The definition used on https://wiki.bbchallenge.org/wiki/Collatz-like for a Collatz-like function is:

> a partial function defined piecewise depending on the remainder of an input modulo some number

(for the best known Collatz conjecture specifically: 2)

and

> A Collatz-like problem is a question about the behavior of iterating a Collatz-like function.

(Regarding the first part, there are other functions that do not use modulo to decide on which "path" to take but some other property of the input number (e.g. its primality: https://mathoverflow.net/questions/205911/a-collatz-like-fun...) that may perhaps also be described as Collatz-like.)

The notation for tape states is documented here: https://wiki.bbchallenge.org/wiki/Directed_head_notation

I think part of the reason why it's so difficult to learn more about this kind of problem is that humanity has simply not found the right language for it yet. And in the case of not only describing, but solving them, a terminology/classification for the "shapes" of halting or non-halting behavior of such systems is also still largely missing.

Some earlier blog posts by the same author give more detail.

https://www.sligocki.com/2021/07/17/bb-collatz.html is one that covers the analysis of a particular machine in great detail, including explaining what definitions like "M(n) = 0^inf < A 1^n 0^inf" mean.

" M(n) = 0^inf < A 1^n 0^inf."

I read that as an infinite string of 0's (0^inf), then the tape head (<), then a string of exactly n 1's (1^n), then infinitely many 0's. So a block of n 1's starting at the tape head, surrounded by endless 0's to the left and right.

OK, you were not fast enough. I asked chatGPT. I think it helped but there is still a lot I do not understand.

  M(n) = 0^inf <A 1^n 0^inf.

  An infinite list of zeros up to the position of the machine pointer followed by some number of ones (maybe zero ones?) followed by another infinite list of zeros.
  

"I wonder if this is something generally true for all BBs"

No. A BB is going to use "all" its states. When you're up into the hundreds, it's not going to be anything recognizably Collatz-like, because Collatz is so simple.

People love what I think of as "squinting until everything blurs and everything is the same" and will probably try to salvage this conjecture for whatever reason people tend to do this, but unless you're going to claim that all Turing Machines are "just" computing some sort of "Collatz-like computation", degenerately, by blurring your definition of "Collatz-like" until the hand-written Turing Machine from school to reverse a string is "Collatz-like" and the hand-written TM to simulate a multitape machine is "Collatz-like", I don't think there's any reason to believe BB will continue to be Collatz-like indefinitely.

To put it another way, we've got a pretty good sense that even some hand-written functions can exceed these Collatz machines pretty handily... we just need enough states to represent them. The odds of a Collatz machine being the optimal machine once you get enough states to represent other ultra-super-exponential functions (e.g., Tree) is basically zero. It's just an artifact of us not being able to test such small Busy Beavers.

The most likely thing that BB(100) "does" is going to be something utterly humanly incomprehensible, not some sort of cute Collatz-like computation that can be characterized in any manner that would make sense to a human.

There's no sign of Collatz-like behaviour in any of the 37 known values of BBλ [1], so it appears specific to Turing Machines, which cannot compute exponentials so easily.

[1] https://oeis.org/A333479

If I were to generalize this in an optimistic view, I'd say that there's a decent chance for open problems in math to be busy beaver candidates. The collatz conjecture is an early family, I'd bet Diophantine equations show up, I think I recall the Golbach conjecture being a record at some point...

And I call this 'optimistic' because it supposes that mathematicians have identified the actual hard problems.