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Replaced by Version 3

This version has been replaced by Beta Function Version 3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 2

There is an error in the notational logic used in Version 1 code for the Beta Function. This version will not fully correct the error but will reduce the size of the error.

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 2 makes it possible to access ordinals in the following range:

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

\(\beta(7.84517,3) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}}(3) = f_{\omega\uparrow\uparrow 3}(3) = f_{\varphi(1,0)}(3)\)

\(\beta(8.69626,3) = f_{(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega}}(3)\)

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

In this example the Beta Function generates undesired values between \(7.84517\) to \(8.999...\) because ordinals of the form:

\((\gamma\uparrow\uparrow T)^{g_E}.g_C + g_x\) are allowed to access values with

\(g_E < \gamma^{\gamma\uparrow\uparrow 1 - \gamma\uparrow\uparrow 0}\) or

\(g_E < \omega^{\omega - 1} = \omega^2\)

Note that this error will not occur for base \(v < 3\) because the ordinal \(g_E\) cannot exceed the desired range of:

\(g_E < \gamma^{(\gamma\uparrow\uparrow (T-1)) - (\gamma\uparrow\uparrow (T-2))}\) by definition.

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

WORK IN PROGRESS


Granularity Examples \(\beta(10.079,4)\) to \(\beta(16,4)\)

When we use base \(v = 4\) we generate more undesired values as in this example:

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

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

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

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

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

\(\beta(15.3955814,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2}}}(4)\)

\(\beta(15.690684873,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}}}(4) = f_{(\omega\uparrow\uparrow 3)^{\omega^{\omega.(\omega^2.3 + \omega.3 + 3)}}}(4) = f_{\varphi(1,0)}(4)\)

\(\beta(15.69488141,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3}}}(4)\)

\(\beta(15.89764036,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3}}}(4)\)

\(\beta(15.999643447,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + \omega.3 + 3}}}(4)\)

\(\beta(15.9998217224,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + \omega.3 + 3}.(\omega)}}(4)\)

\(\beta(15.99994057392,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + \omega.3 + 3}.(\omega^3)}}(4)\)

\(\beta(15.9999950478176,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2)}}(4)\)

\(\beta(15.9999999312198,4) = f_{(\omega\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{\omega^3.3 + \omega^2.3 + \omega.3 + 3}.(\omega^3.3 + \omega^2.3 + \omega.3 + 3)}}(4)\)

\(\beta(16,4) = f_{\varphi(1,0)}(4)\)

In this example the Beta Function generates undesired values between \(15.690684873\) to \(15.999...\) because the maximum value for \(g_E\) is:

\(g_E < \omega^{(\omega\uparrow\uparrow (3-1)) - (\omega\uparrow\uparrow (3-2))} < \omega^{(\omega\uparrow\uparrow 2) - \omega}\)

\(= \omega^{\omega^3.3 + \omega^2.3 + \omega.4 - \omega} = \omega^{\omega^3.3 + \omega^2.3 + \omega.3}\)

\(= (\omega\uparrow\uparrow 2)^{\omega^2.3 + \omega.3 + 3}\)

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

WORK IN PROGRESS


Valid Sequence Counts

WORK IN PROGRESS


Test Bed for Version 2

Below is the test bed and various results using version 2.

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

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

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

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

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

\(\beta(5.9615,3) = f_{(\omega\uparrow\uparrow 2).2}(3)\)

Next attempt - Base v = 3 - 29 Apr 2016

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

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

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

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

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

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

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

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

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

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

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

\(\beta(6.93704825,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.(\omega.2)}(3)\)

\(\beta(6.956925,3) = f_{(\omega\uparrow\uparrow 2)^{\omega}.(\omega^2)}(3)\)

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

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

\(\beta(7.9127944,3) = f_{(\omega\uparrow\uparrow 2)^{\omega^2}.(\omega^2)}(3)\)

WORK IN PROGRESS

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