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Can the number of busy beavers with n states be computed, or would it be necessary to analyze all the machines to count them ?

Peter
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  • To help people who don't have the background, you might consider editing the question to include a definition of a "busy beaver". You can also use dollar signs to enclose mathematical notation such as the $n$; compare n and $n$. – Carl Mummert Dec 13 '13 at 13:18
  • I would be astonished if somebody found a way to count them. – TonyK Dec 13 '13 at 13:19
  • Does anyone know the situation for the known busy beavers ? – Peter Dec 13 '13 at 15:04
  • @Peter: What do you mean, "known busy beavers"? For the very few $n$ where the busy beaver number is known, it is also known exactly which machines realizes that number, so they are easy to count. – hmakholm left over Monica Dec 13 '13 at 16:16
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    Interesting fact: Let $f(n)$ be your number, then we know that $4 \mid f(n)$, because it doesn't matter whether the halting transition is left or right, and it also doesn't matter if we switch left and right completely, since that is only symmetric. Furthermore every TM has at least two transitions so these two modifications can't be exactly the same. Thus $4 \mid f(n)$ and $f(n) \geq 4$ for all natural $n$. – wythagoras Jul 27 '15 at 08:21

1 Answers1

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(A busy beaver is a two-symbol Turing machine that halts when started on a blank tape, and does so in a maximal number of steps among all machines with the same number of states that halt on the blank tape).

It would be extraordinarily surprising if the number of busy beavers of a given size was computable.

Remember that there must be machines where it is independent of the axioms (choose any recursive set of axioms that don't prove arithmetic falsehoods) whether the machine is a busy beaver or not. Given that, it would be very strange if the axioms nevertheless determined how many of them there are -- and unless they do so, we can at least never prove that any given algorithm counts them correctly.

Since we're getting into foundational issues here, the question is not really a yes/no one. There are at least four imaginable resolutions:

  1. Yes, this function is computable, and here is a concrete algorithm that we can prove (from some reasonable axiomatic basis) computes it.

  2. The function is computable, and here is a non-constructive proof that there must be a machine that computes it. We can't ever know which machine that is, though.

  3. It is independent of (insert your favorite axiom system here) whether the function is computable or not.

  4. The function is provably not computable.

Of these (1) and (2) would surprise me, (1) moreso than (2). However, (3) and (4) both sound plausible -- intuitively my money would be on (3), though.

hmakholm left over Monica
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