BEAF, or Bowers Exploding Array function, is an extremely fast-growing
function.
## Definitions
- The "base" (*b*) is the first entry in the array.
- The "prime" (*p*) is the second entry in the array.
- The "pilot" is the first non-1 entry after the prime. It can be as
early as the third entry.
- The "copilot" is the entry immediately before the pilot. The copilot
does not exist if the pilot is the first entry in its row.
- A "structure" is a part of the array that consists of a
lower-dimensional group. This could be an entry (written \(X^0\)),
a row (written \(X^1\)), a plane (\(X^2\)), a realm (\(X^3\)),
or a flune (\(X^4\)), not to mention higher-dimensional structures
(\(X^5\), \(X^6\), etc.) and
<a href="index.php?title=Tetration&action=edit&redlink=1" class="new" title="Tetration (page does not exist)">tetrational</a>
structures, e.g. \(X\uparrow\uparrow 3\). We can also continue
with
<a href="index.php?title=Pentation&action=edit&redlink=1" class="new" title="Pentation (page does not exist)">pentational</a>,
<a href="index.php?title=Hexation&action=edit&redlink=1" class="new" title="Hexation (page does not exist)">hexational</a>,
...,
<a href="index.php?title=Expansion&action=edit&redlink=1" class="new" title="Expansion (page does not exist)">expandal</a>,
... structures.
- A "previous entry" is an entry that occurs before the pilot, but is
on the same row as all other previous entries. A "previous row" is a
row that occurs before the pilot's row, but is on the same plane as
all other previous rows. A "previous plane" is a plane that occurs
before the pilot's plane, but is on the same realm as all other
previous planes, etc. These are called "previous structures."
- A "prime block" of a structure \(S\) is computed by replacing all
instances of \(X\) with \(p\). For example, if \(S = X^3\),
the prime block is \(p^3\), or a cube of side length \(p\). The
prime block of an \(X^X\) structure is \(p^p\), a
\(p\)-hypercube with sidelength \(p\).
- The "airplane" includes the pilot, all previous entries, and the
prime block of all previous structures.
- The "passengers" are the entries in the airplane that are not the
pilot or copilot.
- The value of the array is notated \(v(A)\), where *A* is the
array.
## Rules
1. *Prime rule*: If \(p = 1\), \(v(A) = b\).
2. *Initial rule*: If there is no pilot, \(v(A) = b^p\).
3. *Catastrophic rule*: If neither 1 nor 2 apply, then:
1. pilot decreases by 1,
2. copilot takes on the value of the original array with the prime
decreased by 1,
3. each passenger becomes *b*,
4. and the rest of the array remains unchanged.
## Examples
\begin{eqnarray*} \{3,3,3,3\} &=& \{3,\{3,2,3,3\},2,3\} \\ &=&
\{3,\{3,\{3,1,3,3\},2,3\},2,3\} \\ &=& \{3,\{3,3,2,3\},2,3\}
\\ &=& \{3,\{3,\{3,2,2,3\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,\{3,1,2,3\},1,3\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,3,1,3\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,3,\{3,2,1,3\},2\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,3,\{3,3,\{3,1,1,3\},2\},2\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,3,\{3,3,3,2\},2\},1,3\},2,3\} \\ &=&
\{3,\{3,\{3,3,\{3,\{3,\{3,3,1,2\},1,2\},2,2\},2\},1,3\},2,3\}
\\ &=&
\{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,2,1,2\}\},1,2\},2,2\},2\},1,3\},2,3\}
\\ &=&
\{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,\{3,1,1,2\}\}\},1,2\},2,2\},2\},1,3\},2,3\}
\\ &=&
\{3,\{3,\{3,3,\{3,\{3,\{3,3,\{3,3,3\}\},1,2\},2,2\},2\},1,3\},2,3\}
\\ &=&
\{3,\{3,\{3,3,\{3,\{3,3\uparrow^{3\uparrow\uparrow\uparrow3}3,1,2\},2,2\},2\},1,3\},2,3\}
\end{eqnarray*}
------------------------------------------------------------------------
Using {a, b, ... (1) c, d, ...} to denote
{a, b, ...}
{c, d, ...}
a 2-dimensional array (For example, $\{3,3,3 (1) 3,3,3 (1) 3,3,3\}$
means a 3-by-3 square of threes):
\begin{eqnarray*} \{3,3(1)3,3\} &=& \{3,3,3(1)2,3\} \\ &=&
\{3,\{3,2,3(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,3,2(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,2,2(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,3(1)2,3\}(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,3,3(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,\{3,2,3(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,\{3,3,2(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\} \\ &=&
\{3,\{3,\{3,\{3,\{3,2,2(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3(1)1,3\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,2(1)1,3\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3(1)3,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,3,3(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}(1)2,3\},2(1)2,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,2,3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3,2(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,2,2(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3(1)2\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,3,3\}(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,3\uparrow\uparrow\uparrow3(1)2\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\\ &=&
\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{3,\{3,\{3,\{3,3,3(1)\{3,\{\underbrace{3,\cdots,3}_{3\uparrow\uparrow\uparrow3}\},2(1)2\}\}(1)1,2\},2(1)1,2\}(1)2,3\}(1)2,2\},2\}(1)1,3\},2(1)1,3\}
\end{eqnarray*}
More generally, (n) is typically used as a separator representing a
line/plane/... shift in a n-dimensional array.
BEAF is however formally undefined past (n)-separators. In particular,
the so-called "tetrational" arrays are not defined, contrary to popular
belief.