An important observation is that it doesn't really matter what you start with, because you have two ways to "reduce" each Turing machine to one that can start with all zeroes:
- You can "flip" rules/initial tape values, trying to get a machine that can start with all 0s. This is not always possible. A trivial case where it is possible is a machine that wants to start with all 1s (just flip everything).
- You can insert more rules to set the tape up as needed. This makes it an BBn+x busy beaver candidate, but at least it's not your problem at BBn now.
The last way can get a bit complicated because you need to insert ad-hoc logic, since the tape is infinitely long. So it's not a just a case of inserting setup at the beginning of the program (because you can't setup an infinite tape in finite steps). Also to be fair this second point is a non-trivial assertion and requires proof that it is actually possible to do in every case. It is obviously possible to do for every machine that runs in finite steps though (since it can only consider a finite amount of tape).
Luckily actual computers have finite memory which is generally initialized with 0 anyways.
- You can "flip" rules/initial tape values, trying to get a machine that can start with all 0s. This is not always possible. A trivial case where it is possible is a machine that wants to start with all 1s (just flip everything).
- You can insert more rules to set the tape up as needed. This makes it an BBn+x busy beaver candidate, but at least it's not your problem at BBn now.
The last way can get a bit complicated because you need to insert ad-hoc logic, since the tape is infinitely long. So it's not a just a case of inserting setup at the beginning of the program (because you can't setup an infinite tape in finite steps). Also to be fair this second point is a non-trivial assertion and requires proof that it is actually possible to do in every case. It is obviously possible to do for every machine that runs in finite steps though (since it can only consider a finite amount of tape).
Luckily actual computers have finite memory which is generally initialized with 0 anyways.