| rdfs:comment | - For linear and multidimensional arrays, BAN is the same as BEAF.
* Rule 1. With one or two entries, we have \(\{a\} = a\), \(\{a,b\} = a^b\) (by the way, \(\lbrace a brace = a\) follows from \(\{a,b\} = a^b\), since \(\{a\} = \{a,1\} = a^1 = a\) (inverse of rule 2).
* Rule 2. If the last entry is 1, it can be removed: \(\{\# 1\} = \{\#\}\). (The octothorpe indicates the unchanged remainder of the array.)
* 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: \(\{a,b,1,1,\cdots,1,1,c \#\} = \{a,a,a,a,\cdots,a,\{a,b-1,1,1,\cdots,1,1,c \#\},c-1 \#\}\)
* Rule 5. Otherwise: \(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\)
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| abstract | - For linear and multidimensional arrays, BAN is the same as BEAF.
* Rule 1. With one or two entries, we have \(\{a\} = a\), \(\{a,b\} = a^b\) (by the way, \(\lbrace a brace = a\) follows from \(\{a,b\} = a^b\), since \(\{a\} = \{a,1\} = a^1 = a\) (inverse of rule 2).
* Rule 2. If the last entry is 1, it can be removed: \(\{\# 1\} = \{\#\}\). (The octothorpe indicates the unchanged remainder of the array.)
* 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: \(\{a,b,1,1,\cdots,1,1,c \#\} = \{a,a,a,a,\cdots,a,\{a,b-1,1,1,\cdots,1,1,c \#\},c-1 \#\}\)
* Rule 5. Otherwise: \(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\) With multidimensional arrays, Bird uses \( extrm` a \langle c angle b extrm'\), which is equivalent to Bowers' \(b^c \& a\). These strings are written within the quote signs and have their own specific rules:
* Rule A1. If \(c = 0\), we have \( extrm` a \langle 0 angle b = a extrm'\).
* Rule A2. If \(b = 1\), we have \( extrm` a \langle c angle 1 = a extrm'\).
* Rule A3. Otherwise, \( extrm` a \langle c angle b extrm' = extrm` a \langle c - 1 angle b [c] a \langle c angle (b - 1) extrm'\). The main rules are:
* 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: \(\{a,b [m_1] 1 [m_2] \cdots 1 [m_x] c \#\} = \{a \langle m_1-1 angle b [m_1] a \langle m_2-1 angle b [m_2] \cdots a \langle m_x-1 angle b [m_x] (c-1) \#\}\)
* Rule M5. If there is a non-zero entry after batch of unfilled separators and a 1. \(\{a,b [m_1] 1 [m_2] \cdots 1 [m_x] 1,c \#\} =\) \(\{a \langle m_1-1 angle b [m_1] a \langle m_2-1 angle b [m_2] \cdots a \langle m_x-1 angle b [m_x] \{a,b-1 [m_1] 1 [m_2] \cdots 1 [m_x] 1,c \#\},c-1 \#\}\)
* Rule M6. Rules M1-M6 don't apply. \(\{a,b,c \#\} = \{a,\{a,b-1,c \#\},c-1 \#\}\) For example, \(\{3,3 [3] 1 [2] 1,1,1,4\}\) = \(\{3 \langle 2 angle 3 [3] 3 \langle 1 angle 3 [2] 3,3,\{3,2 [3] 1 [2] 1,1,1,4\},3\}\) Bird uses \([m]\) as a dimensional separator; in Bowers' notation it is equivalent to an \((m - 1)\) separator. This resolves a minor issue in BEAF, where ones are default in the array and zeroes are default in the separators. In the fast-growing hierarchy, linear and multidimensional array notations have limit ordinals \(\omega^\omega\) and \(\omega^{\omega^\omega}\), respectively.
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