Talk:Chaitin's constant

message: The following two sentences don't agree; the first one doesn't make sense.
 It can then be shown that Ω represents the probability that a randomly produced bit string will 
 encode a halting program. This means that if you start flipping coins, always recording a head 
 as a one and a tail as a zero, the probability is Ω that you will eventually reach the encoding of  
 a syntactically correct halting program.
It is true that Omega is the probability that a randomly chosen infinite sequence of 0s and 1s begins with a string that is a halting program. This makes sense, because there is a probability measure (Lebesgue measure) on the set of such infinite sequences, and the set of sequences that begin with a string encoding a halting program is a measurable set.   But there cannot be a  randomly chosen bit string because there is no nontrivial probability measure on the set of finite bit strings. CMummert 04:37, 16 June 2006 (UTC)
replies: text: Yeah, it shold be probability that randomly chosen infinite bit string begins with a proper program which stops after finite time.
I'm not sure will it make things more clear... But the way it is now is correct with some explanations.
After all you can assume that you randomly chose bits of a string until (if ever) you reach a valid program string. In such random experiment
probability of obtaining program that stops is exactly \Omega. Macieksk 09:28, 24 June 2006 (UTC)
thread_title:  On the probability interpretation 

message: The article "Omega (computer science)" is very similar to this one.  I suggest that they be merged.P.L.A.R. 21:41, 29 June 2006 (UTC)
replies: text: See my comments on the other article's discussion page. This page should stay, the other one should go, since this is not the only omega in use in computer science, and perhaps not even the most common. There doesn't seem to be much info on the current Omega (computer science) page that should be brought over here. --Dantheox 23:40, 29 June 2006 (UTC)
thread_title: Merge

message: Shouldn't some reference be made to the fact that one can use Chaitin's constant to determine the answer to any possible phrasable question? Jenigmat429 03:16, 13 February 2007 (UTC)
replies: text: That "fact" is not true.  Chaitin's omega is at level Sigma_1 of the arithmetical hierarchy.  Complete knowledge of it cannot even determine the truth of Pi_2 sentences of Peano arithmetic, much less more complicated questions.  Chaitin's constant is Turing equivalent to the Halting problem, and nothing more. CMummert · talk 03:24, 13 February 2007 (UTC)
replies: text: Ok, more accurately, couldn't you use it to determine the provability of a proposition L in a formal system F? Because, if there's a program that's sufficiently small that will halt if L is provable in F, and if you know enough of chaitin's constant, one could use Chaitin's omega to determine L's provability in F. Or maybe I'm interpreting Chaitin's omega wrong. Jenigmat429 14:16, 13 February 2007 (UTC)
replies: text: Yes, you can use a Chaitin constant to do that.  But this is because these constants are equivalent to the Halting problem, not in particular because of their randomness. CMummert · talk 16:56, 13 February 2007 (UTC)
thread_title:  Properties of Chaitin's constant 

message: I expanded and rewrote the article this morning. Feel free to hack away at it, I'm done for a while. CMummert · talk 17:48, 13 February 2007 (UTC)
thread_title:  rewrite 

message: Currently the article says:
"Each halting probability is a normal and transcendental real number which is definable but not computable, which means that there is no algorithm that enumerates its digits." But there is a non-halting algorithm that does enumerate its digits - Omega can be approximated from below. Eventually the first n digits will be correct. I'll correct this: "which means that there is no halting algorithm that enumerates its digits." Algorithms 20:25, 23 June 2007 (UTC)
thread_title:  more rewrite 

message: One should mention the "Super Omega" of Jürgen Schmidhuber: http://www.idsia.ch/~juergen/kolmogorov.html . The first n bits of Gregory Chaitin's constant Omega are incompressible in the sense that we cannot compute them by a halting algorithm with fewer than n-O(1) bits. Hoewever, consider the short but never halting algorithm which lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first n bits of the output will not change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) on the first n bits of Omega, for any n. In other words, the enumerable first n bits of Omega are highly compressible in the sense that they are limit-computable by a very short algorithm; they are not random with respect to the set of enumerating algorithms. Schmidhuber constructed a Super Omega which is still limit-computable, but more random than Omega, as one cannot significantly compress it  with an enumerating algorithm. Algorithms 15:43, 8 July 2007 (UTC)
thread_title:  Super Omega 

message: For the benefit of people like me, for whom the technical details later in the article are a bit tough to digest, a couple of clarifications in the lead section would be helpful
:In the computer science subfield of algorithmic information theory a Chaitin constant or halting probability is a real number that informally represents the probability that a randomly-chosen program will halt. These numbers are formed from a construction due to Gregory Chaitin.
How is it possible, even in principle, to randomly choose a program, given that "a program" can be of unlimited length? Do you not need to choose randomly from all N-bit programs (or maybe all programs of <= N bits)?
:Although there are infinitely many halting probabilities, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction instead of Chaitin's constant when not referring to any specific encoding.
I don't understand what this is saying. Is it saying:
*There are infinitely many individual halting probabilities because there are infinitely many programs?
*There are infinitely many halting probabilities for a randomly chosen program, one for each of infinitely many encoding schemes?
*There are infinitely many halting probabilities for a randomly chosen program, one for each of infinitely many program lengths?
*Something else?
86.133.243.133 (talk) 03:52, 11 November 2008 (UTC)
thread_title:  Clarification in lead section requested 

message: The section 'Use of Chaitin's constant in proving unsolved problems in number theory' makes no sense to me. How is that different from saying that a program that does that iteration will eventually say if the conjecture is false?. So, how many digits of Omega do we need to know to know wether Goldbach conjecture is true or false?
Chaitin himself though that Fermat Last theorem is "true with no reason", before the proof in the 90's. Did he say also that Omega can help in that problem?  —Preceding unsigned comment added by 76.193.219.199 (talk) 21:17, 9 July 2009 (UTC)
replies: text: I agree. That section is simply the observation that the halting problem can settle any \Pi_1 sentence in number theory.  While true, it has very little to do with the constant.  —Preceding unsigned comment added by 130.195.86.40 (talk) 04:05, 16 October 2009 (UTC)
replies: text: Yes; this is just a consequence of the binary expansion of the constant being Turing equivalent to the Halting problem, nothing more. That section should be reworked to that it doesn't imply this would be a practical method of solution. — Carl (CBM · talk) 12:43, 16 October 2009 (UTC)
replies: text:  Per WP:BB I have partly rewritten  that section and renamed it to Relationship between the knowledge of Chaitin's constant and the halting problem. The first paragraph contains, now in generalized form, the fact that one could use knowledge of the constant to solve the halting problem. The second paragraph mentions the relationship between the number theory problems and the halting problem while dismissing the possibility of solving these problems this way. Hope that helps. X127 (talk) 18:40, 22 June 2011 (UTC)
replies: text:  Yes. This can be supported (I do believe . . .) by Chaitin's book MetaMath!. Take a look at the last sentence, though; it seems to be misconstructed. But this is minor. It's a pleasure to encounter an editor who is coherent and who knows their stuff. (Please register as an official wikipedian. We need all the help we can get.) Bill Wvbailey (talk) 21:59, 22 June 2011 (UTC)
thread_title:  Use of Chaitin's constant in proving unsolved problems in number theory 

message: The Properties section states that all chaitin numbers are normal. How is it that chaitin constants have been proven using their randomness that they're normal and yet we can't use the very random nature of π to prove that π is normal or use the very random nature of the set of prime numbers to prove the twin prime conjecture. π is simpler do define than any of the chaitin numbers and yet it's harder to prove normality of that number. Did any of the refrences in this article prove the normality of chaitin numbers or state their normality, sourcing another source that stated the same thing. I'm sure that any chaitin number can be trivially proven to be normal from being computationally random, but have they really been proven to be computationally random, which is a much harder property to prove than uncomputability. Blackbombchu (talk) 03:14, 16 August 2013 (UTC)
replies: text: This bothers me.  There are a number of similar "numbers" for which it can be proved that many individual bits are non-computable, but that the number is normal.  ("This" number doesn't appear to be "computationally random" in the sense that I would mean it; in fact, with little effort, one could replace Ω with \frac {d+\Omega}{2^n} where d is any nonnegative integer less than 2n.  — Arthur Rubin  (talk) 20:15, 14 January 2014 (UTC)
thread_title:  Possible unproven statement in the article 

message: Good evening,
Am I right : given F, there is no algorithm taking an integer n as argument and returning the n-th digit of \Omega_{F} (or, equivalently, an infinite-time looping algorithm writing all its digits in order), but there is, for all n, an algorithm giving the n-th digit ? 86.196.217.126 (talk) 17:57, 12 October 2013 (UTC)
replies: text: Well, since the nth digit is one of a finite number of possibilities, there is an algorithm producing that number.  There isn't a computable function from n to the nth algorithm, though, as that would compute the number.  Perhaps this might be noted in the article, but I don't really see the point.  — Arthur Rubin  (talk) 20:23, 14 January 2014 (UTC)
thread_title:  Getting the digits of Omega ? 

message: The section 'Relationship to the halting problem' does not explicitly state that if p is a member of the domain of F, this is equivalent to saying that p halts.  This may be confusing for the mathematically-literate general reader.  Would it be possible to add a little more explanation to that section?
My apologies if I have just misunderstood the article - I am a general reader who has not touched computability theory for decades! Booleanmattock (talk) 16:39, 15 March 2014 (UTC)
thread_title:  Clarification for the general reader 

message: Every computer has its Chaitin's constant, does there exist a computer whose Chaitin's constant Ω is exactly 0.03? If no, why not?  — Preceding unsigned comment added by 140.115.189.24 (talk) 06:16, 11 August 2015 (UTC)
replies: text: Nope. See intro: all values of Ω are transcendental. 0.03 is rational. ~ CZeke (talk) 14:08, 29 November 2015 (UTC)
thread_title:  The value 

message: At , the last sentence doesn't seem to make sense (though the subject is way beyond my decades-dusty engineering math ). Is it missing a word or two?
Also, the parenthetical:
:0``` = 0^{(3)}
seems like it should be something like:
:O‴ = O(3)
(3 backticks → triple-prime, zero → letter "O", "^" → superscript) or is it O(3) or ... —[AlanM1(talk)]— 09:49, 14 August 2018 (UTC)
thread_title:  Super Omega section formatting issue 

message: The first sentence of the article is the following:
"In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt." 
But the section Interpretation as a probability appears to make it clear that the Chaitin omega in fact is a probability.
Am I mistaken about this?  If it really is a probability, then it should not be described as representing a probability only "informally speaking". 2601:200:C000:1A0:68A1:236F:AC0B:9D14 (talk) 20:47, 26 August 2022 (UTC)
thread_title:  Why "informally speaking"? 

message: An editor has identified a potential problem with the redirect :Chaitin`s number and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 October 15#Chaitin`s number until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 15:29, 15 October 2022 (UTC)
thread_title:  "Chaitin`s number" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]