All ZBLLs can be done in <R,U> + at most 2 D-layer moves

[qqwref]

From any ZBLL we can easily get to PLL using only 2gen moves. All PLLs are a U, Z, or H perm away from either solved, R perm, or Y perm. Therefore, if we can do R and Y perm with at most 2 D-layer moves, we can do any ZBLL with at most 2 D-layer moves.

So, here is an R perm with exactly 2 D-layer moves:
R U2 R D R' U R D' R' U' R' U R U R' U
And here is a Y perm (but not an efficient one):
R' U' D (R' U' R U)3 D' U R (R U R' U R U2 R')2 U'

I imagine these algs could be very useful for OH - everyone loves 2gen moves, and keeping D turns to a minimum is very helpful. Unfortunately, I don't know of any way to generate optimal algs of this type. Does anyone have an idea?

P.S. You can also do every ZBLL with <R,U> and at most two F moves, although it's less useful IMO. You can prove this with the T perm and this Y perm:
F (R U R' U')3 F' U (R U R' U R U2 R')2 U2

 

[jfly]

This is a neat idea. Nice proof that it can be done, btw.
Almost all cube solvers are built around a graph search, so we just need to define our nodes and edges, and then it's fairly simple to run a search algorithm to find optimal solutions to this 2D2gen problem. One nice thing about cube solvers is that every move is always available. So you basically start at a state, and consider every possible move from there and then every possible move after that, and so on, until you get to solved. With this problem, not every move is always legal, so with each state, I'd keep 2 bits to keep track of the 4 states we can be in: no Ds have been applied, a D has been applied, a D' has been applied, and a D and D' have been applied. The legal moves are different for each of these 4 possiblities.

 

[Lucas Garron]

to this 2D2gen problem.

Did somebody bring up notation there?
How about using generator notation with a limit? Like <U, R, D limit:2>, except hopefully a bit prettier but still distinctive and clear.

Also, any OLL with two flipped edges can be done in <U, R, D limit:2>, right? Any use for that?

This does sound like an interesting thing to implement in a solver; if anyone ever writes a really good solver, it will need a lot of features.

 

[trying-to-speedcube...]

Also, any OLL with two flipped edges can be done in <U, R, D limit:2>, right? Any use for that?

No?

I think you mean <U, R, F limit:2>

 

[DavidWoner]

<R, U, L limit:2> is also possible

Y Perm: R2 U' R2 U' L R U2 R' U' R U2 L' U R U R2

Jperm: L U' R' U L' U2 R U' R' U2 R

 

[qqwref]

Woner's right. You can also prove that without any other algs by flipping the cube around the UR edge and solving F2L, reducing it to the <R, U, D limit:2> problem.

 

[frogmanson]

ive learned 2 sets of zbll and learning 3rd right now and have 4 sets completely generated

most of my algs are RUF RUD RUB and RUL(i just apply z rotation if its better than the RUL case)

 

[LewisJ]

ive learned 2 sets of zbll and learning 3rd right now and have 4 sets completely generated

most of my algs are RUF RUD RUB and RUL(i just apply z rotation if its better than the RUL case)

That covers pretty much all the different kinds of algs. The point of this is that it's possible with RU + only 2 D turns, not that it is possible 3-gen.