Introduction to BEAF

BEAF, or Bowers Exploding Array Function, is a googological function invented by Jonathan Bowers. This article is a walkthrough gradually introducing the function, meant to give an intuitive sense of how it works.

Arrow notation
Before we can understand BEAF, we must understand the extended operators at its core. Arrow notation describes a set of operators invented by :


 * \(a\uparrow b = a^b\).
 * \(a\uparrow\uparrow b = \underbrace{a\uparrow a\uparrow\ldots\uparrow a\uparrow a}_b = \underbrace{a^{a^{a^{.^{.^.}}}}}_b\). Googologists call this tetration; arrows should be solved from right to left.
 * \(a\uparrow\uparrow\uparrow b = \underbrace{a\uparrow\uparrow a\uparrow\uparrow\ldots\uparrow\uparrow a\uparrow\uparrow a}_b\). Also known as pentation.

In general,
 * \(a\uparrow^n b = \underbrace{a\uparrow^{n-1} a\uparrow^{n-1}\ldots\uparrow^{n-1} a\uparrow^{n-1} a}_b\).

Here are some examples:


 * \(3\uparrow 4 = 3^4 = 81\) (3 to the power of 4)
 * \(2\uparrow\uparrow 4 = 2\uparrow 2\uparrow 2\uparrow 2 = 2\uparrow 2\uparrow 4 = 2\uparrow 16 = 65536\) (2 tetrated to 4))
 * \(4\uparrow\uparrow 4 = 2361022671...5261392896\); about \(8.0723\cdot 10^{153}\) digits (4 tetrated to 4)
 * \(3\uparrow\uparrow\uparrow 3 = 3\uparrow\uparrow 3\uparrow\uparrow 3 = 3\uparrow\uparrow 3^{3^3} = 3\uparrow\uparrow 7625597484987\) (3 pentated to 3)

Operator notation
Bowers developed a generalization of arrow notation that he calls operator notation:


 * \(a\ \{n\}\ b = a\uparrow^n b\)
 * i.e.: \(a\ \{1\}\ b = a^b\)
 * and \(a\ \{n\}\ b = \underbrace{a\ \{n - 1\}\ a\ \{n - 1\}\ \ldots\ \{n - 1\}\ a\ \{n - 1\}\ a}_b\)

For example, \(3\ \{4\}\ 5 = 3\uparrow\uparrow\uparrow\uparrow 5\).

This form of operator notation is simply a shorthand for arrow notation. However, Bowers takes it a step further by wrapping n in two pairs of braces instead of one:


 * \(a\ \{\{1\}\}\ b = \underbrace{a\ \{a\ \{\ldots a\ \{a}_b\}\ a\ldots\}\ a\}\ a\)

Bowers calls this a expanded to b. Here, b is the number of "layers" including the outside, or (number of a 's + 1) / 2. (For the computer scientists out there, expansion is not primitive-recursive. You can't program it using only for loops with precalculated limits.)

An example:


 * \(3\ \{\{1\}\}\ 3 = 3\ \{3\ \{3\}\ 3\}\ 3 = 3\ \{3\ \{2\}\ 7625597484987\}\ 3\)

If we change n to 2, we get multiexpansion.
 * \(a\ \{\{2\}\}\ b = \underbrace{a\ \{\{1\}\}\ a\ \{\{1\}\}\ \ldots\ \{\{1\}\}\ a\ \{\{1\}\}\ a}_b\)

Higher values for n can be used:
 * \(a\ \{\{3\}\}\ b = \underbrace{a\ \{\{2\}\}\ a\ \{\{2\}\}\ \ldots\ \{\{2\}\}\ a\ \{\{2\}\}\ a}_b\) (powerexpansion)
 * \(a\ \{\{4\}\}\ b = \underbrace{a\ \{\{3\}\}\ a\ \{\{3\}\}\ \ldots\ \{\{3\}\}\ a\ \{\{3\}\}\ a}_b\) (expandotetration)
 * expandopentation, expandohexation, etc.

All these operators are right-associative, and they work just like the lower hyper-operators.

Using three braces instead of two gives us explosion:


 * \(a\ \{\{\{1\}\}\}\ b = \underbrace{a\ \{\{a\ \{\{\ldots a\ \{\{a\}\}\ a\ldots\}\}\ a\}\}\ a}_b\)
 * \(a\ \{\{\{2\}\}\}\ b = \underbrace{a\ \{\{\{1\}\}\}\ a\ \{\{\{1\}\}\}\ \ldots\ \{\{\{1\}\}\}\ a\ \{\{\{1\}\}\}\ a}_b\) (multiexplosion)
 * \(a\ \{\{\{3\}\}\}\ b = \underbrace{a\ \{\{\{2\}\}\}\ a\ \{\{\{2\}\}\}\ \ldots\ \{\{\{2\}\}\}\ a\ \{\{\{2\}\}\}\ a}_b\) (powerexplosion)
 * \(a\ \{\{\{4\}\}\}\ b = \underbrace{a\ \{\{\{3\}\}\}\ a\ \{\{\{3\}\}\}\ \ldots\ \{\{\{3\}\}\}\ a\ \{\{\{3\}\}\}\ a}_b\) (explodotetration)
 * explodopentation, explodohexation, etc.

Four sets of braces is detonation (multidetonation, powerdetonation, detonotetration, etc.), and five is pentonation (multipentonation, powerpentonation, pentonotetration, etc.) Then we have hexonation, heptonation, etc.

Our operator notation now has four arguments:


 * \(a\ \{1\}\ b = a^b\)
 * \(a\ \{1\}^d\ b = \underbrace{a\ \{a\ \{\ldots a\ \{a\}^{d - 1}\ a\ldots\}^{d - 1}\ a\}^{d - 1}\ a}_b\)
 * \(a\ \{c\}^d\ b = \underbrace{a\ \{c - 1\}^d\ a\ \{c - 1\}^d\ \ldots\ \{c - 1\}^d\ a\ \{c - 1\}^d\ a}_b\)

Here, the superscript d 's indicate the number of curly braces wrapped around c. For example, \(\{\ldots\}^4\) is a shorthand for \(\{\{\{\{\ldots\}\}\}\}\). This particular notation was not used by Jonathan Bowers.

Googologist Chris Bird proved that this 4-argument notation is about as powerful as chained arrow notation. We have barely scratched the surface of BEAF, however!

Linear array notation
Operator notation is starting to burst at the seams. A simpler way to write \(a\ \{c\}^d\ b\) is \(\{a, b, c, d\}\). Our new notation is:


 * \(\{a, b, 1, 1\} = a^b\)
 * \(\{a, b, 1, d\} = \underbrace{\{a, a, \{a, a, \ldots \{a, a, a, d - 1\} \ldots, d - 1\}, d - 1\}}_b\) if \(d > 1\)
 * \(\{a, b, c, d\} = \underbrace{\{a, \{a, \ldots \{a, a, c - 1, d\} \ldots, c - 1, d\}, c - 1, d\}}_b\) if \(c > 1\)

Ones are considered defaults, so we can chop off the end of an array if it consists only of ones. For example, \(\{a, b, 1, 1\}\) can just be written \(\{a, b\} = a^b\).

We can simplify rules 2 and 3 somewhat by relying on their recursive nature:


 * \(\{a, b, 1, d\} = \underbrace{\{a, a, \{a, a, \ldots \{a, a, a, d - 1\} \ldots, d - 1\}, d - 1\}}_b\)
 * \(= \{a, a, \underbrace{\{a, a, \ldots \{a, a, a, d - 1\} \ldots, d - 1\}}_{b - 1}, d - 1\} = \{a, a, \{a, b - 1, 1, d\}, d - 1\}\)


 * \(\{a, b, c, d\} = \underbrace{\{a, \{a, \ldots \{a, a, c - 1, d\} \ldots, c - 1, d\}, c - 1, d\}}_b\)
 * \(= \{a, \underbrace{\{a, \ldots \{a, a, c - 1, d\} \ldots, c - 1, d\}}_{b - 1}, c - 1, d\} = \{a, \{a, b - 1, c, d\}, c - 1, d\}\)

But you'll notice a problem with this refactoring. We've specified an inductive rule, but no base case, so in both cases \(b\) will keep decreasing forever! We supply a rule explaining what happens when \(b\) reaches \(1\):


 * \(\{a, 1, c, d\} = a\)

This step is very important to the definition of BEAF. If it is not clear to you, try working it out on paper.

Five entries and more
Let's recap with our current ruleset:


 * \(\{a, b, 1, 1\} = \{a, b\} = a^b\)
 * \(\{a, 1, c, d\} = a\)
 * \(\{a, b, 1, d\} = \{a, a, \{a, b - 1, 1, d\}, d - 1\}\), \(b,d > 1\)
 * \(\{a, b, c, d\} = \{a, \{a, b - 1, c, d\}, c - 1, d\}\), \(b,c > 1\)

With our new simplification, a pattern is faintly visible in the final two terms. Equipped with this knowledge, we will attempt a fifth argument:


 * \(\{a, b\} = a^b\)
 * \(\{a, 1, c, d, e\} = a\)
 * \(\{a, b, 1, 1, e\} = \{a, a, a, \{a, b-1, 1, 1, e\}, e-1\}\), \(b,e > 1\)
 * \(\{a, b, 1, d, e\} = \{a, a, \{a, b - 1, 1, d, e\}, d - 1, e\}\), \(b,d > 1\)
 * \(\{a, b, c, d, e\} = \{a, \{a, b - 1, c, d, e\}, c - 1, d, e\}\), \(b,c > 1\)

There's a gap in the third rule, but we can mend it by extrapolating back from the fourth and fifth rules:


 * \(\{a, b, 1, 1, e\} = \{a, a, a, \{a, b - 1, 1, 1, e\}, e - 1\}\)

The addition of the fifth entry was mercifully simple. We can continue the pattern and add a sixth:


 * \(\{a, b\} = a^b\)
 * \(\{a, 1, c, d, e, f\} = a\)
 * \(\{a, b, 1, 1, 1, f\} = \{a, a, a, a, \{a, b - 1, 1, 1, 1, f\}, f - 1\}\), \(b,f > 1\)
 * \(\{a, b, 1, 1, e, f\} = \{a, a, a, \{a, b - 1, 1, 1, e, f\}, e - 1, f\}\), \(b,e > 1\)
 * \(\{a, b, 1, d, e, f\} = \{a, a, \{a, b - 1, 1, d, e, f\}, d - 1, e, f\}\), \(b,d > 1\)
 * \(\{a, b, c, d, e, f\} = \{a, \{a, b - 1, c, d, e, f\}, c - 1, d, e, f\}\), \(b,c > 1\)

We should generalize this to an arbitrary number of terms. To keep this concise, we'll introduce some terminology. The first entry \(a\) is the base, and the second \(b\) is the prime. After the prime, the first non-1 entry is the pilot; the entry immediately before it is the copilot, and all entries before that are the passengers. The value of the array is written \(v(A)\). Using these terms, we can completely describe linear array notation.


 * Linear array notation

Let \(b\) be the base and \(p\) the prime.


 * 1) Base rule. If there is no pilot (that is, all entries after the prime are 1), then \(v(A) = b^p\).
 * 2) Prime rule. If the prime is 1, \(v(A) = b\).
 * 3) Catastrophic rule. Otherwise...
 * 4) Replace the copilot with a copy of the original array, but with the prime decreased by one.
 * 5) Decrease the value of the pilot by 1.
 * 6) Set all passengers to \(b\).


 * (end of definition)

This is a re-creation of Bowers' classic array notation, written around 2002. The original used five rules, but we managed to slim it down to three as per the modern BEAF.

Examples
We have completely solved linear arrays now, and now we'll present some examples to get a better intuitive sense.

\begin{eqnarray*} \{3,3,1,1,1,3\} &=& \{3,3,3,3,\{3,2,1,1,1,3\},2\} \\ &=& \{3,3,3,3,\{3,3,3,3,3,2\},2\} \\ &=& \{3,3,3,3,\{3,\{3,2,3,3,3,2\},2,3,3,2\},2\} \\ &=& \{3,3,3,3,\{3,\{3,\{3,1,3,3,3,2\},2,3,3,2\},2,3,3,2\},2\} \\ &=& \{3,3,3,3,\{3,\{3,3,2,3,3,2\},2,3,3,2\},2\} \end{eqnarray*}

If all the entries in an array are the same, we can convert it to a simpler higher-order array:

Three entries converted to four:


 * {a,a,a} = {a,2,1,2}
 * {a,a,{a,a,a}} = {a,3,1,2}
 * {a,a,{a,a,a,{a,a,a}}} = {a,4,1,2} and so on.

Four entries converted to five:


 * {a,a,a,a} = {a,2,1,1,2}
 * {a,a,a,{a,a,a,a}} = {a,3,1,1,2}
 * {a,a,a,{a,a,a,{a,a,a,{a,a,a,{a,a,a,a}}}}} = {a,6,1,1,2}

Five entries converted to six:


 * {a,a,a,a,a} = {a,2,1,1,1,2}
 * {a,a,a,a,a,{a,a,a,a,a}} = {a,3,1,1,1,2}
 * {a,a,a,a,a,{a,a,a,a,a,{a,a,a,a,a}}} = {a,4,1,1,1,2}

In general, {a,b,1,1,...,1,1,2} = {a,a,a,...,a,a,{a,a,a,...a,a,{a,a,a,...,a,a,{a,a,a,...,a,a}...}}} with b layers.

Multidimensional arrays
We'll introduce a new operator, "b array of a":


 * \(b \& a = \underbrace{\{a, a, ..., a, a\}}_b\)

This function "diagonalizes" through everything we've done so far. It's a sizable function; if you're familiar with the fast-growing hierarchy, then \(n \& n\) is about the level of \(f_{\omega^\omega}(n)\).

To continue extending array notation, we need to take a bit of a leap of intuition. \(b \& a\) is vaguely similar to \(a^b = \underbrace{\{a \cdot a \cdots a \cdot a\}}_b\), so let's write an "second-order array notation" with the base rule changed:


 * 1) Base rule. If there is no pilot (that is, all entries after the prime are 1), then \(v(A) = p \& b\).

To indicate second-order array notation, we'll put a subscript number 2 after the array: \(\{a, b\}_2 = p \& b\). We might as well do order 3 while we're at it. Define \(b \&_2 a = \underbrace{\{a, a, ..., a, a\}_2}_b\), and replace \(\&\) with \(\&_2\) in the above base rule. We have third-order array notation. \(\{a,b\}_3\) = \(b \&_2 a\).

In general, \(b \&_c a\) = \(\underbrace{\{a,a...a,a\}_c}_b\) and \(\{a,b\}_c\) = \(b \&_{c-1} a\).

From here the extension should be obvious and a little dull. Let's generalize and create a new ruleset with \(r\) as the order:


 * 1) Base rule. If there is no pilot and \(r = 1\), then \(v(A) = b^p\).
 * 2) Order rule. If there is no pilot and \(r > 1\), then \(v(A) = p \&_{r - 1} b\).
 * 3) The prime rule and catastrophic rule are unchanged.

So how can we take maximum advantage of this extension? By letting \(r\) be the value of an array:


 * \(\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}}\)

Heck, why not nest it \(b\) times?


 * \(\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{._{._.}}}}\)

We'll use \(b \&_{1, 2} a\) to write this, allowing us to define "order-1,2 notation." (Hmmm...why "1,2"?) Next is \(2,2\), which is merely \(\{a, a, \ldots, a, a\}_{1,2}\). We can inductively define order-n,2 in the same way we defined order-n.

Now for 1,3:


 * \(\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{._{._.}},2},2}\)

and in general 1,n for n > 1:


 * \(\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{\{a, a, \ldots, a, a\}_{._{._.}},n - 1},n - 1}\)

Not bad so far. How about 1,1,2? We can see from here that \(r\) is taking on multiple entries, and can start to mirror the way ordinary arrays work. We find the first non-1 entry in \(r\), call that the "order-pilot," replace the order-copilot with a copy of the array with the prime decreased by 1, decrease the order-pilot, and set the first \(p\) entries to \(b\).

But why do we need to segregate the pilot and the order-pilot at all? We only need to care about the order when the array is just the base and the prime, as in \(\{b, p\}_{1,1,2}\). The crucial step here is saying that \(r\) is part of the array, and therefore the order-pilot is the pilot.

Let's revise our notation to put \(r\) within the curly braces. Bowers' fascination with higher dimensions let him to put \(r\) as the second row, e.g.:

\[\left\{ \begin{matrix} b,p \\ 1,1,2 \end{matrix} \right\}\]

Inline, we will write this as \(\{b, p\ (1)\ 1, 1, 2\}\), where the (1) indicates a break between rows. Again, each row is automatically filled with a (countably) infinite number of 1's, so this array is identical to \(\{b, p, 1\ (1)\ 1, 1, 2\}\) or \(\{b, p, 1, 1, 1\ (1)\ 1, 1, 2, 1, 1\}\).

Let's try our new two-row extension on the existing ruleset on, say, \(\{b, p (1) 2\}\). Immediately we see that there is no copilot! We'll need to revise the rules to accommodate copilotless cases. A less trivial problem happens here:


 * \(\{b, p (1) 1, 1, 2\} = \{b, b, b, \ldots (1) b, \{b, p - 1 (1) 1, 1, 2\}, 2\}\)

The problem is the infinite number of b's on the first row; we only want p of them. Our current definition of "passengers" has broken our system, so we'll repair it as well. In summary:


 * The base is the first entry.
 * The prime is the second entry.
 * The pilot is the first non-1 entry after the prime.
 * The copilot is the entry immediately before the pilot. It may not exist if the pilot is on the beginning of the second row.
 * A previous entry is an entry before another one that's still on the same row. (So in \(\{a, b (1) c, d\}\), \(c\) is a previous entry to \(d\) but \(a\) and \(b\) are not.)
 * The prime block of a row is the first \(p\) entries of that row.
 * The airplane is the pilot and all previous entries. If the pilot is on the second row, the airplane also includes the prime block of the previous row.
 * The passengers are all entries in the airplane except for the pilot and copilot (if it exists).

The rules are otherwise the same as linear array notation. No "order rule" is needed since what we originally called the order is now part of the array.

Here's a brief example:

\begin{eqnarray*} \{3,3,3 (1) 2\} &=& \{3,\{3,2,3 (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,\{3,1,3 (1) 2\},2 (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,3,2 (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,\{3,2,2 (1) 2\},1 (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,\{3,2,2 (1) 2\} (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,\{3,\{3,1,2 (1) 2\},1 (1) 2\} (1) 2\} (1) 2\},2 (1) 2\} \\ &=& \{3,\{3,\{3,3 (1) 2\} (1) 2\} (1) 2\},2 (1) 2\} \end{eqnarray*}

More rows
The smallest nontrivial three-row array is \(\{b, p (1) (1) 2\}\), where the second row is only ones. As you might expect, this is equal to \(\{b, b, \ldots, b, b (1) b, b, \ldots, b, b\}\) with \(p\) entries per row. The pilot has now moved to the third row, and we can introduce a copilot as in \(\{b, p (1) (1) 1, 2\} = \{b, p (1) (1) \{b, p - 1 (1) (1) 1, 2\}\}\). More than three rows behave pretty much the same, so let's take the opportunity to generalize.

We will need to alter our definition of the airplane again:


 * The airplane is the pilot, all previous entries, and the prime blocks of all previous rows.

When the pilot is on the first row, there are no previous rows and the airplane is just the previous entries. This takes care of the ugly conditional in our previous airplane.

Now we have completely defined planar array notation. So what is there beyond a row?

Three dimensions
A plane. We'll use the (2) separator to indicate a planar break. The smallest two-plane array is \(\{b, p (2) 2\} = \{b, b, \ldots, b, b (1) b, b, \ldots, b, b (1) \ldots (1) b, b, \ldots b, b (1) b, b, \ldots, b, b\}\), or a \(p \times p\) array of \(b\)'s. (We can also write this as \(p^2 \& b\), more on that later.) Eventually we'll reach multiple planes, which pretty much solve as expected.

Let's define this formally. To aid in generalization, we need to introduce some more abstract terms. One of them is a structure, which is either an entry, a row, or a plane. Now we'll define prime blocks of structures in general.


 * The prime block of a row is the first \(p\) entries, and the prime block of an array is the first \(p \times p\) square of entries (or the prime blocks of the first \(p\) rows).
 * A previous entry to X is an entry before X on the same row. A previous row to X is a row before X on the same plane.
 * The airplane is the pilot, all previous entries, and the prime blocks of all previous structures.

Study that last definition carefully, because it's a core part of how BEAF works.

Four and more
An infinite line of planes, or a three-dimensional space, is a realm, according to Bowers. A four-dimensional space is a flune. The symbols to indicate breaks in realms and flunes are (3) and (4), respectively.

Generally, (n) indicates a break into the next n-dimensional space. A comma is short for (0), since an entry is a 0-dimensional space. A "structure" is any n-dimensional space; we'll notate an n-D structure as \(X^n\).

Let's revise our definitions of "structure," "prime block," "previous structure," and "airplane."


 * A structure is an entry, row, plane, realm, flune, 5-space, ... Inductively, this is defined as an entry or an infinite row of structures.
 * The prime block of a row is the first \(p\) entries; the prime block of a plane is the first \(p \times p\) square; the prime block of a realm is the first \(p \times p \times p\) cube, etc. Inductively, the prime block of of a row is the first \(p\) entries, and the prime block of an \(X^n\) structure is the prime blocks of the first \(p\) \(X^{n-1}\)-structures.
 * A previous structure to A is an \(X^n\) before A on the same \(X^{n + 1}\) structure. Specifically, a previous entry to A is an entry before A on the same row, a previous row to A is a row before A on the same plane, etc.
 * The airplane is the pilot, all previous entries, and the prime blocks of all previous structures.

Up to here we have completely defined dimensional array notation. Recap with full rules:


 * Dimensional array notation

Definitions:


 * The base \(b\) is the first entry.
 * The prime \(p\) is the second entry.
 * The pilot is the first non-1 entry after the prime.
 * The copilot is the entry immediately before the pilot. It may not exist if the pilot is on the beginning of a row.
 * A structure is an entry, row, plane, realm, flune, 5-space, ... Inductively, this is defined as an entry or an infinite row of structures.
 * The prime block of a row is the first \(p\) entries; the prime block of a plane is the first \(p \times p\) square; the prime block of a realm is the first \(p \times p \times p\) cube, etc. Inductively, the prime block of of a row is the first \(p\) entries, and the prime block of an \(X^n\) structure is the prime blocks of the first \(p\) \(X^{n-1}\)-structures.
 * A previous structure to A is an \(X^n\) before A on the same \(X^{n + 1}\) structure. Specifically, a previous entry to A is an entry before A on the same row, a previous row to A is a row before A on the same plane, etc.
 * The airplane is the pilot, all previous entries, and the prime blocks of all previous structures.
 * The passengers are all entries in the airplane excluding the pilot and copilot (if it exists).

Rules:


 * 1) Base rule. If there is no pilot (that is, all entries after the prime are 1), then \(v(A) = b^p\).
 * 2) Prime rule. If the prime is 1, \(v(A) = b\).
 * 3) Catastrophic rule. Otherwise...
 * 4) If the copilot exists, replace the copilot with a copy of the original array, but with the prime decreased by one.
 * 5) Decrease the value of the pilot by 1.
 * 6) Set all passengers to \(b\).


 * (end of definition)

Whew! This is "extended array notation," developed by Bird and Bowers. The original used seven rules, but we have applied the modern BEAF definition to simplify it quite a bit.

Array of operator
The array of operator, notated & (not to be confused with logical "and" operator) returns the array assigned to the structure at the left to &, filled with the number of entries at the right to &. What are these structures? First structures are numbers, and \(n\ \&\ m = \{\underbrace{m,m,m,\cdots,m,m,m}_{\text{n m's}}\}\). The structure above numbers, so called X-structure, should be evaluated to m. So, \(X\ \&\ m = m \&\ m = \{m,m (1) 2\}\).

If we want to specify other number, we can enclose it by brackets: \(X\ \&\ a[b] = b\ \&\ a[b]\). This notation hasn't been used by Bowers himself.

The structures above numbers X are X+1,X+2,X+3,...,2X,2X+1,2X+2,2X+3,...,3X,4X,...,\(X^2\), etc. X+m can be thought as a line of entries with m entries below it, 2X as two lines, 3X as three lines, nX+m as n lines of entries with m entries below it all. Here are some correspondences between our notations above and "array of":

\begin{eqnarray*} 1\ \&\ n &=& \{n\} = n \\ 2\ \&\ n &=& \{n,n\} = n^n \\ 3\ \&\ n &=& \{n,n,n\} = n \uparrow^{n} n \\ 4\ \&\ n &=& \{n,n,n,n\} = n \underbrace{ \{\{\cdots\{\{n\}\}\cdots\}\} }_{n \{\}'s} n \\ m\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{m n's}}\} &=& \{n,m (1) 2\} \\ X\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}}\} &=& \{n,n (1) 2\} \\ X+1\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) n\} \\ X+2\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) n,n\} \\ X+3\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) n,n,n\} \\ X+m\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) \underbrace{n,n,n,\cdots,n,n,n}_{\text{m n's}}\} \\ 2X\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) \underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}}\} \\ 3X\ \&\ n &=& \{\underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) \underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}} (1) \underbrace{n,n,n,\cdots,n,n,n}_{\text{n n's}}\} \\ mX\ \&\ n &=& \{\underbrace{X\ \&\ n(1)X\ \&\ n(1)X\ \&\ n(1)\cdots(1)X\ \&\ n(1)X\ \&\ n(1)X\ \&\ n}_{\text{m X's}}\} &=& \{n,m (2) 2\} \\ X^2\ \&\ n &=& \{\underbrace{X\ \&\ n(1)X\ \&\ n(1)X\ \&\ n(1)\cdots(1)X\ \&\ n(1)X\ \&\ n(1)X\ \&\ n}_{\text{n X's}} &=& \{n,n (2) 2\} \end{eqnarray*}

As can be seen, last three examples can be simplified to \(\{n,3 (2) 2\}, \{n,m (2) 2\}\) and \(\{n,n (2) 2\}\) respectively. Here the last "2" is in the next plane and marks the start of 3-D arrays. Fortunately, we don't have to draw multidimensional arrays in its natural form. The main key to it is using polynomial form: the expression \(a_1X^m+a_2X^{m-1}+a_3X^{m-2}+\cdots+a_{m-1}X+a_m \&\ b[p]\) means that its array is formed by \(a_1\) sections of \(X^m\) (m-D array of b's), then \(a_2\) sections of \(X^{m-1}\), then \(a_3\) sections of \(X^{m-2}\), and so on. The total number of entries can be found by replacing every X to p and solving like normal polynomial. For example, \(X^{100}+X^{99} \&\ 3[4]\) has \(4^{100}+4^{99}\) entries.

Trimensional arrays
Up to here, we have a system at the level of \(f_{\omega^{\omega^{\omega^\omega}}}\) for those familiar with FGH. The next step is not entirely obvious, but it's crucial for continuing the system.


 * \(\{b, p (0, 1) 2\}\)

What does this even mean? This is an \(X^X\) structure, and the prime block of an \(X^X\) structure is a \(p^p\) structure, or a \(\underbrace{p \times p \times \cdots \times p \times p}_p\) hypercube. So it's a 2-by-2 plane, or a 3-by-3-by-3 cube, or a 4-by-4-by-4-by-4 tesseract, etc.


 * \(\{b, p (1, 1) 2\}\)

This is an \(X^{X + 1}\) structure; its prime block is \(p^{p + 1}\) (2x2x2, 3x3x3x3, 4x4x4x4x4, etc.). Generally, an \((n, m)\) separator creates an \(X^{mX + n}\) structure.

We have 3-entry seperators. \((a,b,c)\) seperator creates an \(X^{cX^2 + bX + a}\) structure.

In general, \((a,b,c,...)\) seperator creates an \(X^{a + bX + cX^2 + ...}\) structure.