# Does This Function Terminate?

The function ** G** is defined as follows:

🛃 **Question** 🛃 Does *G(1, 1)* terminate?

🛄 **Answer** 🛄 Yes. *G(1, 1) = 2533…2210 > 18 ^{7003}*.

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:

*H(1, 1)***never**returns an answer.*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).**