Talk:Computably enumerable set

message: What the hell is a tuple??
: An ordered pair. Dysprosia 03:23, 12 Dec 2003 (UTC)
thread_title:  Tuple 

message: Isn't this definition of a computable function: 
A partial function f:\subseteq \mathbb{N} \to \mathbb{N} 
is called computable if the graph of f is a recursively enumerable set.
circular with the definition of a recursively enumerable set:
A countable set S is called recursively enumerable if there exists a partial computable function f : \mathbb{N} \to S such that S is the range of f?
--Michael Stone 00:25, 11 March 2006 (UTC)
replies: text: Good catch. The definitions in computable function should be reworked, and probably computable function and recursive function should be merged. --Trovatore 00:41, 11 March 2006 (UTC)
thread_title:  Circularity of Definitions 

message: This is the invalid definition of recursively enumerable set from the article
:There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if and only if the input is an element of S.
The problem is that there is no formal definition of algorithm (unless you choose the nonstandard definition that identifies algorithms with Turing machines, or something else equally nonstandard).  The definitions of recursively enumerable  set that apear in current texts all make reference to one of the several equivalent formally defined models of computation.  It is only via Church's thesis that they identify the sets that are recursively enumerable with the sets that are enumerable by some “algorithm.”  This confusion about Church's thesis is pervasive in many of the articles about computability.
Several of the comments for the suggested merge of Recursive function and Computable function suffer from the same confusion.  There can be no formal definition of computable function which does not make reference to some model of computation.
CMummert 20:25, 14 July 2006 (UTC)
replies: text: How ya figure? --Trovatore (talk) 07:42, 23 March 2010 (UTC)
thread_title:  Invalid definition of r.e. set 

message: I can't see how both definitions can be considered "equivalent". I can see how, having an algorithm that enumerates the set, I can have another algorithm that, given an input number, eventually halts iff the element is in the set. But I can't see how, having such an algorithm, I could possibly enumerate the elements of the set. -- Anonymous, 13:20, 23 August 2006 (UTC)
replies: text: The definitions aren't considered equivalent -- they are provably equivalent.  If you have an algorithm A that halts on exactly those numbers that are in a set S, you can enumerate S based on how many steps it takes A to halt, as follows.  Run A for one time step on input 0.  Then run A for two time steps on input 0 and for two time steps on input 1.  Then run A for three time steps on each of the inputs 0, 1, and 2.  Whenever you run A long enough on a particular input that A halts, enumerate that input into S.  If you continue methodically in the pattern just listed, you run A arbitrarily long on every input, and thus your enumeration will include every number for which A halts.  CMummert 13:53, 23 August 2006 (UTC)
thread_title:  Equivalent definitions? 

message: 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: s1, s2, s3, ... . 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. 66.127.52.47 (talk) 21:20, 20 March 2010 (UTC)
thread_title: "Algorithm"

message: 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. --Trovatore (talk) 05:53, 29 July 2022 (UTC)
thread_title:  Should move to [[computably enumerable set]] 

message: Detailed here: Wikipedia_talk:WikiProject_Mathematics/Archive/2021/Jun#Proposal: change terminology from "recursive" to "computable". --Dan Polansky (talk) 14:55, 10 October 2023 (UTC)
thread_title:  2021 renaming 

message: 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.) 130.243.94.123 (talk) 16:45, 25 January 2024 (UTC)
replies: text: 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. --Trovatore (talk) 19:17, 25 January 2024 (UTC)
thread_title:  Origin of enumerable = partial?