User blog:B1mb0w/Comparison of Recursive Functions

Comparison of Recursive Functions
This blog will compare Recursive Functions such as Veblen hierarchy and my own Big number and T-Rex functions.

The Big number function is a pair of functions \(B\) and \(g\) which use this simple rule set:

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

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

\(B(g(0), n) = B(n,n)\) and other instances of \(n\) can be substituted with \(g(0)\)

\(g(c + 1) = g(0, c + 1) = B^{g(c)}(g(c)_*,g(c))\)

and

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

\(g(b, c + 1) = B^{g(b,c)}(g(b,c)_*,g(b,c))\)

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

and

\(g = g_0\)

\(g_{a + 1}(0) = g_a(1, 0_{[g_a(0)]})\)

\(g_a(c + 1) = B^{g_a(c)}(g_a(c)_*,g_a(c))\)

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

\(g_a(b, c + 1) = B^{g_a(b,c)}(g_a(b,c)_*,g_a(b,c))\)

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

and

\(B(1, 0, n) = B(g_{g(0)}(0),n)\)

\(B(a + 1, 0, n) = B(a,g_{g(0)}(0),n)\)

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. For example:

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

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

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

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

Growth Rate of the Big number function ... to \(\Gamma_0\)
The Big number function behaves like the FGH function up to a point. Also refer to more detailed explanations in my previous blog The T-Rex Function:

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

\(B(g(0),n) = f_{\omega}(n)\)

\(B(B(1,g(0)),n) = f_{\omega.2}(n)\)

\(B(B(3,g(0)),n) = f_{\varphi(1,0)}(n)\)

\(B(B(g(0),g(0)),n) \approx f_{\varphi(\omega,0)}(n)\)

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

Growth Rate ... to small Veblen ordinal (svo)
The Big number function will easily reach and surpass the small Veblen ordinal (svo):

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

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

\(B(B^2(g(1)),g(2)),n) > f_{\varphi(2,0,0)}(n)\)

\(B(g(3),n) \approx f_{\varphi(3,0,0)}(n)\)

\(B(g^2(0),n) \approx f_{\varphi(\omega,0,0)}(n)\)

\(B(g(1,0),n) \approx f_{\varphi(1,0,0,0)}(n)\)

\(B(g_1(0),n) = B(g_0(1,0_{[g_0(0)]}),n) = B(g(1,0_{[n]}),n) \approx f_{\varphi(1,0_{[n]})}(n) = f_{svo}(n)\)

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

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

\(B(3,g(0)) \approx \epsilon_0 = \varphi(1,0) = \vartheta(1)\)

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

\(g_1(0) \approx svo = \vartheta(\Omega^\omega)\)

\(g_1(1,0_{[g_0(0)]}) \approx \vartheta(\Omega^\omega\omega)\)

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

\(g_2(0) = g_1(1,0_{[g_1(0)]}) \approx \vartheta(\Omega^{\omega+1})\)

\(g_2(1) \approx \vartheta(\Omega^{\omega+2})\)

\(g_2^2(0) \approx \vartheta(\Omega^{\omega.2})\)

\(g_2(1,0) \approx \vartheta(\Omega^{\omega.3})\)

\(g_2(1,0_{[g_0(0)]}) \approx \vartheta(\Omega^{\omega^2})\)

\(g_2(1,0_{[g_1(0)]}) \approx \vartheta(\Omega^{\omega^{\omega}})\)

then

\(g_3(0) = g_2(1,0_{[g_2(0)]}) \approx \vartheta(\Omega^{\omega\uparrow\uparrow\omega}) = \vartheta(\Omega^{\varphi(1,0)})\)

\(g_3(1) \approx \vartheta(\Omega^{\varphi(1,0,0)})\)

\(g_3(g_0(0)) = g_3(g(0)) \approx \vartheta(\Omega^{\Omega})\)

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

\(g_3(g_1(0)) \approx \vartheta(\Omega^{\Omega^2})\)

\(g_3(g_2(0)) \approx \vartheta(\Omega^{\Omega^3})\)

\(g_3^2(0) \approx \vartheta(\Omega^{\Omega^{\omega}})\)

\(g_3(1,0) \approx \vartheta(\Omega\uparrow\uparrow 3)\)

\(g_3(1,0,0) \approx \vartheta(\Omega\uparrow\uparrow 4)\)

\(g_3(1,0_{[g_0(0)]}) \approx \vartheta(\Omega\uparrow\uparrow\omega)\)

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

\(g_3(1,0_{[g_1(0)]}) \approx \vartheta(\varepsilon_{\Omega+2})\)

\(g_3(1,0_{[g_2(0)]}) \approx \vartheta(\varepsilon_{\Omega.2})\)

\(g_4(0) \approx \vartheta(\varepsilon_{\Omega^2})\)

\(g_5(0) \approx \vartheta(\varepsilon_{\Omega^{\omega}})\)

\(B(1,0,n) = B(g_{g(0)}(0),n) \approx f_{\vartheta(\varepsilon_{\Omega^{\omega^{\omega}}})}(n)\)

\(B(g_0(0),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\varphi(1,0)}})}(n)\)

\(B(g_1(0),0,n) \approx f_{\vartheta(\varepsilon_{\Omega^{\Omega}})}(n)\)

\(B(1,0,0,n) \approx f_{\vartheta(\Omega_1)}(n)\)

\(B(1,0_{[g(0)]},n) = B(1,0_{[n]},n) \approx f_{\vartheta(\Omega_\omega)}(n)\)

Bird's U(n) function \(\approx f_{\vartheta(\Omega_\omega)}(n)\)

Bird's S(n) function (new definition) \(\approx f_{\vartheta(\Omega_\Omega)}(n)\)

Some Identities
Some Big number function identities are:

\(B(B(B(a,b)),b) > B(B(a,b),B(a,b))\)

because

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

and

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

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