This is a subspecies of BEAF.

I have wanted an easy notation that I can understand.

I think the easiest way to make a big number is "piling up."

So, I will make a piling-up notation.


PU(a,b,c)=a\uparrow^c b =PU(a,PU(a,b-1,c),c-1)


Now, you need to understand that the default of PU is 1.

In math language, PU(X)=PU(X,1)=PU(X,1,1)=\cdots

I piled up b's. What's next? Piling up c's. Therefore,



Then, PU(a,b,1,2)\simeq a\rightarrow b\rightarrow b\rightarrow 2

Next, I can think the same thing:

PU(a,b,c,2)=PU(a,PU(a,b-1,c,2),c-1,2)\simeq a\rightarrow b\rightarrow b\rightarrow c

(note: this approximation is just shows level. Since I know chain notation, I can think (how big PU is) better.)

Now, I will jump to polynominal. K means some one's.




For example,




I think this is as big as a\rightarrow_2 b\rightarrow_2 2 in extended chain notation, but I don't know how big it is exactly.

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