FANDOM


In this blog post I'll try to make a formal ruleset for BEAF.

Linear Array Notation

1. \(\{a,1,\#\} = a\)

2. \(\{a,b,1,1 \ldots 1,1\} = a^b\)

3. \(\{a,b,1,1 \ldots 1,1,c,\#\} = \{a,a,a,a \ldots a,\{a,b-1,1,1 \ldots 1,1,c,\#\},c-1,\#\}\)

Array of

4. \(a \text{&} b = \{\underbrace{b,b,b,b,\ldots,b,b,b,b}_a\}\)

5. \(\#+1 \text{&} a = \{a,a(\#)a\}\) where # contains only X structures

Explaining the notation used in rule 5:

6. \(\{a,a(\#)2\} = \# \text{&} a\)

7. \(\{a,a(\#)c\} = \{\# \text{&} a(\#)c-1\}\)

8. \(\#+X+a \text{&} b = \{\#+a \text{&} b(\#+X)\#+a \text{&} b\}\)

9. If there are no numbers left solve a X to b

Examples

\(X+1 \text{&} 3 =\)

\(\{3,3(X)3\} =\)

\(\{X \text{&} 3(X)2\} =\)

\(\{3,3,3(X)2\} =\)

\(\{3,3,3(1)2\}\) in normal BEAF


\(X2 \text{&} 3 =\)

\(X+3 \text{&} 3 =\)

\(\{3 \text{&} 3(X)3 \text{&} 3\} =\)

\(\{3,3,3(X)3,3,3\} = \)

\(\{3,3(1)(1)2\}\) in normal BEAF

Separators

\((X) = (1)\)

\((X2) = (1)(1)\)

\((X^2) = (2)\)

\((X^3) = (3)\)

\((X^X) = (0,1)\)

\((X^{X+1}) = (1,1)\)

\((X^{X2}) = (0,2)\)

\((X^{X^2}) = (0,0,1)\)

FGH

Using my definition of &, I can make the following comparisons

FGH BEAF structure
\(\vartheta(\Omega^\Omega)\) \(\{X,X/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega)\omega\) \(\{X,X/2\}+1\text{&}n\)
\(\vartheta(\Omega^\Omega)\omega^2\) \(\{X,X/2\}+2\text{&}n\)
\(\vartheta(\Omega^\Omega)\omega^\omega\) \(\{X,X/2\}+X\text{&}n\)
\(\vartheta(\Omega^\Omega)^2\) \(\{X,X/2\}2\text{&}n\)
\(\vartheta(\Omega^\Omega)^\omega\) \(\{X,X/2\}X\text{&}n\)
\(\vartheta(\Omega^\Omega)^{\varepsilon_0}\) \(\{X,X/2\}X\uparrow\uparrow X\text{&}n\)
\(\vartheta(\Omega^\Omega)^{\vartheta(\Omega^\Omega)}\) \(\{X,X/2\}^2\text{&}n\)
\(\varepsilon_{\vartheta(\Omega^\Omega)+1}\) \(\{X,X/2\}\uparrow\uparrow X\text{&}n\)
\(\zeta_{\vartheta(\Omega^\Omega)+1}\) \(\{X,X/2\}\uparrow\uparrow\uparrow X\text{&}n\)
\(\Gamma_{\vartheta(\Omega^\Omega)+1}\) \(\{\{X,X/2\},X,1,2\}\text{&}n\)
\(\vartheta(\Omega^\Omega,1)\) \(\{X,X2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega,\vartheta(\Omega^\Omega))\) \(\{X,3,2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega,\vartheta(\Omega^\Omega,\vartheta(\Omega^\Omega)))\) \(\{X,4,2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+1)\) \(\{X,X,2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+2)\) \(\{X,X,3/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+\omega)\) \(\{X,X,X/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+\Omega)\) \(\{X,X,1,2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+\Omega^2)\) \(\{X,X,1,1,2/2\}\text{&}n\)
\(\vartheta(\Omega^\Omega+\Omega^\omega)\) \(\{X,X(1)2/2\}\text{&}n\)

\(\vartheta(\Omega^\Omega+\Omega^{\vartheta(\Omega^\Omega)})\)

\(\{X,X/3\}\text{&}n\)

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