User blog:Kyodaisuu/Mashimo function

Mashimo function is a googological function which is intended to work as a scale of various googological numbers. It is now a beta version and it might be updated based on the discussion in this blog post. Mashimo scale is defined with this function in the next blog entry. Mashimo is a character in Sushi Kokuuhen.

Definition of Mashimo function \(M(x)\) is as follows.

For \(x \ge 1\), \[M(x) = max(10^{10x}, ^{x/5}e, H(^{x/20}2,2), H(^{H(x-70,2)}3,3), \\ V(x-72), V(3^{x-80}), BH(x-83), O(x-85), OF(x-86),\\ L(x-90), D(5(x-94),10), \\ FC(x-80), RC(x-120)) \]

For \(0 \le x < 1\), \(M(x) = 10^{10x}\)

For \(x < 0\), \(M(x) = M(-x)^{-1}\)

Here, \(max\) function returns the largest number in the parameters, and several subfunctions of \(H, V, BH, O, OF, L, D, FC, RC\) are defined in this page. \(M(x)\) is a continuous and strictly monotonically increasing function. Extention of tetration to parameters of real numbers is defined with the method of, i.e.,

\begin{eqnarray*} ^{x}a &=& log_a{(^{x+1}a)} &for& x \le -1 \\ ^{x}a &=& x+1 &for& -1 < x \le 0 \\ ^{x}a &=& a^{(^{x-1}a)} &for& 0 < x \\ \end{eqnarray*}

Subfunctions
First of all, for \(x<0\), \(H(x,n) = V(x) = BH(x) = O(x) = OF(x) = L(x) = D(x,n) = FC(x) = RC(x) = 0\).

H function
H function with 2 variables of natural numbers is defined with Hardy function. It is defined as \[H(a,n) = H_\alpha(n)\] where \(a\) is a natural number, \(n \ge 2\) is a natural number, and \(\alpha\) is an ordinal related to \(a\) as follows; write \(a\) with hereditary base-n notation, change \(n\) to \(\omega\), and we get \(\alpha\). For example,

\[H(35,2) = H(2^{2^2+1}+2+1,2) = H_{\omega^{\omega^\omega+1}+\omega+1}(2) \]

Therefore, \(\alpha < \epsilon_0\). Wainer hierarchy is used for fundamental sequence. First parameter of H function is extended to real numbers as follows. Let

\begin{eqnarray*} N &=& floor(a) \\ r &=& a-N \\ H(N,n) &=& H(x,N+1) \\ H(N+1,n) &=& H(y,N+1) \\ \end{eqnarray*}

and define \(H(a,n) = H(N+r,n)\) as

\[H(a,n) = H(x+r(y-x),n+1)\]

For example,

\begin{eqnarray*} H(4,2) &=& H_{\omega^{\omega}}(2) = H_{\omega+2}(2) = H_{\omega+1}(3)\\ &=& H(4,3) \\ H(5,2) &=& H_{\omega^{\omega}+1}(2) = H_{\omega^{\omega}}(3) \\ &=& H(27,3) \\ \end{eqnarray*}

and therefore

\[H(4.3,2) = H(4+0.3(27-4),3) = H(10.9,3) \]

as \(n\) increases, the ordinal in the Hardy function corresponding to the first parameter of H function decreases, and therefore the ordinal finally reaches 0 (this is similar to the proof that Goodstein sequence finally stops). Therefore, by defining

\[H(r,n) = n+r (0 \le r < 1)\]

it is guaranteered that the calculation stops, and \(H(a,n)\) is now defined for \(a \in \mathbb{R}\).

V function
V function is defined with Veblen function and Hardy hierarchy as follows.

\[V(\sum_{k=0}^{n} 3^k a_k) = H_{\phi(..., a_3, a_2, a_1, a_0)}(3) \]

Fundamental sequence of \(\epsilon_1 = lim(\epsilon_0+1, \omega^{\epsilon_0+1}, \omega^{\omega^{\epsilon_0+1}}, ...)\) is used, and it is calculated as follows.

\begin{eqnarray*} V(0) &=& H_{\phi(0,0)}(3) = H_1(3) = 4 \\ V(1) &=& H_{\phi(0,1)}(3) = H_\omega(3) = 6 \\ V(2) &=& H_{\phi(0,2)}(3) = H_{\omega^2}(3) = 24 \\ V(3) &=& H_{\phi(1,0)}(3) = H_{\epsilon_0}(3) = H_{\omega^{\omega^{\omega}}}(3) \\ V(4) &=& H_{\phi(1,1)}(3) = H_{\epsilon_1}(3) = H_{\omega^{\omega^{\epsilon_0+1}}}(3) = H_{\epsilon_0^{\omega}}(3) = H_{\epsilon_0^3}(3) = f_{\epsilon_0 3}(3)\\ V(5) &=& H_{\phi(1,2)}(3) = H_{\epsilon_2}(3) = f_{\epsilon_1 3}(3) \\ V(6) &=& H_{\phi(2,0)}(3) = H_{\zeta_0}(3) \approx f_{\zeta_0}(3) \\ V(7) &=& H_{\phi(2,1)}(3) = H_{\zeta_1}(3) \approx f_{\zeta_1}(3)\\ V(8) &=& H_{\phi(2,2)}(3) = H_{\zeta_2}(3) \approx f_{\zeta_2}(3)\\ V(9) &=& H_{\phi(1,0,0)}(3) = H_{\Gamma_0}(3) \approx f_{\Gamma_0}(3)\\ V(10) &=& H_{\phi(1,0,1)}(3) = H_{\Gamma_1}(3) \approx f_{\Gamma_1}(3)\\ V(27) &=& H_{\phi(1,0,0,0)}(3)\\ V(81) &=& H_{\phi(1,0,0,0,0)}(3)\\ \end{eqnarray*}

Interpolation between \(V(a) = H_{\alpha}(3)\) and \(V(a+h) = H_{\beta}(3)\) is


 * When \(\beta = lim(\beta_i)\) and \(\alpha < \beta_1\), \(V(a+ih/3) = H_{\beta_i}(3)\) for \(i=1,2,3\).
 * When \(\beta = lim(\beta_i)\) and \(\beta_1 \le \alpha < \beta_2\), \(V(a+(i-1)h/2) = H_{\beta_i}(3)\) for \(i=2,3\).
 * When \(\beta = lim(\beta_i)\) and \(\beta_2 \le \alpha\), \(V(a+h) = H_{\beta_3}(3)\).
 * When \(\beta\) is a successor, linear interpolation of \(V(a+nh) = V(a) + (n/h)[V(a+h)-V(a)]\)

For example, interpolation between \(V(5) = \epsilon_2\) and \(V(6) = \zeta_0 = lim(0, \epsilon_0, \epsilon_{\epsilon_0}, \epsilon_{\epsilon_{\epsilon_0}}, ...)\) is

\begin{eqnarray*} V(5.5) &=& H_{\epsilon_{\epsilon_0}}(3)\\ V(6) &=& H_{\epsilon_{\epsilon_{\epsilon_0}}}(3)\\ \end{eqnarray*}

and between these is

\begin{eqnarray*} V(5+4/6) &=& H_{\epsilon_{\epsilon_{\omega}}}(3)\\ V(5+5/6) &=& H_{\epsilon_{\epsilon_{\omega^{\omega}}}}(3)\\ V(6) &=& H_{\epsilon_{\epsilon_{\omega^{\omega^{\omega}}}}}(3)\\ \end{eqnarray*}

Between V(3) and V(4) is interpolated as


 * \(V(3+1/3) = H_{\epsilon_0+1}(3) = H_{\epsilon_0}(4) = f_{\epsilon_0}(3)\)
 * \(V(3+7/18) = H_{\epsilon_0 +\omega}(3) = H_{\epsilon_0 +3}(3) = f_{\epsilon_0}(5)\)
 * \(V(3+8/18) = H_{\epsilon_0 +\omega^{\omega}}(3) \approx f_{\epsilon_0}(3↑↑3)\)
 * \(V(3.5) = H_{\epsilon_0 2}(3) = H_{\epsilon_0 +\omega^{\omega^{\omega}}}(3) \approx f_{\epsilon_0}(A(1,0,0,0,3))\)
 * \(V(3+2/3) = H_{\epsilon_0 \omega}(3) = f_{\epsilon_0 + 1}(3)\)
 * \(V(3+5/6) = H_{\epsilon_0 ^2}(3)\)

BH function
BH function is defined with the Bachmann-Howard ordinal and FGH as follows, using fundamental sequence described in.

\[BH(n) = f_{\psi(\varepsilon_{\Omega+1})}(n)\]

Therefore calculation is;

\begin{eqnarray*} BH(0) &=& f_{\psi(\Omega)}(0) = f_{\psi(0)}(0) = f_{\omega}(0) = 0 \\ BH(1) &=& f_{\psi(\Omega^\Omega)}(1) = f_{\psi(\Omega^{\psi(0)})}(1) = f_{\psi(\Omega^{\omega^{\omega}})}(1) \\ &=& f_{\psi(\Omega)}(1) = f_{\psi(\psi(0))}(1) = f_{\psi(1)}(1) = f_{\psi(0)^{\psi(0)}}(1) \\ &=& f_{1}(1) = 2 \\ BH(2) &=& f_{\psi(\Omega^{\Omega^{\Omega}})}(2) = f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\psi(0)})}}})}(2) \\ &=& f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\omega^{\omega^{\omega}}})}}})}(2) \\ &=& f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\omega^{\omega+2}})}}})}(2) \\ &=& f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\omega^{\omega+1}\omega})}}})}(2) \\ &=& f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\omega^{\omega+1}2})}}})}(2) \\ &=& f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\omega^{\omega+1}}+{\omega^{\omega}}+\omega+2)}}})}(2) \\ &=& ... \end{eqnarray*}

Interpolation method is similar to V function. For example, as

\[BH(2) = f_{\psi(\Omega^{\Omega^{\psi{(\Omega^{\psi(0)})}}})}(2)\]

\(BH(1.5)\) is

\[BH(1.5) = f_{\psi(\Omega^{\Omega^{\psi(0)}})}(2)\]

O function
O function is defined with Ψ₀(Ωω) and FGH as follows.

\[O(n) = f_{\vartheta(\Omega_\omega)}(n)\]

I am not sure about the canonical fundamental sequence up to this ordinal. Definition is complete when fundamental sequence is given.

OF function
OF function is defined with Ψ(ψᵢ(0)) and FGH, using fundamental sequence of \(\psi(0), \psi(\omega), \psi(\Omega_\omega), \psi(\Omega_{\Omega_\omega}), ...  \).

\[OF(n) = f_{\psi(\psi_I(0))}(n)\]

L function
L function is defined with Ln (L2, L3, L4, ...) space of BEAF. It is linearly interpolated.

\[L(n) = \lbrace Ln,10\rbrace_{10,10}\]

D function
2 variable D function is defined with 1 variable D function of loader.c as follows.

\[D(m,n) = D^m(n)\]

It is linearly interpolated.

CKF function
CKF function is a subfunction for defining FC and RC functions. It returns ordinal.

\begin{eqnarray*} CKF(0) &=& 1 \\ CKF(1) &=& 2 \\ CKF(2) &=& 3 \\ CKF(3) &=& \omega \\ CKF(4) &=& \omega+1 \\ CKF(5) &=& \omega 2 \\ CKF(6) &=& \omega^2 \\ CKF(7) &=& \omega^\omega \\ CKF(8) &=& \omega^{\omega^\omega} \\ CKF(9) &=& \epsilon_0 = \phi(1,0) = \psi(0) \\ CKF(10) &=& \epsilon_1 = \phi(1,1) = \psi(1) \\ CKF(11) &=& \zeta_0 = \phi(2,0) = \psi(\Omega) \\ CKF(12) &=& \Gamma_0 = \phi(1,0,0) = \psi(\Omega^{\Omega}) = \vartheta(\Omega) \\ CKF(13) &=& \phi(1,0,0,0) = \psi(\Omega^{\Omega^2}) = \vartheta(\Omega^2) \\ CKF(14) &=& \psi(\Omega^{\Omega^\omega}) = \vartheta(\Omega^\omega) \\ CKF(15) &=& \psi(\Omega^{\Omega^\Omega}) = \vartheta(\Omega^\Omega) \\ CKF(16) &=& \psi(\epsilon_{\Omega+1}) = \vartheta(\varepsilon_{\Omega+1}) \\ CKF(17) &=& \psi_0(\Omega_{\omega}) \\ CKF(18) &=& \psi_0(\varepsilon_{\Omega_\omega + 1}) \\ CKF(19) &=& \psi(\psi_I(0)) \\ \end{eqnarray*}

For \(n \ge 20\), \[CKF(n) = \omega_{CKF(n-20)}^\text{CK}\]

FC function
FC function is defined with the previous CKF function and FGH as

\[FC(x) = f_{CKF(x)}(1000)\]

It is linearly interpolated.

RC function
RC function is defined with CKF function and Rayo hierarchy of Fish number 7 as

\[RC(x) = R_{CKF(x)}(10^{10})\]