Difference between revisions of "Bird's array notation"

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:
$$\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}$$

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

• 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 M7. Rules M1-M6 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, $$\textrm a \langle c \rangle b \textrm' = \textrm a \langle c - 1 \rangle b [c] a \langle c \rangle (b - 1) \textrm'$$.

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*}