6 ms·

The Busy Beaver Problem

64 points by pkrumins 17y ago

techiferous 17y ago
The "tape changes for 4 state busy beaver" image is the Burj Dubai! :)
tocomment 17y ago
I don't understand what the problem is asking.
- jacquesm 17y ago
  The idea is to figure out how many cycles it takes before a 'beaver' halts, if at all (it is a given they should, but after how many cycles is an open problem for more than the smallest numbers of states). They have extremely simple rules, yet take a very long time before they finally do come to a halt. It's an interesting little problem, if such tremendously simple state machines can exhibit such complex behavior it gives you some indication of how hard it would be to answer those questions for more complicated state machines.
- krig 17y ago
  Here is a clearer explanation of the problem: http://www.logique.jussieu.fr/~michel/tmi.html http://www.logique.jussieu.fr/~michel/tmi.html My attempt at explaining it (I'm not sure I've understood it completely, though): Busy Beavers are Turing machines that eventually reach a halting state. The goal is to find the busy beaver, for a given number of states, that takes the longest to do so. This turns out to be a tricky problem for states >= 5. That is, there are solutions to n=5,..., but no one has formally proved that there isn't a busy beaver that will take longer to halt.
- gwern 17y ago
  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...)
- deleted 17y ago
  [deleted]
rw 17y ago
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.
ekarisor 17y ago
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 17y ago
  Ditto here. I think Peter overestimated his audience on this one.
staticshock 17y ago
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 http://www.scottaaronson.com/writings/bignumbers.html
- Torn 17y ago
  Outstanding introduction to this sort of thing, thanks for sharing.
- dangoldin 17y ago
  You should submit this on it's own to HN. That was a great read.
- pbhjpbhj 17y ago
  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.
  - Eliezer 17y ago
    > 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 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.
hegemonicon 17y ago
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.
- eru 17y ago
  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.)
damienkatz 17y ago
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.
btilly 17y ago
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.)

64 points by pkrumins 17y ago

techiferous 17y ago

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

tocomment 17y ago

I don't understand what the problem is asking.

jacquesm 17y ago
The idea is to figure out how many cycles it takes before a 'beaver' halts, if at all (it is a given they should, but after how many cycles is an open problem for more than the smallest numbers of states). They have extremely simple rules, yet take a very long time before they finally do come to a halt. It's an interesting little problem, if such tremendously simple state machines can exhibit such complex behavior it gives you some indication of how hard it would be to answer those questions for more complicated state machines.
krig 17y ago
Here is a clearer explanation of the problem: http://www.logique.jussieu.fr/~michel/tmi.html http://www.logique.jussieu.fr/~michel/tmi.html My attempt at explaining it (I'm not sure I've understood it completely, though): Busy Beavers are Turing machines that eventually reach a halting state. The goal is to find the busy beaver, for a given number of states, that takes the longest to do so. This turns out to be a tricky problem for states >= 5. That is, there are solutions to n=5,..., but no one has formally proved that there isn't a busy beaver that will take longer to halt.
gwern 17y ago
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...)
deleted 17y ago
[deleted]

rw 17y ago

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.

ekarisor 17y ago

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 17y ago
Ditto here. I think Peter overestimated his audience on this one.

staticshock 17y ago

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 http://www.scottaaronson.com/writings/bignumbers.html

Torn 17y ago
Outstanding introduction to this sort of thing, thanks for sharing.
dangoldin 17y ago
You should submit this on it's own to HN. That was a great read.
pbhjpbhj 17y ago
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.
- Eliezer 17y ago
  > 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 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.

hegemonicon 17y ago

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.

eru 17y ago
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.)

damienkatz 17y ago

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.

btilly 17y ago

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.)