User blog:B1mb0w/Notation Explained

Notation Explained
I use notation that is not in general use, but I find very helpful to describe recursive functions. You will see this notation in use on my blogs. This blog will present the exact definition I intend for these notations. Any deviance from the notation here is most likely due to an error in one of my other blogs.

They notations are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).

Parameter Subscript Brackets, where:

\(M(a,0_{[2]}) = M(a,0,0)\)

\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)

Leading Zeros Assumption, where:

\(M(0_{[x]},0_{[2]},b_{[3]},1) = = M(0_{[x + 2]},b_1,b_2,b_3,1) = M(b_1,b_2,b_3,1)\)

Recursion Parameter Subscript \(*\), where:

\(M^2(a) = M^2(a_*) = M(M(a))\) and \(M(a,b_*) = M(a,b)\)

\(M^2(a,b_*) = M(a,M(a,b))\)

\(M^2(a_*,b) = M(M(a,b),b)\)

Decremented Function \(C\), where for any function:

\(M(a_{[b]},c + 1,d_{[e]})\) then \(C = M(a_{[b]},c,d_{[e]})\)

Comparing Veblen and HyperRex functions
For the comparison below, I will take some liberties with the definition of the Veblen Function.

The HyperRex function behaves like the FGH function up to a point. The r function acts like a stronger version of the Veblen function as show here:

\(H^h(g,n_*) = f_g^h(n)\)

\(H(r(0),n) = f_{\omega}(n) = f_{\varphi(1)}(n)\)

\(H(r_c(a_{[b]}),n) >= H(r(a_{[b]}),n) > f_{\varphi(a_{[b]})}(n)\) for any values of \(a, b, c\)

Growth rate of the HyperRex function
The growth rate of the HyperRex function is compared here against my Hyper Function and a calculated lower bound of FGH using Veblen function and collapsing ordinal function.

Further References
Further references to relevant blogs can be found here: User:B1mb0w