User blog:B1mb0w/Comparing Fast Growing Hierarchy Functions

Comparing Fast Growing Hierarchy Functions

This blog is a working page to capture various rules that can be used to compare different combinations of Fast-growing hierarchy Functions.

Basics

\(f_b(a) = f_{b-1}^a(a)\)

\(f_{b-1}^a(a) = f_b(a)\)

\(f_{b-2}^a(a) = f_{b-1}(a)\)

\(f_{b-1}^{a-1}(f_{b-2}^a(a)) = f_{b-1}^{a-1}(f_{b-1}(a)) = f_{b-1}^{a}(a) = f_b(a)\)

Basics with \(\omega\)

\(f_{\omega}(a) = f_a(a)\)

\(f_a(a) = f_{\omega}(a)\)

\(f_a^2(a) = f_a(f_a(a)) = f_a(f_{\omega}(a))\)

\(f_{\omega+1}(a) = f_{\omega}^a(a)\)

\(f_{\omega+b}^c(a) = f_{\omega+b}^{c-1}(f_{\omega+b}(a))\)

Some Comparisons

\(f_b^c(a) = f_{b}^{c-a}(f_{b}^a(a)) = f_{b}^{c-a}(f_{b+1}(a))\) when \(c > a\)

\(f_b^c(a) = f_{b}^{c-a}(f_{b+1}(a)) = f_{b}^{c-a-f_{b+1}(a)}(f_b^{f_{b+1}(a)}(f_{b+1}(a)))\) when \(c > f_{b+1}(a)\)

\(= f_{b}^{c-a-f_{b+1}(a)}(f_{b+1}(f_{b+1}(a))) = f_{b}^{c-a-f_{b+1}(a)}(f_{b+1}^2(a))\)

\(f_b^{f_{b+1}(a)+a}(a) = f_{b+1}^2(a)\)

then

\(f_{b}^{f_{b+1}^2(a).2}(a) = f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}^2(a)}(a)) >> f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)+f_{b+1}(a)}(a)) >> f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)+a}(a))\)

\(= f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)}(f_b^a(a))) = f_{b}^{f_{b+1}^2(a)}(f_{b}^{f_{b+1}(a)}(f_{b+1}(a))) = f_{b}^{f_{b+1}^2(a)}(f_{b+1}(f_{b+1}(a)))\)

\(= f_{b}^{f_{b+1}^2(a)}(f_{b+1}^2(a)) = f_{b+1}(f_{b+1}^2(a)) = f_{b+1}^3(a)\)

and provided n>1

\(f_{b}^{f_{b+1}^n(a).n}(a) >> f_{b+1}^{n+1}(a)\) because \(f_{b+1}^n(a).n >> f_{b+1}^n(a).(n-2) + f_{b+1}^n(a) + f_{b+1}(a) + a\)

and this is enough to successively build \(f_{b+1}^{x+1}(a)\) for x=0 to x=n

then

\(f_{b}^{f_{b+1}^{a-1}(a).(a-1)}(a) >> f_{b+1}^{a-1+1}(a) = f_{b+1}^a(a) = f_{b+2}(a)\)

Some Comparisons with \(\omega\)

\(f_b^c(a) = f_{b}^{c-1}(f_{b}(a)) = f_{b}^{c-1}(f_{\omega}(a))\) when \(b = a\) when \(c > 0\)

\(f_b^c(a) = f_{b}^{c-1}(f_{b}(a)) = f_{b}^{c-1}(f_{f_{\omega}(a)}(a))\) when \(b = f_{\omega}(a)\) and \(c > 0\)

\(= f_{b}^{c-1}(f_{f_{a-1}^a(a)}(a)) >> f_{b}^{c-1}(f_{f_{a-1}^{a-1}(a)+f_{a-1}(a)}(a)) = f_{b}^{c-1}(f_{f_{a-1}^{a-1}(a)+f_{a-2}^a(a)}(a)) >> f_{b}^{c-1}(f_{f_{a-1}^{a-1}(a)+f_{a-2}^{a-1}(a)+a+1}(a))\)

\(= f_{b}^{c-1}(f_{f_{a-1}^{a-1}(a)+f_{a-2}^{a-1}(a)+a}^a(a)) >> f_{b}^{c-1}(f_{f_{a-1}^{a-1}(a)+f_{a-2}^{a-1}(a)+a}^{a-1}(f_{f_{a-1}^{a-1}(a)+f_{a-2}^{a-1}(a)+a}(a)))\)

then

\(f_{f_{\omega}^x(a)}^x(a) >> f_{f_{\omega}^x(a)}^{x-1}(f_{\omega}(a)) >> f_{f_{\omega}^x(a)}^{x-2}(f_{\omega}^2(a)) >> f_{f_{\omega}^x(a)}(f_{\omega}^{x-1}(a)) >> f_{\omega}^{x}(a)\)

and

\(f_{f_{\omega}^a(a)}^a(a) >> f_{f_{\omega}^a(a)}(f_{\omega}^{a-1}(a)) >> f_{\omega}^a(a) = f_{\omega+1}(a)\)

also

\(f_{\omega+1}^{f_{\omega+2}^{a-1}(a).(a-1)}(a) >> f_{\omega+2}^{a-1+1}(a) = f_{\omega+3}(a)\)

Some Comparisons with \(\omega.n\)

\(f_{\omega+a-2}^{f_{\omega+a-1}^{a-1}(a).(a-1)}(a) >> f_{\omega+a-1}^{a-1+1}(a) = f_{\omega+a}(a) = f_{\omega.2}(a)\)

\(f_{\omega+b}(a) = f_{\omega+b-1}^a(a) = f_{\omega+b-1}^{a-1}(f_{\omega+b-1}(a)) = f_{\omega+b-1}^{a-1}(f_{\omega.2}(a))\) if b-1 = a