Bowers Exploding Array Function

BEAF (Bowers Exploding Array Function) is a notation invented by Jonathan Bowers, similar to Chained Arrow Notation (but much stronger). It is a superset of Array Notation and Extended Array Notation (both invented by Bowers).

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 tetrational structures, e.g. \(X\uparrow\uparrow 3\). We can also continue with pentational, hexational, ... ,expandal, ... 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 \(X^n\) is the first \(p^n\) block of entries in that structure.
 * 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:
 * 4) pilot decreases by 1,
 * 5) copilot takes on the value of the original array with the prime decreased by 1,
 * 6) each passenger becomes b,
 * 7) and the rest of the array remains unchanged.

Linear Arrays
Linear arrays are the smallest and simplest type of array. A linear array consists of a one-dimensional row of numbers, e.g. {5,8,7,2,4}. Although they are the smallest of BEAF arrays, linear arrays with more than five entries grow much, much faster than Chained Arrow Notation (a theorem known as Bird's Proof). Positions in linear arrays can be described with a single number, e.g. the fourth entry.

Dimensional Arrays
Dimensional arrays are arrays that need 2 or more dimensions to represent. To write these arrays in a single line, one must use numbers in parentheses in place of commas to indicate breaks in multiple dimensions. (1) means that the following numbers are in the next row, (2) means the next plane, (3) means the next realm (3-space), (4) means the next flune (4-space), and so forth. For example, {3,3,3 (1) 3,3,3 (1) 3,3,3} means a 3-by-3 square of threes. Positions in dimensional arrays require linear arrays to represent. For example, (5,6,8,2) means the fifth entry on the sixth row on the eighth plane in the second realm.

Tetrational Arrays
Tetrational arrays are arrays that require tetrational spaces to represent. Tetrational spaces consist of superdimensional space, trimensional space, quadramensional space, etc.

Superdimensional arrays consists not only dimensional spaces, but also dimensional groups, dimensional spaces of groups, groups of groups, gangs (the next level of structure after the group), etc.

Positions in superdimensional arrays require dimensional arrays to represent, positions in trimensional arrays require superdimensional arrays to represent, etc.

Pentational Arrays
On pentational arrays, the powers will sort into groups, like X^{X^X^X}^{X^X^X}^{{X^X^X}^{X^X^X}^{X^X^X}} or (X^2+2X+1)^^X where X is evaluated at 3. The {} are not to be solved like ordinary parentheses, but are used to group up the exponents into tetrational blocks (so if the prime entry changes, then the number of X's or {X^...^X} on each block will also be changed to the prime entry).

Larger non-legion arrays
There are larger arrays like heptational, octational, nonational, decational, expandal, multiexpandal, powerexpandal, explodal, multiexplodal, detonational, etc.

Jonathan Bowers comments on these arrays, "How to work with these? - Only God knows - but they should form some massive arrays - and utterly unspeakable numbers when solved."

Legions
Before discussing legions, we must first define the array of operator. a array of b, written a & b, is defined as {a, a, a, ...}, in which there are b a 's. If a is an exponent or array, it indicates the dimensions of the resulting array. For example, 32 & 3 = {3, 3, 3 (1) 3, 3, 3 (1) 3, 3, 3} is a 3 by 3 = 32 array of 3's.

Bowers further extends BEAF using legiattic arrays. In the array {a, b, c, ... / 2}, we say that the number 2 is in the second legion. In such a case, we solve the array in the first region normally, but in the initial rule, we say that v(A) = b&b&b&b&... p times, solving from left to right. For example, {3, 3 / 2} = 3&3&3.

In the case {b, p / x}, we say that the pilot x is in the second legion. We can use this knowledge to cover cases in which x > 2:


 * {3, 3 / 3} = {3, {3, 2 / 3} / 2} = {3, {3, {3, 1 / 3} / 2} / 2} = {3, {3, 3 / 2} / 2} = {3, 3&3&3 / 2} = 3&3&3&3&... 3&3&3 times
 * In general, {n, m / 3} = {n, {n, m - 1 / 3} / 2}

This is according to the catastrophic rule of BEAF. Ones are default, so {A / 1} = A.

It is possible for the second legion to contain multiple entries. Either legion can be dimensional, tetrational, pentational, expansional, etc.

An array can have multiple legions, e.g. {3, 4 / 5, 6 / 7, 8}. The structure of the legions can be multidimensional, using (/n) to indicate an n-dimensional legion break, e.g. {3, 4 (/6, 2) 9, 4} is a tetrational legion array.