The function G is defined as follows:

🛃 Question 🛃 Does G(1, 1) terminate?

🛄 Answer 🛄 Yes. G(1, 1) = 2533…2210 > 187003.

That was just a warmup. G is not the function referred to in the title of this post. What we really want to know about is H:

🛃 Question 🛃 Does H(1, 1) terminate?

🛄 Answer 🛄 … ¿ 🛐 ? …

G(1, 1) returns its answer after 28,833 iterations. H(1, 1) does not return an answer after any reasonable number of iterations. There are two possibilities:

1. H(1, 1) never returns an answer.
2. H(1, 1) returns an answer after an unreasonable number of iterations.

Practically speaking, there is no difference – we get no answer either way.

🛃 Question 🛃 Which possibility is more likely?

G and H are examples of iterated Collatz-like functions. The first argument accumulates the answer and the second argument counts how many times to apply the transformation. The countdown argument can get reset, giving these functions an Ackermann flavor.

Notice that we are not asking about whether H is total, and neither are we claiming that G is. Those would be very difficult claims to prove. No, we are asking the severely unambitiious question of whether these functions terminate on just this one particular argument. G(1, 1) terminates, but we can’t answer for H(1, 1).

One way to look at this is to say: G and H have quite similar definitions – they differ at just two parameters. G(1, 1) terminates, so why shouldn’t H(1, 1)? Maybe most functions similar to G terminate, and so we should assume that H does too.

On the other hand, why exactly does G(1, 1) terminate? There’s no obvious reason why it should work. In fact, it seems miraculous that it terminates at all. Maybe most of these functions don’t terminate, and G is a remarkable exception.

🛃 Question 🛃 Why are we talking about these weird functions?

🛄 Answer 🛄 In connection with the Beeping Busy Beaver problem, Turing machine programs have been discovered that implement these functions. The program that implements G is the current 5-state 2-color BBB champion, and the program that implements H is a contender. Both programs are “children” of the so-called Mother of Giants and were discovered by Shawn Ligocki.

Proving the true value of BBB(5) is at least as hard as determining the outcome of H(1, 1).