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 was created by rolling Array Notation and Extended Array Notation (both invented by Bowers) into one notation with a master set of rules.

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, expansional, multiexpansional, explosional, multiexplosional, and detonational 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) If $$p = 1$$, $$v(A) = b$$.
 * 2) If there is no pilot, $$v(A) = b^p$$.
 * 3) 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. Linear arrays consist of a one-dimensional set of numbers, e.g. {5,8,7,2,4}. Although they are the smallest, any arrays five entries or larger are much, much larger than Chained Arrow Notation by 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 in a flune (or tetraspace), 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 consists of superdimensional space, trimensional space, quadrimensional 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 and larger non-legion 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).

There are larger arrays like hexational, heptational, expansional, multiexpansional, explosional, multiexplosional, and detonational, etc. Eventually we creates a really large array that the space its in needs to be represented by array notation (linear, dimensional, tetrational, 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
Legions are portions of the array separated by slashes. Suppose rule 2 was defined as "if there is no pilot, $$v(A) = \underbrace{b\&b\&b\&\ldots}_p$$." (In BEAF, $$m\&n$$ is defined as $$\underbrace{\{n,n,n,\ldots\}}_m$$. This operator is solved from left to right. Note that the value can be different if the form of the left operand is different - for example, 33&3 is dimentri, while 27&3 is ultatri.) Such a modification comes into action if a 2 is placed in the next legion, written $$\{a,b,c,d,\ldots/2\}$$.

One could consider, in $$\{b,p/x\}$$, that the pilot (x) is in the second legion. This defines arrays where x is greater than two. For example, $$\{3,4/3\} = \{3\&3\&3\&3/2\}$$. Note that $$\{A/1\} = \{A\}$$.

An array can have more than two legions, allowing structures like $$\{4,5/3,6/6,7\}$$. Of course, legions need not be arranged in a linear form. Using notations like $$(/n)$$ for breaks, it is possible to create multidimensional, tetrational, and pentational arrays, not to mention expansional, multiexpansional, explosional, and multiexplosional.

And next there would be a double legion array, for example $$\{3,2//2\}$$ is equal to 3&&3, where 3&&3 is $$\{A/A/A\}$$ where A=3&3&3. This could be extended to multipie legion arrays, by repeating the '&' and '/'.

It could be extended further that each legion could make a structure. To define this, Bowers came up a new kind of space called 'legion space', or 'L' for short. It is a space that cannot be reached by using the & operator a constant times.