# Generalized Reduction: An Outline and Expansion on ZZ-b

#### Silky

##### Member
Hi all, Silky here. This is going to be a breakdown of a technique that I've come up with or, more accurately, have created a system of description. This is an expansion/generalization of Phasing described in ZZ-b which reduces ZBLL to ZZLL. Here I'll be describing what I'd consider the 3 main types of phasing and will include some examples for each.

Reduction
Definition: an intuitive transitioning from a lower order cubestate to a higher order cube state.

(1) Another way to describe these cubestates would be through gen reduction since this is generally a consequence of high order cubestates. This however isn't always useful which we will see later. We'll be swapping between these on a case by case basis.

Examples:

(a) HTA: Thistlewaitels Alg is of course the first method that comes to mind since the structure of the method is to continuously phase to higher order cubestates. The cubestates which it goes thru are G1: edge orientation, G2: domino reduction, and G3: Double domino reduction.
(b) ZZ: EO specifically is the most common form of phasing and most notably used in ZZ. This is of course used in several other methdods: Petrus, LEOR, APB, Mehta*, etc
(c) YruRU: again another good example of this is CP+EO cubestate which reduces the moveset to 3-gen and then later to 2-gen.

*Mehta and APB fall under a grey area since, as explained above, this is designed to be an intuitive technique. For example ZBLS/VHLS would not be an example of phasing, where edge control would be. APB is a pretty interesting case since it mixes algs and intuition. I'll go over this in the next section.

Algset Reduction
Definition: phasing which is used specifically to reduce an algset. This is done more specifically right before an algorithmic step.

Examples:

(a) ZZ-b: obviously, as this is the original inspiration of this idea, ZZ-b's edge separation is an example. This reduces the alg count from 493 to 169 and is also known as phasing.
(b) HD-G: here the idea is that you intuitively modify Guimond CO algs to force a V-V case reducing the 6CP algset (called NLL in HD-G) from 47 to 36.
(c) SSC-FR: here leaving out the FR edge edge during WV allows you to instead use 5CO to orient the corners as this can mess up the FR edge. Reduces WV's 27 algs to 5CO's 16 algs (not including OCLL).
(d) APB: as we talked about above APB falls into a relatively ambiguous area. This can be though of as a type of ELS. It sort of WV but creating a pair regardless of orientation and then solving EO and the pair. Probably a better example is solving an oriented pair which then reduces the APB algset. I'll let ya'll debate this one.

Null-Reduction
Definition: Specifically NOT phasing as to maintain flexibility in the step preformed .

(1) This is definitely the most abstract form of generalized reduction so let me know if I can go into more detail/better clarify this technique.

Eamples:

(a) Psuedo Slotting: this is pretty difficult to explain but I'll try my best. The basic idea is that we consider the first step of CFOP, cross, to reduce the moveset from 6-gen to 5-gen as we no longer require D moves. The reason we describe this with gen-reduction is because I can't really think of a way to describe it as a unique cubestate nor do I think think describing is as such would make it any clearer. Anti-phasing would be choosing to NOT reduce the cube to this 5-gen cubestate which is done by not directly solving the cross. This has the benefit of allowing F2L to be solved more efficiently.
(b) ZZ: for ZZ the idea is to ONLY orient cross the E-slice edges, this would be known as partial EO. This makes cross and F2L all 3-gen but resulting in doing OLL instead of OCLL.
(c) SSC: again, partial EO. ONLY orienting E-slice edges allowing for 3-gen psuedo pairs and afterwards sets up for WV pair.
(d) SSC: again this is going to be another hard one to explain especially since I haven't fully developed the idea but I'll try my best. The idea here is that you are going to intuitively modify the WV/5CO algs similarly outlined with SSC-FR. This is also kind of a mix of the previous idea in HD-G. The first part is to ignore creating a WV EO pair, then preforming 5CO, and finally inserting the FR edge. This insertion is done in the middle of the alg. Here is a more specific example to illustrate the idea.

=> Here we are going to be looking at 2 WV cases. WV Case [2]: R U' R' and Case [6]: U' R U' R' U2 R U' R' U2 R U R'. If you look at these cases from the perspective of Guimond Orientation (O5C) then you will notice these are actually the same case (this would be true in 5CO for SSC-FR). If we are ignoring the permutation of the corners, Case [6] then becomes: M U' Rw' U' R'. We then can take this one step further by NOT making the WV pair in the first place. Let's say that the FR edge is oriented and placed in the FD position. We first start our 5CO alg with R and then then perform a U2 M2, completing the WV pair in the middle of the alg, and finishing with U R'. This should reduce our algs from 23 to 16 and allow us to insert the FR edge, finishing the WV pair, from several spots on the cube.

Anyhow, thanks for reading, I know this is a long post.

~TTFN

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#### OreKehStrah

##### Member
No hate or anything, but we've had a concrete definition for what phasing in cubing for over a decade now. I don't think we should be trying to generalize it into something else. It would be better to generalize this idea under reduction systems instead.

#### Silky

##### Member
No hate or anything, but we've had a concrete definition for what phasing in cubing for over a decade now. I don't think we should be trying to generalize it into something else. It would be better to generalize this idea under reduction systems instead.
Can you go into more detail? Exactly what do you mean by a concrete definition of phasing? If your talking about just ZZ-b than that's something that is only ever talked about within the context of that method. The whole point of this post it to expand that idea at least in a descriptive way outside of just ZZ-b.

#### OreKehStrah

##### Member
Can you go into more detail? Exactly what do you mean by a concrete definition of phasing? If your talking about just ZZ-b than that's something that is only ever talked about within the context of that method. The whole point of this post it to expand that idea at least in a descriptive way outside of just ZZ-b.
Phasing has always been defined as placing the edges of the LL such that opposite colors are opposite of each other, but not necessarily permuted. I'm not a fan of changing definitions.

#### Silky

##### Member
I'm not changing the definition, I'm expanding the idea outside of ZZ-b.. aka generalizing it. That's the whole point of the post. If phasing in this context is edge separation I think that we can make a clear connection of this outside of ZZ-b. Pretty directly this has a connection with G3 cubestate however is different in the case of ZZ-b in that it is used to reduce an algset not a moveset. Edge separation is not the only technique of intuitively reducing an algset. Yes its reduction but it's a specific reduction used for a specific purpose which I have defined as Algset Phasing. Do you just want me to rename it to Generalized Reduction? I think here I'm just making the connection between reduction and phasing as phasing is a type of reduction not just that thing you do in ZZ-b.

#### tsmosher

##### Member
I just want to mention that ZZ-r uses phasing as well. To reduce PLL case count. That is all.

#### Silky

##### Member
I just want to mention that ZZ-r uses phasing as well. To reduce PLL case count. That is all.
Must have missed it. Thanks for the addition.

#### OreKehStrah

##### Member
I'm not changing the definition, I'm expanding the idea outside of ZZ-b.. aka generalizing it. That's the whole point of the post. If phasing in this context is edge separation I think that we can make a clear connection of this outside of ZZ-b. Pretty directly this has a connection with G3 cubestate however is different in the case of ZZ-b in that it is used to reduce an algset not a moveset. Edge separation is not the only technique of intuitively reducing an algset. Yes its reduction but it's a specific reduction used for a specific purpose which I have defined as Algset Phasing. Do you just want me to rename it to Generalized Reduction? I think here I'm just making the connection between reduction and phasing as phasing is a type of reduction not just that thing you do in ZZ-b.
Yeah I’m not dismissing the idea. I just think it would be better to call it a general reduction system and leave phasing as just a specific example.

#### tsmosher

##### Member
I also thought anti-phasing was specifically setting up the LL edges to be unphased? i.e., That it truly was the opposite of phasing.

Kinda like CLLEF vs. COLL.

#### Silky

##### Member
Yeah I’m not dismissing the idea. I just think it would be better to call it a general reduction system and leave phasing as just a specific example.
Okay, that makes sense. Basically to make it less confusing. I'll fix it now. Hopefully it will help.

#### Silky

##### Member
I also thought anti-phasing was specifically setting up the LL edges to be unphased? i.e., That it truly was the opposite of phasing.

Kinda like CLLEF vs. COLL.
Well as I said above this is an intuitive technique so it would be specifically misorienting all of the LL edge before solving CLLEF. So it's about setting up for CLLEF not CLLEF itself.

edit: So I guess in this case as it would be anti-reduction as you choosing specifically to not orient edges to force CLLEF. Also anti-reduction isn't the opposite of reduction (like 'un-reducing) but specifically choosing to not reduce

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#### OreKehStrah

##### Member
Well as I said above this is an intuitive technique so it would be specifically misorienting all of the LL edge before solving CLLEF. So it's about setting up for CLLEF not CLLEF itself.

edit: So I guess in this case as it would be anti-reduction as you choosing specifically to not orient edges to force CLLEF. Also anti-reduction isn't the opposite of reduction (like 'un-reducing) but specifically choosing to not reduce
Maybe call it null reduction instead?

#### Silky

##### Member
Maybe call it null reduction instead?
Good idea. And looking back at what @tsmosher said I now see the idea you were saying. Here you'd be reducing from a higher order cubestate to a lower order cubestate. Very interesting.. not something I've ever really heard of. Although this may fall sort of into edge control. I'll add anti-reduction later tonight.