BEAF BAN Notation / Increasing to Nested Arrays
Bowers Extended Array Function's (BEAF) Bird's Array Notation (BAN)
Bowers' Exploding Array Function (BEAF) is a notation for very large numbers invented by Jonathan Bowers, similar to chained arrow notation, but far stronger. It is a superset of array notation and extended array notation, both invented by Bowers.[1] It has become quite famous in googology due to its simplicity and growth speed, not to mention the vast array of whimsically named numbers defined with the function (such as golapulus and the legendary meameamealokkapoowa oompa, one of Bowers' largest numbers). However, there is no agreed-upon definition for the notation above tetrational arrays. Therefore, strictly speaking, BEAF beyond tetrational arrays is ill-defined, although BEAF before tetrational arrays is well-defined.
Although Chris Bird and John Spencer (a friend of Bowers') assisted in the construction of BEAF, Bowers is usually given sole credit for the function.
Sbiis Saibian mentioned that the existence of a notation which fully satisfies Bowers' rules is an open problem in googology. Although he only directly mentions pentational arrays, it probably refers to other levels of BEAF as well.[2]
Definitions
Rules
Prime rule: If \(p = 1\), \(v(A) = b\).
Initial rule: If there is no pilot, \(v(A) = b^p\).
Catastrophic rule: If neither 1 nor 2 apply, then:
pilot decreases by 1,
copilot becomes the original array with the prime decreased by 1,
each passenger becomes b,
and the rest of the array remains unchanged.
Array types
Linear arrays
Main article: Array notation
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 four 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
Main article: Extended Array Notation
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. These structures also can be called exponential arrays.
Tetrational arrays
Tetrational arrays are arrays that require tetrational spaces to represent. They are the largest part of BEAF with an agreed-upon definition in the googology community, and therefore they are arguably the largest well-defined part of BEAF. Tetrational spaces consist of superdimensional space, trimensional space, quadramensional space, etc.
Superdimensional arrays consist of not only dimensional spaces, but also dimensional groups, dimensional spaces of groups, groups of groups, gangs (the next structure level 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 sort into groups, like \(X \uparrow \{X\uparrow X\uparrow X\} \uparrow \{X\uparrow X\uparrow X\} \uparrow \{\{X\uparrow X\uparrow X\} \uparrow \{X\uparrow X\uparrow X\} \uparrow \{X\uparrow X\uparrow X\}\}\) or \(X \uparrow\uparrow (X \uparrow 2+2X+1)\) 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).
The Googology Wiki users Deedlit11[3] and Ikosarakt1[4] have defined the pentational arrays via non-climbing method–each in a separate way–and came to the same results, which agree with Bowers' beginning of this work, but both have a discrepancy with the rule "A&n has A(n) entries" and so can not be considered as valid attempts.
Larger non-legion arrays
There are larger arrays like hexational, heptational, expandal, multiexpandal, powerexpandal, explodal, multiexplodal, detonational, etc. Eventually, we create a really large array that the space its in needs to be represented by array notation (linear, dimensional, tetrational, etc.)