User blog:B1mb0w/The T-Rex Function

The T-Rex Function
The T-Rex function generates very large numbers. It has a growth rate well beyond \(f_{LVO}(n)\).

The Rex Function is a family of functions \(T\), \(r\) and \(e\) and \(x\) which use this simple rule set:

\(R(n) = R(0,n) = n + 1\)

\(R(a + 1, n) = R^n(a,n_*)\)

\(R(x(0), n) = R(n,n)\) and other instances of \(n\) can be substituted with \(x(0)\)

\(x(a + 1) = R^{x(a)}(x(a)_*,x(a))\)

and

\(x(1, 0) = x^{x(0)}(0)\)

\(x(1, a + 1) = x^{x(1, a)}(x(1, a))\)

\(x(b + 1, 0) = x^{x(b, 0)}(b, 0_*)\)

\(x(1, 0, 0) = x^{x(1, 0)}(1_*, 0)\)

and

\(e(0) = x(1, 0_{[x(0)]})\)

\(e(a + 1) = R^{e(a)}(e(a)_*,e(a))\)

\(e(1, 0) = e^{e(0)}(0)\)

\(e(1, a + 1) = e^{e(1, a)}(e(1, a))\)

\(e(b + 1, 0) = e^{e(b, 0)}(b, 0_*)\)

\(e(1, 0, 0) = e^{e(1, 0)}(1_*, 0)\)

and

\(R(1, 0, n) = R(e(1, 0_{[e(0)]}),n)\)

\(R(a + 1, 0, n) = R(a,e(1, 0_{[e(0)]}),n)\)

Notation Explained
I use notation that is not in general use, but I find helpful. They are the \(*\) and parameter subscript brackets.

The \(*\) notation is used to explain nested functions. For example:

\(M(a) = M(a)\)

\(M^2(a) = M(M(a))\)

then let

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

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

Parameter subscript brackets are useful for functions with many parameters:

\(M(a) = M(a)\)

\(M(a,b) = M(a,b)\)

then let

\(M(a,0_{[1]}) = M(a,0)\)

\(M(a,0_{[3]}) = M(a,0,0,0)\)

\(M(a,b_{[2]}) = M(a,b_1,b_2)\)

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

Growth Rate of the Rex Function ... to \(\Gamma_0\)
The R Function behaves like the FGH function up to a point:

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

\(R(x(0),n) = f_{\omega}(n)\)

\(R(R(1,x(0)),n) = f_{\omega.2}(n)\)

\(R(R(2,x(0)),n) = f_{\omega.2^{\omega}}(n)\)

\(R(R(3,x(0)),n) = f_{\varphi(1,0)}(n)\)

\(R(R(x(0),x(0)),n) \approx f_{\varphi(\omega,0)}(n)\)

\(R(x(1),n) = R(R^{x(0)}(x(0)_*,x(0)),n) > R(R^{x(0)}(3_*,x(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)

Growth Rate ... to small Veblen ordinal (svo)
The R Function will eventually reach and surpass the small Veblen ordinal (svo). Also refer to more detailed explanations in my previous blog The R Function:

\(R(R(R^{x(0)}(3_*,x(0)),x(1)),n) \approx f_{\varphi(1,0,1)}(n)\)

\(R(R(R(1,R^{x(0)}(3_*,x(0))),x(1)),n) > f_{\varphi(1,1,0)}(n)\)

\(R(R^2(x(1)_*,x(1)),n) > f_{\varphi(1,2,0)}(n)\)

\(R(R^3(x(1)_*,x(1)),n) > f_{\varphi(1,3,0)}(n)\)

\(R(R^2(x(1)),x(2)),n) > f_{\varphi(2,0,0)}(n)\)

\(R(R^2(x(2)_*,x(2)),n) > f_{\varphi(2,0,1)}(n)\)

\(R(x(3),n) \approx f_{\varphi(3,0,0)}(n)\)

\(R(x^2(0),n) \approx f_{\varphi(\omega,0,0)}(n)\)

\(R(x(1,0),n) \approx f_{\varphi(1,0,0,0)}(n)\)

\(R(e(0),n) = R(x(1,0_{[x(0)]}),n) = R(x(1,0_{[n]}),n) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)

Growth Rate ... to large Veblen ordinal (LVO) and beyond
The Rex Function is one of the Fastest Computable functions where:

\(x(0) \approx \omega = \vartheta(0)\)

\(R(3,x(0)) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)

\(x(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\)

\(e(0) \approx svo = \vartheta(\Omega^\omega)\)

\(e(1,0_{[x(0)]}) \approx \vartheta(\Omega^\omega\omega)\)

TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)

\(R(1,0,n) = R(e(1,0_{[e(0)]}),n) \approx f_{\vartheta(\Omega^{\omega+1})}(n)\)

\(R(1,x(0),n) \approx f_{\vartheta(\Omega^{\omega+2})}(n)\)

\(R(1,x^2(0),n) \approx f_{\vartheta(\Omega^{\omega.2})}(n)\)

\(R(1,x(1,0),n) \approx f_{\vartheta(\Omega^{\omega.3})}(n)\)

\(R(1,e(0),n) \approx f_{\vartheta(\Omega^{\omega^2})}(n)\)

\(R(1,e(1,0),n) \approx f_{\vartheta(\Omega^{\omega^3})}(n)\)

\(R(1,e(1,0_{[x(0)]}),n) \approx f_{\vartheta(\Omega^{\omega^{\omega}})}(n)\)

\(R(2,0,n) \approx f_{\vartheta(\Omega^{\omega^{\omega^{\omega}}})}(n) = f_{\vartheta(\Omega^{\omega\uparrow\uparrow 3})}(n)\)

\(R(3,0,n) \approx f_{\vartheta(\Omega^{\omega\uparrow\uparrow 4})}(n)\)

\(R(x(0),0,n) \approx f_{\vartheta(\Omega^{\omega\uparrow\uparrow\omega})}(n) = f_{\vartheta(\Omega^{\varphi(1,0)})}(n)\)

\(R(R(3,x(0)),0,n) \approx f_{\vartheta(\Omega^{\varphi(1,\varphi(1,0))})}(n) = f_{\vartheta(\Omega^{\varphi^2(1,0_*)})}(n)\)

\(R(R(x(0),x(0)),0,n) \approx f_{\vartheta(\Omega^{\varphi(2,0)})}(n)\)

\(R(x(1),0,n) \approx f_{\vartheta(\Omega^{\varphi(1,0,0)})}(n)\)

\(R(e(0),0,n) \approx f_{\vartheta(\Omega^{\Omega})}(n)\)

Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\)

\(R(e(1),0,n) \approx f_{\vartheta(\Omega^{\Omega^2})}(n)\)

\(R(e(2),0,n) \approx f_{\vartheta(\Omega^{\Omega^3})}(n)\)

\(R(e^2(0),0,n) \approx f_{\vartheta(\Omega^{\Omega^{\omega}})}(n)\)

\(R(e(1,0),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow 3)}(n)\)

\(R(e(1,0,0),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow 4)}(n)\)

\(R(e(1,0_{[x(0)]}),0,n) \approx f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

Bird's H(n) function \(\approx f_{\vartheta(\varepsilon_{\Omega+1})}(n) = f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

\(R(1,0,0,n) = R(e(1,0_{[e(0)]}),0,n) \approx f_{\vartheta(\varepsilon_{\Omega+2})}(n)\)

Some Identities
Some Rex Function identities are:

\(R(R(R(a,b)),b) > R(R(a,b),R(a,b))\)

because

\(R(R(R(a,b)),b) = R^b(R(a,b),b_*) = R(R(a,b),R^{b-1}(R(a,b),b_*))\)

and

\(R^{b-1}(R(a,b),b_*) > R(R(a,b),b) > R(a,b)\)

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