MOTAN 新版定义分析

当n为自然数时，MOTAN的b[n]p跟BEAF的b{n}p是完全相同的，b[n]p=b\(\uparrow^n\)p

3[1<1>2]2=3[1<1>1]3=3[3]3

3[1<1>2]3=3[1<1>1]3[1<1>1]3=3[1<1>1](3[3]3)=3[3[3]3](3[3]3)>3[3[3]3]3>3[4]3=3\(\uparrow\uparrow\uparrow\uparrow\)3=G(1)

3[1<1>2]4>3[1<1>1](G(1))>3[G(1)]3=G(2)

......

3[1<1>2]66>G(64)

MOTAN增长率分析（\(0\) ~ \(\varepsilon_0\)）

MOTAN	FGH
1	\(2\)
2	\(3\)
3	\(4\)
1<1>1	\(\omega\)
1<1>2	\(\omega+1\)
1<1>3	\(\omega+2\)
2<1>1	\(\omega2\)
2<1>2	\(\omega2+1\)
2<1>3	\(\omega2+2\)
3<1>1	\(\omega3\)
1<1>1<1>1	\(\omega^2\)
1<1>1<1>2	\(\omega^2+1\)
1<1>1<1>3	\(\omega^2+2\)
1<1>2<1>1	\(\omega^2+\omega\)
1<1>2<1>2	\(\omega^2+\omega+1\)
1<1>2<1>3	\(\omega^2+\omega+2\)
1<1>3<1>1	\(\omega^2+\omega2\)
2<1>1<1>1	\(\omega^22\)
3<1>1<1>1	\(\omega^23\)
1<1>1<1>1<1>1	\(\omega^3\)
1<2>1	\(\omega^\omega\)
1<2>2	\(\omega^\omega+1\)
1<2>3	\(\omega^\omega+2\)
1<2>1<1>1	\(\omega^\omega+\omega\)
1<2>1<1>2	\(\omega^\omega+\omega+1\)
1<2>1<1>3	\(\omega^\omega+\omega+2\)
1<2>2<1>1	\(\omega^\omega+\omega2\)
1<2>3<1>1	\(\omega^\omega+\omega3\)
1<2>1<1>1<1>1	\(\omega^\omega+\omega^2\)
1<2>1<1>1<1>1<1>1	\(\omega^\omega+\omega^3\)
2<2>1	\(\omega^\omega2\)
3<2>1	\(\omega^\omega3\)
1<1>1<2>1	\(\omega^{\omega+1}\)
1<1>1<2>2	\(\omega^{\omega+1}+1\)
1<1>1<2>3	\(\omega^{\omega+1}+2\)
1<1>1<2>1<1>1	\(\omega^{\omega+1}+\omega\)
1<1>1<2>1<1>2	\(\omega^{\omega+1}+\omega+1\)
1<1>1<2>1<1>3	\(\omega^{\omega+1}+\omega+2\)
1<1>1<2>2<1>1	\(\omega^{\omega+1}+\omega2\)
1<1>1<2>3<1>1	\(\omega^{\omega+1}+\omega3\)
1<1>1<2>1<1>1<1>1	\(\omega^{\omega+1}+\omega^2\)
1<1>1<2>1<1>1<1>1<1>1	\(\omega^{\omega+1}+\omega^3\)
1<1>2<2>1	\(\omega^{\omega+1}+\omega^\omega\)
1<1>3<2>1	\(\omega^{\omega+1}+\omega^\omega2\)
2<1>1<2>1	\(\omega^{\omega+1}2\)
1<1>1<1>1<2>1	\(\omega^{\omega+2}\)
1<1>1<1>1<1>1<2>1	\(\omega^{\omega+3}\)
1<2>1<2>1	\(\omega^{\omega2}\)
1<2>1<2>1<2>1	\(\omega^{\omega3}\)
1<3>1	\(\omega^{\omega^2}\)
1<1<1>1>1	\(\omega^{\omega^\omega}\)
2<1<1>1>1	\(\omega^{\omega^\omega}2\)
1<1<1>1>1<1<1>1>1	\(\omega^{\omega^\omega2}\)
1<1<1>1>1<1<1>1>1<1<1>1>1	\(\omega^{\omega^\omega3}\)
1<1<1>2>1	\(\omega^{\omega^{\omega+1}}\)
1<2<1>1>1	\(\omega^{\omega^{\omega2}}\)
1<1<1>1<1>1>1	\(\omega^{\omega^{\omega^2}}\)
1<1<1>1<1>1<1>1>1	\(\omega^{\omega^{\omega^3}}\)
1<1<2>1>1	\(\omega^{\omega^{\omega^\omega}}\)
1<1<1<1>1>1>1	\(\omega^{\omega^{\omega^{\omega^\omega}}}\)
1<1/1>1	\(\varepsilon_0\)

MOTAN增长率分析（\(\varepsilon_0\) ~ \(\psi(\psi_I(0))\)）

MOTAN	FGH
1<1/1>2	\(\varepsilon_0+1\)
1<1/1>3	\(\varepsilon_0+2\)
1<1/1>1<1>1	\(\varepsilon_0+\omega\)
1<1/1>1<1>2	\(\varepsilon_0+\omega+1\)
1<1/1>1<1>3	\(\varepsilon_0+\omega+2\)
1<1/1>2<1>1	\(\varepsilon_0+\omega2\)
1<1/1>3<1>1	\(\varepsilon_0+\omega3\)
1<1/1>1<1>1<1>1	\(\varepsilon_0+\omega^2\)
1<1/1>1<1>1<1>1<1>1	\(\varepsilon_0+\omega^3\)
1<1/1>1<2>1	\(\varepsilon_0+\omega^\omega\)
1<1/1>1<1<1/1>1>1	\(\varepsilon_02\)
1<1/1>2<1<1/1>1>1	\(\varepsilon_03\)
1<1/1>1<1<1/1>1>1<1<1/1>1>1	\(\varepsilon_0^2\)
1<1/1>1<1<1/1>2>1	\(\varepsilon_0^\omega\)
1<1/1>1<1<1/1>1<1<1/1>1>1>1	\(\varepsilon_0^{\varepsilon_0}\)
1<1/1>1<1<1/1>1<1<1/1>1<1<1/1>1>1>1>1	\(\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}\)
2<1/1>1	\(\varepsilon_1\)
3<1/1>1	\(\varepsilon_2\)
1<1>1<1/1>1	\(\varepsilon_{\omega}\)
1<1<1/1>1>1<1/1>1	\(\varepsilon_{\varepsilon_0}\)
1<1<1<1/1>1>1<1/1>1>1<1/1>1	\(\varepsilon_{\varepsilon_{\varepsilon_0}}\)
1<1/1>1<1/1>1	\(\zeta_0\)
1<1/1>2<1/1>1	\(\varepsilon_{\zeta_0+1}\)
2<1/1>1<1/1>1	\(\zeta_1\)
1<1/1>1<1/1>1<1/1>1	\(\eta_0\)
1<1/2>1	\(\varphi(\omega,0)\)
1<1/1<1<1/1>1>1>1	\(\varphi(\varphi(\omega,0),0)\)
1<1/1<1/1>1>1	\(\Gamma_0\)
1<1/1<1>1<1/1>1>1	\(SVO\)
1<1/1<1/1>1<1/1>1>1	\(LVO\)
1<2/1>1	\(BHO\)
1<2/1>1<1<2/1>1>1	\(\psi(\psi_1(0))2\)
1<2/1>1<1<2/1>2>1	\(\psi(\psi_1(0))^\omega\)
1<2/1>1<1<2/1>1<1<2/1>1>1>1	\(\psi(\psi_1(0))^{\psi(\psi_1(0))}\)
1<2/1>1<1/1>1	\(\psi(\psi_1(0)+1)\)
1<2/1>1<1/1>1<1/1>1	\(\psi(\psi_1(0)+\Omega)\)
1<2/1>1<1/2>1	\(\psi(\psi_1(0)+\Omega^\omega)\)
1<2/1>1<1/1<2/1>1>1	\(\psi(\psi_1(0)2)\)
1<2/1>1<1/1<2/1>2>1	\(\psi(\psi_1(0)^\omega)\)
1<2/1>1<1/1<2/1>1<1<2/1>1>1>1	\(\psi(\psi_1(0)^{\psi(\psi_1(0))})\)
1<2/1>1<1/1<2/1>1<1/1>1>1	\(\psi(\psi_1(0)^\Omega)\)
1<2/1>1<1/1<2/1>1<1/1<2/1>1>1>1	\(\psi(\psi_1(0)^{\psi_1(0)})\)
2<2/1>1	\(\psi(\psi_1(1))\)
3<2/1>1	\(\psi(\psi_1(2))\)
1<1<2/1>1>1<2/1>1	\(\psi(\psi_1(\psi(\psi_1(0))))\)
1<1/1>1<2/1>1	\(\psi(\psi_1(\Omega))\)
1<1/1<2/1>1>1<2/1>1	\(\psi(\psi_1(\psi_1(0)))\)
1<1/1<1/1<2/1>1>1<2/1>1>1<2/1>1	\(\psi(\psi_1(\psi_1(\psi_1(0))))\)
1<2/1>1<2/1>1	\(\psi(\Omega_2)\)
1<2/2>1	\(\psi(\Omega_2^\omega)\)
1<2/1<1/1>1>1	\(\psi(\Omega_2^\Omega)\)
1<2/1<2/1>1>1	\(\psi(\Omega_2^{\Omega_2})\)
1<2/1<2/1<2/1>1>1>1	\(\psi(\Omega_2^{\Omega_2^{\Omega_2}})\)
1<3/1>1	\(\psi(\psi_2(0))\)
1<3/1>1<3/1>1	\(\psi(\Omega_3)\)
1<1<1>1/1>1	\(\psi(\Omega_{\omega})\)
1<1<1>2/1>1	\(\psi(\psi_{\omega}(0))\)
1<1<1/1>1/1>1	\(\psi(\Omega_{\Omega})\)
1<1<1<1/1>1/1>1/1>1	\(\psi(\Omega_{\Omega_{\Omega}})\)
1<1/1/1>1	\(\psi(\psi_I(0))\)

1<1/1>1是1<1<1/1>1>1的省略写法，省略了最外层的1<>1，它们都是\(\varepsilon_0\)。

1<1/1>2跟1<1<1/1>1>2表示同一序数，都是\(\varepsilon_0+1\)。

1<1/1>1<1<1/1>1>1和2<1<1/1>1>1表示同一序数，都是\(\varepsilon_02\)。

1<1/1>1<1>1<1<1/1>1>1和1<1>1<1<1/1>1>1表示同一序数，都是\(\varepsilon_0\omega\)，此时第1个式子开头的1<1/1>1变成了并不影响表达式增长率的“前缀”。

1<1<1/1>2>1和1<1/1>1<1<1/1>2>1表示的都是\(\varepsilon_0^\omega\)

2<1/1>1和1<2<1/1>1>1表示的是同一序数，都是\(\varepsilon_1\)。

这里特别注意：1<1/1>2=1<1<1/1>1>2，而2<1/1>1=1<2<1/1>1>1，其中2所在的位置并不相同！

反射模式分析

1<1>1	1
1<1>1<1>1	1-1
1<1>1<1>1<1>1	1-1-1
1<1/1>1	2
1<1<1>1/1>1	1-2
1<1/1/1>1	2 1-2
1<1/1/1/1>1	2 1-(2 1-2)
1<1{1{1}1}1>1	2-2
1<1{1}1{1{1}1}1>1	2 1-2-2
1<1{1}1{1}1{1{1}1}1>1	2 1-(2 1-2-2)
1<1{1{1}1}1{1{1}1}1>1	2-2 1-2-2
1<1{2{1}1}1>1	2-2-2
1<1{3{1}1}1>1	2-2-2-2
1<1{1{1}1{1}1}1>1	3
1<1{1}1{1{1}1{1}1}1>1	2 1-3
1<1{1}1{1}1{1{1}1{1}1}1>1	2 1-(2 1-3)
1<1{1{1}1}1{1{1}1{1}1}1>1	2-2 1-3
1<1{1}1{1{1}1}1{1{1}1{1}1}1>1	2 1-(2-2 1-3)
1<1{1{1}1}1{1{1}1}1{1{1}1{1}1}1>1	2-2 1-(2-2 1-3)
1<1{2{1}1}1{1{1}1{1}1}1>1	2-2-2 1-3
1<1{1{1}1{1}1}1{1{1}1{1}1}1>1	3 1-3
1<1{1{1}1{1}1}1{1{1}1{1}1}1{1{1}1{1}1}1>1	3 1-(3 1-3)
1<1{1{1}2{1}1}1>1	2-3
1<1{1{1}3{1}1}1>1	2-2-3
1<1{2{1}1{1}1}1>1	3 2-3
1<1{3{1}1{1}1}1>1	3 2-(3 2-3)
1<1{1{1}1{1}1{1}1}1>1	3-3
1<1{1{1}1{1}1{1}1{1}1}1>1	3-3-3
1<1{1{1{1}1}1}1>1	4