# Bird's array notation

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Bird's array notation is a parallel notation to BEAF.

# Linear arrays

• Rule 1. With one or two entries, we have $$\{a\} = a$$, $$\{a,b\} = a^b$$.
• Rule 2. If the last entry is 1, it can be removed: $$\{\#,1\} = \{\#\}$$.
• Rule 3. If the second entry is 1, the value is just the first entry: $$\{a,1 \#\} = a$$.
• Rule 4. If the third entry is 1:
$$\{a,b,\underbrace{1,1,\cdots,1,1}_n,c \#\} = \{\underbrace{a,a,a,a,\cdots,a}_{n+1},\{a,b-1,\underbrace{1,1,\cdots,1,1}_n,c \#\},c-1 \#\}$$
• Rule 5. Otherwise:
$$\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}$$

Limit: $$\{n,n[2]2\}$$ has growth rate $$\omega^\omega$$

## Example

\begin{eqnarray*} \{3,3,1,2\} &=& \{3,3,\{3,2,1,2\},1\} \\ &=& \{3,3,\{3,3,\{3,1,1,2\},1\}\} \\ &=& \{3,3,\{3,3,3\}\} \\ &=& \{3,3,\{3,\{3,2,3\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,1,3\},2\},2\}\} \\ &=& \{3,3,\{3,\{3,3,2\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,2,2\},1\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,\{3,1,2\},1\}\},2\}\} \\ &=& \{3,3,\{3,\{3,\{3,3\}\},2\}\} \\ &=& \{3,3,\{3,7625597484987,2\}\} \\ &=& 3\uparrow^{3\uparrow\uparrow7625597484987}3 \end{eqnarray*}

# Multidimentional arrays

• Rule M1. If there are only two entries, $$\{a, b\} = a^b$$.
• Rule M2. If $$m < n$$, we have $$\{\# [m] 1 [n] \#^*\} = \{\# [n] \#^*\}$$. (This also removes ones from the end of an array.)
• Rule M3. If the second entry is 1, we have $$\{a,1 \#\} = a$$.
• Rule M4. If there is a non-zero entry immediately after batch of unfilled separators:
$$\{a,b [m_1] 1 [m_2] \cdots 1 [m_x] c \#\} = \{a \langle m_1-1 \rangle b [m_1] a \langle m_2-1 \rangle b [m_2] \cdots a \langle m_x-1 \rangle b [m_x] (c-1) \#\}$$
• Rule M5. If there is a non-zero entry after batch of unfilled separators and a 1.
$$\{a,b [m_1] 1 [m_2] \cdots 1 [m_x] 1,c \#\} = \{a \langle m_1-1 \rangle b [m_1] a \langle m_2-1 \rangle b [m_2] \cdots a \langle m_x-1 \rangle b [m_x] \{a,b-1 [m_1] 1 [m_2] \cdots 1 [m_x] 1,c \#\},c-1 \#\}$$
• Rule M6. Rules M1-M5 don't apply.
$$\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}$$
• Rule A1. If $$c = 0$$, we have $$\textrm a \langle 0 \rangle b = a \textrm'$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle c \rangle 1 = a \textrm'$$.
• Rule A3. Otherwise, $$\textrm a \langle c \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \rangle b [c] \cdots [c] a \langle c-1 \rangle b}_b \textrm'$$.

Limit: $$\{n,n[1,2]2\}$$ has growth rate $$\omega^{\omega^\omega}$$

## Example

\begin{eqnarray*} \{3,2[3]2\} &=& \{3 \langle 2 \rangle 2[3]1\} \\ &=& \{3 \langle 1 \rangle 2[2]3 \langle 1 \rangle 2\} \\ &=& \{3,3[2]3,3\} \\ &=& \{3,3,3[2]2,3\} \\ &=& \{3,\{3,2,3[2]2,3\},2[2]2,3\} \\ &=& \{3,\{3,3,2[2]2,3\},2[2]2,3\} \\ &=& \{3,\{3,\{3,2,2[2]2,3\}[2]2,3\},2[2]2,3\} \\ &=& \{3,\{3,\{3,3[2]2,3\}[2]2,3\},2[2]2,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3[2]1,3\}[2]1,3\},2[2]1,3\}[2]2,3\},2[2]2,3\} \end{eqnarray*}

# Hyperdimentional arrays

• Rule M1. If there are only two entries, $$\{a, b\} = a^b$$.
• Rule M2. If $$m < n$$, we have $$\{\# [m] 1 [n] \#^*\} = \{\# [n] \#^*\}$$. (This also removes ones from the end of an array.)
• Rule M3. If the second entry is 1, we have $$\{a,1 \#\} = a$$.
• Rule M4. If there is a non-zero entry immediately after batch of unfilled separators:
$$\{a,b [m_1 \#_1] 1 [m_2 \#_2] \cdots 1 [m_x \#_x] c \#\} = \{a \langle m_1-1 \#_1 \rangle b [m_1 \#_1] a \langle m_2-1 \#_2 \rangle b [m_2 \#_2] \cdots a \langle m_x-1 \#_x \rangle b [m_x \#_x] (c-1) \#\}$$
• Rule M5. If there is a non-zero entry after batch of unfilled separators and a 1.
$$\{a,b [m_1 \#_1] 1 [m_2 \#_2] \cdots 1 [m_x \#_x] 1,c \#\}$$
$$= \{a \langle m_1-1 \#_1 \rangle b [m_1 \#_1] a \langle m_2-1 \#_2 \rangle b [m_2 \#_2] \cdots a \langle m_x-1 \#_x \rangle b [m_x \#_x] \{a,b-1 [m_1 \#_1] 1 [m_2 \#_2] \cdots 1 [m_x \#_x] 1,c \#\},c-1 \#\}$$
• Rule M6. Rules M1-M5 don't apply.
$$\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}$$
• Rule A1. If $$c = 0$$, we have $$\textrm a \langle 0 \rangle b = a \textrm'$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle A \rangle 1 = a \textrm'$$.
• Rule A3. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it:
$$\textrm a \langle 0,\underbrace{1,1,\cdots,1,1}_n,c \# \rangle b \textrm' = \textrm a \langle \underbrace{b,b,b,\cdots,b,b}_{n+1},c-1 \# \rangle b \textrm'$$
• Rule A4. Otherwise, $$\textrm a \langle c \# \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \# \rangle b [c \#] \cdots [c \#] a \langle c-1 \# \rangle b}_b \textrm'$$.

Limit: $$\{n,n[1[2]2]2\}$$ has growth rate $$^4 \omega$$

## Example

\begin{eqnarray*} \{3,2[1,1,2]2\} &=& \{3 \langle 0,1,2 \rangle 2\} \\ &=& \{3 \langle 2,2 \rangle 2\} \\ &=& \{3 \langle 1,2 \rangle 2[2,2]3 \langle 1,2 \rangle 2\} \\ &=& \{3,3[2]3,3[1,2]3,3[2]3,3[2,2]3,3[2]3,3[1,2]3,3[2]3,3\} \end{eqnarray*}

# Nested arrays

Main rules will remain the same forever.

• Rule A1. If $$c = 0$$, we have $$\textrm a \langle 0 \rangle b = a \textrm'$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle A \rangle 1 = a \textrm'$$.
• Rule A3. If $$[A] < [B]$$, $$\textrm a \langle \# [A] 1 [B] \#^* \rangle b \textrm' = \textrm a \langle \# [B] \#^* \rangle b \textrm'$$.
• Rule A4. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it:
$$\textrm a \langle 0 [x_1 \#_1] 1 [x_2 \#_2] \cdots 1 [x_n \#_n] c \# \rangle b \textrm' = \textrm a \langle b \langle x_1-1 \#_1 \rangle b [x_1 \#_1] b \langle x_2-1 \#_2 \rangle b [x_2 \#_2] \cdots b \langle x_n-1 \#_n \rangle b [x_n \#_n] c-1 \# \rangle b \textrm'$$
• Rule A5. Otherwise, $$\textrm a \langle c \# \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \# \rangle b [c \#] \cdots [c \#] a \langle c-1 \# \rangle b}_b \textrm'$$.

Limit: $$\{n,n[1\backslash2]2\}$$ has growth rate $$\varepsilon_0$$

## Example

\begin{eqnarray*} \{3,2[1[2]2]2\} &=& \{3 \langle 0[2]2 \rangle 2\} \\ &=& \{3 \langle 2 \langle 1 \rangle 2 \rangle 2\} \\ &=& \{3 \langle 2,2 \rangle 2\} \\ &=& \{3,3[2]3,3[1,2]3,3[2]3,3[2,2]3,3[2]3,3[1,2]3,3[2]3,3\} \end{eqnarray*}

# Hyper-Nested arrays

• Rule A1. If $$c = 0$$, we have $$\textrm a \langle 0 \rangle b \textrm' = \textrm a \textrm'$$ and $$\textrm a \langle 0 \rangle \backslash b \textrm' = \textrm a \textrm'$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle A \rangle 1 \textrm' = \textrm a \textrm'$$ and $$\textrm a \langle A \rangle \backslash 1 \textrm' = \textrm a \textrm'$$.
• Rule A3. If $$[A] < [B]$$, $$\textrm a \langle \# [A] 1 [B] \#^* \rangle b \textrm' = \textrm a \langle \# [B] \#^* \rangle b \textrm'$$ and $$\textrm a \langle \# [A] 1 [B] \#^* \rangle \backslash b \textrm' = \textrm a \langle \# [B] \#^* \rangle \backslash b \textrm'$$.
• Rule A4. If the first entry in the angle brackets is zero, and there exists a non-zero entry after it:
$$\textrm a \langle 0 [x_1 \#_1]\backslash 1 [x_2 \#_2]\backslash \cdots 1 [x_m \#_m]\backslash 1 [y_1 \#^*_1] 1 [y_2 \#^*_2] \cdots [y_n \#^*_n] c \# \rangle b \textrm'$$
$$= \textrm a \langle b \langle x_1-1 \#_1 \rangle \backslash b [x_1 \#_1]\backslash b \langle x_2-1 \#_2 \rangle \backslash b [x_2 \#_2]\backslash \cdots b \langle x_n-1 \#_n \rangle \backslash b [x_n \#_n]\backslash b \langle y_1-1 \#^*_1 \rangle b [y_1 \#^*_1] b \langle y_2-1 \#^*_2 \rangle b [y_2 \#^*_2] \cdots b \langle y_n-1 \#^*_n \rangle b [y_n \#^*_n] c-1 \# \rangle b \textrm'$$.
$$\textrm a \langle 0 [x_1 \#_1]\backslash 1 [x_2 \#_2]\backslash \cdots 1 [x_m \#_m]\backslash 1 [y_1 \#^*_1] 1 [y_2 \#^*_2] \cdots [y_n \#^*_n] c \# \rangle \backslash b \textrm'$$
$$= \textrm a \langle b \langle x_1-1 \#_1 \rangle \backslash b [x_1 \#_1]\backslash b \langle x_2-1 \#_2 \rangle \backslash b [x_2 \#_2]\backslash \cdots b \langle x_n-1 \#_n \rangle \backslash b [x_n \#_n]\backslash b \langle y_1-1 \#^*_1 \rangle b [y_1 \#^*_1] b \langle y_2-1 \#^*_2 \rangle b [y_2 \#^*_2] \cdots b \langle y_n-1 \#^*_n \rangle b [y_n \#^*_n] c-1 \# \rangle \backslash b \textrm'$$.
• Rule A5. First non-zero entry is prior to a single backslash:
$$\textrm a \langle 0 [x_1 \#_1]\backslash 1 [x_2 \#_2]\backslash \cdots 1 [x_n \#_n]\backslash 1 \backslash c \# \rangle b\textrm'$$
$$= \textrma \langle b \langle A_1' \rangle\backslash b [A_1]\backslash b \langle A_2' \rangle\backslash b [A_2]\backslash \cdots b \langle A_n' \rangle\backslash b [A_n]\backslash R_b \backslash c-1 \# \rangle b\textrm'$$;
$$\textrm a \langle 0 [x_1 \#_1]\backslash 1 [x_2 \#_2]\backslash \cdots 1 [x_n \#_n]\backslash 1 \backslash c \# \rangle \backslash b\textrm'$$
$$= \textrma \langle b \langle A_1' \rangle\backslash b [A_1]\backslash b \langle A_2' \rangle\backslash b [A_2]\backslash \cdots b \langle A_n' \rangle\backslash b [A_n]\backslash R_b \backslash c-1 \# \rangle \backslash b\textrm'$$.
• Rule A6. Otherwise, $$\textrm a \langle c \# \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \# \rangle b [c \#] \cdots [c \#] a \langle c-1 \# \rangle b}_b \textrm'$$ and $$\textrm a \langle c \# \rangle \backslash b \textrm' = \textrm \underbrace{a \langle c-1 \# \rangle \backslash b [c \#]\backslash \cdots [c \#]\backslash a \langle c-1 \# \rangle \backslash b}_b \textrm'$$.

Limit: $$\{n,n[1/2]2\}$$ has growth rate $$\Gamma_0$$.

## Examples

\begin{eqnarray*} \{3 \langle 0 \backslash 2 \rangle 2\} &=& \{3 \langle R_2 \rangle 2\} \\ &=& \{3 \langle 2 \rangle 2\} \\ &=& \{3,3[2]3,3\} \\ &=& \{3,\{3,\{3,\{3,\{3,3,3[2]\{3,\{3,\{3,\{3,\{3,3,3[2]\{3,\{\underbrace{3,\cdots,3}_{3\uparrow\uparrow\uparrow3}\},2[2]2\}\}[2]1,2\},2[2]1,2\}[2]2,3\}[2]2,2\},2\}[2]1,3\},2[2]1,3\} \end{eqnarray*}

\begin{eqnarray*} \{3 \langle 0 [2]\backslash 2 \rangle 2\} &=& \{3 \langle 2 \backslash 2 \rangle 2\} \\ &=& \{3,3[2]3,3[1 \backslash 2]3,3[2]3,3[2 \backslash 2]3,3[2]3,3[1 \backslash 2]3,3[2]3,3\} \end{eqnarray*}

# Nested Hyper-Nested arrays

This part consists of 2 parts.

## Negations

• Rule A1. $$\textrm a \langle 0 \neg \# \rangle b \textrm' = \textrm a \textrm'$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle A \neg B \rangle 1 \textrm' = \textrm a \textrm'$$.
• Rule A3. If $$[A] < [B]$$, $$\textrm a \langle \# [A] 1 [B] \#^* \neg \% \rangle b \textrm' = \textrm a \langle \# [B] \#^* \neg \% \rangle b \textrm'$$.
• Rule A4. If the first entry in the angle brackets is zero:
$$\textrm a \langle 0 [A_{1,1}] 1 [A_{1,2}] \cdots [A_{1,p_1}] c_1 \#_1 \neg \#^* \rangle b \textrm' = \textrm a \langle S_1 \neg \#^* \rangle b \textrm'$$. Set i to 1.
• Rule A4a. $$[A_{i,p_i}] = [1 \neg 1 [A_{i+1,1}] 1 [A_{i+1,2}] \cdots [A_{i+1,p_{i+1}}] c_i \#_i]$$:
$$S_i = \textrm b \langle A_{i,1}' \rangle b [A_{i,1}] b \langle A_{i,2}' \rangle b [A_{i,2}] \cdots [A_{i,p_i-1}] b \langle b \neg S_{i+1} \rangle b [A_{i,p_i}] c_i \#_i \textrm'$$. Increase i by 1 and follow Rules A4a~d again.
• Rule A4b. $$[A_{i,p_i}] = \backslash$$:
$$S_i = \textrm R_b \textrm'$$,
$$R_n = \textrm b \langle A_{i,1}' \rangle b [A_{i,1}] b \langle A_{i,2}' \rangle b [A_{i,2}] \cdots [A_{i,p_i-1}] b \langle R_{n-1} \rangle b [A_{i,p_i}] c_i-1 \#_i \textrm'$$,
$$R_1 = \textrm 0 \textrm'$$.
• Rule A4c. Rule A4b don't apply, $$[A_{i,p_i}] = [1 \neg k \#_{i+1}] (k>1)$$:
$$S_i = \textrm b \langle A_{i,1}' \rangle b [A_{i,1}] b \langle A_{i,2}' \rangle b [A_{i,2}] \cdots [A_{i,p_i-1}] b \langle R_b \rangle b [A_{i,p_i}] c_i-1 \#_i \textrm'$$,
$$R_n = \textrm b \langle A_{i,1}' \rangle b [A_{i,1}] b \langle A_{i,2}' \rangle b [A_{i,2}] \cdots [A_{i,p_i-1}] b \langle R_{n-1} \rangle b [A_{i,p_i}] c_i-1 \#_i \neg k-1 \#_{i+1} \textrm'$$,
$$R_1 = \textrm 0 \textrm'$$.
• Rule A4d. Otherwise:
$$S_i = \textrm b \langle A_{i,1}' \rangle b [A_{i,1}] b \langle A_{i,2}' \rangle b [A_{i,2}] \cdots [A_{i,p_i-1}] b \langle A_{i,p_i}' \rangle b [A_{i,p_i}] c_i-1 \#_i \textrm'$$.
• Rule A5. Otherwise, $$\textrm a \langle c \# \neg \#^* \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \# \neg \#^* \rangle b [c \#] \cdots [c \#] a \langle c-1 \# \neg \#^* \rangle b}_b \textrm'$$.

Note: $$\#, \#^*, and \%$$ does not contain $$\neg$$s.

Limit: $$\{n,n[1[1\neg1\neg2]2]2\}$$ has growth rate $$\theta(\Omega^\Omega)$$.

### Examples

\begin{eqnarray*} \{3 \langle 0 [1 \neg 1 \backslash 2] 2 \rangle 3\} &=& \{3 \langle 3 \langle 3 \neg 3,3[2]3,3[3]3,3[2]3,3 \rangle 3 \rangle 3\} \end{eqnarray*}

\begin{eqnarray*} \{3 \langle 0 [1 \neg 4] 2 \rangle 3\} &=& \{3 \langle 3 \langle 3 \langle 3 \neg 3 \rangle 3 \neg 3 \rangle 3 \rangle 3\} \end{eqnarray*}

## Hierarchal backslashes

• Rule A1. $$\textrm a \langle 0 \backslash_n \# \rangle b \textrm' = \textrm a \textrm', n>1$$.
• Rule A2. If $$b = 1$$, we have $$\textrm a \langle \# \rangle 1 \textrm' = \textrm a \textrm'$$.
• Rule A3. If $$[A] < [B]$$, $$\textrm a \langle \# [A] 1 [B] \#^* \rangle b \textrm' = \textrm a \langle \# [B] \#^* \neg \% \rangle b \textrm'$$.
• Rule A4. If the first entry in the angle brackets is zero:
$$\textrm a \langle 0 [A_{1,1,1}] 1 [A_{1,1,2}] \cdots [A_{1,1,p_{1,1}}] c_{1,1} \#_{1,1} \backslash_n \#^* \rangle b \textrm' = \textrm a \langle S_{1,1} \backslash_n \#^* \rangle b \textrm', n>1$$. Set i and j to 1.
• Rule A4a. $$[A_{i,1,p_{i,1}}] = \backslash$$:
$$S_{i,1} = \textrm R_b \textrm'$$,
$$R_n = \textrm b \langle A_{i,1,1}' \rangle b [A_{i,1,1}] b \langle A_{i,1,2}' \rangle b [A_{i,1,2}] \cdots [A_{i,1,p_{i,1}-1}] b \langle R_{n-1} \rangle b [A_{i,1,p_{i,1}}] c_{i,1}-1 \#_{i,1} \textrm'$$,
$$R_1 = \textrm 0 \textrm'$$.
• Rule A4b. $$[A_{i,j,p_{i,1}}] = \backslash_j, j>1$$:
$$S_{i,j} = \textrm R_{b,j-1} \textrm'$$,
$$R_{n,j-1}$$

$$= \textrm b \langle A_{i,j,1}' \rangle b [A_{i,j,1}] b \langle A_{i,j,2}' \rangle b [A_{i,j,2}] \cdots b \langle A_{i,j,p_{i,j}-1}' \rangle b [A_{i,j,p_{i,j}-1}] b \langle A_{i,1,1}' \rangle b [A_{i,1,1}] b \langle A_{i,1,2}' \rangle b [A_{i,1,2}] \cdots b \langle A_{i,1,p_{i,1}-1}' \rangle b [A_{i,1,p_{i,1}-1}] b \langle R_{n-1,1} \rangle b [A_{i,1,p_{i,1}}] c_{i,1}-1 \#_{i,1} \backslash_j c_{i,j}-1 \#_{i,j} \textrm'$$,

$$R_{n,k} = \textrm b \langle A_{i,k+1,1}' \rangle b [A_{i,k+1,1}] b \langle A_{i,k+1,2}' \rangle b [A_{i,k+1,2}] \cdots b \langle A_{i,k+1,p_{i,k+1}-1}' \rangle b [A_{i,k+1,p_{i,k+1}-1}] b \langle R_{n,k+1} \rangle b [A_{i,k+1,p_{i,k+1}}] c_{i,k+1}-1 \#_{i,k+1} \textrm'$$
$$R_{1,1} = \textrm 0 \textrm'$$.
• Rule A4e. Otherwise:
$$S_i = \textrm b \langle A_{i,1,1}' \rangle b [A_{i,1,1}] b \langle A_{i,1,2}' \rangle b [A_{i,1,2}] \cdots [A_{i,1,p_{i,1}-1}] b \langle A_{i,1,p_{i,1}}' \rangle b [A_{i,1,p_{i,1}}] c_{i,1}-1 \#_{i,1} \textrm'$$.
• Rule A5. Otherwise, $$\textrm a \langle c \# \rangle b \textrm' = \textrm \underbrace{a \langle c-1 \# \rangle b [c \#] \cdots [c \#] a \langle c-1 \# \rangle b}_b \textrm'$$.