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== Tuple ==
What the hell is a tuple??
: An ordered pair. [[User:Dysprosia|Dysprosia]] 03:23, 12 Dec 2003 (UTC)
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No: All ordered pairs are tuples, but many tuples are not ordered pairs; some are ordered triples, etc. [[User:Michael Hardy|Michael Hardy]] 21:10, 12 Feb 2004 (UTC)
Ehm.. [[Tuple]] --[[User:NavarroJ|NavarroJ]] 12:59, 20 August 2007 (UTC)
== Is this correct? ==
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[[User:CMummert|CMummert]] 20:25, 14 July 2006 (UTC)
To add to the the above, the first paragraph in the "Remarks" section seems to contradict the opening definition of recursively enumerable sets.
[[User:Bandar|kapil]] ([[User talk:Bandar|talk]]) 06:53, 23 March 2010 (UTC)
:How ya figure? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 07:42, 23 March 2010 (UTC)
== Equivalent definitions? ==
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:::You are right. The range of a recursive function has to be a set of natural numbers, so an r.e. set has to be a set of nat num's too. This makes me think though that maybe the comment about goedel numberings should be edited to reflect the fact that a goedel numbering function, while clearly intuitively effectively computable, cannot be, strictly speaking, recursive--? Also, does this mean that the Church-Turing thesis must also be restricted to functions from (n-tuples of) nat. num's to nat. num's--since clearly there are effectively computable functions (e.g. goedel numberings) that are not recursive? --[[User:Futonchild|Futonchild]] 21:52, 28 February 2007 (UTC)
::::Yes, it is true that the things called Godel numberings in computability theory, like the Godel numbering of the sentences of a formal language, cannot literally be "computable functions" unless their ___domain and range are included in the set of legal inputs/outputs of whatever model of computation we use. I used the word "corresponds" in this article precisely for that reason. Unfortunately the WP article on Godel numberings is currently in very bad shape. Feel free to edit either article if you would like to improve them.
::::And yes, the Church-Turing thesis is limited to functions whose ___domain and range are included in the set of possible inputs/outputs of the model of computation. For example, the successor operation on [[ordinal]]s, although in some sense effective, can't be computable because there is no model of computation that can represent all the ordinals as inputs. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 23:14, 28 February 2007 (UTC)
== Dumb Questions ==
QUOTING from the article:
"A set S of natural numbers is called recursively enumerable if there is a partial recursive (i.e. computable) function whose ___domain (co-range) is exactly S, meaning that the function is defined if and only if its input is a member of S."
I'm starting pretty far back in trying to comprehend this. May I please have answers to these questions:
* Is "partial recursive function" synonymous with "computable function"? (Partial recursive and computable are not defined or linked. Are they regarded as obvious in meaning?)
* I know the terms ___domain and range. But why "co-range"? Is there any good reason to introduce this alternate and confusing term here?
* The test of whether the set S is recursively enumerable relates to the "___domain" of functions. Does the "range" of functions have any relevance to the matter?
* After pondering my way through the introduction to the article, I was pleased to see that there is a section of "Examples". I eagerly went to it, and was deeply disappointed to see that it contains no (IMO) real examples of functions; only what I would call examples of types of functions. Are there any simple examples of what this article is about?
* Is the set of all integers that are multiples of 7 recursively enumerable?. If so, what function satisfies the test?
Thanks very much. [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 14:13, 8 August 2008 (UTC)
:# Yes, "partial recursive function" is the same as "partial computable function". Logically it should really be "recursive partial function" but this is the traditional name. They should be defined or linked.
:# I think "co-range" is kind of silly.
:# An r.e. set may be defined either as the ___domain or as the range of a partial recursive function. The two definitions are equivalent, but one may sometimes be more ocnvenient than the other.
:# Sure, there are tons of simple examples. Whether it's useful to list them is something that might want some thought....
:# Yes, the multiples of 7 form an r.e. set. Write a program that tests the input to see if it's a multiple of 7, and if so outputs 0; otherwise, it goes into an infinite loop. The partial recursive function defined by that program witnesses that the set is r.e. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 18:45, 8 August 2008 (UTC)
::Thank you very much. That helps a bit. Encouraged by that, I'm going to ask another question.
::* Suppose I now have my program that tests the input to see if it is a multiple of 7. If it is, the output is 0. If it is not, the output is 41. BUT the ___domain of this function is all positive integers, not all integers that are multiples of 7. ???
::I didn't mean to suggest including tons of examples but I think a few examples would help a lot. Thanks. [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 19:10, 8 August 2008 (UTC)
:::The answer to your question is "yes" (well, except I'd quibble on "positive", since we seem to be allowing 0), and your program witnesses that the set of all natural numbers is r.e. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:14, 8 August 2008 (UTC)
::::Thanks. Can you suggest a program that witnesses that the set of natural numbers that are multiples of 7 are r.e., because the ___domain of the function is that set? [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 19:28, 8 August 2008 (UTC)
:::::Trovatore's original function witnesses that. If the input is a multiple of 7, the program halts. Otherwise, it does not halt. So the ___domain (set of values for which it halts) is exactly the set of multiples of 7. This is a [[partial function]].
:::::The term "co-range" is not used in recursion theory; it's a category-theory term. I think it was added here in an attempt to clarify for category theorists which meaning of "___domain" is intended. See [[___domain (mathematics)]] — Carl <small>([[User:CBM|CBM]] · [[User talk:CBM|talk]])</small> 19:46, 8 August 2008 (UTC)
(unindent) The word "___domain" is ambiguous. According to [[Binary relation]], "''A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset of the Cartesian product X × Y. The sets X and Y are called the ___domain and codomain, respectively, of the relation, and G is called its graph.''". Sometimes it is used to mean the set of things which might be considered as inputs to the function — in our case, the ___domain in that sense of a partial recursive function is always the natural numbers. Other times, ___domain (second sense) is used to mean the set of things within the ___domain (first sense) which when input to the function yields a value. This is what I was calling the "co-range" (for want of a better word) because the "range" of a function is the set of values which it outputs while the "co-___domain" is the set of things within which the range (image of the ___domain) is located, that is, in our case, the natural numbers (again). [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 06:00, 9 August 2008 (UTC)
:Sure, ''___domain'' is ambiguous in general, but it's not ambiguous in recursion theory or descriptive set theory. In those fields it has a standard meaning. On the other hand ''co-range'' I have never heard of — Carl says it's a category-theory term, which is possible, but in my limited experience with category theory I've never come across it. We aren't supposed to make up language here. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 07:22, 9 August 2008 (UTC)
::Thanks to all of you for your help in uderstanding this. As I said, I'm starting pretty far back.
::As a general comment, I think complex terminology is a huge barrier to comprehension of this subject area. Both the article itself and some of the above discussion illustrate that.
::For example, in the first sentence of the article, even before an r.e. set is defined, the reader is given an alternate name for the subject (i.e., computability theory, traditionally called recursion theory) and four alternatives for the term r.e. (i.e., computably enumerable, semidecidable, provable and Turing-recognizable). IMO this is packing too much information into one sentence.
::Thanks, [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 18:53, 9 August 2008 (UTC)
:::It is hard to avoid doing that. We need to identify the subject of the article quickly by whatever name a reader might know it. Once you realize that these are synonyms, you can ignore the ones you do not recognize on your first reading of the article. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 06:41, 10 August 2008 (UTC)
::::Thanks. I wonder if an initial statement in italics might be better, by taking the synonyms out of the defining sentence. E.g.,
::::''(Computably enumerable, semidecidable, provable and Turing-recognizable are synonyms for recursively enumerable.)''
::::Another question -- suppose a lookup table is created by picking a million numbers from a random number table, and an algorithm is created that halts only when given an input that is in the lookup table. Would that algorithm determine an r.e. set? [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 15:45, 10 August 2008 (UTC)
:::::Yes. Every finite (or co-finite) set of natural numbers is recursively enumerable. Indeed, they are recursive. [[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 16:13, 10 August 2008 (UTC)
=="Algorithm"==
Article says:
:* There is an algorithm that enumerates the members of S. That means that its output is simply a list of the members of S: s<sub>1</sub>, s<sub>2</sub>, s<sub>3</sub>, ... . If necessary, this algorithm may run forever.
I have requested discussion at
:[[Talk:Algorithm characterizations#can_an_algorithm_produce_infinite_output.3F]]
of whether this is a legitimate use of the word "algorithm", which I thought had to produce at most finite output. [[Special:Contributions/66.127.52.47|66.127.52.47]] ([[User talk:66.127.52.47|talk]]) 21:20, 20 March 2010 (UTC)
== Universal recursively enumerable set ==
Strangely, I do not see here a universal recursively enumerable set. [[User:Tsirel|Boris Tsirelson]] ([[User talk:Tsirel|talk]]) 17:52, 17 February 2018 (UTC)
:It is not defined here (and I have forgotten the definition), but I believe that these two examples listed in the section on examples would qualify:
:* Given a [[Gödel numbering]] <math>\phi</math> of the computable functions, the set <math>\{\langle i,x \rangle \mid \phi_i(x) \downarrow \}</math> (where <math>\langle i,x \rangle</math> is the [[Cantor pairing function]] and <math>\phi_i(x)\downarrow</math> indicates <math>\phi_i(x)</math> is defined) is recursively enumerable (cf. picture for a fixed ''x''). This set encodes the [[halting problem]] as it describes the input parameters for which each [[Turing machine]] halts.
:* Given a Gödel numbering <math>\phi</math> of the computable functions, the set <math>\lbrace \left \langle x, y, z \right \rangle \mid \phi_x(y)=z \rbrace</math> is recursively enumerable. This set encodes the problem of deciding a function value.
:[[User:JRSpriggs|JRSpriggs]] ([[User talk:JRSpriggs|talk]]) 12:39, 18 February 2018 (UTC)
== Should move to [[computably enumerable set]] ==
I have tagged [[computably enumerable set]] as G6 -- this page should be moved there per [[WP:NOUN]]. I think this is pretty cut and dried; I hope we don't need an RM. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 05:53, 29 July 2022 (UTC)
== 2021 renaming ==
Detailed here: [[Wikipedia_talk:WikiProject_Mathematics/Archive/2021/Jun#Proposal: change terminology from "recursive" to "computable"]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|talk]]) 14:55, 10 October 2023 (UTC)
== Origin of enumerable = partial? ==
Instinctively, I would expect ’enumerable’ to mean that I can make an enumeration, in which case ‘computably enumerable’ ought to mean that there is some algorithm which computes the next item in the sequence for me. Instead it seems to mean the opposite: adding this attribute to a term ''weakens'' it so that I can never be sure what is the next item in a sequence, unless it is just the numerical successor! ‘Partial’ I can see why it should be like that, but for ‘enumerable’ it comes across as a mystery. What is the historical background for this choice of terminology? (That would be a useful addition to the article.) [[Special:Contributions/130.243.94.123|130.243.94.123]] ([[User talk:130.243.94.123|talk]]) 16:45, 25 January 2024 (UTC)
Okay, found the explanation — in the alternative definition "The set S is the range of a total computable function, or empty.", which in at least {{cite book |last1=Mendelson |first1=Elliott |title=Introduction to Mathematical Logic |date=1987 |publisher=Wadsworth & Brooks/Cole |isbn=0-534-06624-0 |page=259 |edition=3rd}} is taken as the primary definition. The proof that this implies "is the ___domain of a partial recursive function" is straightforward, but the converse is '''very''' difficult. [[Special:Contributions/130.243.94.123|130.243.94.123]] ([[User talk:130.243.94.123|talk]]) 17:33, 25 January 2024 (UTC)
:This isn't the place to discuss it, but actually the proof of the converse is not conceptually difficult. There might be a few details to get past, but the idea is straightforward. If you want to know more, please ask a question on [[WP:RD/Math]]. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:17, 25 January 2024 (UTC)
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