

The Busy Beaver Problem - pkrumins
http://www.catonmat.net/blog/busy-beaver/

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staticshock
This reminds me of an essay I really like (it covers what the busy beaver
numbers are, and what their utility is):
<http://www.scottaaronson.com/writings/bignumbers.html>

~~~
pbhjpbhj
_But do people fear big numbers? Certainly they do._

Apathy !== fear.

 _comforting smallness of mysticism?_

In what way is mysticism small? It is bigger by most definitions than anything
we're capable of being consciously aware of.

On topic, excellent; but the off-the-cuff remarks could have used a little
more thought IMO.

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Eliezer
> In what way is mysticism small? It is bigger by most definitions

Saying "infinite" creates a vague sensation of bigness that isn't anywhere
near as big as, say, 3^^^3 (<http://en.wikipedia.org/wiki/Knuth%27s_up-
arrow_notation>) to say nothing of really large numbers.

The idea of the night sky as "infinite" is nowhere near as shocking as the
idea of galaxies a septillion meters away. It's like the difference between
Thor, versus the machine superintelligences of Iain Banks's Culture or the
Powers from Vernor Vinge's _A Fire Upon the Deep_. You can imagine the
"infinite" Jehovah demanding the worship of Israelite tribespeople, but it's
rather harder to visualize a mind 10^50 ops/sec large bothering to do so.

If you think of mystics as primitive amateur philosophers and science fiction
writers who demand that their work be protected from ordinary criticism, you
can see there's no way they'd be able to keep up with modern science fiction
writers or modern mathematicians. In this light, words like "infinite" or
"unimaginable" are just words - showing instead of telling, mere adjectives, a
demand that we be impressed which isn't backed up by any actual ability to
conceive of impressive things.

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pbhjpbhj
_it's rather harder to visualize a mind 10^50 ops/sec large bothering to do
so_

How about a more Leibnizian idea of the Creator having designed into the
primordial energy event all of the consequences of existence - some slight
fluctuation or variation being ultimately responsible for this internet post
... is that a valid activity for your concept of a mystical supermind. Or
consider that the realist approach to many worlds is true - is the awareness
of every subatomic interaction in every one of an infinite array of possible
worlds, is that enough processing for you? How about if that mystical being is
responsible for an infinite number of such worlds?

You have a very large concept of the infinite aleph-0^^^...^^aleph-0 (I doubt
you can verily comprehend those numbers though) or whatever but some
numerological mystics would say those concepts are discovered from ideas
embedded in the created order or inherited from a prime movers creative mind.

Your view of mysticism is narrow.

~~~
pwmanagerdied
Your view of mysticism is wrong.

~~~
pbhjpbhj
In what way?

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rw
This problem is what originally got me interested in computational complexity.
Everybody knows about the halting problem; but the Busy Beaver, now that's a
clever and weird idea! The growth rate of Sigma(n) is ridiculous. Studying
this illuminated just how _vast_ a problem-space can be (especially concerning
the lack of knowledge we have by knowing a particular machine's configuration:
we don't learn much by inspection, we just have to run the damn thing). I see
the Busy Beaver "competition" as our more-interesting equivalent to searching
for large primes.

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hegemonicon
Worth pointing out that busy beavers have the potential to be alot more than
just 'interesting'. If you knew what the busy beaver numbers were for a given
number of states, it could be used to solve various unsolved mathematical
problems.

For example, Goldbach's conjecture states that every integer greater than 2 is
the sum of 2 primes. So far it has been resistant to being proved. But if you
could convert it to a computer program with say, 500 states, and you knew the
busy beaver number for 500, you could check it by running the program that
number of cycles. If you did and the program didn't halt, then it would NEVER
halt, and Goldbach's conjecture would be proven true.

Of course, BB(500) would certainly be an unfathomably huge number, so there
would be practical problems running a computer program that number of cycles.
But if you had sufficient computing power, knowing the busy beaver numbers
would be an amazing mathematical tool.

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eru
Yes. (Though you just proved that knowing the busy beaver numbers is as least
as heard as knowing the truth-values of all those famous conjectures.)

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ekarisor
I read the first paragraph, stopped, read the wikipedia page (in particular
the section Examples of busy beaver Turing machines), came back and enjoyed
the article.

~~~
Confusion
Ditto here. I think Peter overestimated his audience on this one.

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tocomment
I don't understand what the problem is asking.

~~~
gwern
It's basically asking, 'what's the biggest number a particular Turing Machine
can compute?' (expressed in unary).

This is interesting as an upper bound - for example, if we have a Turing
Machine and we know its Busy Beaver number _m_, we can run any of its programs
and if the program fills more than _m_ while running, we know the program
doesn't terminate! (Because if it did ever terminate, we would have a
different Busy Beaver number, which is a contradiction.) In other words, the
Busy Beaver lets us solve the Halting Problem for a specific Turing Machine;
doesn't violate Turing, but is still very interesting.

As Wikipedia says, "This machine attains the limits on the amount of time and
space that a halting Turing machine of the same class can consume." (This is
terribly worded, so I'm going to go edit it now...)

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btilly
Here is a fun thought.

There is no possible consistent axiom system understandable by humans in which
it is possible to prove a specific number to be an upper bound for
BB(1000000)!

Here is why. Suppose you have an axiom system. We can write a Turing machine
that searches for proofs and disproofs of the consistency of this axiom
system. If the axiom system is consistent the program will run forever. But if
it is proven that BB(size of machine) has an upper limit then the program will
eventually prove that the limit exists, and it has run past it, and therefore
it will never halt, and therefore it will never find an inconsistency, and
therefore the axiom system is consistent.

For the rest of my claim I claim that an axiom system that can be understood
by humans can always be encoded in a Turing machine of size at most a million.
I obviously have no proof of that, but it seems extremely likely to me.

For another fun thought, there is a school of mathematical philosophy in which
BB(n) is not a well-defined function at all! (Look up the constructivists.)

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techiferous
The "tape changes for 4 state busy beaver" image is the Burj Dubai! :)

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damienkatz
I was confused at first by the first paragraph since it didn't say the beavers
_must_ stop. Once I figured that out the rest made sense.

