User blog:MonkeyProphet/Big and massive functions

The Big Function 

{B(x),(y)}= x^x^...x with x raised to itself y^y^...y with y y's

{B(x),(0)}=0

{B(x),(1)}=x

{B(x),(2)}=x^x^x^x  2^2 x's

{B(x),(3)}=x^x^...x with 3^3^3 x's

{B(x),(4)}=x^x^...x with 4^4^4^4 x's

etc.

The Massive Function

([M],n,x)=({B{B{...{B(x),(x)}...}} With ([M].n-1,x) iterated {B}'s where ([M],0,x)= {B,(x),(x)}

Massive Modifiers

(p,[M],n,x) = ([M],([M],...([M],n,x),...,x),x) where ([M],n,x) is nested ([M],n,x) times repeated p steps.

(ω,ω-1,...,p,[M],n,x)=(ω-1,...,p,[M],n,x) where ω-1 now equals [M],(ω-1,...,p,[M],n,x),x)

[Mψ]x

[Mψ]x=([M],n,x),([M],n,x),...,([M],n,x) with ([M],n,x) iterations where n =([M],0,x)

 Expanding{B,(x),(y)} 

{B,(x),(y)}

=(x)↑↑(y~...~y~y~y) solve right to left, replicating the value to the immediate left of the ~ by the rightmost value on the right and removing that value from the right.

=(x)↑↑(y~...~y~y(y,y,...,y))

=(x)↑↑(y~...~y(y)~y-1(y,y...y))

=(x)↑↑(y~...~y(y(y))~y-2(y,y...y)) repeat until all arguments have been collapsed to the leftmost one

=(x)↑↑(y(y(y....(y)...))) Multiply all values together

Examples for y=2 and y=3

{B,(x),(2)}

=(x)↑↑(2~2)

=(x)↑↑(2,2)

=(x)↑↑4

{B,(x),(3)}

=(x)↑↑(3~3~3)

=(x)↑↑(3-(3,3,3))

=(x)↑↑(3,3,3~(3,3))

=(x)↑↑(3,3,3,3,3,3,3,3,3~(3))

=(x)↑↑3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

=(x)↑↑7625597484987

More to follow when i'm not in bed......