
Chaitin's Constant - mgreenleaf
https://en.wikipedia.org/wiki/Chaitin%27s_constant
======
fxj
He even did more:

Mathematician GregoryChaitin defines elegance in computer programming in this
way: A computer program written in a given language is elegant if no smaller
program written in the same language has the same output. He goes on to prove
that it is impossible to prove that a given program above a certain very low
level of complexity is elegant.

[https://wiki.c2.com/?ChaitinElegance](https://wiki.c2.com/?ChaitinElegance)

And he gave some examples in his book.

[http://jillian.rootaction.net/~jillian/science/chaitin/www.c...](http://jillian.rootaction.net/~jillian/science/chaitin/www.cs.umaine.edu/chaitin/unknowable/index.html)

See the example code in LISP here:

[https://github.com/darobin/chaitin-lisp](https://github.com/darobin/chaitin-
lisp)

~~~
OskarS
The general principle here, which is quite foundational in computer science
and the theory of complexity, is that of "Kolmogorov complexity". The
Kolmogorov complexity of a certain string (roughly speaking) is the shortest
computer program needed to produce that string. Much like Turing's halting
problem or Church's beta-equivalence, it is impossible to write a program to
calculate the Kolmogorov complexity in general. It is "non-computable".

~~~
whatshisface
> _it is impossible to write a program to calculate the Kolmogorov complexity
> in general. It is "non-computable"._

I don't get it, why couldn't you "simply" iterate through every program
shorter than the string you're trying to compress, compiling and running each
while discarding the ones with the wrong output and halting when you either
find a working program or run out of shorter strings to try?

Edit: Oh, I get it now, you can't tell whether or not a given long-running
program will eventually produce the target string without solving its halting
problem.

~~~
OskarS
> Edit: Oh, I get it now, you can't tell whether or not a given long-running
> program will eventually produce the target string without solving its
> halting problem.

Exactly right! I kinda think that these non-computable problems are all sort-
of the same problem, just viewed from different angles.

------
jaymzcampbell
If you want to read a bit more from the mathematician himself on this very
topic he wrote an accessible "pop-math" book about it, "Meta Math!: The Quest
for Omega" though you'll need to look beyond the author's rather strange
choices of metaphor
([https://www.goodreads.com/book/show/249849.Meta_Math_](https://www.goodreads.com/book/show/249849.Meta_Math_)).

~~~
alanbernstein
It's been a while, but I assume you're referring to the similarities drawn
between information theory and sex. This was pretty offputting to me, until I
realized the connection was deeper than I recognized - DNA is an apt
comparison.

I've read plenty of "pop math" books, and this one stands out as somewhat odd.
It's also a quick read and, somewhat uncommonly, written by a person closely
connected to the topic - so I'd recommend it.

------
opengrave
I just want to drop this playlist here [https://www.youtube.com/watch?v=HLPO-
RTFU2o&list=PL86ECDEDE3...](https://www.youtube.com/watch?v=HLPO-
RTFU2o&list=PL86ECDEDE3FA8D8D1) as its one of my fav lectures

Gregory Chaitin Lecture at Carnegie-Mellon University in 2000, he gives a bit
of history of parts of math/computing that leads up to him talking about
qualities of random. He touches on Cantor, Bertrand Russell, Hilbert, Gödel
and Turing.

~~~
jdkee
Thanks for the link. I just watched that lecture and it really concretized a
number of concepts from the literature. Kudos to Gregory Chaitin.

------
deepnotderp
Afaict Chaitin independently came up with the concept of Kolmogorov
complexity... as a teenager!

------
daxfohl
The surprising thing to me was, following the link to "normal numbers", that
this is called out as one of the only proved irrational normal numbers, even
though it is proven that the set of irrational numbers is normal almost
everywhere.

------
me_me_me
Can somebody explain to me how is this useful? Is it used for anything?

Or is it pure theoretical concept with interesting emerging properties.

~~~
tialaramex
The constants themselves would perhaps be useful but they're non-computable so
we can't find out what they are anyway.

The idea is useful yes.

~~~
me_me_me
> The idea is useful yes.

Well that's the crux of my question, how can it be useful wen its completely
undefined.

~~~
jerf
The idea is mathematically useful.

If you're asking about it's practical use, it has none.

------
KboPAacDA3
Numberphile has a video explaining the relationship between the number
categories, and includes a brief discussion of where Chaitin's Constant
belongs.

[https://www.youtube.com/watch?v=5TkIe60y2GI](https://www.youtube.com/watch?v=5TkIe60y2GI)

------
Chris2048
In contrast: [https://www.jamesrmeyer.com/topics/chaitins-
omega.html](https://www.jamesrmeyer.com/topics/chaitins-omega.html)

------
benji-york
I'm a fan of Fulton's constant.

Fulton's constant is any number of 9s. E.g., 9, 999, or 999999.

------
dboreham
Not to be confused with his Croissant.

