Talk:Turing degree

message: This article is listed as needing cleanup.  Since it is such a well-known topic, not just in mathematics but in computer science, some discussion before editing is probably worthwhile.  Here is a possible outline for an improved version.
* Intro paragraph
* Definition of Turing reducibility and Turing degree.  C.e. degrees.
* Semilattice operations; ideals.
* Survey of structural properties.  With references to nontrivial results.
* Post's problem; the priority method.  Nontechnical overview of this classical topic.
* Important results.   Statements only.  References to published surveys.
* Other degree structures.  
* References.
I also think we could redirect [Post's problem] to this page, after the edit.
I hope that someone else has some opinions about what should go in this article.   CMummert 18:44, 13 June 2006 (UTC)
thread_title:  plans  for clean up 

message: Thanks to CMummert for rewriting this article!  I very much enjoyed reading it.  One minor point: could you clarify what is meant by "Not all finite lattices can be embeded in the r.e. degrees." (naïvely, one might wonder whether the "embedding" is supposed to preserve meets and joins or just the partial order).  --Gro-Tsen 15:39, 8 July 2006 (UTC)
thread_title: Great work!

message: I think this article could use a brief introduction explaining in simple mathematical language what precisely is being described here. I am hardly math- illiterate, but this was very difficult to understand. I would appreciate it if this article could be clarified and introduced better. Thanks RSido 03:20, 30 January 2007 (UTC)
replies: text: Thanks for the comment.  I still don't see quite which sections of the article you're asking about ("the whole thing" isn't helpful).  Is the first section, entitled Turing equivalence, readable? If not, why not? Is there another section you had in mind? CMummert · talk 03:34, 30 January 2007 (UTC)
replies: text: I edited the introduction.  Absent any more specific comments, I'll remove the  template in a few days. CMummert · talk 04:39, 1 February 2007 (UTC)
thread_title:  Confusing 

message: On Computable number page it is said that decision problem "Given Turing machine, determine whatever it computes total function" is Turing degree 0″. Is it true? Maybe it is truly degree 0′ 83.23.20.209 (talk) 16:56, 19 March 2012 (UTC)
replies: text: [Be careful about using double-prime in Wiki markup, as it is considered to be a mark of italics.  I fixed your question.]Yes, it is true.  Determining whether a Turing machine computes a total function is a ∀∃ (i.e., Π2) type question, because one must decide whether for all input there exists an execution trace showing that the machine halts, and it is pretty obvious that any ∀∃ question can be reduced to this kind; now the halting problem for 0′, that is 0″, is ∃∀, because to say that a 0′ machine halts means that there exists an execution trace showing that for all "does not halt" reply the halting oracle gave the corresponding Turing machine indeed does not halt (for all time).  So this halting problem can be reduced to asking whether a certain Turing machine does not compute a total function.  There is probably a more intuitive way of saying it which makes the Turing machine more explicit, but 0″ is undoubtedly the correct degree. --Gro-Tsen (talk) 12:01, 20 March 2012 (UTC)
thread_title:  0″ 

message: The article currently states There are pairs of degrees with no greatest lower bound. Silly me, I don't understand. Isn't every degree greater than zero? User: Linas (talk) 03:05, 3 September 2013 (UTC)
replies: text: 0 is always a lower bound, but in general not the greatest lower bound (=infemum). --Gro-Tsen (talk) 11:53, 5 September 2013 (UTC)
replies: text: Dohh, thanks. I wonder how this could be clarified; it seems like an important point. User: Linas (talk) 17:32, 2 November 2013 (UTC)
thread_title:  Not a lattice? 

message: Who did prove that and when? --Jobu0101 (talk) 18:01, 28 April 2015 (UTC)
thread_title:  Not every degree less than 0′ is an r.e. degree. 

message: * https://cs.stackexchange.com/questions/92503/how-strong-consistent-guessing-problem-is
* http://people.csail.mit.edu/andyd/cg.pdf
* https://www.scottaaronson.com/blog/?p=710
If so, should that be placed on the page as an example of such a problem that has intermediate Turing degree? --70.31.88.62 (talk) 22:08, 3 March 2019 (UTC)
replies: text: The consistent guessing problem is a set of languages; some of these languages have intermediate degree, but others have the same degree as the halting problem. So probably not worth mentioning as an example of intermediate degree, although it might be worth mentioning as a "zero injury" proof. (Actually, it's not clear to me that the language in question is computably enumerable, or even that it is reducible to the halting problem. So it might not have intermediate degree.) Ben Standeven (talk) 08:31, 21 December 2020 (UTC)
thread_title:  Is this an example of intermediate turing degree problem? 

message: I might be wrong but I think I have found a result refuting a sentence from the Order properties section. The article says "no infinite, strictly increasing sequence of degrees has a least upper bound", but supposedly Sacks's Degrees of Unsolvability (1963) proves that any countable set of Turing degrees has a minimal upper bound. (I don't have a copy of the original, I got this citation via a reference from Maximal Chains in the Turing Degrees.) If we were to take some countable strictly increasing chain of degrees, and look at the set of members of that sequence, by Sacks wouldn't it have a least upper bound? C7XWiki (talk) 02:26, 12 October 2022 (UTC)
replies: text: For an infinite strictly increasing sequence of Turing degrees, a minimal upper bound always exists, but is never unique, hence there is no least upper bound. I added a sentence to that effect to the article. Dmytro (talk) 03:53, 12 October 2022 (UTC)
thread_title: Result contradicting quote in article?