The T-Rex Function[]
The T-Rex function generates very large numbers. It has a growth rate approaching \(f_{LVO}(n)\).
The T-Rex Function is a family of functions \(T()\), \(r()\) and \(e()\) and \(x()\) which use this simple rule set:
\(T(n) = T(0,n) = n + 1\)
\(T(a + 1, n) = T^n(a,n_*)\)
\(T(x(0), n) = T(n,n)\) and other instances of \(n\) can be substituted with \(x(0)\)
\(x(a + 1) = T^{x(a)}(x(a)_*,x(a))\)
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
\(e(0) = x(1, 0_{[x(0)]})\)
\(e(a + 1) = T^{e(a)}(e(a)_*,e(a))\)
\(e(1, 0) = e^{e(0)}(0)\)
\(e(1, a + 1) = e^{e(1, a)}(e(1, a))\)
\(e(b + 1, 0) = e^{e(b, 0)}(b, 0_*)\)
\(e(1, 0, 0) = e^{e(1, 0)}(1_*, 0)\)
and
\(r(0) = e(1, 0_{[e(0)]})\)
\(r(a + 1) = T^{r(a)}(r(a)_*,r(a))\)
\(r(1, 0) = r^{r(0)}(0)\)
\(r(1, a + 1) = r^{r(1, a)}(r(1, a))\)
\(r(b + 1, 0) = r^{r(b, 0)}(b, 0_*)\)
\(r(1, 0, 0) = r^{r(1, 0)}(1_*, 0)\)
and
\(T(1, 0, n) = T(r(1, 0_{[r(0)]}),n)\)
\(T(a + 1, 0, n) = T(a,r(1, 0_{[r(0)]}),n)\)
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 T-Rex Function ... to \(\Gamma_0\)[]
The T-Rex Function behaves like the FGH function up to a point. Also refer to more detailed explanations in my previous blog The Rex Function:
\(T^h(g,n_*) = f_g^h(n)\)
\(T(x(0),n) = f_{\omega}(n)\)
\(T(T(1,x(0)),n) = f_{\omega.2}(n)\)
\(T(T(3,x(0)),n) = f_{\varphi(1,0)}(n)\)
\(T(T(x(0),x(0)),n) \approx f_{\varphi(\omega,0)}(n)\)
\(T(x(1),n) = T(T^{x(0)}(x(0)_*,x(0)),n) > T(T^{x(0)}(3_*,x(0)),n) \approx f_{\varphi(1,0,0)}(n) = f_{\Gamma_0}(n)\)
Growth Rate ... to small Veblen ordinal (svo)[]
The T-Rex Function will eventually reach and surpass the small Veblen ordinal (svo):
\(T(T(T(1,T^{x(0)}(3_*,x(0))),x(1)),n) > f_{\varphi(1,1,0)}(n)\)
\(T(T^2(x(1)_*,x(1)),n) > f_{\varphi(1,2,0)}(n)\)
\(T(T^2(x(1)),x(2)),n) > f_{\varphi(2,0,0)}(n)\)
\(T(x(3),n) \approx f_{\varphi(3,0,0)}(n)\)
\(T(x^2(0),n) \approx f_{\varphi(\omega,0,0)}(n)\)
\(T(x(1,0),n) \approx f_{\varphi(1,0,0,0)}(n)\)
\(T(e(0),n) = T(x(1,0_{[x(0)]}),n) = T(x(1,0_{[n]}),n) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)
Growth Rate ... to large Veblen ordinal (LVO) and beyond[]
The T-Rex Function is one of the Fastest Computable functions where:
\(x(0) \approx \omega = \vartheta(0)\)
\(T(3,x(0)) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)
\(x(1) \approx \Gamma_0 = \varphi(1,0,0) = \vartheta(\Omega^2)\)
\(e(0) \approx svo = \vartheta(\Omega^\omega)\)
\(e(1,0_{[x(0)]}) \approx \vartheta(\Omega^\omega\omega)\)
TREE(n) function \(≥ f_{\vartheta(\Omega^\omega\omega)}(n)\)
\(r(0) = e(1,0_{[e(0)]}) \approx \vartheta(\Omega^{\omega.2})\)
\(r(1,0_{[e(0)]}) \approx f_{\vartheta(\Omega^{\omega.3})}(n)\)
\(T(1,0,n) = T(r(1,0_{[r(0)]}),n) \approx f_{\vartheta(\Omega^{\omega.4})}(n)\)
\(T(1,0,0,n) = T(r(1,0_{[r(0)]}),0,n) \approx f_{\vartheta(\Omega^{\omega.6})}(n)\)
\(T(1,0_{[n]},n) \approx f_{\vartheta(\Omega^{\omega^2.2})}(n)\)
Large Veblen ordinal \(LVO ≥ f_{\vartheta(\Omega^\Omega)}(n)\)
Some Identities[]
Some T-Rex Function identities are:
\(T(T(T(a,b)),b) > T(T(a,b),T(a,b))\)
because
\(T(T(T(a,b)),b) = T^b(T(a,b),b_*) = T(T(a,b),T^{b-1}(T(a,b),b_*))\)
and
\(T^{b-1}(T(a,b),b_*) > T(T(a,b),b) > T(a,b)\)
Further References[]
Further references to relevant blogs can be found here: User:B1mb0w