User:Wythagoras/Dollar function/Linear Array Notation

Linear Array Notation is the third part of Dollar Function.

Formal definition
\(\bullet\) can be anything

\(\circ\) is a group of brackets. 1. If there is nothing after the $, the array is solved. The value of the array is the number before the $.

2. \(a\$b\bullet=(a+b)\$\bullet\)

3. \(a\$\circ[0]\bullet\circ=a\$\circ a\bullet\circ\)

4. \(a\$\circ[\bullet+1]_c\bullet\circ=a\$\circ[\bullet]_c[\bullet]_c...[\bullet]_c[\bullet]_c\bullet\circ\) with a \(\bullet\)'s

5. If the bracket contains a zero and the bracket has other content, you can remove the zero.

6. If the active bracket has level k and a zero in it, search for the least nested bracket with level (k-1) with the same array in it, nest that bracket a times in the place of the level k bracket and put [0] on the place where the active bracket was.

7. \( b\bullet,c = [[0,c-1]_{[b-1\bullet,c][b-1\bullet,c]}]\)

8. \( \diamond,b\bullet,c,\bullet = [[\diamond,[\diamond,b\bullet,c-1,\bullet]_{[\diamond,b-1\bullet,c,\bullet]},c-1,\bullet]\)

9. \([0,c,\bullet] = [0]\)

S1. The outermost bracket is always level 1

S2. If there is no bracket with level (k-1), add it directly after the level k bracket.

S3. Zeroes at the and of the array must be removed

The active bracket is the bracket with the lowest level. The brackets can be ordered by level in FGH, and then removing outermost bracket. Or you look to: smallest bracket type, least number of nestings, smallest number inside the bracket.

Examples
\([0],1 = a,1 = [[0]_{[a-1,1][a-1,1]}]\)

\([0]1,1 = [[0]_{[[0],1][[0],1]}]\) and is therefore much more, it takes a long expanding before reaching the second \(\omega\)

Analysis
\([0],1\) has level \(\psi(\psi_I(0))\)

\([0][0],1\) has level \(\psi(\psi_I(1))\)

\([1],1\) has level \(\psi(\psi_I(\omega))\)

\([[0]_2],1\) has level \(\psi(\psi_I(\varepsilon_0))\)

\([[[0]_2]_2],1\) has level \(\psi(\psi_I(\zeta_0))\)

\([[[0]_3]_2],1\) has level \(\psi(\psi_I(\varphi(\omega,0)))\)

\([[[0],1,1]]\) has level \(\psi(\psi_I(\psi(\psi_I(0))))\)

\([[[[[0],1,1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(\psi(\psi_I(0))))))\)

\([0]_2,1\) has level \(\psi(\psi_I(\Omega))\)

\([0]_{[0]},1\) has level \(\psi(\psi_I(\Omega_\omega))\)

\([[0],1],1\) has level \(\psi(\psi_I(\psi_I(0)))\)

\([0,1]_2,1\) has level \(\psi(\psi_I(I))\)

\([1,1]_2,1\) has level \(\psi(\psi_I(I\omega))\)

\([[0]_2,1]_2,1\) has level \(\psi(\psi_I(I\Omega))\)

\([[0,1],1]_2,1\) has level \(\psi(\psi_I(I\psi_I(0)))\)

\([[0,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^2))\)

\([[1,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^\omega))\)

\([[[0,1]_2,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^I))\)

\([0,1]_3,1\) has level \(\psi(\psi_I(\varepsilon_{I+1}))\)

\([[[0],1]_3,1]_3,1\) has level \(\psi(\psi_I(\varphi(\omega,I+1)))\)

\([[[0,1]_3,1]_3,1]_3,1\) has level \(\psi(\psi_I(\Omega_{I+1}))\)

\([0],2\) has level \(\psi(\psi_{I_2}(0))\)

\([0],[0]\) has level \(\psi(\psi_{I_\omega}(0))\)

\([0],[0]_2\) has level \(\psi(\psi_{I_\Omega}(0))\)

\([0],[0,1]\) has level \(\psi(\psi_{I_{\psi_I(0)}}(0))\)

\([0],[0,2]\) has level \(\psi(\psi_{I_{\psi_{I_2}(0)}}(0))\)

\([0],[0,1]_2\) has level \(\psi(\psi_{I_{I}}(0))\)

\(0,[0],1\) has level \(\psi(\psi_{\chi(1)}(0))\)

\(0,[0][0],1\) has level \(\psi(\psi_{\chi(1)}(1))\)

\(0,[0]_2,1\) has level \(\psi(\psi_{\chi(1)}(\Omega))\)

\(0,[0,0,1]_2,1\) has level \(\psi(\psi_{\chi(1)}(\chi(1)))\)

\( 0,0,2 \) has level \(\psi(\psi_{\chi(2)}(0))\)

\( 0,0,3 \) has level \(\psi(\psi_{\chi(3)}(0))\)

\( 0,0,[0] \) has level \(\psi(\psi_{\chi(\omega)}(0))\)

\( 0,0,[0]_2 \) has level \(\psi(\psi_{\chi(\Omega)}(0))\)

\( 0,0,[0,1] \) has level \(\psi(\psi_{\chi(\psi_I(0))}(0))\)

\( 0,0,[0,0,1]_2 \) has level \(\psi(\psi_{\chi(M)}(0))\)

\( 0,0,[1,0,1]_2 \) has level \(\psi(\psi_{\chi(M\omega)}(0))\)

\( 0,0,[0,0,1]_3 \) has level \(\psi(\psi_{\chi(M_2)}(0))\)

\(0,0,[0],1\) has level \(\psi(\Psi_{\Xi(3,0)}(0))\)

\(0,0,0,[0],1\) has level \(\psi(\Psi_{\Xi(4,0)}(0))\)

The limit of linear arrays is \(\psi(\Psi_{\Xi(\omega,0)}(0))\)