Isn’t it a true coincidence? Collatz-like problems are one of the simplest (in operation), no surprise that BB(low n) chooses them. It’s probably more surprising that we found it before analyzing BBs.
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.
Thank you for the link. I have been wondering whether the Lambda Calculus has a BB like values. I do not know much about the Lambda Calculus and this might be a good reason for me to learn more.
The values shown in the OEIS link look different than the values for the TM BBs. Am I just reading it wrong?
"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.
I wouldn't put it so strongly: often a long-running TM can be analyzed in several layers with their own independent rules, each of which simulates the next. E.g., at the lowest level, most nontrivial TMs look like a list of unary or binary counters interacting with each other. I wouldn't be surprised if at least some of the lower layers of the longest-running machines included one or more Collatz-like components. After all, it is a cheap way to pad out the tape when setting up the parameters of some greater process! It's just not going to be the entirety of what it does, since no simple Collatz-like process will practically get you past an exponential number of steps.
Thank you for providing an example of exactly what I mean by "yeah, but if we just squint harder..."
It is generally considered very unlikely that BBs will just compose together lower level BBs in any comprehensible manner because that very human-comprehensible composibility is itself going to be a waste of states that could be used to do something simply incomprehensible. It is very, very unlikely that BB(100) will consist even of parts a human can understand. It will be a monolithic whole of impenetrable incomprehensibility.
When you say TMs can often be analyzed in terms of smaller ones, you are talking about human-built TMs. We have to build that way. We can't work with the raw chaos. You can see this in our design in every field; we have to work in modules. BBs do not and will not.
I'm not talking about human-built TMs: I specifically have in mind some of the 3x3 holdouts I have looked into. Every last one of those can be seen as simulating a higher-level process, up to a point where it potentially breaks down (either halting or transitioning to another process); we just can't tell whether that breakdown ever occurs in reality.
Just because a machine is incomprehensible as a whole, doesn't mean that every singular aspect is incomprehensible! The thing with TMs is, there is a finite (or very slow-growing) number of meaningful things a machine can do with raw symbols on the tape. "Raw chaos" printed directly on the tape would just cause it to either halt nearly immediately, or enter some trivial non-halting configuration. Compare this with Conway's Game of Life, where 99.99999999+% of random soups create only trivial output, despite the CA itself being universal.
Note that I'm not at all saying that large machines aren't doing wonderful and terrible things, immediately past the lowest level. But the formalism itself restricts them from being too wild right away, at least not without sacrificing dozens of valuable states to constantly fix things up. For a machine to be nontrivial, it must syntactically maintain a very fine line between halting and non-halting, so that we can't just easily decide it one way or another.
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.
Is there a more "popular science" description of what is being said here. I would like to understand what is meant by "Collatz-like". Is it a function like:
f(n) = {n/2 : for even n,
3n+1 : for odd n}
Can I sit round for days making trivial calculations looking for patterns?
What does
M(n) = 0^inf < A 1^n 0^inf.
mean?
I have really enjoyed the Busy Beaver stuff and play with simple code to try and learn what I can but when I read about it I run into the brick wall of math reading comprehension. Numberphile on youtube is pretty good sometimes with explanations but is not a reference. I do not know where to turn (I might ask chatGPT just to see what happens).
> 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.)
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.
Interesting - if any computer program can be rewritten in the form of a Collatz-like function, is it at all surprising that the BB problems appear in this form?
The interesting part is how simple the resulting Collatz-like functions are to describe, suggesting that the Collatz-like form is the most 'natural' form for hard problems given the TM formalism. E.g., any program can also be rewritten in the form of a Diophantine equation, but most of those would be monstrously large.
The wikipedia article doesn't do full justice perhaps though the essence is there. The reason the halting problem is equivalent to solving collatz like problems is because you can rewrite any turing machine in terms of a generalized collatz function. The Fractran wikipedia page has more info.
(the original conway article itself is also very readable: FRACTRAN: A SIMPLE UNIVERSAL PROGRAMMING LANGUAGE FOR ARITHMETIC)
Thank you. I have been looking a little at your github Busy_Beaver repository. I think I will stick with my on code writing for a little while (amateurish but I am learning a little bit writing it).
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.
Is, in pseudo math/English, some Turing machine in state n is equal to
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 think at this point chatGPT gets something wrong but at this point I hit the Free plan limit for GPT-4o and will have to wait till 2:13PM.
A turing machine has a position on a tape and an internal state
The <A means the machine is in state A and facing left. To its left are infinitely many zeroes (0^inf), to its right are n 1s, and then infinitely many zeroes (the initial tape is infinite, and starts filled with zeroes)
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.
reply