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For any two functions that have different asymptotic growth rates, there must be a point after which they no longer cross. In other words, if f(x) grows faster than g(x), there is some y such that f(x) > g(x) for all x > y.

I'll single out four "interesting" functions to ask about here: The busy beaver function BB(x), Rayo's function Rayo(x), Graham's function G(x), and the TREE function TREE(x). I'm curious about the values of the final crossover points between these functions.

For example, the final crossover point between TREE(x) and G(x) is in between 2 and 3. TREE(2) < G(2) while TREE(3) > G(3). (Neither function is defined on the nonintegers, so we don't get a single exact point.)

The uncomputability of BB() and Rayo() doesn't prevent us from establishing upper and lower bounds on their final crossover points with other functions, nor does it even necessarily prevent us from finding out the exact value. For example, the final crossover point between BB(x) and 2^x is in between 4 and 5, since BB(4) = 13 < 2^4 = 16, while BB(5) >= 4098 > 2^5 = 32.

This is a little trickier than it seems, since there might be any number of times the two functions cross each other before the final one. I've asserted above that the final crossover point between G() and TREE() is in the range 2-3, but all I actually know for sure is that there is a crossover point there. It's conceivable that they cross again, and maybe TREE(x) is smaller than G(x) for some extremely large x. (But I really doubt it.)

So my challenge to you is: What are the best upper and lower bounds you can prove for the final crossover point of any combination of two of these functions?

I'll edit in a list of the best bounds as people refine them.

Best lower bounds so far:

  • G & TREE = ?
  • G & BB = ?
  • G & Rayo = ?
  • TREE & BB = ?
  • TREE & Rayo = ?
  • BB & Rayo = ?

Best upper bounds so far:

  • G & TREE = ?
  • G & BB = ?
  • G & Rayo = ?
  • TREE & BB = ?
  • TREE & Rayo = ?
  • BB & Rayo = ?
Isaac King
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1 Answers1

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I'm going to weigh in with a couple bounds that I'm reasonably certain of, using values of these functions from around the internet.

G & TREE

$TREE(2) = 3 \ll G(2)$, so a lower bound on the crossover is 2.

$TREE(3) \gg G(64) \gg G(3)$, so an upper bound is 3. Nice and simple.

G & BB

$BB(6) \geq 10\uparrow\uparrow15 \ll G(6)$, so I would conjecture that $BB(6) \leq G(6)$, but unfortunately I do not have the means to prove this. As such, until the values of $BB(5)$ or $BB(6)$ are proven the lower bound must be set at 4.

Thanks to this blog post a Turing machine with 18 states has been found that prints more than $G(64)$ ones. As such, $BB(18) \gt G(64) \gg G(18)$, so the upper bound is at 18.

G & Rayo

As far as I can tell, the highest Rayo number proven is $Rayo(29) = 1 \therefore Rayo(29)\ll G(29)$, putting a lower bound at 29.

As for an upper bound, $Rayo(7339) > S(2^{65536}-1) \geq BB(2^{65536}-1) \gg G(2^{65536}-1) \gg G(7339)$, so the upper bound for the crossover point is 7339.

Summary

Crossover for $G(n), Tree(n)$ has bounds $2<n<3$.

Crossover for $G(n), BB(n)$ has bounds $4<n<18$

Crossover for $G(n), Rayo(n)$ has bounds $29<n<7339$

Shophaune
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    Unfortunately, as the OP points out, seeing the inequality between two functions change at two specific points doesn't guarantee that there are no other inequality changes later on. – Greg Martin Jul 23 '23 at 01:09
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    @GregMartin I do not believe that's a concern for any but the G & TREE case - if there's another inequality change for BB it's impossible to prove, as doing so inevitably would mean finding the values of all lower BB(n), which are uncomputable. And as it can be shown that Rayo outgrows BB, there cannot be a further inequality change between G and Rayo unless there is one between G and BB. – Shophaune Jul 23 '23 at 01:15
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    As for the G & TREE case, I believe it would be easiest to show that the weak tree(n) function outgrows the G function, and then that the strong TREE function outgrows the weak tree, and that both dominate beyond whatever n G(n)=TREE(3) at – Shophaune Jul 23 '23 at 01:59