User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35870936-20180813215517

First of all, you should be using UNOCF for comparisons with BMS, because UNOCF expressions below psi(W_w) DIRECTLY translate into BMS expressions (I will explain this more down below). Don't listen to PsiCubed when he says we don't know how powerful pair sequence BM2/3 is. We do know. It's psi(W_w), ''without any shadow of a doubt. ''And it's also important to note that BM2.x and BM3.x all expand the exact same way in pair sequence. They only start to expand differently in trio sequence.

Anyway, there is a very nice correspondence between a UNOCF expression below psi(W_w) and a BM2 pair sequence expression.

If you have a term (a,b) in BMS, then that term corresponds to a psi_b function at a depth of a in UNOCF.

Here are some examples of what I mean:

(0,0)(1,1)(2,2) = psi(psi_1(psi_2(0))) = psi(psi_1(W_2)) = BHO

(0,0)(1,1)(2,2)(0,0) = psi(psi_1(psi_2(0)))+psi(0) <--- the psi(0) term is at the 0th layer in this case.

(0,0)(1,1)(2,2)(1,0) = psi(psi_1(psi_2(0))+psi(0)) <--- notice how the psi(0) moved up one layer

(0,0)(1,1)(2,2)(2,0) = psi(psi_1(psi_2(0)+psi(0))) <--- moved up to the 2nd layer

(0,0)(1,1)(2,2)(3,0) = psi(psi_1(psi_2(psi(0)))) <--- moved up to the 3rd layer, which is the innermost layer in this case

(0,0)(1,1)(2,2)(3,1) = psi(psi_1(psi_2(psi_1(0)))) <--- since the term is (3,1), we have a psi_1 function in the 3rd layer

(0,0)(1,1)(2,2)(3,2) = psi(psi_1(psi_2(psi_2(0)))) <--- now we have a psi_2 function

(0,0)(1,1)(2,2)(3,3) = psi(psi_1(psi_2(psi_3(0))))<--- now we have a psi_3 function

If you don't believe these examples, look at the analysis people have done of BMS below the BHO. It matches perfectly with the correspondence I explained here.

And finally, there is also a correspondence between UNOCF and standard OCFs that would be useful to you. I will use U(x) to denote UNOCF and psi(x) to denote standard OCF's here.

psi_a(0) = U_a(W_a)

psi_a(1) = U_a(W_a*2)

psi_a(2) = U_a(W_a*3)

psi_a(w) = U_a(W_a*w)

So as you can see, adding 1 inside the psi function in standard OCFs corresponds to adding W_a inside UNOCF. Also, U_a(x) catches up to psi_a(x) at psi_a(W_a^w).

I hope this helps you to study pair sequence!