Social Scientist ... Microfiche Complexity Links IEEE Conference on Computational Complexity Electronic Colloquium on Computational Complexity BEATCS Computational Complexity Column Complexity Zoo ... Favorite Complexity Books Weblogs Andy Drucker Ars Mathematica Computing Research Policy D. Sivakumar ... Terence Tao Other Links DMANET FYI Nielsen's Principles of Research Parberry's TCS Guides ... Theorynet Discussion Groups Computer Science Theory Theory Edge
##### Thursday, December 07, 2006

12. [quant-ph/0402128] Computable Functions, The Church-Turing Thesis And The Quantu
If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the churchturing thesis finitude of the degree of
http://arxiv.org/abs/quant-ph/0402128
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##### Title: Computable Functions, the Church-Turing Thesis and the Quantum Measurement Problem
Authors: R. Srikanth (Submitted on 18 Feb 2004 ( ), last revised 18 Feb 2004 (this version, v2)) Abstract: It is possible in principle to construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolution, would contradict the Church-Turing thesis, which lies at the foundation of computer science. Elsewhere we have argued that the quantum measurement problem implies a finite, computational model of the measurement and evolution of quantum states. If correct, this approach helps to identify the key feature that can reconcile quantum mechanics with the Church-Turing thesis: finitude of the degree of fine-graining of Hilbert space. This suggests that the Church-Turing thesis constrains the physical universe and thereby highlights a surprising connection between purely logical and algorithmic considerations on the one hand and physical reality on the other. Comments: 5 pages, no figures, REVTeX4

13. From Fred Galvin Galvin@math.ukans.edu Subject Re
The churchturing thesis is the proposition that the vague primordial notion The church-turing thesis can t be proved mathematically because it asserts
http://www.math.niu.edu/~rusin/known-math/99/church
 From: Fred Galvin Subject: Re: countability and computability Date: 23 Dec 1999 22:12:34 -0600 Newsgroups: sci.math Jeremy Boden writes: >Why is the Church-Turing thesis always described as a "thesis"? I would >infer from this that it is not a theorem in any reasonable logical >system. It connects the non-formal and intuitive notion of "computable" with something strict and formal (the Turing machine). So it could not be a theorem. Today many people take CT as actually being the definition of computable, which makes CT a definition, and makes the notion of computable formal or formalizable, instead of being intuitive. > I can appreciate that the halting problem is "quite tricky" but >a thesis is generally understood to be just a defensible opinion rather >than an accepted theorem, axiom or postulate. The trickiness of the halting problem really is not involved. It is just that a theorem cannot bridge the gap between the intuitive and the formal. ============================================================================== From: Gareth Rees Subject: Re: countability and computability Date: Fri, 24 Dec 1999 10:25:47 GMT Newsgroups: sci.math Jeremy Boden

 14. Church-Turing Thesis@Everything2.com A lot of people harbor misconceptions about the churchturing thesis. While it is a very significant statement about mathematics, its scope is not as broadhttp://www.everything2.com/index.pl?node_id=720661

15. The Church-Turing Thesis (Stanford Encyclopedia Of Philosophy)
setis.library.usyd.edu.au/ stanford/entries/churchturing/ - Similar pages The church-turing thesis (ResearchIndex)This paper and independently of the matter of considering Church s and Turing s thesis as de nitions in any philosophical sense or as empirical or
http://setis.library.usyd.edu.au/stanford/entries/church-turing/
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##### The Church-Turing Thesis
First published Wed Jan 8, 1997; substantive revision Mon Aug 19, 2002 There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind.
##### The Thesis and its History
The Church-Turing thesis concerns the notion of an effective or mechanical
• M is set out in terms of a finite number of exact instructions (each instruction being expressed by means of a finite number of symbols); M will, if carried out without error, produce the desired result in a finite number of steps; M can (in practice or in principle) be carried out by a human being unaided by any machinery save paper and pencil; M demands no insight or ingenuity on the part of the human being carrying it out.
•  16. Church-Turing Thesis Anima Ex Machina Archives. December 2007 November 2007 October 2007 March 2007 February 2007 January 2007 December 2006 November 2006 October 2006http://www.mathrix.org/liquid/tag/church-turing-thesis

17. Hypercomputation And The Physical Church-Turing Thesis -- Cotogno 54 (2): 181 --
A version of the churchturing thesis states that every effectively realizable physical system can be defined by Turing Machines (ÂThesis PÂ);
http://bjps.oxfordjournals.org/cgi/content/abstract/54/2/181
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##### Hypercomputation and the Physical Church-Turing Thesis
Paolo Cotogno A version of the Church-Turing Thesis states that every effectively realizable physical system can be defined by Turing Machines an empirical, more than a logico-mathematical, proposition. We review the main approaches to computation beyond Turing definability analog, quantum, and retrocausal computation. These models depend

18. Churchs Thesis, Or Church-Turing Thesis, Or ChurchÂs Theorem (mathematics) --Â
The churchturing thesis asserts that the informal notion of calculability is completely captured by the formal notion of recursive functions and hence,
http://www.britannica.com/eb/topic-117351/Churchs-thesis
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##### or Church-Turing thesis, or
A selection of articles discussing this topic. a principle formulated by the 20th-century American logician Alonzo Church, stating that the recursive functions are the only functions that can be mechanically calculated. The theorem implies that the procedures of arithmetic cannot be used to decide the consistency of statements formulated in accordance with the laws of arithmetic.
statement
• ##### statement (in logic, philosophy of: Logic and computability)
...have turned out to coincide with each other. The claim that recursive functions exhaust the class of all functions that are effectively calculable (in some intuitive informal sense) is known as Church's thesis (named after the American logician Alonzo Church).
##### statement (in Turing, Alan M.: Early life and career)

19. Disinfotainment: A Response To Kevin Marks' Anti-DRM Argument
Firstly, the churchturing thesis, one of the basic tenets of Computer Science, Unfortunately, Marks has completely misstated the church-turing thesis.
http://weblog.ceicher.com/archives/2006/01/a_response_to_kevin_marks_anti.html
Main
##### A Response to Kevin Marks' Anti-DRM Argument
Kevin Marks recently posted an argument against Digital Rights Management on his weblog and apparently has submitted it to a working group in the British House of Parliament. When I read his argument, I was astounded. The entire argument is founded on an error, a miscomprehension of a fundamental theorem of Computer Science.
I could summarize Marks' statement into two basic arguments:
1. DRM is futile, it can always be broken.
2. DRM is a perversion of justice.
Marks opens his argument with a huge misstatement of facts:
Firstly, the Church-Turing thesis, one of the basic tenets of Computer Science, which states that any general purpose computing device can solve the same problems as any other. The practical consequences of this are key - it means that a computer can emulate any other computer, so a program has no way of knowing what it is really running on. This is not theory, but something we all use every day, whether it is Java virtual machines, or Pentiums emulating older processors for software compatibility.
How does this apply to DRM? It means that any protection can be removed. For a concrete example, consider MAME - the Multi Arcade Machine Emulator - which will run almost any video game from the last 30 years. It's hard to imagine a more complete DRM solution than custom hardware with a coin slot on the front, yet in MAME you just have to press the 5 key to tell it you have paid.

20. A Modest Expansion Of The Scope Of The Church-Turing Thesis Â« Apperceptual
The churchturing thesis is that every function that would naturally be regarded as computable can be computed by a Turing machine. This thesis cannot be
http://apperceptual.wordpress.com/2007/01/07/a-modest-expansion-of-the-scope-of-
##### Apperceptual
Apperception: the process whereby perceived qualities of an object are related to past experience. Attributes and Relations: Redder than Red Democracy 2.0
##### A Modest Expansion of the Scope of the Church-Turing Thesis
The Church-Turing thesis is that every function that would naturally be regarded as computable can be computed by a Turing machine . This thesis cannot be proven rigorously, because it relates an informal notion (naturally be regarded as computable) to a formal notion (Turing machine). However, there is very strong evidence for the Church-Turing thesis, since every proposed formal treatment of computation ( Church Lambda calculus Markov algorithms , the Game of Life , and even quantum computers ) has been shown to be equivalent to Turing machines. real reality , in this context? That question is what I want to explore here. Turing developed the idea of Turing machines while working on Hilbert Entscheidungsproblem (decision problem). Informally, Hilbert wondered whether it was possible to automate mathematics. Could there be an automatic procedure that could take any mathematical theorem as input and produce as output a proof that the theorem is true or that it is false? Turning showed that there could be no such procedure. More precisely, he showed that there are mathematical questions that Turing machines cannot answer. A Turing machine is essentially a very abstract model of a mathematician Several people have argued that the physical universe may be a Turing machine ( Wolfram Poundstone Deutsch ). Others have argued that the human mind may be a Turing machine (

21. Church-Turing Thesis - Wikipedia
The churchturing thesis states in its most common form that every effective computation or algorithm can be carried out by a Turing machine.
http://nostalgia.wikipedia.org/wiki/Church-Turing_thesis
##### Church-Turing thesis
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22. Brains: The Physical Church-Turing Thesis: Modest Of Bold?
The Physical churchturing thesis Modest of Bold? This is the title of my talk at the Eastern APA, in the session on Classical Computation and
http://brainbrain.blogspot.com/2005/12/physical-church-turing-thesis-modest.html
##### Brains
On Mind and Related Matter
##### The Physical Church-Turing Thesis: Modest of Bold?
This is the title of my talk at the Eastern APA, in the session on Classical Computation and Hypercomputation (see previous post). The other presenter is Oron Shagrir and the commentator is Jack Copeland . Jack is probably the most distinguished philosopher of AI and computation, and Oron is one of the best philosophers in this area. Both are probably less well known in the US than they deserve. They are, respectively, from Israel and New Zealand. If you are based in the US, you may not have many other opportunities to see them in action. So if you are in NYC next week, you should consider coming to our session on Wednesday. posted by gualtiero piccinini at 4:58:00 PM

posted by
Name: gualtiero piccinini Location: St. Louis, Missouri, United States

23. Church-Turing Thesis Is Almost Equivalent To Zuse-Fredkin Thesis
In the present brief article we speculate about the mutual equivalence of ChurchTuring and Zuse-Fredkin theses. Since church-turing thesis is widely
http://digitalphysics.org/Publications/Petrov/Pet02a1/Pet02a1.htm
##### Church-Turing Thesis is Almost Equivalent to Zuse-Fredkin Thesis (An Argument in Support of Zuse-Fredkin Thesis)
ppetrov@digitalphysics.org Abstract: In the present brief article we speculate about the mutual equivalence of Church-Turing and Zuse-Fredkin theses. Since the Church-Turing thesis is widely accepted while the Zuse-Fredkin thesis is not, we propose their "near-equivalence" as a strong argument in support of the Zuse-Fredkin thesis. Last revised: September 6, 2003, 2:15 AM
##### 1. Introduction
Somewhere in 1936-37, shortly after the appearance of the fundamental work of Turing [ ] several researchers, among them Alonzo Church, Alan Turing himself, and (according to some sources) Emil Post as well independently came to the conclusion that all mathematical functions conceivable by the human mind can be also computed on a computational device now known as Universal Turing Machine (UTM) This famous proposition is usually referred to as "Church-Turing(-Post) thesis"; for brevity, we shall denote it below simply as the C-T thesis. There are various equivalent formulations of the C-T thesis; the one to be used here is:

24. The Argument About The Church-Turing Thesis
In their references, the authors listed Copeland s entry on The churchturing thesis in the Stanford Encyclopedia. In the summer of 1999, I circulated an
http://www.turing.org.uk/philosophy/stanford.html
Philosophy Area Alan Turing in the
Stanford Encyclopedia
of Philosophy:
##### A Thesis and an Antithesis
The origin of my article lies in the appearance of Copeland and Proudfoot's feature article in Scientific American, April 1999. This preposterous paper, as described on another page, suggested that Turing was the prophet of 'hypercomputation'. In their references, the authors listed Copeland's entry on 'The Church-Turing thesis' in the Stanford Encyclopedia. In the summer of 1999, I circulated an open letter criticising the Scientific American article. I included criticism of this Encyclopedia entry. This was forwarded (by Prof. Sol Feferman) to Prof. Ed Zalta, editor of the Encyclopedia, and after some discussion he invited me to submit an entry on 'Alan Turing.' My entry, which appeared in 2002, stands on its own as a discussion of Turing's life and thought, but it is also constructed as a corrective to Copeland's arguments. Whether anyone has ever noticed this I do not know, as the editors did not wish to see contributors' differences accentuated and they would probably escape any but a really expert reader. The point of this introduction is to highlight those differences. You can then read and judge for yourself. Copeland's entry is focussed on the claim that the Church-Turing thesis was never meant to apply to machines. It was a thesis ONLY about what a human being, working to a rule, could do. The thesis that anything a machine can do is computable, is called 'Thesis M' (following the logician

25. Science: Microsoft Researchers Prove The Church-Turing Thesis From Simpler Axiom
Microsoft researchers prove the churchturing thesis from simpler axioms (research.microsoft.com). 60 points posted 5 months ago by rspeer25 comments

 26. Church-Turing Thesis @ Computer-Dictionary-Online.org churchturing thesis @ Computer Dictionary Online. Computer terminology definitions including hardware, software, equipment, devices, jargon abbreviationshttp://www.computer-dictionary-online.org/?q=Church-Turing thesis

27. Digitalphysics.org Church-Turing Thesis Is Almost Equivalent To Zuse-Fredkin The
Below is a small excerpt of StartAid members bookmark description, for more information on churchturing thesis is Almost Equivalent to Zuse-Fredkin Thesis
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##### Church-Turing Thesis is Almost Equivalent to Zuse-Fredkin Thesis
Below is a small excerpt of StartAid members bookmark description, for more information on Church-Turing Thesis is Almost Equivalent to Zuse-Fredkin Thesis please visit the original webisite :
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them Alonzo Church, Alan Turing himself,and (according to some sources) Emil Post as well independentlycame to the conclusion that all mathematical functions conceivableby the human mind can be also computed on a computational devicenow known as Universal Turing Machine (UTM)1.This famous propositi
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28. PHYS771 Lecture 4: Minds And Machines
Alright, the main philosophical idea underlying computability is what s called the churchturing thesis. It s named after Turing and his adviser Alonzo
http://www.scottaaronson.com/democritus/lec4.html
 Lecture 4: Minds and Machines Scott Aaronson Today we're going to launch into something I know you've all been waiting for: a philosophical foodfight about minds, machines, and intelligence! First, though, let's finish talking about computability. One concept we'll need again and again in this class is that of an oracle. The idea is a pretty obvious one: we assume we have a "black box" or "oracle" that immediately solves some hard computational problem, and then see what the consequences are! (When I was a freshman, I once started talking to my professor about the consequences of a hypothetical "NP-completeness fairy": a being that would instantly tell you whether a given Boolean formula was satisfiable or not. The professor had to correct me: they're not called "fairies"; they're called "oracles." Much more professional!) Oracles were apparently first studied by Turing, in his 1938 PhD thesis. Obviously, anyone who could write a whole thesis about these fictitious entities would have to be an extremely pure theorist, someone who wouldn't be caught dead doing anything relevant. This was certainly true in Turing's case indeed he spent the years after his PhD, from 1939 to 1943, studying certain abstruse symmetry transformations on a 26-letter alphabet.

 29. Michael Nielsen Â» Interesting Problems: The Church-Turing-Deutsch Principle The CTD Principle is a descendant of a famous idea known as the churchturing thesis, taught to all computer scientists early in their degrees.http://michaelnielsen.org/blog/?p=71

30. BletchleyPark.net - Church-Turing Thesis
The churchturing thesis, although is not a proof of algorithm, it remains true in the philosophical sense, since it hasn t been disapproved.
http://bletchleypark.net/algorithms/Church_Turing.html
 Church-Turing Thesis "An envisioned computation is said to be feasible if it would consume only a quantity of resources that does not exceed what is likely to be available to the computational process." - R. Gregory Taylor. Introduction 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 more precise definition. Turing defined a machine showing that this machine, which is now called a Turing machine, can compute anything that can be intuitively computed. In 1931, Kurt GÂdel first introduced a set of functions that he called recursive. In 1934, Stephen Kleene was able to generalize GÂdelÃs recursive functions. Then in 1936, Church made a similar connection to Turing, in that anything that can be intuitively computed can be expressed in terms of general recursive functions. Thus, Church and Turing both related the philosophical notion of effectively computable to the formal notion of Turing and Recursively computable. There are actually three computability paradigms that are connected together, but are not provably connected, hence the term thesis. We have the (1) machines paradigm from

31. CS 4311 Schedule Fall 2006
F, 10/27, Chapter 3 The churchturing thesis Turing machines formal definition configuration of a TM definition of Turing-recognizable
http://www.cs.mtu.edu/~nilufer/classes/cs4311/2006-fall/schedule.html
##### Schedule for CS4311 Introduction to Computation Theory Fall 2006
Week Date Topic/Read Before Class To be assigned To be collected M, 09/04 Labor day: no class W, 09/06 Course information, go over the syllabus,
Ch. Introduction
sets
sequences hw1: warm-up F, 09/08 K-day: no class M, 09/11 Ch. Introduction (cont'd)
functions
total functions, one-to-one functions
proof techniques
proof by construction W, 09/13 Solve homework 1
Ch. Introduction (cont'd) proof techniques proof by contradiction proof by induction Section 4.2 proof by diagonalization the set of even numbers is countable the set of odd numbers is countable the set of pairs is countable the set of real numbers is uncountable the power set of N is uncountable hw2: countability F, 9/15 Section 4.2 proof by diagonalization the set of pairs is countable the set of real numbers is uncountable the set of functions is uncountable the power set of N is uncountable M, 09/18 Section 4.2 proof by diagonalization the power set of N is uncountable the set of integers (positive and negative) is countable the set of repeating functions is uncountable W, 9/20

32. Phys. Rev. Lett. 79 (1997): M. A. Nielsen - Computable Functions, Quantum Measur
We conclude that either the churchturing thesis needs revision, The church-turing thesis 2,3 of computer science states that this class of functions
 Physical Review Online Archive Physical Review Online Archive AMERICAN PHYSICAL SOCIETY Home Browse Search Members ... Help Abstract/title Author: Full Record: Full Text: Title: Abstract: Cited Author: Collaboration: Affiliation: PACS: Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Phys. Rev. ST PER Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article: MyArticles: View Collection Help (Click on the to add an article.) Phys. Rev. Lett. 79, 2915 - 2918 (1997) Previous article Next article Issue 15 View PDF (87 kB) or Buy this Article Use Article Pack Export Citation: BibTeX EndNote (RIS) Computable Functions, Quantum Measurements, and Quantum Dynamics M. A. Nielsen Center for Advanced Studies, Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131-1156 and Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125 Received 4 June 1997 We construct quantum mechanical observables and unitary operators which, if implemented in physical systems as measurements and dynamical evolutions, would contradict the Church-Turing thesis which lies at the foundation of computer science. We conclude that either the Church-Turing thesis needs revision, or that only restricted classes of observables may be realized, in principle, as measurements, and that only restricted classes of unitary operators may be realized, in principle, as dynamics. URL: http://link.aps.org/abstract/PRL/v79/p2915

 33. Presentation-Wen12-14669-lecture33-Church-Turing-Thesis-Machines churchturing thesis church-turing thesis ChurchÃ¢ÂÂs original (1935) Lambda calculus is equivalent to real world computers (can compute any computablehttp://www.authorstream.com/Presentation/Wen12-14669-lecture33-Church-Turing-The

 34. The Church-Turing Thesis Breaking The Myth. D2R Server The churchturing thesis Breaking the Myth. Resource URI http//www4.wiwiss.fu-berlin.de/dblp/resource/record/conf/cie/GoldinW05. Home Example Recordshttp://www4.wiwiss.fu-berlin.de/dblp/resource/record/conf/cie/GoldinW05

35. Universality
The churchturing thesis implies a universality among different models of . These are not known to violate the church-turing thesis because we do not
http://www.cs.princeton.edu/introcs/75universality/
• Intro to Programming
• 1. Elements of Programming
##### 7.5 Universality
This section under major construction. One of the crowning scientific achievements of the 20th century was formalizing the notion of computation . In this section we address the fundamental question of what is computable in this Universe. Surprising revelation of 20th century is that a general purpose computer is capable of performing any computation that any other computer can. Perhaps the most important idea in all of computer science. Many different types of computational devices: Cray supercomputer, Dell PC, iMac, Palm Pilot, XBox, Tivo, Turing machine, TOY machine, Java programming language, Microsoft Excel, Java cell phone, quantum computer, Perl programming language. Is there any fundamental different between what these things can do and what a Gaggia espresso maker can do? Turing machines are equivalent in power to TOY and Java. Can simulate any Turing machine with a Java program, can simulate TOY with a Turing machine, can simulate, Java with a TOY machine. Same idea works for C, C++, C#, Python, Excel, Outlook. Also Mac, PC, Cray, ENIAC, Konrad Zuse's Z3 (but not proved until 1998), Palm pilot. And TiVo, Xbox, Java cell phone/ But not DFA, Gaggia espresso maker, or this Tinker Toy computer that MIT students built to play Tic-Tac-Toe.

36. The {C}hurch-{T}uring Thesis As A Guiding Principle For Physics
For just two such attempts Thus, it comes as no surprise that the churchturing thesis is under52 and Hogarth 24,25.. reasonable statistics`` time on
http://tph.tuwien.ac.at/~svozil/publ/ct.htm
##### Karl Svozil
Abstract Two aspects of the physical side of the Church-Turing thesis are discussed. The first issue is a variant of the Eleatic argument against motion, dealing with Zeno squeezed time cycles of computers. The second argument reviews the issue of one-to-one computation, that is, the bijective (unique and reversible) evolution of computations and its relation to the measurement process.
##### Introduction
It is reasonable to require from a ``useful'' theory of computation that any capacity and feature of physical systems (interpretable as ``computing machines'') should be reflected therein and vice versa. The recognition of the physical aspect of the Church-Turing thesis-the postulated equivalence between the informal notion of ``mechanical computation'' (algorithm) and recursive function theory as its formalized counterpart-is not new [ ]. In particular Landauer points out that computers are physical systems, that computations are physical processes and therefore are subject to the laws of physics [ ]. As Deutsch puts it [

37. The Doctrine Of Equivalents And The Church-Turing Thesis
I also understand that the churchturing thesis is that any algorithm that can be carried out on one computer can equivalently be carried out on any Turing
http://legalminds.lp.findlaw.com/list/cyberia-l/msg46869.html
##### The Doctrine of Equivalents and the Church-Turing Thesis
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38. [FOM] Re: On Hypercomputation
A weak hypercomputer is compatible with physical churchturing thesis because its operation can be modeled by a Turing machine.
http://cs.nyu.edu/pipermail/fom/2004-March/008009.html
##### [FOM] Re: On Hypercomputation
Dmytro Taranovsky dmytro at mit.edu
Mon Mar 15 22:59:06 EST 2004 dmytro at mit.edu Surely this is not an argument against hypercomputation. By definition, a Turing machine must also accept input of arbitrary length. I don't think you intend to say that this is one of the strongest arguments against the Turing machine. I don't know what exactly you mean by the "physical Church-Turing thesis" ... Your "weak hypercomputer" could presumably solve this instance of the halting problem and it is difficult to see in what sense this is "compatible with the physical Church-Turing thesis" Let's put it this way: Suppose one builds an alleged "weak hypercomputer" that can tell us whether an ordinary Turing machine that searches for an inconsistency in ZFC+MC (MC = measurable cardinals exist) halts. How exactly do you propose to verify that this alleged weak hypercomputer is really a weak hypercomputer and correctly answers that question http://web.mit.edu/dmytro/www/main.htm

39. ICHIM 7-11 September
This statement is one version of the socalled church-turing thesis, The church-turing thesis is not mathematically provable (although it is refutable)
http://www.leeds.ac.uk/cedars/pubconf/papers/ichim01SG.html
 Abstract This paper considers the use of emulation for digital preservation. After describing the theoretical background provided by mathematical logic, the paper moves on to consider practical issues in the use of emulation for digital preservation. Keywords emulation, digital preservation, Turing machines, the Church-Turing thesis Introduction: the problem of digital preservation The Task Force Report To date the most widely advocated solution to these problems has been that of migration. Migration is the process of converting a digital object that runs on one platform so that it will run on another (non-obsolete) platform. But migration has its problems and disadvantages: the process of conversion runs the risk of losing data, the 'look and feel' of a digital object and/or its functionality; it is a time consuming and therefore costly process; it is not obvious how or indeed whether all digital objects can be migrated (think of computer programs). For these and other reasons some believe that emulation may, sometimes at least, provide a better solution. In the CAMiLEON project we have reached the conclusion that emulation does have a valuable role to play in digital preservation [3] Caveat : this paper explores the possible role of emulation in digital preservation, it does not purport to explain how to write an emulator.

 40. JSTOR The Legacy Of Alan Turing, Volumes 1 And 2. Volume 1 Antony Galton s title is The churchturing thesis Its Nature and Status . The church-turing thesis properly so called is the assertion that everyhttp://links.jstor.org/sici?sici=0026-4423(199901)2:108:429<187:TLOATV>2.0.CO;2-

41. Church-Turing Thesis: Information And Much More From Answers.com
churchturing thesis Information and Much More from Answers.com.
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42. Languages And Machines
Chapter 11 Decision Problems and the churchturing thesis 11.4 The church-turing thesis. 11.5 A Universal Turing Machine. Chapter 12 Undecidability
http://www.cs.wright.edu/~tsudkamp/book.htm
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##### An Introduction to the Theory of Computer Science
THIRD EDITION Addison-Wesley Publishing Co. The primary objective of the book Languages and Machines is to give a mathematically sound presentation of the theory of computing at a level suitable for junior and senior level computer science majors. The topics covered include the theory of formal languages and automata, computability, computational complexity, and the deterministic parsing of context-free languages. To make these topics accessible to the undergraduate student, no special mathematical prerequisites are assumed. Rather, the mathematical tools of the theory of computing, naive set theory, recursive definitions, and proof by mathematical induction, are introduced in the course of the presentation. The presentation of formal language theory and automata develops the relationships between the grammars and abstract machines of the Chomsky hierarchy. Parsing context-free languages is introduced via standard graph-searching algorithms to make it accessible to students having taken a data structures course. Finite-state automata and Turing machines provide the framework for the study of effective computation. Topics covered include decidability, the Church-Turing thesis, and the equivalence of Turing computability and

43. Generation 5: Artificial Intelligence Repository - Turing Machines
This machine is (by the churchturing thesis) capable of making any computation. This is not a provable theorem (it has yet to be disproved) nor a strictly
http://library.thinkquest.org/18242/turing.shtml
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##### Turing Machines
During the 1930s-1950s many researchers debated over what was computable, and what wasn't. Many had argued over formal approaches to computability. In 1937, Alan Turing, a British mathematician who is now considered the father of computing and artificial intelligence sought to seek an answer to this dilemna. He constructed the theory of a Turing machine. His theorem (the Church-Turing thesis) states that Any effective procedure (or algorithm) can be implemented through a Turing machine.
So what are Turing machines? Turing machines are abstract mathematical entities that are composed of a tape , a read-write head , and a finite-state machine . The head can either read or write symbols onto the tape , basically an input-output device. The head can change its position, by either moving left or right. The finite state machine is a memory/central processor that keeps track of which of finitely many states it is currently in. By knowing which state it is currently in, the

44. Logic Matters: May 2007
Both papers are about what Fitz calls the Physical churchturing thesis (a function is effectively computable by a physical system iff it is Turing machine
http://logicmatters.blogspot.com/2007_05_01_archive.html
##### Logic Matters
Logical reflections and prejudices: enthusiasms and sceptical thoughts
##### Hand engraved examination scripts
What strikes me as I wade through Part II tripos papers (well, the thing that strikes me that it wouldn't be out of place to comment on, here and now) is that students are tending to write less than they used to. And I suspect that part of the explanation is this: few students actually ever write
But of course the shorter an answer, the more difficult it is to shine (especially if an answer starts with a bit of routine exposition). Heaven knows what we do about this: but we are surely sooner rather than later going to have to come up with a system that doesn't quite so favour those who have happened to acquire the philosophically irrelevant antique skill of being able to write fast. Posted by Peter Smith at 9:23 PM 6 comments
##### Tuesday, May 29, 2007
Posted by Peter Smith at 9:00 PM 1 comments Labels: Logic
##### Monday, May 28, 2007

45. Brains
If TuringÂs thesis i.e., the churchturing thesis is correct, stored-program computers can perform any computation (until they run out of memory) and can
http://philosophyofbrains.com/2006/01/05/did-i-commit-the-churchturing-fallacy.a
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##### Did I Commit the Church-Turing Fallacy?
This entry was posted on 1/5/2006 8:36 AM and is filed under Computation and Logic
Today I received my complimentary copy of The Philosophy of Science: An Encyclopedia , edited by Sahotra Sarkar and Jessica Pfeifer, Routledge, 2006. I wrote the entry on artificial intelligence. To my astonishment, the entry reads as follows:
If TuringÃ¢ÂÂs thesis [i.e., the Church-Turing thesis] is correct, stored-program computers can perform any computation (until they run out of memory) and can reproduce mental processes (p. 27). The italicized part is a perfect example of what Jack Copeland calls the Church-Turing fallacy, namely, the mistake of supposing that the computational theory of mind, or the view that mental processes are computational (and more specifically, that they are computable by Turing machines) follows from the Church-Turing thesis. Sadly, the Church-Turing fallacy is common among philosophers. Even more sadly, it is now firmly inserted in the entry on AI in the Routledge Encyclopedia of Philosophy of Science. Worse for me is, my name is at the bottom of that entry!
The most upsetting part of the story for me is that the offending statement was not in the original article that I submitted to the editors. The original text, which I wrote, read:

46. CiteULike: Classical Physics And The Church--Turing Thesis
In this article, we observe that there is fundamental tension between the Extended ChurchTuring thesis and the existence of numerous seemingly intractable
http://www.citeulike.org/user/djhda/article/504932
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