Talk:Church–Turing thesis

message: I just made my best effort to rewrite the introduction. Judging from the comments below, the introduction has been in need of revision by an expert for quite some time, i.e., someone who could rewrite it with confidence, rather than just trying to sort of preserve and polish what was there. Still, I tried to stay true to the spirit of the original as far as possible, and re-used as much as possible --- I also borrowed at least one sentence from the article on computable functions.
I got rid of some pseudo-terminology such as "combined hypothesis", which does not mean anything as far as I was able to determine, except perhaps in statistics. I tried to be slightly more precise in the description of the three formally defined classes of computable functions (lambda definable, Turing definable, and general recursive) - in particular, the first two of these classes require an encoding of the natural numbers (either as lambda terms, such as the Church numerals, or as symbols on a tape of the Turing machine). The old intro had glossed over this a bit too much in my opinion --- although certainly I am also only alluding to it and not giving fully formal definitions. I tried to strike a balance between concision and precision. To see the full and precise definitions, the readers will have to follow the links to the respective articles.
I also renamed the next section "Statement in Church's and Turing's words", because that is exactly what the section contains. The old title, "Formal Statement", didn't make sense, because by its nature, the Church-Turing thesis is an informal statement that can never be formalized. Selinger (talk) 00:17, 25 August 2014 (UTC)
replies: text: Thank you Selinger, the article is much improved. Turing completeness badly needs similar treatment. --TedColes (talk) 07:57, 25 August 2014 (UTC)
thread_title:  Rewrote introduction 

message: I'm not sure why we even bother with a link to nonsense about Burgin's supposed refutation of the thesis. If anyone is in for a good laugh, skim through the appallingly poorly written The rise and fall of the Church-Turing Thesis. To quote his modest conclusion:
:We may compare these three approaches to what we have in our everyday life. From this  perspective, DNA computing is like a new model of a car. Quantum computations are like planes at the stage when people did not have them. At the same time, super-recursive computations are like rockets, which can take people beyond the “Church-Turing Earth”. Thus, it is possible to say that we have rockets that might take us to the Moon and even to other planets of the Solar system if we know how to navigate them. However, we will need new physical ideas for realization of super-recursive algorithms to a full extent. Using our metaphor, we may say that spaceships that will take us to stars are now only in perspective. 
...riiiight... His work is almost universally regarded, at best, as something between a misunderstanding, a new word for an old concept and a marketing fraud. See for instance Martin Davis' unkind review of Burgin's book  and his description of Burgin's misunderstanding . Pascal.Tesson (talk) 19:34, 7 February 2008 (UTC)
replies: text: Most likely this should be discussed in conjunction with hypercomputers; I need to pick up Burgin's book from the library to make sure. It's well known that if you eliminate the any of the requirements {discrete, finitistic, deterministic} from the definition of algorithm then you can get a broader class of functions. So It's not nonsense; but all the previous work I have seen relies on a nonstandard reading of Church's thesis to argue that Church's thesis is incorrect. — Carl (CBM · talk) 19:44, 7 February 2008 (UTC)

text: It appears from this arxiv paper that Burgin is interested in, among other things, limit recursive functions. It is very well known that there are limit recursive (i.e. \Delta^0_2) functions that are not computable; the common belief is that this does not impinge on Church's thesis. — Carl (CBM · talk) 19:50, 7 February 2008 (UTC)
thread_title:  Burgin's theory 

message: The term Church's thesis (CT) is used in intuitionistic logic to describe an additional axiom, saying that all functions are computable. There should be either a pointer in the introduction or a disambiguation page.
--Thorsten (talk) 15:59, 6 March 2008 (UTC)
replies: text: I am familiar with this sort of axiom, but I don't think we have an article on it. If we do, then a note at the top of this article would be very appropriate. — Carl (CBM · talk) 16:43, 6 March 2008 (UTC)
thread_title:  Church's thesis in Intuitionistic Logic 

message: To keep the discussion together in one place, I will remove this duplicate section and point to the discussion on this exact topic at Talk:Algorithm#A_paradoxical_situation. — Carl (CBM · talk) 20:38, 19 March 2008 (UTC)
thread_title:  A paradoxical situation 

message: Is it just me, or are there other mathematicians out there that regard the Church-Turing thesis as nonsense? The phrase'effectively calculable' is only defined in terms of other English words, which are themselves not defined. The author then seems to lose interest in the pretence of rigour by not defining the word 'machine'.  —Preceding unsigned comment added by 86.9.138.200 (talk) 20:02, 7 October 2008 (UTC)
replies: text: Yes, the terms "effectively calculable" and "algorithm" are often left as undefined terms. Then the thesis states that these terms apply to the same functions as the formal definition of "computable function" does. From that point of view, the thesis could be viewed as a definition of "effectively calculable", except that this term already has an informal definition in English. — Carl (CBM · talk) 21:24, 7 October 2008 (UTC)
thread_title:  The Emporer's Clothes 

message: I'm a little troubled by the formatting here. Shouldn't the title have a hyphen-minus in it instead of an en-width dash? And, also, full spaces around the em-width dashes in the article body are not so good. 66.102.80.246 (talk) 03:04, 13 October 2008 (UTC)
replies: text: The title should have an en dash. Please feel free to remove the spaces around em dashes in the text. — Carl (CBM · talk) 11:25, 13 October 2008 (UTC)

text: I agree, "effectively calculable", with emphasis on the word "calculable", has the broader meaning of "calculable by any means (any sort agent using any sort of tools) whatever"; this includes hand calculation, as opposed to "computation", i.e. by a computing-machine. Your suggestion looks good to me; sometimes less is more.  BillWvbailey (talk) 17:11, 2 May 2014 (UTC)
thread_title:  Article Cleanup 

message: In my opinion, this article is very hard to understand. Most of the questions I had before coming here are still unanswered after reading it, and worst: now I'm confused. Someone with a strong background in this area, please rewrite this article in a more clear, helpful way.
* We need less footnotes.
* References seem all messed up.
* I expected to find the original unmodified statement followed perhaps by a more modern re-statement of the thesis.
* Please add a link that explains the difference between a thesis, an hypothesis and a theorem.
* The part about Boundedness, Locality and Determinance needs a clarification.  —Preceding unsigned comment added by 90.80.39.42 (talk) 12:29, 31 August 2009 (UTC)
replies: text: I greatly shortened the lead-in paragraphs and moved the cutout stuff into the article. I pulled out the content of some footnotes that seemed useful commentary and put it into the text, and fleshed out "The part about Boundedness, Locality and Determinancy". I fiddled with quotes and text here and there. Bill Wvbailey (talk) 20:08, 31 August 2009 (UTC)
thread_title:  Hard to understand, poorly formatted 

message: I am going to work on rewriting the lede, and moving much material out of it ot he main article. The lede is meant to be a summary of the article in plain language, so that even someone who can't read the whole article can get a sense of what's going on from the lede. The present lede has 15 footnotes and focuses primarily on historical terminology; it needs to be more concise and direct. — Carl (CBM · talk) 12:50, 24 October 2008 (UTC)
replies: text: I greatly shortened the lead-in paragraphs and moved the cutout stuff into the article. I pulled out the content of some footnotes that seemed useful commentary and put it into the text, and fleshed out "The part about Boundedness, Locality and Determinancy". I fiddled with quotes and text here and there. Bill Wvbailey (talk) 20:08, 31 August 2009 (UTC)
thread_title:  lede 

message: :(moved from above) 
What appears to be a rewrite of the wiki Church-Turing thesis article appears here: http://www.fact-archive.com/encyclopedia/Church-Turing_thesis. It's interesting to see their crack at it. Wvbailey (talk) 21:47, 15 September 2009 (UTC)
replies: text: This is actually just an old version of our article. I didn't look up the exact revision, but this one is quite close. — Carl (CBM · talk) 02:21, 16 September 2009 (UTC)
thread_title:  old version 

message: Humph. Given certain assumptions about the answers to certain unsolved physics problems leads to the ability to solve the halting problem for the Turing machine. Therefore, this thesis is not proved. 75.45.17.132 (talk) 02:54, 6 December 2009 (UTC)
replies: text: No one has claimed that this thesis has been "proved". I'm not even sure if that makes sense. How could one prove something like that? --Robin (talk) 04:08, 6 December 2009 (UTC)
thread_title:  Physical CTT 

message: This change replaced Strong Church Turing  Thesis" with "Complexity-theoretic Turing  Thesis" claiming that "strong" is a weasel word. I've reverted it because WP:WEASEL does not apply to terminology quoted as such. Pcap ping 04:53, 8 December 2009 (UTC)
replies: text: I agree with Pcap. "Strong" is part of the name of the thesis, not an adjective we're adding to make it sound cool. --Robin (talk) 13:43, 8 December 2009 (UTC)

text: This is not widely accepted terminology.  On the other hand, the adjective "Complexity Theoretic" is factually accurate and therefore does not require supporting citations. AshtonBenson (talk) 18:42, 8 December 2009 (UTC)
replies: text:  "factually accurate and therefore does not require supporting citations" is WP:OR. Pcap ping 21:01, 8 December 2009 (UTC)
replies: text:  I think this is a no-brainer. Google shows 2 hits for "Complexity-Theoretic Church-Turing Thesis" but 16800 hits for "Strong Church-Turing Thesis." What more can I say than that? --Robin (talk) 22:38, 8 December 2009 (UTC)

text: It's a bit more than that. It's an assertion that not only is the standard deterministic TM universal, but it also simulates all other "reasonable" or "realistic" models of computation with at most polynomial slowdown. The usual hypothesis, without the word "strong", doesn't say anything about the slowdown. --Robin (talk) 22:32, 8 December 2009 (UTC)
replies: text:  That's what I was saying. The article here talks about probabilistic machines, while I thought the strong thesis was about ordinary machines. — Carl (CBM · talk) 01:03, 9 December 2009 (UTC)
replies: text: Oh, I see. I'm not too sure which one is the proper version of the strong CTT, but it seems the probabilistic TM statement is backed up by the textbook by Kaye/Laflamme/Mosca. Most complexity theorists probably wouldn't mind either formulation, since it is believed that polytime deterministic TMs can simulate polytime probabilistic TMs. --Robin (talk) 03:27, 9 December 2009 (UTC)

text: See #"If quantum computers are physically possible" for this. Pcap ping 23:28, 8 December 2009 (UTC)
replies: text:  Oh, thanks! — Miym (talk) 23:34, 8 December 2009 (UTC)
thread_title:  This is no weasel 

message: By the way, for some reason I have got the impression is that only quantum computing people have ever claimed that there exists a "strong CT thesis", and the only reason for them to define it is to show how to refute it. Did the strong CT thesis really exist before the quantum computing era? Did someone present it before they knew how to refute it? Using a quantum computing book as a source does not really help to convince me... — Miym (talk) 23:00, 8 December 2009 (UTC)
replies: text: See #"If quantum computers are physically possible" for this. Pcap ping 23:28, 8 December 2009 (UTC)
replies: text:  Oh, thanks! — Miym (talk) 23:34, 8 December 2009 (UTC)
thread_title:  By the way 

message: The Church Turing thesis has an important history in computer science.  See What is computation? Concurrency versus Turing's Model from the article published in Arxiv (a somewhat competitor to Wikipedia that is often censored here).76.254.235.105 (talk) 23:13, 28 November 2010 (UTC)
replies: text: ArXiv is not a competitor to Wikipedia in any way. It is an uncensored article repository. Wikipedia is an encyclopedia. There is no similarity. Please refer to Wikipedia:What Wikipedia is not. AmirOnWiki (talk) 14:47, 14 October 2011 (UTC)
thread_title: Church Turing Thesis in Computer Science

message: I have tried to arrange the messy paragraph involving super-recursive, or hyper-Turing, computation. I moved the claims attributed to Eberbach and Wegner to an earlier section, and rephrased them following their article. I removed the claims attributed to Peter Kugel since his article, cited in the reference section, does not support was has been written in his name. I did not touch the text referring to Mark Burgin's claims as I do not have his book at hand, but it should be removed if it cannot be verified. AmirOnWiki (talk) 14:42, 14 October 2011 (UTC)
thread_title:  a little cleanup 

message: The article is perfectly fine. That said, the Church-Turing thesis is nothing but a normal math definition, like any other, e.g. the epsilon-delta definition of a continuous function. Maybe this should be mentioned more clearly in the article. See e.g. R. I. Soare, 1996. Computability and recursion, Bulletin of Symbolic Logic v. 2 pp. 284–321(?).  — Preceding unsigned comment added by DanielVallstrom (talk • contribs) 18:04, 29 January 2012 (UTC)
replies: text: Take a look at History of the Church-Turing thesis where the issue of whether it's a definition (Turing), Kleene 1952 (two proofs from formal systems leading to a single "Church-Turing" proof) or Post (a natural law) is debated. Along your line of reasoning, Goedel makes an observation (cf Goedel 1946 Remarks before the Princeton Bicentennial conference on problems in mathematics) that "in the concept of general recursiveness (or Turing's computability . . . one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e. one not depending on the formalism chosen" (Feferman et. al. 1990 [Volume II]:150). But he doesn't take it further. As for my opinion, I'm with Post (it's a natural scientific law determined by repeated observation but still open to counter-evidence). Here's why: there's still these niggling little problems of "analog computation" and "quantum mechanics" and "consciousness", so I wouldn't be too hasty to jump to the conclusion "the thesis" is just a matter of definition. Bill Wvbailey (talk) 00:17, 30 January 2012 (UTC)
thread_title:  Church-Turing thesis is nothing but a normal (correct) mathematical definition 

message: There is an incorrect definition of the Church Turing thesis here Turing_(disambiguation)--Mongreilf (talk) 21:37, 18 June 2012 (UTC)
thread_title:  Incorrect definition on disambig 

message: Is the Physical Church–Turing thesis (PCTT) equivalent to the Church–Turing–Deutsch principle?
Bayle Shanks (talk) 08:20, 26 May 2013 (UTC)
replies: text: The physical Church-Turing thesis actually comes to two broad flavours: bold and weak.  The CT-Deutsch is a form of the bold PCTT.  See http://plato.stanford.edu/entries/computation-physicalsystems/#PhyChuTurTheBol    Honestly, the section on PCTT in this article isn't well written. After reading the Stanford article above, you may be able to do better.  :)  Ross Fraser (talk) 23:59, 28 May 2013 (UTC)
thread_title:  PCTT == Church–Turing–Deutsch principle? 

message: What is the appropriate term, a dissertation or a thesis? UK convention is that a thesis is to obtain a Doctorate degree and dissertation to complete a Master's. The US convention is the reverse – a dissertation is written for a PhD and a thesis for a Master’s.
Turing was British but obtained his PhD from Harvard in the US. --TedColes (talk) 08:54, 19 February 2014 (UTC)
thread_title:  Turing's doctoral work 

message: There are also an "extended form" of the thesis.
As comment by recently by P. Shadbolt et all (in DOI:10.1038/NPHYS2931)
"Famously, the extended Church–Turing thesis (ECT) says that all computational problems that are efficiently solvable with realistic physical systems can be efficiently solved with a classical machine — a statement clearly in conflict with our hopes for the capabilities of quantum computers", Shor, P. W. in Proc. 35th Ann. Symp. Found. Comput. Sci. 124–134 (IEEE, 1994).  --Krauss (talk) 03:09, 14 December 2014 (UTC)
thread_title:  extended Church–Turing thesis 

message: SMmoto is correct that by 1939 both Church and Turing were well aware of each other's work -- but to accept Turing's paper (1936) Church was the only possible referee, and he had already published (paper appeared) on 15 April, pre-empting Turing by . To quote Hodges: "A new idea had found its way into two human minds simultaneously and independently" (Hodges 1983:111). For Hodges' take on the entire sequence of events see Hodges 1983:111-113. I also remember reading somewhere that a "certification process" had to occur after which the paper was accepted on 28 May 1936 to the London Mathematical Society. Hodges isn't so clear on this. If I find out what happened, I'll add it here. BillWvbailey (talk) 17:19, 2 February 2015 (UTC)
thread_title:  Good catch by SMmoto 

message: For references on the relationship between Turing's thesis and human computers, here are three good sources:
* Section 9 of Turing's original paper  
* The Stanford Encyclopedia article by Jack Copeland 
* Section 3 of Soare's paper Computability and Recursion 
The short summary is that, at the time Turing was writing, the word "computer" itself referred to a human computer, and that this notion was what Turing was seeking to analyze. Turing's analysis of the Entscheidungsproblem relied on his analysis of what a human could compute, and his argument that this notion of computability was captured by a Turing machine. Indeed, as Copeland writes in the SEP article linked above, "The Turing machine is a model, idealised in certain respects, of a human being calculating in accordance with an effective procedure". — Carl (CBM · talk) 13:18, 20 June 2015 (UTC)
thread_title:  Human computer 

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thread_title:  External links modified 

message: The philosophical implications paragraph begins with this sentence:    The only cite is a somewhat vaguely-worded bit of prose that says   This gives the impression that the sentence was written by an editor who then pointed to arguments they personally felt were based on errors or misappropriation, which is obviously original research.  Do we have any sources specifically saying that the Church-Turing thesis has been frequently misunderstood by philosophers? --Aquillion (talk) 01:57, 11 October 2017 (UTC)
thread_title:  Philosophical implications paragraph. 

message: I removed the following paragraph from the text:
"Kleene himself never stated that Turing had made a mistake in his paper, important in its own right for helping to establish the unsolvability of problems in group theoretic computations, although corrections to Turing's paper were also made later by Boone who originally pointed out "points in the proof require clarification, which can be given"[34] and Turing's only PhD student, Robin Gandy. That Kleene doesn't mention this mistake in the body of his textbook where his presents his work on Turing machines but buried the fact he was correcting Alan Turing in the appendix was appreciated by Turing himself can be surmised from the ending of Turing's last publication "Solvable and Unsolvable Problems" which ends not with a bibliography but the words,"
It wasn't clear at all from the context what the mistake being referred to was. The paragraph also ends mid-sentence. If anyone can find the context and conclusion, feel free to add it back. (Assuming it's worthy of inclusion.)--Rxtreme (talk) 04:57, 14 July 2018 (UTC)
thread_title:  Turing's Mistake 

message: I see the paragraph "One can formally define functions that are not computable. A well-known example of such a function is the Busy Beaver function. This function takes an input n and returns the largest number of symbols that a Turing machine with n states can print before halting, when run with no input. Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines. Since the busy beaver function cannot be computed by Turing machines, the Church–Turing thesis states that this function cannot be effectively computed by any method."
But this is trivial - the largest possible loop is n-1 states (-1 because one of the states is the halting state).  therefore if it doesn't loop it is n or less states long and the final state is the halting state.  The state transitions depend entirely on the initial state. And a turing machine that can run it as a virtual machine can determine if it halts (from whatever initial state it starts in) in N or less steps of the virtual machine. less if it repeats a state - because if and only if it repeats a state does it loop. Kevin Baastalk 18:59, 3 December 2021 (UTC)
replies: text: In your argument, you didn't consider the tape. When the machine re-enters a state again, the tape contents is usually different, and hence so is the further maschine behavior. Examples are shown e.g. at Busy_beaver#Examples, including an animation of a 3-state machine that halts after 14 steps (see :File:Busy Beaver 3-state 2-symbol Run.gif for a version expanded in time). - Jochen Burghardt (talk) 22:15, 3 December 2021 (UTC)
replies: text: No, I do consider the tape.  Unless you are using an infinite tape - in which case you have an infinite state turing machine, which is equavelent to all other infinite state turing machines.  Finite state can only be taken to mean finite tape lenght as well.  if it's an infinite tape turing machine then it's a universal turing machine.  Am i misunderstanding something here?  Maybe the problem is poorly posed.  if it's turing complete then with an infinite tape (or you can think of it as infinite memory)... maybe the term "state" is poorly defined here.  i take "state" to mean all of the data on the stack and the tape, put together.  so if the stack space is 16 bytes and the tape is say 256 bytes, then the number of states is 2 to the power of 8*(256+16), and at least one of those states is the halting state.  so either it repeats before going through all of the states, in which case it never halts, or it eventually reaches one of those halting states, which occures in at most 2 to the power of 8*(256+16) state changes.  Because it can't go through any more state changes than that without hitting a state it's already been in.  And if it hits a state it's already been in then it's stuck in an infinite loop. Kevin Baastalk 18:24, 20 December 2021 (UTC)
replies: text: Your understanding of "state" (although it does make sense) deviates from the usual jargon in the definition of a Turing machine (on which the article is based). The latter assumes a potentially infinite tape, separated from a finite-state control. I guess, this is the misunderstanding here. I agree that your above argument is valid if the tape is restricted to a fixed finite length. See finite automaton for the usual definition of what you (apparently) have in mind. - Jochen Burghardt (talk) 20:33, 20 December 2021 (UTC)
thread_title:  an n-state turing machine terminates in n or less states or does not terminate.