Talk:Recursively enumerable set
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What the hell is a tuple??
- An ordered pair. Dysprosia 03:23, 12 Dec 2003 (UTC)
No: All ordered pairs are tuples, but many tuples are not ordered pairs; some are ordered triples, etc. Michael Hardy 21:10, 12 Feb 2004 (UTC)
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[edit] Is this correct?
"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 it is a member of S and otherwise runs forever." I've changed this sentence because all recursive sets are also recursively enumerable. Algorithms for RE sets are simply not guaranteed to halt if the input is not a member of the set. Also notice that the above sentence contradicts one of the examples.
- Both versions are correct. But perhaps your version is clearer. If I have an algorithm A for a recursive set S which terminates if an element e is not in S I can create a new algorithm A' which is the same as A but instead of terminating if e is not in S it goes into an infinite loop. Using this construction I can covert all terminating algorithms into non-terminating ones.
- P.S.If you post messages on talk pages please sign your message with ~~~~.MathMartin 19:19, 5 Apr 2005 (UTC)
- Actually, Matiyasevich's Theorem says the converse: Every recursively enumerable set is Diophantine. Jim
[edit] "if" versus "if and only if"
- I can not agree with the wording "... eventually halts if and only if the input is an element of S ...". If it was so, recursive set could not be recursively enumerable, as the algurithm stops for input not being from S. I have changed the wording to "... eventually halts if the input is an element of S ...". This will allow recursive sets being also recursively enumerable. Zde 15:09, 3 January 2006 (UTC).
- Unfortunately your definition allows any set whatsoever to be recursively enumerable. (Think about it.)
- Given a recursive set, it's true that there's an algorithm that always halts and says whether the input is or is not in the set. But there's another algorithm that halts if and only if the input is in the set. For example, take your first algorithm, and rewrite it so that whenever it would have halted saying the element is not in the set, it instead goes into an infinite loop. (This is actually how you prove that recursive sets are recursively enumerable, or one way to prove it anyway.)
- Accordingly, I'm reverting your edit (don't take it personally; it's a simple matter of correctness). --Trovatore 18:40, 3 January 2006 (UTC)
[edit] Circularity of Definitions
Isn't this definition of a computable function:
A partial function 
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 
such that S is the range of f? --Michael Stone 00:25, 11 March 2006 (UTC)
- 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)
[edit] Invalid definition of r.e. set
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)
[edit] Equivalent definitions?
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)
- 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)
