Talk:Computability theory

message: I was checking out this disambiguation page with a view towards disambiguating the links, and I wonder: should this page exist?  I am sure that there are some set theorists who have used the term "computability theory" to describe recursion theory, but I'm willing to bet that if I were to go through the links to this page all, or almost all, should point to computability theory (computer science).  Now, of course I could be wrong which is why I am asking here.  Any comments?   --Deville (Talk) 12:33, 8 February 2006 (UTC)
replies: text: I'd prefer an unequal disambiguation to computability theory (computer science), but only if User:Trovatore is fine with it as well. —Ruud 12:45, 8 February 2006 (UTC)
replies: text: Really I think they should probably never have been split; I'd prefer a re-merge at some point. At the time of the split the article was written too much from the computer science perspective. That ought to have been corrected, but it's tricky to do right. --Trovatore 13:59, 8 February 2006 (UTC)
replies: text: This really should just be one page, there's no point in seperating the two pages as they discuss the same topic just from different points of view.  —Preceding unsigned comment added by 85.180.67.188 (talk) 00:50, 11 January 2008 (UTC)

text: Of course you're right. This is largely my fault (the other part of the fault would belong to those who wrote the computability theory article before I got to it). The article at that time was very boring and had a lot of repetitive stuff (duplicative of other articles, to boot) about what was computable and what wasn't; I didn't feel comfortable weeding it out the way it needed to be weeded, to make room for more interesting material. Maybe you'd like to take a crack at it? --Trovatore 23:51, 13 September 2006 (UTC)
thread_title:  Should this page even exist? 

message: There's about 100 "ambiguous" links to this page. Maybe one of you could take a look and see how they should be resolved? Here's the list. Thanks! Ewlyahoocom 06:16, 9 October 2007 (UTC)
thread_title: Ambiguous links to this page

message: Why is this page a double dab? Shouldn't computable be its own dab page? Pcap ping 17:14, 13 August 2009 (UTC)
thread_title:  Double dab? 

message: The page Recursion theory was moved here, as part of a larger reorganization. Centralized discussion is at Talk:Recursion theory#Reorganize the Computability articles. — Carl (CBM · talk) 14:53, 22 August 2009 (UTC)
replies: text: I have started the reorganization. Let's keep the discussion in the same location, Talk:Recursion theory#Reorganize the Computability articles, even though recursion theory is a redirect now. — Carl (CBM · talk) 15:20, 22 August 2009 (UTC)
thread_title:  Reorganization 

message: Computability theorists have an odd sense of "weak" and "strong".  
* First, there is a very old usage of "strong reducibility" to mean "many-one reducibility", due to Shapiro. I think this has vanished now, but it is in Davis 1958. 
* Second, there is a classical use documented in Rogers 1967, p 138, where Turing and truth-table reductions are "the weak reducibilities" and many-one and one-one reductions are "the strong reducibilities". 
In modern computability terminology, if ≤A implies ≤B then A is said to be stronger than B. Equivalently, as  Odifreddi, "Strong reducibilities", p. 43, says:
:"Every degree of a weaker reducibility is the union of degrees of stronger reducibilities" 
This may seem backwards if you think of reducibilities as equivalence relations first; but if you think of them as computational processes, a stronger reducibility carries with it more information than a weaker reducibility. So from weak to strong, we have
: arithmetical ← Turing ← truth-table ← many-one
I tried to fix this in the article just now, but I also have to think about the terminology to get it right. One difficulty is that computability theorists generally study particular reducibilities, rather than the general theory of reducibilites. I found a paper online that gave a general definition of "stronger reducibility"  but I don't think it's a very good reference for this article. I would look in Odifreddi's book but I don't have a copy at hand. — Carl (CBM · talk) 22:09, 6 April 2010 (UTC)
replies: text: PS There are also "strong" and "weak" reducibilites corresponding to Medvedev degrees and Muchnik degrees of sets of reals, respectively. This is a different setting; I just wanted to complete my taxonomy of the terms. — Carl (CBM · talk) 22:12, 6 April 2010 (UTC)
thread_title:  Weak and strong 

message: The second paragraph of the article begins with:
       The basic questions addressed by recursion theory are "What does it mean for a function from the natural numbers to themselves to be computable?"  ...
We seem to have a grammatical glitch here. What is the intent?  — Preceding unsigned comment added by Toddcs (talk • contribs) 19:07, 31 March 2011 (UTC)
replies: text: What is the glitch? — Carl (CBM · talk) 20:19, 31 March 2011 (UTC)
thread_title:  Say what? 

message: Apart from the usefulness of alphabetization, it seems strange to include a paper as a separate entity that can be found in a collection. In this instanceThe Undecidable is the collection. If the refs were to list the papers under the collection, then they would look like the following:
* Burgin, M. and Klinger, A. "Experience, Generations, and Limits in Machine Learning." Theoretical Computer Science v. 317, No. 1/3, 2004, pp. 71–91
* M. Davis, ed., 1965.  The Undecidable—Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven, New York. Reprint, Dover, 2004. 
:* A. Church, 1936a. "An unsolvable problem of elementary number theory." American Journal of Mathematics v. 58, pp. 345–363.  Reprinted in "The Undecidable", M. Davis ed., 1965. 
:* A. Church, 1936b. "A note on the Entscheidungsproblem." Journal of Symbolic Logic v. 1, n. 1, and v. 3, n. 3. Reprinted in "The Undecidable", M. Davis ed., 1965. 
:*E. Post, 1944, "Recursively enumerable sets of positive integers and their decision problems", Bulletin of the American Mathematical Society, volume 50, pages 284–316.
:*E. Post, 1947. "Recursive unsolvability of a problem of Thue." Journal of Symbolic Logic  v. 12, pp. 1–11. Reprinted in "The Undecidable", M. Davis ed., 1965.
:* A. Turing, 1937. "On computable numbers, with an application to the Entscheidungsproblem." Proceedings of the London Mathematics Society, ser. 2 v. 42, pp. 230–265.  Corrections ibid. v. 43 (1937) pp. 544–546.  Reprinted in "The Undecidable", M. Davis ed., 1965. PDF from comlab.ox.ac.uk 
:* A. Turing, 1939. "Systems of logic based on ordinals." Proceedings of the London Mathematics Society, ser. 2 v. 45, pp. 161–228. Reprinted in "The Undecidable", M. Davis ed., 1965.
* R. M. Friedberg, 1958. "Three theorems on recursive enumeration: I. Decomposition, II. Maximal Set, III. Enumeration without repetition." The Journal of Symbolic Logic, v. 23, pp. 309–316.
* E. M. Gold, 1967. "Language identification in the limit". Information and Control, volume 10, pages 447–474.
*L. Harrington and R. I. Soare, 1991. "Post's Program and incomplete recursively enumerable sets", Proceedings of the National Academy of Sciences of the USA, volume 88, pages 10242—10246.
* C. Jockusch jr, "Semirecursive sets and positive reducibility", Trans. Amer. Math. Soc. 137 (1968) 420-436
* S. C. Kleene and E. L. Post, 1954. "The upper semi-lattice of degrees of recursive unsolvability." Annals of Mathematics v. 2 n. 59, 379–407.
*J. Myhill, 1956. "The lattice of recursively enumerable sets." The Journal of Symbolic Logic, v. 21, pp. 215–220.
* 
*T. Slaman and W. H. Woodin, 1986. "Definability in the Turing degrees." Illinois J. Math. v. 30 n. 2, pp. 320–334. 
* R. I. Soare, 1974. "Automorphisms of the lattice of recursively enumerable sets, Part I: Maximal sets." Annals of Mathematics, v. 100, pp. 80–120.
To make this good, I'd have to go through the various papers and find out where they're located in The Undecidable (not a problem). (My apologies to all if this has been argued over already with a consensus to just list alphabetically). BillWvbailey (talk) 23:06, 23 April 2011 (UTC)
replies: text: It's quite unusual to group articles by the collection/journal/etc. they have been published in, in bibliogrpahies. In particular, this collection seems to be a collection of reprints of previously published papers. This makes the approach somewhat infeasible as it would not form a tree-structure. In these cases I would list either the whole collection, or some/all of the papers contained within it separately, or perhaps even both, but not in such a hierarchy. —Ruud 23:20, 23 April 2011 (UTC)

text: I prefer to cite the individual papers when the citation is for a scientific reason, that is, when it's actually to convey information about the subject. I feel less strongly about citing collections like The Undecidable, but for the sake of an encyclopedia article I can see some benefit in listing it for readers who may realize it's a useful purchase. Still, I like to have the papers alphabetized by author, not by editor of a collection. For example, if I cited Enderton's article from the Handbook of Mathematical Logic I would want it to sort under "E", not under "B" for the editor Jon Barwise. — Carl (CBM · talk) 00:53, 24 April 2011 (UTC)
thread_title: References located in collections

message: This article uses the following reference style (I should know, I established it):
* Print references use Harvard references
* Parenthetical remarks and non-print references (e.g. mailing lists, conference webpages, etc.) use footnotes.
This is a style used by some academic journals, as well, which will not allow "unpublished" sources to be in the references section (since they will not be in libraries), but will allow them to be mentioned in footnotes. — Carl (CBM · talk) 22:16, 2 April 2013 (UTC)
replies: text: This archaic citation style is no longer allowed in WP since 2020 (per WP:PAREN). I have therefore started to convert it to sfn-style. --Matthiaspaul (talk) 17:45, 2 August 2022 (UTC)
replies: text:  I recognize the unfortunate outcome of the RFC, but it really isn't necessary to call it "archaic".  --Trovatore (talk) 17:46, 2 August 2022 (UTC)
thread_title:  Reference style 

message: To find a way to make this article useful to students, I have given up. Lede contains content that appears superficially, and only in the lede. Paragraph after paragraph, section after section, content appears that if sourced, is ambiguous (misstating year, lacking page numbers, etc.), but is more often unsourced synthesis, or possible OR. I am very sorry to hear that this is considered "B+ class". There is nothing to be done, but to return to the reliable EB. 
71.239.87.100 (talk) 05:51, 2 February 2015 (UTC)
thread_title: After trying for an hour

message: …to this article, "A essay written in largest part by User:CBM and User:Frank_Stephan between June 2006 and May 2007." so that readers can address its veracity based on authorship, since it is essentially unverifiable content for lack of inline citations, page numbers of books referenced, etc. Or, perhaps there is a tag that says "Trust Us", that we could append to the top, for that is what is expected of this and other articles that lack the scholarly sourcing that would allow readers to check individual ideas against sources. Do as you see fit. The article is in any case, useless for any serious educational purpose. 71.239.87.100 (talk) 06:52, 2 February 2015 (UTC)
thread_title:  Please consider adding the byline 

message: Terms like "very recent" will become dated, and you will never know when. JMK (talk) 11:59, 6 January 2016 (UTC)
thread_title:  Very recent 

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thread_title:  External links modified 

message: There was consensus on the WikiProject Math (discussion here) that "computable" is a preferable alternative to the outdated, ambiguous "recursive" (in reference to computability/recursion theory). Therefore, articles in this category should generally use the term "computable X" rather than "recursive X". Exceptions should be made when "recursive" is still the popular name for the concept, e.g. Primitive recursive, μ-recursive, Kleene's recursion theorem. --Jordan Mitchell Barrett (talk) 22:54, 26 April 2021 (UTC)
thread_title:  Change in terminology: "recursive" to "computable"