User blog:B1mb0w/Beta Function Code Version 3

Beta Function - Sequence Generating Code
The Beta Function has been defined using program code shown below.

A separate blog will be written to explain how Sequence Generator Code is compiled and executed using a normal programming language ... Work in Progress.

Sequence Generating Code Version 3
The error identified in Version 2 code for the Beta Function can be corrected using this logic.

Let \(\lambda\uparrow\uparrow (T+1) = (\lambda\uparrow\uparrow T)^{\delta(T)}\)

then

\(\delta(1) = \lambda\) by definition

\(\delta(2) = \lambda^{\lambda - 1} = \lambda^{\delta(1) - 1}\)

\(\delta(3) = (\lambda\uparrow\uparrow 2)^{\delta(2) - 1}\)

and

\(\delta(T+1) = (\lambda\uparrow\uparrow T)^{\delta(T) - 1}\)

The proof for this is:

\(\lambda\uparrow\uparrow (T+1) = (\lambda\uparrow\uparrow T)^{\delta(T)}\)

\(= (\lambda\uparrow\uparrow T)^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1}}\)

\(= (\lambda^{\lambda\uparrow\uparrow (T-1)})^{(\lambda\uparrow\uparrow (T-1))^{\delta(T-1) - 1}}\)

Sequence Generating Ruleset (Version 1)

The Beta Function is equivalent to a sequence of the form:

\(\beta(r,v) == (v,h_0)\)

using this Sequence Generating RuleSet:
 * \(h_x = (d<2,d(0:x<v-u(x),1:(f_0<1,P_h = 1)))\)
 * \(f_x = (g_x,g_x((0,0,0):h_u,(h_U,(f_{x+1}<g_x,P_h=d-h_U)))\)
 * \(g_x = (q<v+1,g_{[q]},n_0<q,t_x)\)
 * \(n_x = (Q,t_A,g_{[Q]},n_{x+1})\)
 * \(t_x = (h_T,g_E,g_C,g_x)\)

The ruleset is correct. But there are constraints imposed on the sequences to guarantee only finite sequences can be generated and converted in notational literal strings of Veblen ordinals and FGH functions:
 * \(h_x = (d<2,d(0:x<v-u(x),1:(f_0<1,P_h = 1)))\) - TO BE EXPLAINED FURTHER - in my blog on Sequence Generator Code ... Work in Progress

The constraints imposed on the \(t_x\) sequence function in Version 1 code are:
 * \(t_x = (h_T<P_h,(g_E<q-n_0,0,0,1),(g_C<q-h_T),(g_x<q-g_E)\) - TO BE EXPLAINED FURTHER - in my blog on Sequence Generator Code ... Work in Progress

Without explaining the syntax of this constraint the relevant logic is \(t_x\) is meant to access all ordinal values around an arbitrary limit ordinal \(\gamma\), usually a Veblen function, such that:

\(\gamma ... (\gamma\uparrow\uparrow T)^{g_E}.g_C + g_x ... \) until the next limit ordinal.

The constraint in Version 1 code intends to limit:
 * \(g_E < \gamma\) if \(T = 1\) and
 * \(g_E < \gamma^{\gamma}\) if \(T > 1\)
 * \(g_C < \gamma\uparrow\uparrow T\)

The \(g_E\) constraints are excessive and the correct limits can be calculated as follows:

Let \(\gamma\uparrow\uparrow (T+1) = (\gamma\uparrow\uparrow T)^x\)

\(= \gamma^{\gamma\uparrow\uparrow T} = \gamma^{(\gamma\uparrow\uparrow (T-1)).x}\)

Then \(\gamma\uparrow\uparrow T = (\gamma\uparrow\uparrow (T-1)).x\)

And \(x = (\gamma\uparrow\uparrow T) / (\gamma\uparrow\uparrow (T-1)) = \gamma^{(\gamma\uparrow\uparrow (T-1)) - (\gamma\uparrow\uparrow (T-2))}\)

Therefore a precise constraint exists for the maximum allowed value for \(g_E\), unfortunately I am unable to make the sequence generating code handle constraints involving the difference between two numbers. Instead the Version 2 code has been changed to (over-)compensate for the error. The allowable range for \(g_E\) will be any value up to \(\gamma^{(\gamma\uparrow\uparrow (T-1)) - 0} = \gamma\uparrow\uparrow T\) which will generate a number of undesired values but will be a significant improvement on the size of the error in the Version 1 code, because it will generate a much larger number of desired values.

The constraint in Version 2 code will be changed to:
 * \(g_E < \gamma\uparrow\uparrow T\) for all \(T\)

The error will remain for now and I will try to correct it in a future version of the code.

WORK IN PROGRESS

Granularity Examples \(\beta(6.838,3)\) to \(\beta(9,3)\)
Version 3 makes it possible to access ordinals in the following range:

WORK IN PROGRESS

Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)
When we use base \(v = 4\) we generate more undesired values as in this example:

WORK IN PROGRESS

Valid Sequence Counts
WORK IN PROGRESS

Test Bed for Version 3
Below is the test bed and various results using version 3.

\(\beta(3.141,3) = f_{\omega + 1}(3)\)

\(\beta(3.4417,3) = f_{\omega.2}(3)\)

\(\beta(3.9485,3) = f_{\omega^2}(3)\)

\(\beta(4.53,3) = f_{\omega^2.2}(3)\)

\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)

\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)

\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)

\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)

\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)

\(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\)

\(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\)

\(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\)

\(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\)

\(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\)

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)

\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)

\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)

\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)

\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)

\(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\)

\(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\)

\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\)

\(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\)

\(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega}}}(4)\)

\(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2}}}(4)\)

\(\beta(15.69488145,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3}}}(4)\)

\(\beta(15.89764036,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.2}}}(4)\)

\(\beta(15.94873806,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3}}}(4)\)

\(\beta(15.9871691,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}}}(4)\)

\(\beta(15.995721886,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3}}(4)\)

\(\beta(15.9973264,4) = f_{1}^{6}(f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}}(4))\)

\(\beta(15.99759342,4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega^2.3 + \omega.3 + 3}.3 + 3}.(\omega)}(4)\)

\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)

Next attempt - 2 May 2016

\(\beta(3.141,3) = f_{\omega + 1}(3)\)

\(\beta(3.4417,3) = f_{\omega.2}(3)\)

\(\beta(3.9485,3) = f_{\omega^2}(3)\)

\(\beta(4.53,3) = f_{\omega^2.2}(3)\)

\(\beta(5.1963,3) = f_{(\omega\uparrow\uparrow 2)}(3)\)

\(\beta(5.3777,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)

\(\beta(5.5655,3) = f_{(\omega\uparrow\uparrow 2).(\omega)}(3)\)

\(\beta(5.9612,3) = f_{(\omega\uparrow\uparrow 2)^2}(3)\)

\(\beta(6.1694,3) = f_{(\omega\uparrow\uparrow 2)^2.2}(3)\)

\(\beta(6.83855,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(3)\)

\(\beta(6.917229885,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.2}(3)\)

\(\beta(7.324573,3) = f_{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega)}(3)\)

\(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(3)\)

\(\beta(8.5974,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}}(3)\)

\(\beta(8.79635,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega)}(3)\)

\(\beta(8.94865,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2)}(3)\)

\(\beta(8.99574,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2)}(3).2\)

\(\beta(8.99892723,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)}(3)\)

\(\beta(8.999463585,3) = f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega}}(3)\)

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(6.34962,4) = f_{(\omega\uparrow\uparrow 2)}(4)\)

\(\beta(7.407,4) = f_{(\omega\uparrow\uparrow 2)^3}(4)\)

\(\beta(8,4) = f_{(\omega\uparrow\uparrow 2)^{\omega}}(4)\)

\(\beta(8.314075,4) = f_{(\omega\uparrow\uparrow 2)^{\omega.2}}(4)\)

\(\beta(8.979697,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(4)\)

\(\beta(9.698609,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3}}(4)\)

\(\beta(9.887156,4) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega}}(4)\)

\(\beta(10.06323,4) = f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}(4))\)

\(\beta(10.07937,4) = f_{(\omega\uparrow\uparrow 3)}(4)\)

\(\beta(11.75788,4) = f_{(\omega\uparrow\uparrow 3)^3}(4)\)

\(\beta(12.699209,4) = f_{(\omega\uparrow\uparrow 3)^{\omega}}(4)\)

\(\beta(14.254379491,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)}}(4)\)

\(\beta(15.101989005,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega}}}(4)\)

\(\beta(15.69488142,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.2}}}(4)\)

\(\beta(15.99358326,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}}(4)\)

\(\beta(15.9967913075,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega)}}(4)\)

\(\beta(15.9996434468,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3)}}(4)\)

\(\beta(15.9999987619545,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3)}}(4)\)

\(\beta(16.0000001,4) = f_{\varphi(1,0)}(4)\)

WORK IN PROGRESS