Exploding Array Function

original Array Notation by Jonathan Bowers
comments and algorithms by Giga Gerard

My Exploding Array Function is the generalized form of the various array notations that I have developed a few years ago, such as (linear) array notation, extended (or dimensional) array notation, and tetrational array notation, etc with a few modifications. Originally the various array notations had several rules - 5 for the linear, 7 for the extended, and for the tetrational I was still trying to work it out - now, there's only three rules and it takes care of all arrays, even way past tetrational arrays.

How it Started

Over the years, I have been interested in large numbers, even to the point of coining names for several, see my "illion numbers" page. One day (about 20 years ago) I read a book (don't remember the name) which discussed operators beyond powers, operators such as tetration, pentation, hexation, etc. This got me interested in investigating more extreme numbers as well as extreme spaces beyond dimensional space (spaces that I now call, super dimensional, trimensional, quadramentional, etc). As time went on, I started to develop an extention to the operators seen in that book, these I will refer to as extended operators. Later after looking into known large number functions such as Ackermann's Function and Conway's Chained Arrow Notation, I could compare the extended operators with these well known functions. The extended operators far surpassed Ackermann's Function, and could actually be compared to Conway's Chained Arrow - check out M. Rob's large numbers page for details. Later I took on the task of extending the extended operators - I later discovered that extended operators could be defined by four numbers, and the next extention by five. This is the beginning of the development of my Array Notation.

Array Notation First row, first dimension in Bean or Beaf

The five rules for the original Array Notation were as follows:

Recursion 1 over a function row

Rules to evaluate arrays U of a single row,
with wildcards in upper case for arbitrary sequences,
and integer variables n = 1..:n>0 as usual in lower case.
Rep :n is to repeat the left selected word in the expression n times.

  1. {a,b} = a*..1 :b = a^b
  2. {X,1} = {X}
  3. {a,1,Z} = a
  4. {a,b1,1,..11Z} 1,:k = {a,..V,1Z} a,:k1
  5. {a,b1,c1,Z} = {a,V,c,Z}
  6. let U = {a,b1,Y}  then V = {a,b,Y}

As such Bowers defines his systems Bean and Beaf for the initial row.

Various upload systems can be created by bending rule 4.
For example, Eat could put value b1 in place of the last 1 and nothing more, so that after iteration of its rule 4 value b is assigned to all but the lowest entries (eat the row), in {a,b1,b1,/b,:k-/Z} that is.
Rule 4 of system Beat would upload value b1 to all prior entries at once. And Beate as an extra substitutes V as above, which makes her the fastest function we've ever seen on the first row.
For the same expression Beat results in smaller and Beate in larger numbers than Bowers' function. Note that none of these systems is significantly faster than the other, because the difference expressed in any last parameter z is less than 1 really.

Rule 1: Condition - only 1 or 2 entries - {a} = a, {a,b} = a+b Chris Bird has suggested changing this to a^b for two reasons - first, so it can be easier to compare with Conway's Chained Arrow, second so it will jibe with rule 2 better (note that a^1=a where a+1 is not), I decided to make the change.

Rule 2: Condition - last entry is 1 - {a,b,c,...,k,1} = {a,b,c,...,k}

Rule 3: Condition - 2nd entry is 1 - {a,1,c,d,..,k} = a

Rule 4: Condition - 3rd entry is 1 - {a,b,1,..,1,d,e,..,k} = {a,a,a,..,{a,b-1,1,..,1,d,e,..,k},d-1,e,..,k} - the ".." between the 1's represent 1's - there can be any number of ones, from 1 "1" (3rd entry alone) to a string of 1s - the last 1 of this string is what becomes {a,b-1,1,..,1,d,e,..,k} all entries prior becomes "a". For an array like this {3,2,1,1,1,5,1,2} the last 1 in the string is the one before the 5 (not the one after - since it is a different string of 1s).

Rule 5: Condition - Rules 1-4 doesn't apply - {a,b,c,d,...,k} = {a,{a,b-1,c,d,...,k},c-1,d,..,k}

Extended Operators Sidestep operator notation

Following shows how the operators work as well as my extension on those operators.

a {c} b = {a,b,c}, where c=1,2,3,4,5 etc represents adding, multiplying, exponentiation, tetration, pentation, etc. (After Bird's suggestion, I decided to let a {1} b = a^b instead of a+b to match the modified array notation, so the results are different now).

Tetration is also known as power towers, it is also represented as a^^b = a {4} b (under old version) = a {2} b (under new version) = a^(a^(a^(a^.....^a)...)) a to the power of itself b times.

Pentation is a tetrated to itself b times = a^^^b = a{5}b (old version) = a(3)b (new version), from now on lets use the new version.

Now for the extended operators.

a {{1}} b = a { a { a....a { a { a } a } a.... a } a } a (b a's from center out) - I call this a expanded to b

a {{2}} b = a expanded to itself b times (or a multiexpanded to b) = a {{1}} (a {{1}} (a {{1}} (a .... (b times)

a {{3}} b = a multiexpanded to itself b times (or a powerexpanded to b)

a {{4}} b = a powerexpanded to itself b times (or a expandotetrated to b)


Evaluate, with reps :n: of words selected by dots to both sides:

{a,b1,1,2} = {a,a,{a,b,1,2}}

          == {a,a,..1.} :b1:

           = {a,a,..a.} :b: = a{{1}}b1

{a,b1,2,2} = {a,{a,b,1,2},1,2}

          == {a,..1.,1,2} :b1:

           = {a,..a.,1,2} :b: = a{{2}}b1

{a,b1,c1,2} = {a,{a,b,c1,2},c,2}

           == {a,..1.,c,2} :b1:

            = {a,..a.,c,2} :b: = a{{c1}}b1

Here's how this compares to Array Notation: a {{c}} b = {a,b,c,2}

a {{{1}}} b = a {{ a {{ a ... a {{ a {{ a }} a }} a ... a }} a }} a ( b a's from center out) - I call this a exploded to b.

a {{{2}}} b = a exploded to itself b times (a multiexploded to b)

a {{{3}}} b = a multiexploded to itself b times (a powerexploded to b)

a {{{4}}} b = a powerexploded to itself b times (a explodotetrated to b)


Again, here it is compared with Array Notation: a {{{c}}} b = {a,b,c,3} etc.

{a,b,1,3} == {a,a,..1.,2} :b: = a{{{1}}}b

{a,b,2,3} == {a,..1.,1,3} :b: = a{{{2}}}b

{a,b,c1,3} == {a,..1.,c,3} :b: = a{{{c1}}}b

{a,b,c1,4} == {a,..1.,c,4} :b: = a{{{{c1}}}}b

These operators can be notated more naturally:

{a,b,c,5} = a{{{{{c}}}}}b = a{c,5}b

{a,b,c,d} = a{..c.}b {:d:} = a{c,d}b

a {{{{1}}}} b = a detonated to b = {a,b,1,4}

a {{{{{1}}}}} b = a pentonated to b = {a,b,1,5}

{a,b,1,1,2} = {a,a,a,{a,b-1,1,1,2}} = a {{{{{{{....{{{a}}}....}}}}}}} (where there are a {{{{{...{{a}}...}}}}} a brackets (where there are a {{{{..{{a}}..}}}} a brackets ( where there are .......b times ..... (where there are a brackets)))))) - YIKES!!

Chris Bird has shown in a proof (which we can call Bird's proof) that four entry arrays are comparible to Conway's Chained Arrow Notation and five entry arrays far surpass it.

No real difference between operators and array function here.

{a,b,1,1,2} = {a,a,a,..1.} :b: = a{1,1,2}b

{a,b,2,1,2} = {a,..1.,1,1,2} :b: = a{2,1,2}b

{a,b,c1,1,2} = {a,..1.,c,1,2} :b: = a{c1,1,2}b

{a,b,1,2,2} = {a,a,..1.,1,2} :b: = a{1,2,2}b

{a,b,c,2,2} = a{c,2,2}b

{a,b,c,d,e} = a{c,d,e}b

{a,b,c,d,e,f,g,h} = a{c,d,e,f,g,h}b

{a,b,Z} = a{Z}b

Currently I have operator notations for up to 8 entry arrays (i.e. {a,b,c,d,e,f,g,h}), however they will be quite cumbersome to type out - a,b, and c are shown in numeric form, d is represented by brackets (as seen above), e is shown by [ ] like brackets, but rotated 90 degrees, where the brackets are above and below (uses e-1 bracket sets), f is shown by drawing f-1 Saturn like rings around it, g is shown by drawing g-1 X-wing brackets around it, while h is shown by sandwiching all this in between h-1 3-D versions of [ ] brackets (above and below) which look like square plates with short side walls facing inwards. - one of these days, I'll add a jpeg.

Extending Array Notation Bean system

Later I decided to extend array notation (which used a 1-D array) to allow for many dimensions in the array - I called it Extended Array Notation. Since multidimensional arrays are difficult to type out, Bird suggested using bracketed numbers to represent separating n-D structures, example: {3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3} represents a 3-D array - a 3^3 array of 3's to be exact, which when solved will give the enormous number "dimentri".

Following are the rules for Extended Array Notation using Bird's inline technique:

Rule 1: Condition - only 1 or 2 entries, only 1 dimension - {a} = a, {a,b} = a^b was originally a+b

Rule 2: Condition - rows ending with 1. - {# (n) a,b,c,...,k,1 (m)#} = {# (n) a,b,c,...,k (m) #} - # denotes remainder of array.

Rule 3: Condition - 2nd entry of main row (which is the very first row) is 1 - {a,1,c,d,..,k #} = a

Rule 4: Condition - 3rd entry is 1, next non-1 entry in same line - {a,b,1,..,1,d,e,..,k #} = {a,a,a,..,{a,b-1,1,..,1,d,e,..,k #},d-1,e,..,k #}

Rule 5: Condition - Rules 1-4 doesn't apply, there are at least 3 entries on main row - {a,b,c,d,...,k #} = {a,{a,b-1,c,d,...,k #},c-1,d,..,k #}

Recursion 2 over dimension types

Rules to evaluate expressions {U} in multiple array dimensions.
In every rule the left expression is reduced = to the right, and == is to repeat this rule.
A colon : applies a term-rewrite to a word, and := is one way substitution (right for left).

Signs in /cyan/ are syntax of our own Regular Expressions flavour, with reps :n notated inside now, directly after the selected word.
Dimensional separators are indexed or subscripted commas, so instead of inline (S) we write ,[S] inside arrays and equate ,[0] to ,[] to ,
Brackets [] open a new level in the array, following a comma ,[ they open a separator level ], else [ a rewrite array function is opened ] on a deeper level, which we precede by a D to denote a dimensional array.
We leave out the outer brackets {} of U for reasons of clarity.

  1. a,b := D[a,b] = a^b (initial power)
    1. D[a,1]D[a] = /1:a/ = a
    2. D[a,b1]/D[a,b]:a/
  2. X,[S]1/(,[1T]Z)?/ = X/_0 (drop trailing entry 1)
    so  X,[S]1 = X  where ,[],
        X,[S]1,[1T]Z = X,[1T]Z
  3. a,1,Z = a (inner collapse)
  4. a,b1,/1,:k/11Z = /a,:k1/{V},1Z (upload on row)
      if U = a,b1,Y
    then V = a,b,Y (precursor of U)
  5. a,b1,11Z = a,{V},1Z (value motor)
            == {a,..1.,1Z} :b1:
             = {a,..a.,1Z} :b:
  6. a,b/,[Si]:s/11Z = /D[a,b,Si],[Si]:s/1Z
    (dimension motor)
    1. D[a,b,1]/a,:b/1
    2. D[a,b,2T]/D[a,b,1T],[1T]:b/1
    3. else D[U]D[U'] (upload step in Bean)
      and V' := D[V] (decreasing precursor)
             if D[a,1,Z]D[a]a
       or V' := V0 (fixed precursor in Beaf)
              = a,b-/,[Si]:s/2Z
  7. a,b/,[Si]:s//1,:k1/11Z =
    (upload over dimensions)
    1. D[a,b,0]a,a
      (to cover rule 4. by S1=0 and s=1)

Physical evaluation issues

In Beaf subrule 6.3 the substituted V0 is fixed for all evaluations of new block series D[Xj],[Tj].. It is important to keep the same outer expression (and fixed precursor), if we want to refer outside of these dimension blocks.
For suppose you'd reduce one block D[a,b,1] on the left, and (before resolving the many blocks D in the middle) restart the motor rule of the outer expression a,a,a,Z. Then when you stop the motor at any step m this would result in varying outer Vm (for example if the second entry is down to 2 the projected Vm=a is smallest).
So if these erratic Vm were to substitute in the blocks D still waiting, the whole expression would be randomized (yet reducible to an extremely random number).

This happens when we work away dimensions D from left to right – the wrong direction! All (sub)expressions should be resolved from right to left.
So in Beaf we assign the precursor V0 of the original expression U0 to all substituted V' in the D then issued, and evaluate all block series of D from right to left.
But if the separator subscript S in a,b,[S]Z contains a similar array, for all next level D1[U] the precursor V1 must be derived and stored separately. We have to remember one Vn for each uploaded Dn level – and in our record number there are 104.

A less abstruse problem is that we usually have to substitute many words V before a number result is reached. If we work from the top down in these algorithms, the amount of simultaneous nestings of V keeps growing. Keeping track of the memory addresses to store those V alone produces a heap error on any computer.
To avoid this overhead, you'd best apply the counterintuitive method of evaluating from the bottom up – which in effect means building a complete system.
Also the Beat method of substituting the 2nd parameter value (instead of V) for all entries 1 to the right can be used to simplify rule 4 and 7. Because b is not significantly smaller than V (argumented below) this system is as powerful as Beaf.

Bowers had no business with it, so we call our system Beat Beate's Express Array Train and dedicate it to Beate Völker of Utrecht University, with Godspeed!
System Beat for a,b,1,..Z uploads b to all entries 1 along the way (instead of a). In the system Beate we keep the Extra substitution of V0 for the rightmost 1.

Eventually (...) any expression will be reduced fully, resulting in a Big number 1..
But usually the amount of dedicated supercomputer-filling universes to be able to perform such a calculation is not significantly less than the resulting number itself.

Rule 6: Condition - two entries on main row, next non-1 entry begins a row - (a,b (n)...(k) c #} = {b^n array of a's (n)...(k) c-1 #} - this is hard to describe, but the (n)...(k) could be anything from going to the next row (i.e. (1)) to something like this: (4)(4)(4)(4)(4)(3)(3)(3)(3)(2)(2)(2)(2)(2)(2)(2)(2) - which means going into the next 5 4-D regions, then into the next 4 realms (3-D), then into the next 8 planes - this example will be solved as {b^4 array of a's (4) b^4 array of a's (4) b^4 array of a's (4) b^4 array of a's (4) b^4 array of a's (4) b^3 array of a's (3) b^3 array of a's (3) b^3 array of a's (3) b^3 array of a's (3) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) b^2 array of a's (2) c-1 #}.

Rule 7: Condition - two entries on main row, next non-1 entry is preceded by 1's in it's row - {a,b (n) ... (k) 1,1,...1,1,c #} = {b^n array of a's (n) ... (k) a,a,...a{a,b-1 (n) ... (k) 1,1,...1,1,c #},c-1 #} - same thing with the (n) ... (k)

As you can see - the rules are getting very complicated to write down

Yet one can keep going into higher spaces, higher than dimensional space - consider the tetrational spaces (which includes super dimensions, trimensions, quadramensions, etc.) - what is this exactly - dimensional spaces can be represented as X^n R^n = {a,b,[n]2} where X represents a line, X^2 would then be a plane, X^100 would be 100 dimensional. Super dimensions are of this order - X^(X^n), trimensions, quadramensions, etc are as follows: X^(X^(X^n)), X^(X^(X^(X^n))), etc. - these kind of arrays are possible - so array notation can be addapted to these arrays, but the rules get trickier. Using Bird's inline technique we can represent super dimensional space as follows: {# (a,b,c,....,k) #} which represents going to the next a'th dimension of the b'th second level of dimensions of the c'th third level of dimensions .... of the k'th n'th level of dimensions - notice this is only super dimensions, it gets crazier in trimensions on up.

Let schematic expressions X^Y denote dimensional structures.

{3,2,[1,2]2} = D[3,2,1,2]

             = D[3,3,3] = {3,3,[3]2}

             = D[3,3,2],[2]D[3,3,2],[2]D[3,3,2] = R^3

Superdimensional Bean:

{a,2,[1,2]2} = D[a,2,1,2] = D[a,a,a] = R^a

{a,3,[1,2]2} = D[a,3,1,2] = D[a,a,{D[a,a,a]}] = R^R^a

{a,b,[1,2]2} = D[a,b,1,2] = R^^b

{a,3,2,[1,2]2} = {a,{a,a,[1,2]2},[1,2]2} = R^^R^^a

Super-pentational Bean: 

{a,b,2,[1,2]2} = R^^^b 

{a,2,c,[1,2]2} = R^..a ^:c

{a,b,c,[1,2]2} = R^..b ^:c1

But Arrays can go even further - there are also pentational arrays such as an X^^^6 array which would be an X^^(X^^(X^^(X^^(X^^X)))) array where X^^X alone would be an X^(X^(X^(....^(X^X))....)))) array. Bird has attempted to write rules for extending past dimensional arrays and appears to have captured rules that fit tetration and possibly pentation - but they get quite tedious.

Below is a picture showing a 2 superdimensional array that leads to the number "dulatri" - the yellow brackets represent dimensional groups - inside could take on any number of dimensions, they represent X^X structures, the red brackets represents an X^X array of X^X arrays - or an X^(2X) array.


Exploding Array Function Beaf principles

Finally, after a bit of tweeking on the array notations, I came up with a much simpler way of defining the master function behind it all, I call it the Exploding Array Function, feel free to call it the Bowers's Exploding Array Function (puts a new meaning to the concept of BEAFing up something). Before we get to the three rules of the BEAF, here are some definitions:


Array - a structured set of entries (the array will be represented as A)

Entry - a position in the array that holds a positive integer value (1 is default - so any entry not represented automatically has the value of 1)

Base - first entry (value represented by b)

Prime entry - second entry (value represented by p)

Pilot - first non-1 entry after prime entry (can be as early as 3rd entry, can be in a different dimension or beyond)

Copilot - entry right before pilot (if pilot is the first entry on its row, then there's no copilot. Copilot might be the prime entry itself, if the pilot is 3rd entry.

Structures - structures are sub-arrays of the following types: entry (X^0), row (X^1), plane (X^2), realm (X^3), flune (X^4),..,X^n,..,dimensional group (X^X)=X^^2,..,n-D array of dimensional groups (X^(X+n)),..dimensional group of dimensional groups (X^(2X)),...X^(2X+n),...m-D array of dim.group of dim.group of.....dim.groups n times (X^(nX+m)),...dimensional gang (X^(X^2)),...X^(polynomial),...X^(X^X)=X^^3,....X^(X^X+polynomial),........X^^4,....X^^n,...X^^X,......X^^^X,....X^^^^````^^^X,...etc.

Previous Structures - previous entries are entries before the pilot but are on the same row, previous rows are rows before the pilot's row, but are in the same plane, previous n-spaces are n-spaces in the same n+1-space that are before the pilot's n-space, etc. They are structures before the pilot's version of that structure, but on the same structure that is of higher order.

Prime Block - the prime block of an X^n structure is the first p^n block of entries left of an n-dimensional separator (n) in that structure, - this generalizes to all structures, turning the X's to p's, where p is the prime. - so if p=10, and the structure is X^(3X^5+2X^4+3X+6), the prime block is exploded to the first 10^(3*10^5+2*10^4+3*10+6) set of entries.

Airplane - a set of entries that include the pilot, all previous entries, and the prime block of all previous structures.

Passengers - any entry in the airplane that is not the pilot or copilot. This will always include the base.

Allowance: Default entries (entries that have a 1 value) can be chopped off as long as none of the entries change their positions.


Rule 1 - the prime rule - Condition: default prime (p = 1) - Result: v(A)=b. where v(A) means value of array.

Rule 2 - the initial rule - Condition: no pilot - Result: v(A)=b^p.

Rule 3 - the catastrophic rule - Condition: first two rules don't apply - Result: Array changes in the following way:

  1. pilot decreases by 1.
  2. copilot (if exists) becomes original array, but with its prime entry reduced by 1.
  3. (*Prime Blocks of) Previous Structures X^n are exploded to sizes p^n & b values, so all passengers take on base value.
  4. remainder of array remains unchanged.

That's it!! - much simpler - and even more powerful, in that it can be continued as long as the structures can be defined.

Notational rule 0 - finalize trailing entries - Allowance to drop any default (dead) and last (right) entries with value 1 together with their preceding (left) separator if it is trailing too. This is not strictly necessary (for further evaluation), but cleans up the expression.
Provided the signifying (highest) right separator of (*the Prime Block of) every Previous Structure remains, we can choose to drop the default entries with their (lower) left separators, because rule 3c (or 3c*) will overwrite all defaulted substructures with sizes very much larger than the current ones.
A left separator (n) of an entry 1 is considered "trailing" (and redundant), when it has a larger (*or equal) separator (m) directly on the right.

The difference in rule 3c - between exploding only the Prime Blocks (3c*) of Previous Structures, or keeping and exploding all Previous Structures up to the pilot entry - is often negligible. When higher separators occur further on the right (past the pilot) this rule difference is provably insignificant for the resulting Big numbers. The development of higher structures dominates the algorithm.
Nevertheless we feel that keeping all separators signifying Previous Structures (*not just Prime Blocks) in rule 3c would be the natural method for Beaf. Natural, because Bowers allows multiple b,p (n).. (n):m to explode to m series of (n) separated structures of sizes p^n & b values. This is pretty obviously maximal, when applied to the explosion of run down Previous Structures.

Why high structures need deep uploads

In Bean dimensional structures X^Y are formed by the last of Bowers' rules 6 and 7. Here in Beaf and in Bowers' later Hypernomials and Array Sizes these (second level) structures X^Y are treated as a brand new notation, decoupled from their (first level) method of construction.
This is a slow deal, crippling the overall algorithm, and is contrary to the spirit of what we consider Jonathan Bowers' great discovery – the upload principle.

To say that an upload rule dominates the algorithm is an understatement. Take for instance Bowers' upload to the counted down entry c – this in itself is so powerful that, where Conway needs a whole row of parameters, Bowers requires just 4 to express an equally Big number.
Combined with a proper motor rule, the power of uploading its accumulated value is maximal. The upload rule takes this value b prepared all the way at the start (left) of the function array and sends it up to the highest available parameter, dimension size or (indeed!) structure.

But if we detach any higher structures from the original algorithm in which these are embedded, the feedback from the primary accumulation in b is interrupted.
When dimensional structures would build up from separate set of rules, as Bowers seems to imply, those new rules have to start iterating all over again. And the earlier motor-upload cascade would be broken.
Therefore the idea of simplification as presented above by Bowers is detrimental for his higher structures, which from then on increase scale at a subliminal (lower level) rate.

Value b can rival subexpression V

Bowers' algorithm maximizes the value of b before it is uploaded to the next entry c. The prime value motor of the Bean machine – rule 5 – is successively applied Big c number of times. This repetition dwarfs the difference between subexpression V and the value b, which amounts to a single application of rule 5 or only one c- countdown.

So the substitution of a maximal subexpression V in Bean may look impressive, but generally we can upload b instead, to get an almost equally fast increasing function.
Nevertheless we keep the rules of Bean as originally conceived by Bowers. And we add an option to subrule 6.3 to define Beaf so that its upload to extended dimensions will get feedback from original level V0 instead of reduced level D[V] substitutions.

In the next sections we look at the way that V or equally b are applied to upload or upgrade dimensions and structures in Bean rule 6 and 7.

Beaf remains reducible under deep recursion

In our definition of Beaf the dimension motor rule 6.3 substitutes the V0 subexpression from the original via dimension blocks into comma subscripts that quantify dimension types (formerly Bird's bracketed numbers). This happens over and over.
You may wonder if such an unfathomable construction maintains reducibility to natural number? Here we present an argument that it does, for the whole list of rules.

Rule 1,2,3 are trivial, they just simplify the expression by letting characters drop.
Rule 4,5,7 substitute a subexpression V in a position left of a higher iterator. In V entry b is reduced (by 1) and because the iterator entry is counted down (by 1) the expression as a whole is reduced.
This is a common method in Big number land – we insert subexpression V that reduces a parameter from the original U, and it goes. But to put V = U- would be a mistake – for example a*b = a(a*b)- is irreducible except for a=1 with 1.. {1:b} as a solution.

Rule 6,7 both count down the dominant entry right of the subscripted ,[S] which is kept (and not repeated). Now the newly substituted dimensions should have a smaller coefficient in S' for the expression as a whole to remain reducible.
Subrule 6.1 is trivial. Repetition times :b is huge, but systematically reducible.
Subrule 6.2 counts down a dimension before repeating it, no problem there.
Subrule 6.3 in Bean evaluates just the type of the comma subscript, following the rules above, which have already proven to maintain reducibility.

Subrule 6.3 in Beaf also just evaluates the type of the comma subscript, but it pastes the subexpression V0 from the original or top level into the level of the subscript. And it does so persistently, but always as a substituted value, never as an inserted word.
We carefully check that there is no automatic nesting of subscripts in subscripts taking place, which could happen at a more advanced stage of the algorithm.
This holds for any subscripted comma in a dimension block D which will be redistributed within commas in between blocks – because it surely lived on the same depth in the original expression before. And this holds for the subexpressions V0 – each is to be evaluated individually to Big number.

In our experience, the further the algorithm is expanded, the smaller the relative gap between the accumulated b and V becomes. So to simplify the issue we may fill in b in place of V0 in place of V' in subrule 6.3 of Beaf, and look again.
This transformed dimensional expression X' can be transposed into an expression of Beat, by replacing all inserted a by b. This expression is larger, but similar to Bean, and therefore reducible in place in D the same way as it is on the original level.
This solves the problem – all expressions of Beaf (and likewise of Beate) are reducible to natural number.

Is Beaf significantly greater than Bean?

Generally, all exploded dimensions D are dominated by the size b of their highest dimension (in the reduced expression followed by the original ,[S] separator type).
The newly exploded dimensions are huge, but the right part with the higher parameters in the original (that are missing from the dimension blocks D) is more significant. That is because all dimensions D are constructed simply by one iteration on the right – an angel's feather – the right part can outlast any lower dimension…

So the substitution of the longer sequence V0 in subrule 6.3 of Beaf recursively wins over the upload of a shortening D[V] inside the dimension block of Bean. But the application of this rule is a rare (though systemic) occassion. We doubt if this effect is stronger than the size b inflating the dominant dimension directly in both systems.

We found that value b can rival subexpression V (as V is equivalent to a single countdown of Big c). So in theory both V' could well be replaced by the same b.
And yes, in each block D we find the original b in place, but once it is uploaded in subrule 6.3 Bowers puts value a in second position. Now in top level expressions the next b will be Bigger than before, but D[a,a,c,Z] has no motor rule to increase the second entry.
So Bean's subrule 6.3 from the second call onwards will upload the fixed value of a. And subrule 6.2 just counts off c to spread out dimensions of size a which is small, but not significantly, as the highest dimension size b got out first and it still dominates.

A good working motor-upload cascade produces the Biggest numbers. But the inner motor in D is turned outward, it does not accumulate and the high upload value – the dimensional subexpression D[V] – is increasingly ineffective against the original V0.
This results in a subscript algorithm in Bean which is weaker than that of Beaf.
Still, the greatest dimensional gain is made at the start, and the first V' is strongest because it contains the original b (comparable to V0).

Our conjecture is that Beaf is NOT significantly faster than Bean [put to the test later!].

The systems Eat and Beat we introduced earlier don't replace second entry b in the upload, and b is comparable to V, so our systems are not significantly slower than Beaf. This must hold even when higher structures get involved.
On the brighter side Beate runs ahead of Beaf, but not much. The only difference lies in their upload rules with its rare substitution of insignificant parameters by b instead of a.

On creating world record numbers

This is not the place to eat any more Beaf, because there's no need to produce any Bigger numbers in what follows. After the superexponential array dimensions Bowers conjures up a cornuscopia of characters and carefree notations, that are more inspirational than practical, and that do not help your Bigness grow very fast.

It's a difficult job, but we must find out what Jonathan Bowers once intended.
Using the Bean rules listed above we continue at a leasurely pace to meet with Bowers's world record number Meameamealokkapoowa Oompa! and calculate it rigourously (present proof by estimation) for the very first time.

Woman skiing on a snowy hill From that Beate's version will follow, already markedly larger. Superduperseded by her system's highest *** Beate's Mountain Top *** the new largest number in the world !!!
(A good thing! Juggler-mathematician Ronald Graham recommends having a number on your name, no matter if you invented it or what ;-)

And there Beate goes, off piste from that snowy mountain, descending with great speed, like forever… while she keeps on wondering,  “Why can't I stop?”

After this article was written an email came in from Chris Bird's mother, that Chris had updated his array functions, specially the Beyond Bird's Nested Arrays articles I,II & III, which (in their new form) give rise to much larger numbers.
So Beate's Mountain Top may have been the best defined largest number in the world for 20 days, but it is no more (or ever was?) a world record.

Types of Arrays


Linear arrays consists of one row, ex: {3,5,6,1,3,4,2} - these are the smallest and simplest arrays - even though the arrays are small, the value of the arrays can be so huge that not even Conway's Chained Arrow notation can keep up (an array with only 5 entries will clobber Chained Arrow!!). The positions on the array can be represented by a positive integer, i.e. the 17th position.


Dimensional arrays have a multidimensional structure - these are the second smallest arrays. These arrays consists of rows, planes, realms, flunes (4-spaces), and various n-spaces. The rules above can easily handle dimensional arrays - although the results seem to nearly reach infinity. The positions in the array can be described with a linear array, i.e. (4,5,3,1,2) represents the 4th entry on 5th row on 3rd plane on 1st realm on 2nd flune. Dimensional arrays can be represented as follows: {4,2,5,7 (2) 5,6,1,2 (2) 5,4 (1) 6 (3)(3)(2) 2,5,6 (1) 1,1,1,7 (2) 6,8,5,2 (5) 7,5,6,8}. Here (2) represents going to next second dimension (or going to next plane), (3) is next realm, (5) is next 5-space.


These arrays go way beyond dimensional arrays, and take on structures that require tetrational spaces. They consist not only of rows, planes, realms, etc, but also dimensional groups, rows of groups, planes of groups, groups of groups, groups of groups of groups of groups, gangs, rows of gangs, groups of gangs, gangs of gangs, realms of groups of groups of gangs of gangs of gangs, etc, etc, etc, superdimensional groups, trimensional groups, etc,etc. These are the third smallest arrays (and the largest type that I have fully grasped how they work). Tetrational arrays consist of super dimensional arrays, trimensional, quadramensional, etc in the same way dimensional arrays consists of planar, realmic, flunic, etc arrays.

Positions of superdimensional arrays can be described by a dimensional array, i.e. (5,6,3 (1) 3, 5 (1) 6,7 (2) 5) represents the 5th entry of the 6th row of the 3rd plane of the 3rd dimensional group, of the 5th row of groups, of the 6th group of groups, of the 7th row of group of groups of the 5th dimensional gang. To represent a 3^(3^2) array, let A represent the 3^3 dimensional group {3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3 (2) 3,3,3 (1) 3,3,3 (1) 3,3,3}, now a 3^(3^2) array of 3's would be {A,A,A (1,1) A,A,A (1,1) A,A,A (2,1) A,A,A (1,1) A,A,A (1,1) A,A,A (2,1) A,A,A (1,1) A,A,A (1,1) A,A,A} To get a 3^(3^3) array let A be equal to the group of groups mentioned above (the 3^(3^2) array that is) and let (1,1) and (2,1) change into (1,2) and (2,2) respectively.

Positions of trimensional arrays can be described by a superdimensional array, positions of a quadramensional array can be described by a trimensional array, etc.


Next in line are pentational arrays, these require pentational spaces. Trying to understand how these work might give someone a migrane. One would need to keep up with tetration levels which can also get out of hand. There will be tetrational groups that show up here, X^^X arrays are more tidy examples of tetrational groups. a X^X^X^X^X array is a tetrational one, quintamensional to be exact (keep in mind that powers are solved in reverse) - for pentational arrays, the powers will sort into groups, i.e. an X^X^X^X^{X^X^X^X^X}^{X^X^X^X^X}^{{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}^{X^X^X^X^X}} array, the {} are not to be solved like ordinary parenthesis, but are used to group up the exponents into tetrational blocks - the number of entries in this array can be found out by removing the {} and solving - ie. X^^39 in this case - and this is barely scratching the beginning of pentationals.

Beyond Pentationals:

Hexational, heptational, expandal, multiexpandal, explosal, powerexplosal arrays, etc. arrays so huge that the space its in needs to be represented by array notation (linear, dimensional, tetrational, and so forth) - how to work with these? - Only God knows - but they should form some massive arrays - and utterly unspeakable numbers when solved.

After superdimensions come the Next Legion arrays:

{3,3,1,2,[1,2]2} = {3,3,{v},[1,2]2} = 3^..3 ^:v

    v = 3,2,1,2,[1,2]2 = 3,3,3,[1,2]2 = 3^^^^3 

{a,3,1,2,[1,2]2} = a^..a ^:v ≅ a1&a&a&a

    v = a,a,a,[1,2]2 = a^..a ^:a1 ≅ a1&a&a

Bowers devises new notations for old style recursions: 

a,b1,1,2,[1,2]2 = {a,a,..a.,[1,2]2} :b:

    ≅ a1.&a.. :b1 ~ {a1,b2 /2} = Â

Big boowa example:

3,3,3 /2 =

  3,v,2 /2 = 3,..1. /2 :v:

    v = 3,2,3 /2 = 3,3,2 /2 = 3,3&3&3 /2  <~

3,3,3,2,[1,2]2 =

  3,v',2,2,[1,2]2 = 3,..1.,1,2,[1,2]2 :v':

    v' = 3,2,3,2,[1,2]2 = 3,3,2,2,[1,2]2

       = 3,{3,3,{3,3,3,[1,2]2},[1,2]2},1,2,[1,2]2

Bowers' 2nd legion defined on the row:

a,b,c,2,[1,2]2 ≅ {a,b,c /2}

a,b,c,d1,[1,2]2 ≅ {a,b,c,d /2}

a,b,c,d1,R,[1,2]2 ≅ {a,b,c,d,R /2}

Structures preceding the 3d legion:

a,2,[1,2]3 = D[a,a,a],[1,2]2 = a^a /2

a,b,[1,2]3 = D[a,b,1,2],[1,2]2 = a^^b /2

a,3,2,[1,2]3 = a,{a,a,[1,2]3},[1,2]3 = a^^^3 /2

a,b,c,[1,2]3 = a^..b /2 ^:c1 ≅ c1&b&a /2

Next legions defined on the row: 

a,b,1,2,[1,2]f1 ≅ {a1.&a.. /f} :b ~ {a,b1 /f1}

a,b,c,1R,[1,2]f ≅ {a,b,c,R /f}

The Next Legion:

Let & represent "array of", so 3 & 3 = {3,3,3} = tritri = 3 to the power of itself 3^27 times - now consider 3&3&3 (solved frontwards) this is a {3,3,3} array of 3's which is a 3^^^3 array of 3's - now imagine a new array notation such that the second rule changes to this:

Condition - no pilot, result - v(A)=b&b&b&b&.....&b&b -- p times - welcome to the next level of array notation.

Another way to represent this is to simply add an entry with value 2 in a second "legion", where the first legion can take on any of the previous arrays - {3,3 / 2} = 3&3&3, the big boowa = {3,3,3 / 2} = 3&3&3&3&3&............&3 - where there are 3&3&3.......&3 3's arrayed to each other - where there are 3&3&3.....3 3's arrayed to each other - where there are 3&3&3.....&3 3's arrayed to each other - ... - ... - ... - - - - this is said 3&3&3&3&......&3 times - where there are 3&3&3 3's arrayed to each other here before getting to the end (which is "where there are 3 3's arrayed to each other")!!! - keep in mind that in {3,3,3 / 2} 3,3,3 are in the first legion and 2 is in the second legion - this 2 represents "second level of array notation".

One can consider {b,p / 2} as if the pilot is in the 2nd legion, and in this case the airplane is defined to be a b&b&b...&b p times - array in the first legion - of course you can go to the 75th level of array notation if you like - {b,p / 75} = {b&b&b....&b - p times - array / 74}. Note that {A / 1} = A, so 1's are still default.

Linear Legion Arrays:

{A1 / A2 / A3 / A4 / ..... An} - here each of the An's are arrays (can be linear, dimensional, tetrational, pentational, etc - up to but not including legion arrays) - nuf said!!!

Linear legion arrays resolved from right to left:

D[a,b,c],[1,2]f  = Â /f

a,b,1,2,[1,2]2,2 == a,v,[1,2]1,2  with v = Â

     = D[a,v,1,2],[1,2]{a,v-,[1,2]1,2}

    == D[a,v,1,2],[1,2]{..1.} :v:

     = R^^v,[1,2]{..a.} :v:

     = Â,[1,2]{..a.} :v:

     = Â /..a  ~ Â(/2)Â = Â(/3)2

Arrays of Bean translated to Bowers' dimensional legions:

a,b,2,2,[1,2]2,2 = Â(/2)Â(/2)Â = Â(/3)3

a,b,c,2,[1,2]2,2 = Â(/3)c1

a,2,1,3,[1,2]2,2 = a,a,a,2,[1,2]2,2 = Â(/3)a1

a,3,1,3,[1,2]2,2 = Â(/3)Â(/3)a1 ~ Â(/4)3

a,b,1,3,[1,2]2,2 = Â(/4)b

a,b,2,3,[1,2]2,2 = Â(/5)b

a,b,c,3,[1,2]2,2 = Â(/c3)b

Further array structures are ill-defined, we imagine that:

a,b,R,[1,2]2,2 = Â(/R)b

a,b,[1,2]3,2 = Â(/Â)b ~ Â//2 (Legion legions)

a,b,R,[1,2]3,2 = Â(/Â)Â(/R)b

a,b,[1,2]c1,2 = Â//c ~ Â(//2)2

a,b,[1,2]c,3 = Â///c ~ Â(///2)2

a,b,[1,2]c,d = Â/..c /:d

a,b,[1,2]c,d,2 = Â/../(1)c /:d

a,b,[1,2]c,d,e1 = Â/../(1)..c /:d /(1):e ~ Â/(2)e

a,b,[1,2]ci,..z ci,:n ~ Â/(n)z 

a,b,[1,2],[1,2]2 ~ Â/(Â)b ≅ L (Legion space) 

Dimensional Legions, Tetrational Legions, Legion Legions, and beyond:

One can take arrays further to include things like the following:

{A / A / A / A (/2) A / A / A (/3,4,5,2) A } - this is a tetrational legion array.

{3,3 // 2} - a legion legion array - this is equal to 3&&3&&3, where 3&&3 = {A / A / A} where A = 3&3&3 array (in other words 3&&3 is a 3 legion array of 3's), 3&&3&&3 = 3&&3 legion-array of 3's. - note that a mere 3^3 legion array of 3's would be {A / A / A (/1) A / A / A (/1) A / A / A (/2) A / A / A (/1) A / A / A (/1) A / A / A (/2) A / A / A (/1) A / A / A (/1) A / A / A} - which is nothing compared to a 3&&3 legion array of 3's

{3,3 /// 3} - a legion legion legion array - which equals 3&&&3&&&3, where 3&&&3 = {A // A // A} where A = 3&&3&&3 array of 3's

{3,3 //////////////// 2 } - use your imagination.

{3,3 ///(1)///(1)///(2)///(1)///(1)///(2)///(1)///(1)/// 2 } - the next step would be to give the legions themselves an added structure - {3,3 (1)/ 2} (the legion mark is in the second row of legion marks) - but how would we define this, first lets come up with a new kind of space called "legion space" - we can think of it as the highest extent of space before we encounter the legions, in the same way that X represents the line, and X^X is above all dimensional space, legion space "L" is above all of the non-legion array notation spaces - that means it is beyond tetration {X,n,2}, pentational {X,n,3}, linear array-spaces {X,X,X,X,X,X,...,X}, dimensional array-space 100^100 & X's, and even X&X&X&X&X&5 - the prime block of a legion space would be b&b&b&b....&b - p times (b=base). Legion of legions describe the double legion marks //, while a legion "of itself" 50 times would have 50 legion marks - so lets define {L,n}b,p as legion of legion of legion of....legion - n times using b and p as the base and prime. {L,1}b,p = {b,p / 2} = b&b&b&b&b... p times, {L,2}b,p = {b,p // 2} = b&&b&&b&&b... p times - which is the prime block of a legion of legions. {L,X}b,p = {b,p (1)/ 2} = {b,p //////....///// 2} - p /'s - notice how the previous row of /'s is the prime block of the row of /'s - so we can still use that rule here. Next we can do {L,A} - where A is an array-space - this will lead to the legion marks themselves taking on an array structure, not just the separate legions. We can continue with {L,L} - here the legion marks will take on legion space - and it gets worse, for {L,L} = {L,2,2}, so lets go for {L,3,2} - this would be {L,{L,2,2},1} = {L,{L,L}} - the legion marks taking on a legion of legions - can you feel the tension. Next would come things like {L,X,2}, {L,A,2}, {L,X,6}, {L,L,L,L,L,L,L,L}, 100^100 array of L's - this is array notation on the legion space - all this will lead to the next group - "lugions"

In Bowers' gallery of numbers a Mark Lugo steals the show:

a,b,ci..,[1,2],[1,2]2  ,ci:n  ≅ {L,..1} L,:n2

a,b,[1,2],[1,2]3  ≅ L2

a,b,[1,2],[1,2]n1  ≅ Ln

Approximately Bowers' last batch:

a,b,[1,2],[1,2],[1,2]2  ≅ LL

a,b.,[1,2]..2  :n1  ≅ L.. :n

a,b,[2,2]2  ≅ M 

A plausible upper bound to Bowers' final fantasy: 

a,b,[13,5,2]2  ≅ MEA!

a,20,[1,[1]2]2 ~> MEAMEAMEALOKKAPOOWA!

10,20,[15,15,13,16,2,[1]2]2 ~>


Lugions, Lagions, Ligions, and beyond

We will use the following to describe lugions, the lugion mark is \, "legiattic array of" mark is @ (symbols were suggested by John Spencer), and finally Lugion space will be L2. {L2,1} b,p = {b,p \ 2} = b@b@b@b....@b - p times - where n@m = {L,L,L,.....,L}n,m - there are m L's - so a 3@3 legiattic array is {L,L,L}3,3 - a 3@3@3 legiattic array would be {L,..a....,L}3,3 where a = {L,L,L}3,3. We can continue with double lugions (or lugions of lugions) {L2,2} cases which lead to double lugion marks in the arrays "\\". Look what can be done to the lugiattic space arrays - {L2,X}, {L2,A}, {L2,L}, {L2,B} - B is a legiattic array, {L2,L2} = {L2,2,2}, {L2,L2,L2,L2,L2,L2}, 100^100 array of L2's, legiattic array of L2's - and finally the lagions L3 - we can do similar things there. Lagion marks are |'s, we can use % to denote "lugiattic array of". L4 space is the ligion space, we can use "-" for ligion marks, and # for "lagiattic array of". Of course we can continue to L5,L6,...L100,...L(gongulus),......

Going even further

Now imagine LX space where the X can change to p when finding the prime blocks of the space, L(X+1), L(2X), L(X^2), LA, LL, LL2, LLL, LLLLLLLLLLL, LLLL(1)LLLL(1)LLLL (3 rows of L's) - 10^100 array of L's (no comma's between them this time) - just use the imagination to where this could lead . . . MEAMEAMEALOKKAPOOWA OOMPA!

Beate is the acronym for the algorithm Beate's Express Array Train Extra

and its rules are discussed above as a variation on our definition of Beaf,

with series of the (accumulated) b uploaded left of V (instead of items a).

Inspired upon our restoration of the ancient Indian Asamkhyeya

– the incalculable number 10^(5*2^104) (a tower of squares).

    Beate's Mountain Top =

    10,5,[.. .]2  :104:  in Beate  ,[ ] ≡ ,

Algorithmically larger than the highest of Bowers' numbers, but still below infinity

– for a higher type of countable infinity, replace the coefficients above by omega ω.