bigΨ.I. Number operations

Ψ.3. Chained arrows versus subscripting

# 3.1. Subscript mixing

§3.1.1. Mixed majority arrows?

Our formula for ^R; subscripted arrows did not cover mixed subscripts ^m;^n; {m≠n}
For a sequence of majority arrows with competing unequal subscripts the question is which one to evaluate next. We solve this issue by extending the fixity rules for arrow operations – namely right associative majority precedence – to include subscripts.
Majority precedence will apply externally between two subscripted operations, as well as internally in a single operation, where it chooses the reduction to apply first.

The right associative order of evaluation is only decisive, when subscripts are of equal size, in other cases size determines which operator is evaluated first. We do not use brackets in subscripts.
Subscripts live in a complicated multicultural environment, and the benefit from wielding such an expressive tool is increased notational resolution (more numbers can be expressed).
We won't cover all the combinations, but explain the most essential mixed arrows in a few examples. The smaller arrow in the operator reminds us of the postponed backslashed operation – it waits its turn.

a^1;^;b = a^;...b {^#b1} = a*1;b11\--
a^m1;^n;b {0<n<m1} = a^n;^m1;b = a^n;^m;... {a#b}

After a certain reduction step the companion arrows ^n put on hold because of smaller size, will merge with the new, equally subscripted operators ^m; {m=n}.

Because the mixing of operators resolved by majority precedence doesn't add significantly to the size of the resulting numbers (compared to the value of an operator's coefficients), these should be treated as an obscure variant. In our book of Big numbers the defining rules never allow for mixing of majority operators.

§3.1.2. Mixed minority stars

Also in the definition of *R; subscripted star operations we extend their fixity rules – namely right associative minority precedence. These decide which reduction step to apply to a combination of mixed star subscripts.
Subscript inequality unfolds as a great tool in the construction of Big numbers. Given our observation that mixing of unequal arrow subscripts ^R is insignificant, we'll make a bigO comparison between the two subscript systems of minority stars and majority arrows.
The sign <~ means the left expression is algorithmically close(st!?) to the expression on the right, yet smaller.

a*1;b = a*..b {*#b} = a^..b {^#b-} <~ a^1;b <~ a*1;b1
a*1;*b = a*1;.. {a#b} <~ a^1;.. {a#b} = a^1;^1;b
a*1;**b = a*1;*.. {a#b} <~ a^1;^1;.. {a#b} = a^1;^1;^1;b
a*1;*1;b = a*1;*..b {*#b} <~ a^1;..b {^1;#b1} <~ a^2;(b+1)
a*1;*1;*b = a*1;*1;.. {a#b} <~ (a+1)^2;.. {(a+1)#b} = (a+1)^2;^2;b
a*1;*1;**b = a*1;*1;*.. {a#b} <~ (a+1)^2;^2;^2;b
a*1;*1;*1;b = a*1;*1;*..b {*#b} <~ (a+1)^2;..b {^2;#b2} <~ a^3;(b+2)

Star subscripts are way ahead – when they put value d=2 arrows need a 2nd parameter e.
And when stars append the next iterator e, arrow subscripts come to the end of their first row.

a*2;b = a*1;..b {*1;#b} <~ a^b;(b+b) <~ a^1,1;(b+1)
a*2;*b = a*2;.. {a#b} <~ (a+1)^1,1;.. {(a+1)#b} = (a+1)^1,1;^1,1;b
a*2;*1;b = a*2;*..b {*#b} <~ (a+1)^1,1;..b {^1,1;#b1} <~ a^2,1;(b+1)
a*2;*2;b = a*2;*1;..b {*1;#b} <~ a^b,1;(b+b) <~ a^1,2;(b+1)
a*3;b = a*2;..b {*2;#b} <~ a^1,b;(b+1) <~ a^1,1,1;(b+1)
a*1,1;b = a*b;..b {*b;#b} = a*b1;b <~ a^1,..;(b+1) {1#b1}

The progression of mixed stars versus unmixed arrows gives us a tool to study similar systems at work on different levels. Minority stars are comparable to majority arrows, yet the stars as defined by the first row of parameters in bigI travel on a higher function level past the 4th parameter (compared to the arrows of bigE).
The general definition of subscripted minority star operations has a somewhat broader scope.

  • aS*..b1 {*#c1} = aS*..(aS*..b) {*#c *#c1}
                  == aS*.. ... {*#c a#b1}
    where S = *Tj;;.. .:. {*Tj;;#kj; *Tj;;..#m}
    where Tj; = tjp,tjpi,.. {tjpi#n tjq≠0} and Tj; > Tjd; {d>0}
  • aS*1,..,1R;b {1#+k} = aS*b,..,R;...b {b#k *b,..,R;#b}
    same S with Tj; > 1,..,1R
  • aS*1R;...b {*1R;#c1} = aS*1R;...*R;...b {*1R;#c *R;#b}
    same S with Tj; > 1R

The order of star subscripts *T; is quite intuitive, even if the conditional comparisons T > R are difficult to define exactly. In the reduction train of bigI to natural number this is usually not a problem – when we start from the evaluation of an expression a*R;..b {*#c} all subsequent mixed subscripts will satisfy these conditions.

# 3.2. Pluses in row context

Disk of green jade with centre hole and stylized animal motives on the edge

Sun-faced buddhas, moon-faced buddhas. The three sovereigns and the five emperors. How pale they look…
For 20 years I've had bitter experiences. I have descended into the Green Dragon's Cave. For whom? I cannot tell the depth of it.
Anybody here with a clear eye? Do not be careless about this!

– Ma-tsu in The Blue Cliff Records 3.

§3.2.1. Number plus subscripts

The venerable plus character snatched new meaning when the left associative unary operator + of bigA was defined with a single subscript a+c;... {+c;#b} in chapter 2.7.
Left unary pluses are ruled by very simple concepts. As we saw in the 0th dimensional case + equals 1. Such pluses function as subscriptable ones.

Another way to look at this + is as an opening sign for a deeper nesting, virtually delimited by a closing semi-colon, as in +1..; {1#c} where if c=0 the initial +; is (reduced to) the unit 1.
Any which view on + you take – operator, number or bracket – its reduction rules and results stay the same.

  • a+0;Z = a+Z = a1Z (exchange)
  • a+c1;Z = a+c;...Z {+c;#a}

Brackets () aren't necessary in the reduction process of left unary pluses, because the leftmost operators +c; right after the number operand a are evaluated first. But brackets are allowed to help write out bigA expressions.
Now try to fit a 4th coefficient d into this evaluation system, modelling it after parameter d of majority arrows.

  • a+++;; = a++1;; = a+1,1; = a+a1; = a+a;... {+a;#a}
  • a+++;;... {+++;;#b1} = (a+1,1;)+1,1;... {+1,1;#b}
  • a+++;...; {++;#c1} = a+c1,1; = a+c,1;... {+c,1;#a}
  • a+++...;; {+#d1} = a+1,d1; = a+a1,d; = a+a,d;... {+a,d;#a}

And continue to a row of equally subscripted +R; operators, comparable to a row of majority arrows.
By the last rule below, with k=0, expansion from single to multiple pluses is expressed.

  • a+1,..; {1#k1} = a+a,..;... {a#k +a,..;#a}
  • a+1R; = a+R;... {+R;#a}
  • a+R;... {+R;#b1} = (a+R;)+R;... {+R;#b} (left associativity)
  • a+1,..,1R; {1#+k} = a+a,..,R;... {a#k +a,..,R;#a} (upload)

The upload axiom maximizes the number k of subsequences 1, because its meta-repetition sign is greedy. In case R is empty 0 we must apply the generic post-countdown axiom +A,; = +A; for upload to work.
With exchange and upload for axioms, the only possible way to reduce expressions with pluses is left associatively. In fact if you're happy to keep the resulting amount in + currency, then the exchange to 1.. type numbers isn't necessary, not even for operand a.
All in all it's remarkable that a single non-generic rewrite axiom is enough to create some uncanny large numbers.

Show an example how a computer program that writes to a standard line can keep track of distinctive nesting levels.

11111;;; = 1111;1;1;; = 1111;1;;11;1;; = 1111;;11;;11;1;;
         = 111111;;11;1;; = 1111111111;1;; = 1... {1#2048}

+++++ = 2++1;; = 2+3; = 2+2;+2; = 2+1;+1;+2;
      = 2+++1;+2; = 4+1;+2; = 4+++++2; = 8+2;
      = 8+1;... {+1;#8}
= 8*2^8 = 2048

So the constructs 1; and +; are isomorphic following the rules of operatorial bigA. Though subscriptable + seem to function as left unary operators, in the left associative context of words these can better be characterized as unaries (just that) or as extendible units 1.

§3.2.2. Can pluses be mixed?

The question arises whether we can successfully mix unequal subscripts into the general definition of these unary plus operators? Does the possibility of a+m+n {m≠n} work out naturally into a tremendously fast operator system, parallel to the way mixed subscripts for minority stars followed up on superpowers?

Because multiple pluses are left associative, as determined above, the answer seems no. And indeed, if mixed pluses were ruled by minority precedence we could reduce such a minority pluses system by simple recursion, mostly expanding and eventually cashing in pluses from the left.

a+R;... {+R;#b} == (.a+R;).. {#b#+R;}

But when we agree to apply majority precedence (with the operator sign plux ×), a larger subscripted ×R; operator can extend the range from operand a to include all the smaller subscripted ×R'; operators in between.
Given an expression with mixed pluxes, the largest subscripted (leftmost) operator is evaluated first and all the mathematical terms on its left become subject to repetition (by character duplication).

Because a plux has a single left operand there's a nasty problem – for the greatest impact the iterator value to count these repetitions has to be derived from the repeated sequence itself. To achieve this the left sequence has to be reduced to natural number first. And then the original unreduced left characters are to be iterated over.
This cannot happen in place, but requires two separate programme states. Enumeration of states of computation is a complication issue we will not dive into in this article. Our focus is on general recursion and not computability, though we'll miss out on some truly Big numbers.

With unary operators mixed sequences such as in the general formula of mixed minority stars, if possible, will have to occupy some secondary space for evaluation. For instance, a plux at the right end could still employ the empty character space on its right to perform a calculation, but this extension of operator sequences to derive an iterator value essentially uses temporary (nested) right operands.
These alternatives of countable programme states and virtual extra operands are not truly what operators are about. Such constructions are better set in the framework of functions and their operatorial extensions in part two.

We will forget about majority pluxes and happily raise bigA on the foundation of subscriptable pluses, which look so simple but are to our surprise more powerful than the arrowheads ^.. of operatorial bigE. This is proved in chapter 4.2.



chapter 3.3, edit 0.X.X
for John Conway who loves cats

# 3.3. Arrows quarters

Under 2011 Construction

§3.3.1. Repainting up-arrows

The definition of Conway's chained arrows is apt to draw its own bow → skip the basic trio a→b→c = a^...b {*#c} and derive superpowers directly from its axioms for the first row. That is, provided the duo is defined separately as exponentiation a→b = a^b, which slows the row expansion down by a parameter position, compared to a more rigorous application of the scheme of chained arrows.

We will define a right arrow function using the sign as a parameter separator. On the operator axis a new algorithm for what we'll still call up-arrows (not Knuth's, but painted anew) forms the basis.
Our notation applies the convention of majority precedence for arrow operators, leaving out brackets when possible.
Let's walk through the different forms of up-arrows, following the trail of an expanding row of parameters to the right.

  1. An operator .. {#z} = a without operands to operate upon is strange, but natural z = 1.. {#a}
  2. The initial up-arrows a;00;.. are unary a↑.. {↑#z} with z counting their series.
  3. Then come the still unsubscripted yet doubly iterable binary up-arrows a↑..y {↑#z}
  4. Continue with subscripted up-arrows a↑R;..y {↑R;#z} which are unmixable, so you can either write for example a↑x;x;x;y or a↑x;↑↑y or a↑↑↑x;y as {↑x;#3} is what's intended.

Within a series of up-arrows different subscripts will not mix – describe this property in a logical statement.

a;SiRi;..y {;SiRi;#z} => Ri = R0 & Si = S0

We've painted the new zubscripts magenta (not red), to indicate that they shift to the right – with every final parameter z as an operator counter (instead of the fixed red c) and penultimate parameter y as the main iterator (instead of a fixed b).
For instance the operation a↑b,x;..y {↑b,x;#z} equals the expression a→b→x→y→z.
This isn't just cosmetic – we'll position those parameters that most increase function speed (create the Biggest numbers) to the right of function arrays. In the chained arrows algorithm the double iterator z functions as operator counter and has the greatest impact on the result, since there is no uploading of accumulated iteration values.

§3.3.2. Redrawing right arrows

Here we define a right arrow function with navy coloured separators to distinguish it from chained arrows.
Right arrows ai→... {ai#n} are roughly similar to chained arrows, but have three major differences.

  1. Of value importance the a↑R;.. operator counter z is counted down (virtually) to 0 (and dropped) instead of dropping 1, in support of our general notion of operatorial functions.
  2. Of value importance when penultimate iterator y counts down to 0 and collapses, it is revived by the pristine value of x. Only one parameter drops off, whereas chained arrows drops both last positions.
  3. Of parametric importance the main axiom of the R→y→z scheme applies directly to the first two parameters a→z past {a≠1 z≠1} the initial cases.

Because a→b→c = a^..b {^#c} = a^0;..b {^0;#c} = a→0→b→c seems to make sense, in principle any parameter value zero in between could simply collapse its position. Plausibly even those at the start.

  • X→0 = X
  • X→0→Z = X→Z
  • 0→Z = Z

Helped by our collapsing insight, only two axioms – the initial and the main axiom – suffice to define the row function of right arrows, together with its zubscripted up-arrow operators.
If however stubborn, we cannot accept axioms with zero values 0 up front as item or in between as iterator, then we must adapt our rules that count down from 1 to express the same state of affairs. We list them below.

  • a→1 = a↑ = a1
  • 1→b1 = (0→b1)→b = b1→b
         = 1↑.. {↑#b1} = b1↑.. {↑#b} ~> 2*..b3 {*#b--}
  • a1→b1 = (a→b1)→b
          = a1↑.. {↑#b1} = (a↑..)↑.. {↑#b1 ↑#b} ~> 2*..a3 {*#b-}
  • R→1→z1 = R→(R→0→z1)→z = R→(R→z1)→z
         {R:a}= a↑..1 {↑#z1} = a↑..(a↑..) {↑#z ↑#z1}
       {R:a→x}= a↑x;..1 {↑x;#z1} = a↑x;..(a↑..x) {↑x;#z ↑#z1}
     {R:a→Q→x}= a↑Q,x;..1 {↑Q,x;#z1} = a↑Q,x;..(a↑Q;..x) {↑Q,x;#z ↑Q;#z1}
  • R→y1→1 = R→(R→y→1)
          == R→(..R→1→1.) {#y#} = R→(..R→1.) {#y1#}
         {R:a}= a↑y1 = a↑.. {↑#z z:a↑y}
       {R:a→x}= a↑x;y1 = a↑..x {↑#z z:a↑x;y}
     {R:a→Q→x}= a↑Q,x;y1 = a↑Q;..x {↑Q;#z z:a↑Q,x;y}
  • R→y1→z1 = R→(R→y→z1)→z
           == R→(..R→z1.)→z {#y1#}
       {R:a}= a↑..y1 {↑#z1} = a↑..a↑..y {↑#z ↑#z1}
     {R:a→Q}= a↑Q;..y1 {↑Q;#z1} = a↑Q;..a↑Q;..y {↑Q;#z ↑Q;#z1}

The translation from these new up-arrows to right arrows doesn't look so smooth, but it's workable. The numbers created by our right arrows are a single parameter Bigger than Conway's chained arrows which is not terribly significant on a grand scale, where parameter rows R can be stretched to arbitrary length.
If you'd like to see how the above estimate a→b2 ~ 2*..a2 {*#b} of two parameter superpowers was reached, click here!

  • a→2 = (a-→2)→1 = (a-→2)1 == (0→2)a = a2
  • a→3 = (a-→3)→2 = (a-→3)2 == (0→3)(a*2) = (2*a2)-
  • a→4 = (a-→4)→3 ~> (a-→4)*2 == (0→4)*2^a ~ 2^a2
  • a→5 = (a-→5)→4 ~> 2^(a-→5) == 2^^a\^(0→5) ~> 2^^a2
  • a→b {a>1} = (a-→b)→b- ~> 2^..(a-→b) {^#b-4}
             == 2*..a\*..(0→b) {*#b-2 *#b-3} ~> 2*..a2 {*#b--}

If we'd fill in the extreme value 1→b1 in the main rule, our approximation of the two parameter case turns out rather crude.

1→b1 = (0→b1)→b = b1→b ~> 2*..b3 {*#b--}
     ~ 2*..3 {*#b-} = 2*..4 {*#b--} => b~1

§3.3.3. Mixing down arrows

Under 2011 Construction

Initially arrow expressions are without multiple subscripted operators – the opportunity for mixing didn't occur yet – this is where up-arrows and down-arrows do not differ in function and result.

  • a←y←z = a↓..y {↓#z} = a↑..y {↑#z} = a→y→z
  • a←x←1←1 = a↓x;1 = a←x←(a←x←0←1) = a↓..x {↓#z z:a↓x}
            = a↑..x {↑#z} = a→x→(a→x→1) = a→x→1→1 = a↑x;1
  • a←x←y1←1 = a↓x;y1 = a←x←(a←x←y←1) = a↓..x {↓#z z:a↓x;y}
             = a↑..x {↑#z} = a→x→z = a↑x;y1 = a→x→y1→1

Obviously the final parameter z stays unmixed, because it expresses the number of possible mixes. Less obviously the penultimate parameter y is reserved to contain a waitor subexpression and shouldn't be subscripted or mixed.
Follow the translation rules for revolving arrows around the clock.

  • a←R←x←0←z1 = a↓R,x;..0 {↓R,x;#z1} = a↓R,x;..R;x {↓R,x;#z}
  • a←1←1←2 = a↓1;1;1 = a←1←(a←1←0←2)←1 = a↓1;a↓1;↓1
            = a↑1;a↑↑1;1 = a→1→(a→1→1→2)→1 = a→1→2→2 = a↑↑1;2
  • a←x←1←2 = a↓x;x;1 = a←x←(a←x←0←2)←1 = a↓x;a↓x;↓x
            = a↑x;a↑↑x;x = a→x→(a→x→x→2)→1 = a→x→x1→2 = a↑↑x;x1
  • a←x←2←2 = a↓x;x;2 = a←x←(a←x←1←2)←1 = a↓x;a↓x;x;1
            = a→x→(a→x→x1→2)→1 = a→x→x2→2 = a↑↑x;x2
  • a←x←y1←2 = a↓x;x;y1 = a←x←(a←x←y←2)←1 = a↓x;a↓x;x;y
             = a→x→(a→x→xy→2)→1 = a→x→xy1→2 = a↑↑x;xy1
  • a←x←1←3 = a↓x;x;x;1 = a←x←(a←x←0←3)←2 = a↓x;x;a↓x;x;↓x
            = a→x→(a→x→xy→3)→1 = a→x→xy1→3 = a↑↑x;xy1
  • a←x←1←z1 = a←x←(a←x←0←z1)←z
             = a↓x;..1 {↓x;#z1} = a↓x;..a↓x;..↓x {↓x;#z ↓x;#z}
             = a↑x;a↑↑x;x = a→x→(a→x→x→z1)→z = a→x→x1→z1 = a↑↑x;x1
  • a←1←y1←z1 = a↓1;..y1 {↓#z1}
              = a↓1;....y1 {↓1;#z ↓#v} where v = a↓1;..y {↓1;#z1}
  • a←x1←y1←z1 = a↓x1;..y1 {↓#z1}
               = a↓x1;..x;..y1 {↓x1;#z ↓x;#v} where v = a↓x1;..y {↓x1;#z1}
  • a←R←y1←z1 = a↓R;..y1 {↓#z1} = a↓R;..R-;..y1 {↓R;#z ↓R-;#y1}

More arrow quarters round the clock.

a↓1;↓b = a↑↑1;b = a→1→b→2 = a↑1;a↑↑1;b-
a↓1;..b {↓#z} = a↑1;..b {↑1;#z1}
              = a→1→b→z1 = a↑1;..a↑1;..b- {↑1;#z ↑1;#z1}

a↓1;1;b = a←1←b←2 = a←1←(a←1←b-←2)←1 = a↓1;a↓1;1;b-
       = a←1←b←b1 = a↑1;..b {↑1;#b1}
a←1←b←2 = a↓1;1;b = a↓1;..b {↓#b}
        = a←1←b←b1 = a↑1;..b {↑1;#b1}

# 3.4. Recursive arrow mixing

Under 2011 Construction

§3.4.1. Revolving the axis

There's a beautiful story recalled by Richard D. Gill, how John H. Conway derived a power tower of omegas ω on a small blackboard, that he rotated 90° clockwise to create a much larger infinite expression involving epsilons εεε..

Using ω→2→3 = ω^^ω = ε0 to express ε0→2→4 is not so impressive as it seems, considering the chained row of omegas ω→.. {ω#ω} that must have been on Conway's mind when he invented his chained arrows.
We don't want to spoil Gill's story here. It gains interest when we find a way to turn clockwise the very arrows Knuth and Conway started with! This involves a revolution of the arrow's axis. Of course the first turn is Conway's…

§3.4.2. The next revolution

bigΨ.II. Function sizes

part 2, Work draft
publishes: 2011

§II.0. Summary

In the chapters below you'll find a showcase of 5 different operatorials bigO* – created from their corresponding subscripted operators, which we've introduced in the first part of this paper.

You've witnessed Conway's chained arrows in action before – this is an operatorial function in disguise – and we'll work it over in the first row of bigY.
So you already have an idea of what is coming – after the basic operations the first row of parameters will be defined and then we start counting and iterating over array lengths and dimensions and over extra-dimensional types.

The challenge is to iterate in a smart way, to recognize and grasp each opportunity to move the algorithm higher by feedback of iterators from down below – thereby creating Big numbers undreamed of before.
In this 2nd part of the book the operatorials pass through 4 different levels of function classes together. To each operatorial level we'll devote a separate chapter.

* Note on: bigO*, generic operatorial function O

The nomenclature is confusing here – the names "bigO" (bigOh) and O() and "bigOmega" and "Omega" are already in use as measures for algorithmic complexity or function speed. The primitive bigOh notion of algorithmic order just accounts for the fact that the size of the numbers an operatorial function produces is determined by the value of its highest positioned parameter.

We feel our operatorials are entitled to the name bigO, because of the shared property that all parameters are put in order of algorithmic weight. This allows us to compare a given operatorial on a whole range of parameters. To decide which operatorial bigO increases faster we first check the lengths of the parameter rows, and then the values of the rightmost differing parameter.

We originally opted for the generic function prefix O() because our operatorials are constructed on top of a system of operators. Also, a numerical property of our operatorials O(X,z) is that final iterators z are count down to 0 (zero) and then dropped.
Related Big number functions such as Conway's chained arrows may be called operator functions.

Ψ.4. Superpowers of bigO

§4.0.1. Generic rules

Stone sculpture of a stylized animal

In those days the multitude being very great, and having nothing to eat, Jesus called his disciples unto him, and saith unto them, I have compassion on the multitude, because they have now been with me three days, and have nothing to eat: And if I send them away fasting to their own houses, they will faint by the way: for divers of them came from far.

And his disciples answered him, From whence can a man satisfy these men with bread here in the wilderness? And he asked them, How many loaves have ye? And they said, Seven.
And he commanded the people to sit down on the ground: and he took the seven loaves, and gave thanks, and brake, and gave to his disciples to set before them; and they did set them before the people.

And they had a few small fishes: and he blessed, and commanded to set them also before them. So they did eat, and were filled: and they took up of the broken meat that was left seven baskets.
And they that had eaten were about four thousand: and he sent them away.

Mark 8.

An operatorial O is a container O(T) with an array of parameters O(a,..,z), where every next term is an iteration counter, modelled after and expanding on previous terms, and based on an initial case O(a,1).
The term basic refers to the primitive recursive function class or the superpower level of construction. A related subclass is Kalmár's elementary, which covers all functions up to iterations of multiplication.

To evaluate an expression of an operatorial completely, it is reduced to the smallest possible number of operations on units. Without other numeric units than 1 the final reduction of an operatorial must always deliver a natural number. This is what's meant with reducibility, and every new rule should preserve it!

Creating operatorials for Big numbers is more of an art than a science. Our bigE should capture all recursive functions under the hood (if yours accelerates faster we'll catch up later), by extrapolating the principles of natural operators to higher parameter dimensions and beyond.
We'd like simplicity to be our guide, but this is alas not trivial. Contrary to popular belief it is not the higher which rules the lower from above. Operatorial higher constructs facilitate the lowest to iterate over all other constructs below the higher. In a fast algorithm the highest intervention – characterized by z countdown – will happen very very rarely.

The post-reduction and post-countdown destructors lay the foundation for all operatorials. Both rules are marked here in the generic context for bigE.
The post-reduction axiom maps a single item (such as a natural number) to itself. Post-reduction in the context of bigI can also be incorporated by its addition rule.
The generic post-countdown axiom is most fundamental for bigO as it puts a lower limit to all iterations, but it can perhaps be overruled in a special case of z=0 such as via the substitution definition of multiplication in bigE.

  • O(a) = a
    so  O() =   = 0  if a = 1.. {1#a}
  • O(X,) = O(X)
    so  O(X,0) = O(X)

§4.0.2. Basic bigE

Traditionally multiplication is defined by repeated addition – recursively in two steps. Notice that the initial step of unity is already contained via the two generic axioms E(a,0) = a, so the zero-arrow definition of bigE is not a complete fit, because it does not cover multiplication by zero, which is unreachable as the iterator drops at b=1.

  • E(a,1) = E(a) = a
  • E(a,b1) = E(a E(a,b)) {b>0}
           == E(a... E(a,1)) {a#b} = E(a...) {a#b1}
            = a+... {a#b1} = a(b+1) {0^} = a*b1 {0*}

You might like to leave multiplication untouched, there's something to say for that – just drop the single separator and collect the result E(a,b) = ab = a*b
For bigE we are tempted to choose the substitution definition of multiplication, where the value b=0 is allowed to overrule the generic post-countdown axiom for once. Unusual here is that a functions as an iterator over b, while bigE's version of the lonely separator completely replaces all units 1 in a by the value b

  • E(a,b) = E(1...) {1#a 1:b}
           = b+... {b#a} = ba {0^} = b*a

Now how is this suddenly possible? As the last single separator is dropped, the algorithm of bigE should select the highest remaining separators to grab its parameters from. At the lowest level dwell the empty separators which add the unit parameters that compose numbers – so that finger counts 1 become available now… Contrived?

Naturally an iterator countdown is a zero-level process too – dropping what seems to be a 0 number of separators , in between z1 -> z. Consider this a context where zero space 0, = 0* =  
We can take the hair-splitting yolk to its amniotic extreme and define addition in bigE by assuming separators of size 0 and then iterating over the two virtually separated values of a and b. The result is the same as without the empty separator, but that's what you always have with addition.

  • E(a,...b1) {,#0} = E(a1,...b) {,#0}
                    == E(a1...,...1) {1#b ,#0}
                     = E(a1...) {1#b1}
                     = E(ab1) = a+b+1 = ab1 {0*}

Recursive scheme Nº0 for bigE with 1 parameter.

  1. 0.1. Function destruction: post-reduction
  2. 0.2. Empty recursion: addition
  3. 0.0. Separator destruction: post-countdown

Basic recursive definition of bigE with 3 parameters – first an example (exponentiation), then two (temporary) axioms.

  • E(a,b1,1) = E(a,E(a,b,1)) == E(a,..a.) {#b#}
              = a... {a#b1 0^} = a^(b+1) = a**b1
  • E(a,1,c1) = E(a,1,c) == E(a,1) = a
  • E(a,b1,c1) = E(a,E(a,b,c1),c) == E(a,..a.,c) {#b#}
               = a^.. ... {^#c a#b1} = a^...(b+1) {^#c1}

Recursive scheme Nº1 for bigE with 2 to 3 parameters.

  1. 1.1. Root recursion: multiplication [defined by repeated addition]
  2. 1.2. Recursion example: exponentiation <= 1.4 + 0.0
  3. 1.3. Parameter destruction: second iterator collapse
  4. 1.4. Double iteration: superpowers

§4.0.3. Basic bigI

Follows the basic abc of operatorial function bigI, called the big Iteration. Compared to the big Expansion of bigE, an advantage of bigI could be that it incorporates addition ab with its initial pair of parameters.
The definition list of bigI covers superpowers with the 3d parameter c counting the number of stars *.. of the corresponding superpower operation, multiplication represented by a single star.
Here too, multiplication by zero cannot be reached by reduction of any expression of bigI.

  • I(a,1) = I(a1) = a1
  • I(a,b1) = I(I(a,1),b) == I(ab,1) = ab1
  • I(a,1,1) = I(a) = a
  • I(a,b1,1) = I(a,I(a,b,1)) == a... {a#b1 0*} = a*b1
  • I(a,1,c1) = I(a,1,c) {c>0} == I(a,1,1) = a
  • I(a,b1,2) = I(a,I(a,b,2),1) == a*... {a#b1} = a**b1
  • I(a,b1,c1) = I(a,I(a,b,c1),c) == a*.. ... {*#c a#b1}

Recursive scheme Nº1 for bigI with 3 parameters.

  1. 0.1. Initial destruction: post-reduction => 1.1
  2. 1.0. Initial case: succession => 1.1
  3. 1.1. Root recursion: addition   [Drop axiom: lonely separator I(a,b) = ab]
  4. 1.2. Destruction case: identity => 1.4
  5. 1.3. Primary recursion: multiplication <= 1.5
  6. 1.4. Parameter destruction: double iterator collapse
  7. 1.5b. Recursion example: exponentiation <= 1.5
  8. 1.5. Double iteration: superpowers

We'll continue modelling the operatorial bigI after mixed star operators, expanding the above lists to a row of parameters in chapter 5.3. As it turns out bigI is the big brother of bigE, whose first row is the next construction, building upon the basic parameters defined in this section.

§4.0.4. Axioms of addition

The recursive definition of addition in bigI is a threefold loop – starting off as an induction step on two parameters, it drops second parameters b=1 by succession, which in turn destructs the function post-reduction.
The case of b=0 must then be resolved post-countdown (not by inverse construction) to completely define addition as we like it.

I(a,0) = I(a,) = I(a) = a = a0
so  I(a,b) = ab

But addition I(a,b) is also the natural choice for an initial axiom of bigI. It is then defined as a rewrite rule — to drop the lonely separator comma , together with the function I brackets.
As was demonstrated early in the book the operator of addition is empty and redundant. Succession is surpassed by addition of 1, the next rule in bigI's definition list. And the impractical rule of post-reduction of one unreduced parameter a is derivable by inverse application of the post-countdown axiom.

I(a) = I(a,) = I(a,0) = a0 = a
so  I() = I(0) = I(0,0) = 0 =  

Therefore an addition which drops the lonely separator can function perfectly as a stand-alone axiom to deal with the final reduction of expressions in a superpower operatorial.

Compare the axiom of addition by zero separators in bigE. We feel that countable separators are an indispensable property of operatorials, and that bigE applies this more rigorously than bigI.
However, with the introduction of waitors to accommodate mixed minority stars in parameters on the front row, the single value c=0 wasted on addition becomes the hallmark of bigI's row expansion.

§4.0.5. Definition lists and function hierarchy

A function definition is an equation existing of a new rule A=B, defining a reduction from A to B. With often appended a conclusion B=..Z derived by iteration == of the rule and/or from the preceding definition context.
This context has 3 types of scope – from generic rules, from earlier lists, or from earlier definitions in the same list.

The four types of list markers (none, circle, disc, square) for definition lists rate the axiomatic strength of each new or rehearsed rule by placing it in the function's definition context (as printed in this article).

  • salient = example, non-essential case dependent on a next rule in the same list.
  • transient = recursive rule, part of the conclusion of a next rule in the same list.
  • provisional = recursive rule, surpassed by a rule in a following definition list.
  • fundamental = axiom, recursive rule necessary for all following definitions.

Within the same definition list provisional and fundamental rules prescribe a certain order of application. For example, the rule I(a,1,c1) is listed before superpowers I(a,b1,c1), prescribing that in case b=0 the former rule takes precedence, even if the latter rule is declared without specifying that b>0.
When a rule is not fundamental (an axiom), it will be overruled by a more general definition in a following definition list. Our job is to abstract new rules, create Bigger numbers and keep the context expanding forever…

A definition list should cover an (intuitive notion of an) operatorial hierarchy – in the case of bigI the *... {*#n} operators. The definition lists in this section present the basic context of superpowers, so that each basic operatorial covers all subclasses of the primitive recursive Grzegorczyk hierarchy.

Doctorat honoris causa Andrzej Grzegorczyk
Grzegorczyk
Auvergne 2008

# Grzegorczyk's subclasses

The number-theoretical space occupied by bigE, bigI and bigA with 3 parameters is known as the Grzegorczyk hierarchy {K.1.c} of classes of functions closed under the operation of limited recursion.
Limited recursion over class {K.1.1} for example, means that you cannot step from exponentiation to power towers a^a^a directly. From this Grzegorczyk proved that every primitive recursive function belongs to a separate subclass c.
We take addition as level {K.0} while Grzegorczyk calls this class ε1.

Grzegorczyk in 1953 devised his own primitive recursive function family to investigate his super-exponential hierarchy. The operatorial function Grz is essentially his (except at *notes).

  • Grz(a,b,-) = a1 (*Grz reversed a <—> b with c1 as function subscript)
  • Grz(a,b,0) = ab (Grz's addition axiom, here enumerated by c=0)
  • Grz(a,b,1) = a1*b1 (Grz's extra multipligation axiom)
  • Grz(a,1,c1) = Grz(a1,a1,c) (*Grz put 0 in parameter b on the left)
  • Grz(a,b1,c) = Grz(Grz(a,b,c),b,c) (the Grzegorczyk hierarchy)

This is a 3 parameter, multiple substitution, non-nested, non-alternating, primitive recursive function.
Suppose we'd let drop the addition and the multipligation axiom, and then rely on the rest of the definition list Grz(X) => Gr(X) to accelerate the cases c=0 and c=1 and thereby our new Gr expansion.
These functions stay superpowers – they do not escape from primitive recursion like Ackermann did.

  • Gr(a,1,0) = Gr(a1,a1,-) = a11 = a+2
  • Gr(a,b,0) = Gr(Gr(a,b-,0),b-,0) == a(11**b) = a+2^b
  • Gr(a,1,1) = Gr(a1,a1,0) = a1(11**a1) = 2^(a+1)+a+1
  • Gr(a,b,1) ~ 11***b\**a1 ~ 2^^(b+1) (say multipligration in Gr)
  • Gr(a,b,c) ~ a*...b {*#c11}

# 4.1. Inverse iteration

Because operatorials build natural numbers 1.. they shun the number 0 from their definitions. Any awkward expression with an inner parameter value of 0 should be resolved by inverse recursion, which iterates backwards from an initial value 1, as is done here for the familiar 0 values of superpower iterators.

I(a,1,1) = I(a,I(a,0,1)) = a I(a,0,1) = a
      => I(a,0,1) = 0

I(a,1,c1) {c>0} = I(a,I(a,0,c1),c) = I(a,1,c)
      => I(a,0,c) {c>1} = 1

We try out inverse recursion to determine how negative operators work – many solutions are the same and extract a successor function on b. However, there are second solutions, can you see?

I(a,b1) = I(a,I(a,b),-) = I(a,ab,-) = ab1
     => I(a,b,-) = b1

I(a,b1,-) = I(a,I(a,b,-),--) = I(a,b1,--)
     => I(a,b,-*n) {n>0} = b1

An aesthetic solution, but Peano will have to wait for negative eternity for the ambiguity about true negative functions to go away – it is only the probability of his successor function that rises…
A few hints for future research:

or  I(a,b,-) = (b*a**-)a1 = b/a+a+1  {a≠0}
    I(a,b,--) = b(a**-) = b+1/a  or  = (b-a-1)×a+2
I(a,b,-*m) {m>2} == b1 = b+1    or  ad infinitum

Considering the fundamental role of the successor function we may well be hallucinating here.
It's alright to define I(a,b,-*n) {n>0} = b1 as an axiom. There's an overwhelming amount of evidence to show that the operation that precedes addition need not be difficult – compare Gr or Fa- above and the translation of Rózsa Péter's recursion to bigI format below. Because the Big Lebowski calls this zeration and thinks highly of it, we've restated its law in double Dutch for all Dudes.
The inverse separator reminds us of the relations between inverse points, counted by Newton's negative binomial.

a**b     -> a*b    -> ab     -> a*..b {*#-}    = b+1
E(a,,b) --> E(a,b) -> E(ab) --> E(a,..b) {,#-} = ?

The formula for primary negative iterators b by application of inverse recursion, is given as an exercise.

I(a,-,111) = 0  <=  I(a,-*m1,n11) = -*m {m<n}

# 4.2. From bigA to fractation

Smooth abstract Carrara marble sculpture

There is something unborn, unoriginated, uncreated and unformed.
If there was not, escape from the world of the born, the originated, the created, the formed, would not be possible.

But since the unborn, unoriginated, uncreated, unformed is real, there exists a way out of this world of the born, the originated, the created and the formed.

Persistent in their meditation, resolute and strong, the wise reach out to conquer Nirvana, the highest happiness.

Buddha

§4.2.1. Basic bigA and affiliates

Translate the rules for unary pluses to the basic parameters of a new operatorial function bigA.

  • A(a,1) = a+ = a1
  • A(a,b1) = a+... {+#b1} = ab1
  • A(a,1,c1) = a+c1; = a+c;... {+c;#a} = A(a,a,c)
  • A(a,2,c1) = A(A(a,a,c),A(a,a,c),c)
  • A(a,b1,c) = a+c;... {+c;#b1} = (a+c;)+c;... {+c;#b}
              = A(A(a,1,c),b,c)

We've described the case Rmin(0;a,b,c) of unbracketed minority stars in section 2.5.4. We'll derive a slightly faster function Fb = a×...b {×#c} by unbracketing the function family Fac from section 2.0.2.
It will then become clear that the binary function Fb() and the unary operatorial A() are the same, at least in the basic three parameters.

First restate the rules for function family Fac(a,b) to Fa(a,b,c) in operatorial format.

  • Fa(a,b1) = Fa(a,b)1 == Fa(a)b1 = ab1
  • Fa(a,1,c1) = Fa(a,a,c)
  • Fa(a,b1,c1) = a×...b1 {×#c1} = a×...(a×...b) {×#c ×#c1}
                = Fa(a,Fa(a,b,c1),c)

Assume the operator × of Fa and Fb is ruled by right minority precedence.
Unbracket Fa by removing brackets ( -> (0 in the last rule of the above definition.

  • Fb(a,b) = ab
  • Fb(a,1,c1) = Fb(a,a,c)
  • Fb(a,b1,c1) = a×...b1 {×#c1} = (a×...a)×...b {×#c ×#c1}
                = Fb(Fb(a,a,c),b,c1)

Because the two step reduction A(a,b1,c1) = A(A(a,a,c),b,c1) matches the main rule of Fb and the previous two rules are the same, operatorial A equals function Fb in the superpower context.
From this we conclude that for 3 parameters, bigA increases faster than bigI on two points – because Fa starts squaring sooner than bigI at b=1 and because its unbracketing to function Fb mostly produces notably bigger numbers, as it feeds the subexpression to the larger operator (this was explained in section 2.5.4).

But how fast is bigA approximately compared to bigE?
The dagger sign is used to cut calculations short.

  • A(a,1,1) = A(a,a) = aa = a*2
    A(a,b,1) == A(.a.,1,1).. {#b#} = a*2^b
  • A(a,1,2) = A(a,a,1) = a*2^a
    A(a,2,2) = A(A(a,1,2),1,2) = a*2^(a*(2^a+1)) ~> 2^2^a †
    A(a,3,2) = A(A(a,1,2),2,2) ~> 2^2^(a*2^a) ~> 2^2^2^a †
    A(a,b,2) == A(.a.,1,2).. {#b#} ~> 2^^b\^a †
  • A(a,1,3) = A(a,a,2) ~> 2^^a†
    A(a,b,3) == A(.a.,1,3).. {#b#} ~> 2^^^b\^^a†
  • A(a,1,c) = A(a,a,c-) ~> 2^..a† {^#c-}
    A(a,b,c) == A(.a.,1,c).. {#b#} ~> 2^..b\^..a† {^#c ^#c-}

Unlike unbracketed stars versus arrowheads, when c>1 early riser bigA is always slashly ahead of bigE.
This is because: a^^b < 2^^b\^a in which the final exponent a < 2^a is dominant.

§4.2.2. Half-speed superpowers

Getting your hands dirty and making mistakes is the practice of a recursionist.
In chapter 2.2 we asked if there exists a continuum of algorithms in the gap between superpower counts of c of which we've so far only met the integer values. This is a problem Stephen Wolfram posed recently.
Natural candidates for the position E(a,b,½) halfway between multiplication a*b and powers a^b for example, are the binary exponential function a*2^b or the factorial a! perhaps.
Today in our algorithmic laboratory we stumbled across the following approximate evaluations for tryB.

  • B(a,b,1) = B(aa,b-,1) == (.a.*2).. {#b#} = a*2^b
  • B(a,2,2) = B(B(a,a,1),a,1) = a*2^(a*2)
    B(a,b,2) == B(.a.,a,1).. {#b#} = a*2^(a*b)
  • B(a,2,3) == B(B(a,a,2),a,2) ~> 2^(a^2*2^a^2)
    B(a,b,3) == B(.a.,a,2).. {#b#} ~> 2^^b\^(a^2)
  • B(a,2,4) == B(B(a,a,3),a,3) ~> 2^^a\^(2^^a\^(a^2))
    B(a,b,4) == B(.a.,a,3).. {#b#} ~> 2^^(a*b)
  • B(a,b,5) == B(.a.,a,4).. {#b#} ~> 2^^^b\^^(a^2)

This function tryB expands pairwise and traverses the superpower hierarchy at half the speed of bigA!
But how does one construct the algorithm? Should we supply an extra coefficient to store left parameter a temporarily, or can we retrieve the appropriate variable a from within the depth of all nests?
The first option would require us to remember these a somehow as waiting on a particular c too.
We choose a nested notation, which keeps expressions substituting for the first parameter unresolved, until the outer iteration counts down to value b=1. Nested subexpressions put on hold like this are called waitors.

We use the waitor mechanism to reduce superpower function speed to a fraction of one half. But tryB is not dependent on its special rules for waitors w. The whole mechanism can be taken care of by the nested recursion in the conclusion of tryB's main axiom (in the last line).
To indicate that tryB's special waitor rules can be represented alternatively as nested recursion these are marked as transient (with left a small circle). But when you'd insist on having a construction in the simplest of steps, the waitor rules should be considered fundamental.

Define our algorithm for tryB. The substitute inner expressions w are waitors B(v,1,c) nested to arbitrary depth. Variables a and v can be either natural numbers n or waitors, where v is selected for reduction by unnesting.

  • B(1..) {1#n} = n
  • B(a,1) = a1
  • B(a,b) = B(B(a,1),b-) == B(.a.,1).. {#b#} = ab
  • B(a,1,1) = B(a,a) = a*2
  • B(a,b,1) = B(B(a,1,1),b-,1)
            == B(B(a,b-,1),1,1) = B(a,b-,1)*2 == a*2^b
  • B(v,1,c) =: B(v,B(v),c-)
  • B(x,B(w),c) {w:B[v,1,cd] c>0} = B(x,B(v),c) {d<2}
                                or  B(x,w,c) {d≥2}
  • B(a,b,c) = (w,b-,c) {w:B[a,1,c]} == B(.a.,1,c) {#b#}
             = B(B(a,b-,c),1,c) =: B(B(a,b-,c),a,c-)
             == B(B(B(a,b-,c),a-,c-),B(a,b-,c),c--)
            == B(.w.,a,c-).. {#b-#} = B(.a.,a,c-).. {#b#}

This fractional waitor algorithm has the following features.

  1. A waitor is a subexpression in first parameter position that has an inner parameter value b=1 with an iterable (non-initial) value in the outside expression's parameter c (here c>1 is iterable).
    Waitors w cannot be resolved in place, instead they must wait for the outer expression to be iterated down to b=1. We've talked about the application of outside precedence to functions before.
  2. There's an initial rule at a fixed low value of c (here at c=0 and c=1) which resolves inner as well as outer expressions B(v,1,c) directly – unconditionally and without unnesting (unlike waitors).
  3. A selection rule which cannot be applied in waitor subexpressions, as indicated by the special equal =: sign. This postpones the reduction of nested waitors until the outer expression is resolved.
    Officially each waitor w must wait for the value b=1 to occur in its outer expression, so that it is selected with the guarantee that its inner parameter c is still intact for comparison by the next rule.
  4. A waitor reduction fork which is conditionally evaluated, dependent on the difference d between the inner and outer value of c. It either selects the next inner first parameter or just unselects the waitor so it can be reduced separately as a normal expression B to become the next iterator.
    If d<f (tryB has f=2 and d<2 means d=1) this reduction rule digs up the inner first parameter from the nested waitor. For nested subexpressions w = B(.a.,1,c).. the reduction is recursive:
    B(w,1,c) =: B(w,B(w),c-) == B(w,a,c-)
    Else if d≥f the waiting subexpression is substituted directly in iterator position, where the ex-waitor is reduced with priority, so it can subsequently be counted down by the main rule. This branch works like the more familiar superpower mechanism we met in bigA.
  5. A main introduction rule which substitutes first parameter a for a waitor subexpression as it counts down parameter b. During the course of the iteration over b waitors are nested to depth b-
  6. An unofficial short cut, overruling the default evaluation direction (from right to left).
    Further on in the reduction of fractional superpowers (when the outer c is counted down so that d=f) waitors can't be reduced by unnesting any more – essentially the waiting stops. Then we can decide to resolve these subexpressions as normal fractations, even when substituted in first parameter position.
    In tryB the original parameter a nested in the waitor becomes unreachable as soon as d=2 and it does no harm to resolve those ex-waitors, like we did in the last line of tryB's main rule.

It is sometimes difficult to realize we are still dealing with numbers here. For two examples of step by step evaluation of numerical expressions in tryB, click here!

B(3,1,3) =: B(3,B(3),2) = B(3,3,2) = B(w,2,2) {w:B[3,1,2]}
= B(w1,1,2) {w1:B[w,1,2]} =: B(w1,B(w1),1) = B(w1,B(w),1) =
B(w1,3,1) = B(w2,2,1) {w2:B[w1,1,1]} = B(w3,1,1) {w3:B[w2,1,1]}
= B(w3,w3) = w3*2 = w2*4 = w1*2^3 = w*2^(3*2) = 3*2^3^2 = 1536.

B(3,2,3) = B(w,1,3) {w:B[3,1,3]} =: B(w,B(w),2) = B(w,3,2)
= B(w1,2,2) {w1:B[w,1,2]} = B(w2,1,2) {w2:B[w1,1,2)} =:
B(w2,B(w2),1) = B(w2,B(w1),1) = B(w2,B(w),1) = B(w2,w,1)
= B(w2,1536,1) == B(.w2.,1,1).. {#1536#} = w2*2^1536,
w2 = B(w1,1,2) =: B(w1,B(w1),1) = B(w1,w,1) = w1*2^1536,
w1 = B(w,1,2) =: B(w,B(w),1) = B(w,w,1) = 1536*2^1536,
B(3,2,3) = 1536*(2^1536)^3 = 1536*2^4608 = 3*2^(3^2*(2^3^2+1))

Prelude> 1536*2^4608 =
21508555048174302465876949175324489276722303317367625962613911393023535708309001
78680983067662471539315804775104840474528205150793642557204649350532414481362407
98199460785373402205046566762413511868871486946075012367733934401906436482634209
59419069734176249215449135598234647461283588582007188817294549851994894346936544
65285376410883677697566682011933691337122602607789253333664335939653210088009086
99770469576158631315246172393814114168729552291134778046293682760898673771130164
23519319173790600630511245345170358550201216597797145343678870759570666042935838
73996122460887992355390472801144634846876389929620929491842384582172483641434952
95188793678333958000373946494030710435917675778261498598834095731097965721642497
52594115967964661223700523122229635070400058206614546673319630344229139366308243
81426551272291648373590148626915038305230061956894736445177133599840169934060686
86541722487468972954615413466637605741404272180714146589918068237526297031550033
73427622995380922980483996219588881729412826516433562381873184009838669561476556
84391421837876739456716595598295903592305799885521657744249913875158756304330407
10054312458814812162316829404987960391540162046020325373999458283755441089542197
42529876945845356666207899983789154092325303066851863276617976497269531860721327
95718743734826255856889194790563371505928800555981765807151134341884599237347359
1937166512672488746198778249216.

A more precise mathematician may prove tryB's set of rules with waitors cannot be derived by primitive recursion, as the existence of the Grzegorczyk hierarchy suggests (else his concept of hierarchic function classes is in reality less strict). To us the above steps in the definition of tryB look quite primitive, but to prove that they are (as we conjecture) is not crucial for our discussion.
The striking feature of tryB is that it distinguishes between even and odd values of c in order to subdivide Grzegorczyk's enumeration of superpower functions in two.

Follows our proof that tryB increases at half the speed of superpowers. We take all reduction steps from the outside in – with no resort to the unofficial 6th rule for waitors – but with many short cuts. Expressions of tryB are approximately translated to arrows by means of backslashes \.
To show the whole construction of our proof by example, click here.

  • B(v,b,1) = B(B(v,1,1),b-,1) == B(.v.,1,1).. {#b#}
             = B(v,b-,1)*2 == B(v,1,1)*2^b- = v*2^b
  • B(v,b,c) = B(B(v,1,c),b-,c) == B(.v.,1,c).. {#b#}
             = B(B(v,b-,c),1,c)) = B(B(v,b-,c),B(v),c-)
  • B(a,1,2) = B(a,a,1) = a*2^a
  • B(a,b,2) =: B(B(a,b-,2),a,1) = B(a,b-,2)*2^a
             == B(a,1,2)*(2^a)^b- = a*2^(a*b)
  • B(a,1,3) = B(a,a,2) = a*2^a^2
  • B(a,b,3) = B(w,1,3) {w:B[a,b-,3]} =: B(w,a,2)
             = B(B(w,a-,2),1,2) =: B(B(w,a-,2),w,1)
             = B(w,a-,2)*2^w == w*2^(w*a)
             ~ 2^(a*B(a,b-,3)) ~ 2^..B(a,1,3) {2^#b-}
           ~ 2^^b-\^(a^2*2^a^2) ~ 2^^b\^a^2
  • B(a,1,4) = B(a,a,3) ~ 2^^a\^a^2
  • B(a,b,4) =: B(w4,a,3) {w4:B[a,b-,4]}
             =: B(w3,w4,2) {w3:B[w4,a-,3]}
              = w3*2^(w3*w4) ~ 2^B(w4,a-,3)
             == 2^^a-\^B(w4,1,3) =: 2^^a-\^B(w4,w4,2)
              = 2^^a-\^w4*2^w4^2 ~ 2^^a\^B(a,b-,4)^2
             == 2^^a\^(..B(a,1,4).)^2 {#b-#}
              ~ 2^^(a*b-)\^(2^^a\^a^2) ~ 2^^(a*b+2)
  • B(a,1,c1) = B(a,a,c)
        {c%2=0} ~ 2^..(a^2+2) {^#c/2}.
        {c%2=1} ~ 2^..a\^..a^2 {^#c1/2 ^#c-/2}.
  • B(a,2,c1) = B(w0,a,c) {w0:B[a,1,c1]}
              = B(w1,w0,c-) {w1:B[w0,a-,c]}
        {c%2=1} ~ 2^..(w1*w0) ~ 2^..w1 {^#c-/2}
               == 2^..a-\^..B(w0,1,c) {^#c1/2 ^#c-/2}
                ~ 2^..a-\^..(2^..w0) {^#c1/2 ^#c-/2 ^#c-/2}
                ~ 2^..a\^..w0 {^#c1/2 ^#c-/2}
                ~ 2^..(a*2)\^..a {^#c1/2 ^#c-/2}.
        {c%2=0} ~ 2^..w0\^..w1 {^#c/2 ^#c--/2}
               == 2^..w0\^..(..B(w0,1,c).) {^#c/2 ^#c--/2 #a-#}
                ~ 2^..(w0*a-)\^..2^..w0 {^#c/2 ^#c--/2 ^#c--/2}
                ~ 2^..(w0*a) {^#c/2} ~ 2^..w0 {^#c/2}
                ~ 2^..2^..(a^2) {^#c/2 ^#c/2}.
  • B(a,b,c1) = B(w,a,c) {w:B[a,b-,c1]}
              = B(B(w,a-,c),w,c-)
        {c%2=1} ~ 2^..a-\^..B(w,1,c) {^#c1/2 ^#c-/2}
                ~ 2^..a\^..B(a,b-,c1) {^#c1/2 ^#c-/2}
               == 2^..a\^..(..B(a,1,c1).) {^#c1/2 ^#c-/2 #b-#}
                ~ 2^..(a*b-)\^..2^..a\^..a^2
                      {^#c1/2 ^#c-/2 ^#c-/2 ^#c---/2}
                ~ 2^..(a*b+2) {^#c1/2}.
        {c%2=0} ~ 2^..w\^..(..B(w,1,c).) {^#c/2 ^#c--/2 #a-#}
                ~ 2^..B(a,b-,c1) {^#c/2}
               == 2^..(..B(a,1,c1).) {^#c/2 #b-#}
                ~ 2^..b-\^..2^..(a^2+2) {^#c2/2 ^#c/2 ^#c/2}
                ~ 2^..b\^..(a^2) {^#c2/2 ^#c/2}.

The numbers expressed by bigA and tryB always lie near superpower multiples of two. Factor a is dwarfed by the repetition of items 2 produced by double iteration over b and c as the numbers get bigger. Therefore tryA and tryB are said to express a binary-exponential order.
The difference is that tryB moves at half the speed of tryA, so its resolution (the number range covered by 3 parameters in tryB) is twice as good as superpowers.

Under 2011 Construction

# 4.3. Super-factorial bigU

§4.3.1. Extending the factorial

Oil on canvas - Man standing with scarlet robe and black skull-cap
John Wallis
Oxford 1701

The factorial operation n! = 1×2×3×..×n hasn't received a proper update since John Wallis published Arithmetica Infinitorum (1656) and his Algebra (1685).
We propose a new definition for an enumerable sign !X as the operator of the upcoming operatorial bigU. Because the general function is so fast, we've nicknamed it the fastorial.
You can forget about the slow superfactorials Sloan & Plouffe and Pickover defined in 1995, those are both exponential hybrids. For Big number lovers the factorial nature of the new super-factorial a!b operation – where b sits on the fastorial front row – is paramount!

Our super-factorial operators !b are always subscripted with a coefficient, starting with !1 (the good old a! factorial) defined by i*... {i#a} using an incrementor i.
Two factorials in a row a!1!1 = (a!1)!1 will be evaluated from left to right. Likewise a series of mixable (not necessarily the same) super-factorial operators.

a!bi... {bi#n} = (.a.!bi).. {#n#}

All fastorials !X are left associative. You can compare this to the coefficient used in the short cut formula for majority arrows, for which mixing unequally subscripted operators was deemed insignificant.
The super-factorial !b is a unary operator. Unary operations enjoy a higher precedence than binary operations, so for example m*n!1 = n*(n!1) evaluates the factorial with priority.
The b in a!b functions as a binary operand on a, but because we allow the subscript to contain more than a row of iterators, we've practically done away with the notion of n-ary operators. The operatorial function bigU comes instead.

Keep in mind that a number variable a is a series 1... {1#a} of characters one. Then a single 1 counted down to 0 will be a series of zero ones, nothing but an empty space.
Applying the generic operatorial axioms to bigU below, the unsubscripted !0 = ! would equal 0 if it was allowed.
The third general axiom will be applied to counting down U(1,b) and is unique for bigU.

  • U(a) = a (bracket drop at a number)
  • U(A,) = U(A) (right comma drop at z=0)
      so  U(a,0) = a!0 = a
  • U(,Z) = U(Z) (left comma drop at a=0)
      so  U(0,b) = 0!b = b

Follows the definition of the basic two super-factorial parameters of bigU.
The evaluation of numbers with zero space {0*} in between follows the star convention, so in a- = a-1 and a1!1 = (a+1)! an immediate form of addition takes place (variables and units combine with precedence).
Work out a few cases with the new super-factorial !2 click here!

  • U(1,1) = U(U(0,1),0) = U(U(1)) = 1
  • U(a,1) = U(1...) {1#a 1:U(a-,1)} = U(U(a-,1)*a)
          == 1*i... {*i#a} = a!1 = a! (factorial)
  • U(2,2) = U(U(2,1)*2,1) = 4!
    U(3,2) = U(24*3,1) = 72! ~ 6.12345*10^103
    U(4,2) ~ U(6E103*4,1) ~ 2E104!
    U(5,2) ~ U(2E104!*5,1) ~ 2E104!!
  • U(a,2) = U(v*a,1) {v:U(a-,2)} == (.2.*i)!1.. {#a#}
           = a!2 ~ 2E104!.. {!#a-3 a>3}
  • U(2,3) = U(3!2*2,2) ~ 1E104!2 ~ 2E104!.. {!#1E104}
    U(3,3) = 3!3 ~ 1E104!2!2
    U(a1,3) ~ u!2.. {!2#a} & u = ((4!1*3)!1*2) ~ 10^104 <~ 5!!
  • U(2,4) = U(4!3*2,3) ~ 4!3!3 ~ u!2!2!2!3
    U(a1,4) ~ 4!3.. {!3#a1}
    ~ u!2!2!2!3.. {!3#a}
  • U(2,5) ~ 5!4!4 ~ 5!1!1!2!2!2!3!3!3!3!4
    U(a1,5) ~ 5!1..!2..!3..!4.. {!1#2 !2#3 !3#4 !4#a}
  • U(1,b1) = U(b1,b) = b1!b
  • U(2,b1) = U(U(b1,b)*2,b) ~ b1!b!b {b≥2}
  • U(a1,b1) = U(v*a1,b) {v:U(a,b1)}
            == (.b1.*i)!b.. {#a1#} = a1!b1 (super-factorial)
                                   ~ b1!b;.. {!b;#a1}

             ~ 5!i...:.!b;.. {!i#i1 !i..#b !b;#a}

So you create a maximal predecessor subexpression v by counting down the first entry a in an array of U. Then retake the original expression U and count down the leftmost available iterator (value not 1 unless it's the final entry) right of a (here that's b which can be dropped) to form outer expression. And substitute all units in the parameter(s) left of the outer iterator (here each 1 of a) for inner expressions v.

§4.3.2. Speed of the super-factorial

The principle of bigU is to consequently replace every increasable construct by the maximal predecessor v. Just like a length is increased when we substitute its units of length, a number variable is seen not as a whole number, but as a series of units 1 (separated as it were by 0 commas), which isn't increased directly but through its units. This depends on the function of the single separator , for when the outer iterator lies to the right of a double comma ,, the number of parameters is increased too.

In general only the rightmost increasable construct is significant, which when substituted for v delivers a function increasing approximately just as fast as bigU. Simply replacing the whole variable a as in Rózsa Péter's recursion has comparable strength. (though she misses the factorial nature on our planet)~:
However, the problem lies in recognizing constructs, we only (try to think like a Vegan:~) expand units 1 and , in expressions of bigU.

Two reasons why bigU's basic pair of super-factorial parameters is the perfect Hînayâna motorcycle.
The original ideas are – substitution of the smallest dimensions (the units 1) and multiple alternating recursions.

  1. Instead of substituting parameters, bigU replaces the lesser dimensions 1 of its free parameter values. Here this amounts merely to multiplication, but it immediately kick-starts the engine.
  2. Fundamentally bigU is powered by alternating recursive rules or wheels. Every front wheel U(a,b1) depends on an existing rear wheel U(1,b1) which in turn relies on the front wheel U(a,b) for recursion.
    Rózsa Péter calls this double recursion and proves it increases as fast as a 3-wheel superpower function.

Because the algorithm is so intertwined, you'll literally have no idea what you're looking at when you see a number as a function of U() and this is a good thing, for all operatorials are heinously fast. When creatures with logarithmic senses feel comfortable about Big numbers, the golden brown Buddhas of Oblivion only if ever frown |:–|

The algorithm for bigU is slightly faster than having a!b1 count factorial signs a!b.. by {!b#a}
But in the end it converges to this double recursion. And with this bigU achieves superpower speed in two parameters instead of the usual three.

a!1 ~ a^a/2^a {a≥6} <~ E(a,2,2)
a!2 ~ a^^a/a^^(a-1) <~ E(a,2,3)
a!b ~ a^..a {^#b} or E(a,2,b1)

And so the question, whether the super-factorial !b grows with step 1 on the superpower speed-indicator ^.. {^#c} of bigE, succumbing to the same c++ beat as our other recursive operatorials towards the end, is vindicated with a big affirmative.

# 4.4. Operation Babelfish

In the Northern Ocean there is a fish called Kun which is many thousand li in size.
It changes into a bird named Peng whose back is many thousand li in breadth.
When it rises and flies, its wings are like clouds filling the sky.

Chuang Tsu 1.

§4.4.1. Péter's recursion

Chinese blue bowl with rows of decorative stripes around a stylized character

Inspired by Ackermann's proof that primitive recursion is not the final word, Rózsa Péter continued work on the classification of recursive functions. Ackermann himself used a foundation of 3 parameters – essentially an inflated superpower function bigI – but from her hand is a basic 2-parameter algorithm, which she built to put an Ackermann φ penthouse on top (it's sometimes presented as the Ackermann recursion, though that's not historical), and which employs the same alternating double recursion as super-factorial bigU.
In this subchapter we'll try to generalize Péter's Ackermann formula by putting it in bigI format. It feels like Rózsa would have sought such a translation herself, but it's not in her book.

The first thing we'll do is to reverse the direction of parameters in the functions which Péter and other recursion theorists like to use. For an intuitive understanding of the strength of parameters, we'll append every higher iterator to the right end, not to the start of the array. The last parameters create the Bigger numbers.

Pé(a,0) = a1 = a+1
Pé(0,b1) = Pé(1,b)
Pé(a1,b1) = Pé(Pé(a,b1),b)

Here Péter's algorithm has the same recursive substitute Pé(a,b1) as we saw in the basic pair of bigU. But in bigU every parameter unit 1 in a is to be substituted (i.e. the parameter is multiplied a*v), and Péter replaces her primary parameter in one go.
Therefore Péter must at least add n=1 to the single parameter a. Else if she had put Pé(a,0) = a then every reduction Pé(a,b) {a>0} would have the same result m=1.

§4.4.2. Prefix Pe a parameter

Our operatorials do not change single parameters. The principle is that a single variable has nothing to relate to.
We'll generalize Péter's formula so that any value n can be added to its initial case. This creates a recursive function in bigO style. Because variable n has a minor effect on the result it is positioned left.

Pe(n,a,0) = Pe(n,a) = Pe(an) = an = a+n
Pe(n,1,b1) = Pe(n,Pe(n,0,b1),b)
           = Pe(n,v,b) {v:Pe(n,1,b)}
Pe(n,a1,b1) = Pe(n,v,b) {v:Pe(n,a,b1)}

In expression Pé(1,b1) Péter's algorithm substituted v = Pé(1,b) where in bigU v = U(0,b) reduces to the smaller value b (the difference is not so significant – it doesn't happen often).
We could generalize the 2nd item in the above formula to contain all cases v = Pe(n,m,b) by prefixing another coefficient m to Pe's parameter array. Still m would remain constant, while the substitute expression in bigU varies during the course of the reduction.
Moreover, for a clean result m is dependent on the value of n as shown below.

First we rename the variables. The n of Pe should get the name a – note that a is not counted down.
Formula Pa is the operatorial form of Péter's recursion.

  • Pa(a,b) = Pa(ab) = ab
  • Pa(1,1,c1) = Pa(1,Pa(1,1,c),c)
  • Pa(a,1,c1) {a>1} = Pa(a,Pa(a,m,c),c) {m:Ma(a,c)=a-k}
  • Pa(a,b1,c1) = Pa(a,Pa(a,b,c1),c)

The 3d item in this definition list is newly introduced and provisional.
In the reduction of expressions Pa where a>1 and b=1 the nested inner iterator m is replaced by a-kc where kc<a and so 0<m<a simplifies the expression.
We still have to find out about m and leave it open for the time being. It's time to listen to the fish…

§4.4.3. Pa gone fishing

In the following 3-parameter translation Rózsa Péter's function behaves perfectly primitive recursive, as it finds expression in the superpowers of bigI.

Pa(1,b) = 1b = b+1 = I(2,b3,-)---
Pa(1,b,1) == Pa(1,..1.) {#b1#} = b+2 = I(2,b3)---
Pa(1,b,2) == Pa(1,..1.,1) {#b1#} = b×2+3 = I(2,b3,1)---
Pa(1,b,3) == Pa(1,..1.,2) {#b1#} = 2^(b+3)-3 = I(2,b3,2)---
Pa(1,b,c1) == Pa(1,..1.,c) {#b1#} = I(2,b3,c)---

We wrote the first parameter as a=m+k because it's not obvious what a good value for k is. (It says somewhere in our lost calculations that if a>2 then k=2 …and all will be well?!)
Keep things open with a provisional inner substitute mc:=Ma(a,c) to be determined by variable c.

Pa(a,1,1) = Pa(a,Pa(a,m1;)) = aam1;

Pa(a,b,1) = Pa(a,Pa(a,b-,1)) == Pa(a,..m1;.) {#b1#}
          = (a*b1)m1; = I(a,bx,1)-(a×(x-1)-m1;)

Pa(a,1,2) = Pa(a,Pa(a,m2;,1),1) = (a*(a*m2;1)m1;1)m1;

Pa(a,b,2) = Pa(a,Pa(a,b-,2),1) == Pa(a,..m2;.,1) {#b1#}
          = (a*..m2;.1)m1; {#b1#}
          = (m2;+1)×a^(b+1) + (m1;+1)×(a^i+...+1)-1 {a^i#b}

It looks like we're at a loss, but it's not too bad. To continue, consider case a=2 and put mc=1 as before.

Pa(2,b,2) = (m+1)×2^(b+2)-m-2 {m:1}
          = 2^(b+3)-3 = I(2,b3,2)---
Pa(2,1,3) = Pa(2,Pa(2,m3;,2),2) {m3;:1} = 2^2^4-3
Pa(2,b,3) = 2^^(b+3)-3 = I(2,b3,3)---
Pa(2,b,c) = 2^..(b+3)-3 {^#c-} = I(2,b3,c)---

§4.4.4. The catch-all

Fill in mc=a-2 to translate expressions of Pa where a>2 to superpower format.

Pa(a,b,1) {a>2} = a×(b+1)+m1; {m1;:a--} = (a*b2)--
Pa(a,b,2) {a>2} = (m2;+2)×a^(b+1)-2 {m2;:a--} = (a**b2)--
Pa(a,1,3) {a>2} = Pa(a,Pa(a,m3;,2),2) = a^(a^(m3;+2))-2
Pa(a,b,3) {a>2} == Pa(a,..m3;.,2) {#b1# m3;:a--}
                = a^^(b+2)-2 = (a***b2)--
Pa(a,b,c) {a>2 mc;:a--} = (a*..b2)-- {*#c} = I(a,b2,c)--

This works for all parameters c, which means we can completely translate our extension of Péter's recursion to a primitive recursive definition in bigI or bigE.
For all n>0 in Péter's initial successor formula Pé(a)=a+n the recursion keeps increasing as fast as superpowers and therefore stays below Ackermann-Conway level.

Rewrite the basic definition of our operatorial version of Péter's recursion.

  • Pa(X,0) = Pa(X)
  • Pa(a) = a
  • Pa(a,b) = Pa(ab) = ab
  • Pa(a,1,c1) {a=1 | a=2} = Pa(1,Pa(1,1,c),c)
  • Pa(a11,1,c1) {a>0} = Pa(a,Pa(a,a,c),c)
  • Pa(a,b,c) = Pa(a,Pa(a,b-,c),c-) = I(2,b3,c-)--- {a=1}
                                    = I(2,b3,c)--- {a=2}
                                    = I(a,b2,c)-- {a>2}

Under 2011 Construction

Ψ.5. First row of parameters

§5.0.1. Crafting by grafting

Ceramic hand with mathematical inscription

And the Cyclopes then gave Zeus thunder and lightning and a thunderbolt, and on Pluto they bestowed a helmet and on Poseidon a trident.
Armed with these weapons the gods overcame the Titans, shut them up in Tartarus, and appointed the Hundred-handers their guards. But they themselves cast lots for the sovereignty, and to Zeus was allotted the dominion of the sky, to Poseidon the dominion of the sea, and to Pluto the dominion in Hades...

– Apollodorus, The Library

To expand the superpower operatorials defined in the previous chapter to a row of parameters – or generally to move from one class level of functions to the next – there are two approaches.
You can either keep the basic rules and supplement them with a different set of axioms for the rest of the row – this we call grafting – or you can construct the whole row or higher level by extending the earlier axioms, so that for example the rule for bigE's parameter c is now included in bigE's row formula for the c,..,z range of parameters.

In this chapter the basic parameter definition of operatorials will be expanded in a handful of illustrative ways, where it's often a matter of taste which rules apply naturally.
With E(a,b,c) = I(a,b,c1) bigE and bigI both covered superpowers in the first three parameters. The rest of bigE's front row will be modelled after unmixed majority arrows, while bigI implements mixed minority stars which cover a whole row of arrow subscripts in a single parameter d with the help of waitors.

A function that resembles grafting is Conway's chained arrow notation, which expands superpowers to a row of iterators of arbitrary length. Chained arrows are very much the consequence of the basic 3 parameters, but Conway designed them carefully to stand alone.

When Van Novaloka first became fascinated with the subject of Big numbers, totally oblivious, she used grafting to scratch at the Ackermann numbers (her focus was less on functions).
The invention of the so called Anchorman, Barman, Caveman, Doorman, etc. numbers M corresponded directly to the value and position of final parameters in the following recursive function O(R,z).
Here's a review of the formulas for the first row in Van Novaloka's colourful essay.

O(a,b,c1) = a*..b {*#c}
M(p,1,r) = O(p,...) {p#r}
M(p,q1,r) = O(v,...) {v#r v:M(p,q,r)}
O(pi;,...,q) {pi;#r r>2} = O(v,...) {v#r v:M(pi;,q,r)}

The practice to put several functions in a set of equations is confusing, we agree, but perhaps you've noticed the nested substitution of numbers M for parameters in O. Without recourse to a helper function, a similar set of equations can be expressed by a single operatorial Om and some nifty meta-statements.
In the example rules for functions with r=4 parameters you learn how Om is nested to depth d before the multitude of 3^d deepest subexpressions drop their final parameter to become superpowers.

  • Om(a,b,c) = I(a,b,c) [Om shifts addition back to c=0]
  • Om(a,b,c,1) = Om(Om(a,a,a),Om(b,b,b),Om(c,c,c))
  • Om(a,b,c,d1) = Om(Om(a,a,a,d),Om(b,b,b,d),Om(c,c,c,d))
  • Om(ai,..,z1) {ai#r r>2} =
       Om(Om(ai,..,z),.:.,h) {ai#r Om(ai,..,z)#r h=0}

The last rule is awesome, as every free parameter is replaced by an iteration of Ackermann numbers. Note that the 4-dot illipsis sign .:. owns and increments the i for every subexpression Om(ai,..,z) here.
Obviously outer iterator h need not be 0 (and dropped). It can be given any value as long as the reducibility of the expression is preserved – any outer iterator 0<h<z1 could turn Om' into a faster function.
But, because we'd like to avoid a choice-type of functional subexpression, an original operatorial with an iterator z1 evaluates naturally to an outer iterator of either h=0 or h=z in the reduced expression.

§5.0.2. General recursive baby steps

Later at Van Novaloka's old weblog in another function O' grafted on superpowers she took predecessor h=z as the outer iterator, thereby increasing the size of the numbers expressed by the same parameter values.
Here we'll call that particular function On and compare it with Om' where h=z (which we call Omz) to see if these functions result (on average) in equally large numbers, given the same superpower stock a,b,c.

  • On(a,b,c,1) = On(On(a,b,c),On(a,b,c),On(a,b,c))
  • On(a,b,c,d1) = On(On(a,b,c,d),On(a,b,c,d),On(a,b,c,d),d)
  • On(a,b,c,d,e1) = On(v,v,v,v,e) {v:On(a,b,c,d,e)}
  • On(ai,..,z1) {ai#r r>2} =
       On(On(ai,.:.,z),..,h) {ai#r On(ai,..,z)#r h=z}

Adapt the main rule of Om to create a function Omz = Om' {h=z} with the same maximal iterator h as in On. Then to gauge Omz versus On we have to compare the final subexpression Omz(ar;,..,z) {ar;#r} with On(ai,..,z) {ai#r} because these have decisive bigO strength.
In case of equal parameters a,.. both functions do not differ, but otherwise the last substituted expression in On has the value x=ar-; in its penultimate parameter, while the same substitute in Omz holds a sequence of undiluted y=ar; so that the difference between Omz and On boils down to the relative size of the rightmost unequal two parameters akak1 – in the example below the x and y to start with differ.
Therefore both these functions have an equal chance of producing roughly equally large numbers.

Omz(a,..,z) {a#r} = On(a,..,z) {a#r}

   x<y <=> On(R,x,y,z) < Omz(R,x,y,z)
&  x>y <=> On(R,x,y,z) > Omz(R,x,y,z)

The same reasoning applies to the simplified function Ono = On' {h=0} versus function Om.
If x<y then Ono < Om so that numbers expressed by Om and Ono have roughly the same size.

Nevertheless, some recursive functions are significantly slower than function Om in the first row. As an example we'll graft Ob {h=0} on superpowers, defining Ob so it stops growing much further.

  • Ob(a,b,c) = a*..b {*#c}
  • Ob(a,b,R,z1) = Ob(a,Ob(a,b,R,z),R)
  • Ob(a,b,c,1) = a*..a*..b {*#c *#c} ~ b*..3 {*#c1 a~b}
  • Ob(a,b,c,d) ~ b*..d2 {*#c1 a~b}
  • Ob(a,b,c,d,1) ~ b*..d2*..d2 {*#c1 *#c1 a~b}
  • Ob(a,b,c,d,e) ~ d2*..e2 {*#c2 a~b~d2}
  • Ob(a,b,c,ti;,..z) {ti;,#r} =
       Ob(a,..b.,c) {#u#} where u = ti;1*..z1 {ti;1*#r}
      ~ tr;2*..z2 {*#cr1 a~b~ti;2 i<r}

You can see that row length in Ob merely adds to the number of stars *.. counted by c. Such functions Obr still belong to the superpower class of functions, below the first Ackermann jump.
Subchapter 5.5 generalizes the reason why Ob is so slow – higher parameters than b are never iterated over.

# A walk on the Ackermann-Conway level

Any function rule which limits the substitution position to some fixed receptor, even though higher than Ob's b, must be structurally slower than Om, because Om keeps expanding its row of receptors to the right.
We count these limit positions as our first superpower-like steps {K.2.0.c} past the Ackermann jump, and each step takes us further from the basic primitive recursive level {K.1} of functions.

Functions {Om,Ono,Og} belong to the same class as {Omz,On} which covers the first general recursive subclass {K.2.0} in the first row, waiting for the second Ackermann jump.
Note that the superpower stock Om and On were grafted on, does not play a major role in smoothing out the differences between them. Pivotal is that both original definitions (before grafting) already require a full row to fill the Grzegorczyk hierarchy. The next section and later Og0 confirms this explanation.

Conway's chained arrows express every step {K.2.0.c} of the row of Om with their 4th parameter, and so they make the 2nd Ackermann jump by the 4th chained arrow (and next parameter).
By recursive grafting on Og we shall prove, that the full row of chained arrows cover the enumeration of Ackermann jumps and sublevels {K.2.d} totally.

§5.0.3. Multiple substitution is futile

Young woman with dark hair and a squint
Rósa Politzer
Budapest ±1923

In this section we'll show the meagre consequences of multiple substitution by means of a function Nn, which looks fast in the beginning, but actually requires a whole parameter row to cover only the Grzegorczyk subclasses.
It was Rózsa Péter (read her Recursive Functions) who first gave a rigorous proof that unnested multiple recursion does not lead out of the class of primitive recursive functions.

Example function Nn applies the rules of On from the start – after Nn(a,1) actually. Because Nn is incapable of making the Ackermann jump in its first row, this proves that a whole row of parameters in On cannot take us to a higher subclass, and that On must stay firmly within the class {K.2.0} of Ackermann functions. This despite the fact that Om was grafted on Ackermann numbers and that On increases faster than Om (as shown above).
Remember that the Ackermann function Ack_ψ belongs to a higher class than the superpowers of Ack_φ, because it jumped from primitive recursion. By contrast, Nn is incapable of transcending primitive recursion in the first row.

This example function is pretty straightforward – students can easily check that Nn maintains reducibility. The weak point of Nn is its induction step, which is not so efficient because the final parameter z is reduced twofold in each step – counting down z in both the outer and the inner expression.
The generous substitution of free parameters in the outer expression of Nn seems to compensate for this, but still multiple substitution remains comparatively slow.

  • Nn(a,1) = Nn(1..) {1#a 1:a}
            = Nn(a..) {a#a} = a*a
  • Nn(a,b1) = Nn(v,b) {v:Nn(a,b)} = a**11**11**b  ~ 2^^3\^(b+1)
  • Nn(a,b,1) = Nn(v,v) {v:Nn(a,b)}  ~ 2^^6\^b
  • Nn(a,b,c1) = Nn(v,v,c) {v:Nn(a,b,c)}  ~ 2^^(3×(c+2))\^b
  • Nn(R,y,z1) = Nn(v,..,z) {v#r1 v:Nn(R,y,z)}  where R = ti;,.. {ti;#r}
      ~ 2^..(2^(z+1))\^..(2^y)† {^#r1 ^#r}  ~ 11*..(11**z1)111† {*#r11}

In the formula for Nn the generic rules of bigO apply and addition happens naturally when variables are adjoined. The variable v that substitutes for every free parameter is determined by the meta-statements that follow.
We use backslashed \ operators to enhance notation precision and a dagger † to cut the calculation short.

By the 5th parameter Nn progresses practically on a par with the superpower ***** of bigI. According to our last calculation, at the end of its first row Nn will converge close between two superpower limits.

I(2,I(2,tn;,2),n) <~ Nn(t1;,..,tn;) <~ I(3,3,n1) {n↑∞}

The point to pay attention to is that all parameters in Nn's first row stay well inside superpower limits, so that Nn will only reach Ackermann functionality past the first dimension.
While Nn is groping for emancipation at the right edge of its parameter row, Conway's chained arrows as well as bigA, bigE and bigI become Ackermann functions in their 4th parameter, and bigU already in its 3d.

The main difference between Om and On lies, as shown above, in the value h of the outer iterator. But this doesn't even count as a decisive step in our classification system.
Suppose we'd remove Om from its superpower stock to define a function Mm.
Let Mm(a,b) = a*b and Mm(a,b,c1) = Mm(Mm(a,a,c),Mm(b,b,c)) then the first 3 parameters of Mm remain in Kalmár's elementary class of functions, well below the power towers pointing to tetration.
Define the rest of Mm(R,z) like we defined Om before and Mm will take the whole row to match superpowers.

Mm(a,b,1) = a^2*b^2
Mm(a,b,c) = a^2^c*b^2^c
Mm(a,b,c,1) ~ ab^2^c^2^c1
Mm(a,b,c,d) ~ ab^2^c^2^..c1 {#d1} ~ abc***4*d1
Mm(a,b,c,d,e) = abc***..4*d1 {#e1} ~ abcd****e2
Mm(ti;,..z) {ti;,#n} ~ t*..z2 {*#n}

The notation with an algorithmic mean t or ab.. indicates, that when parameters ti;.. are common non-extreme integers these are less relevant to the result.
The first row of Mm increases with superpower speed, like Nn above (albeit one count of c slower), and evolves on a par with the simpler ur-function Og0 of Graham's Og below. In fact Mm reduces to the same approximate expression as Ôg0 towards the row limit – a clear example that multiple substitution cannot help you in the end.

Generally multiple substitution over a range of parameters does not create significantly larger numbers than single substitution applied to its highest parameter. Because multiple substitution is a tedious, unnatural and ineffective procedure, we'd best choose another policy to optimize operatorials.

# 5.1. Graham's Og

§5.1.1. Graham's number

Ron juggling with four balls in front of blackboard
Ron Graham
Juggler of Og

In the next section we'll create an operatorial function Ôg that enables us to give a simple and exact expression of the famous number of Ronald Graham.
What we call Graham's Og is the simpler brother of Ono, which uses multiple substitution and is derived above from On. All Og-type functions only substitute the penultimate parameter, which is simpler and not significantly less fast. On top of that, Graham's Ôg rises up a little earlier with the power a^b at the count of c=1 (hence the circumflex on the Ô).

Now about that number. Weinstein's Encyclopedia tells us that Graham's number is the smallest dimension of a hypercube such that, if the edges (lines) joining all the pairs of vertices (corners) are two-coloured (either red or blue), a planar complete graph K4 of one colour (on its 6 edges) will be forced.

So multidimensional cubes can be kept free from monochromatic K4 complete squares in every dimension below a certain maximum, for which the estimate is Graham's number.
It is not a very good estimate and neither is it the upper bound Graham published in his 1971 article with Rothschild. What we know as Graham's number was first described by Martin Gardner in an article in Scientific American in 1977. (but Ron is pleased to have a number named after him, he says he recommends it ;-)
David Wells lists Graham's number in his popular Dictionary of curious and interesting numbers as the final entry. The following quotation explains its cult status.

The World Champion largest number, listed in the Guinness Book of Records (1980 ed.), is an upper bound, derived by R.L. Graham from a problem in a part of combinatorics called Ramsey theory.
Graham's number cannot be expressed using the conventional notation of powers.
Consider...

The largish number v1; = 3^...3 {^#v} with v = 3^^^^3 arrows.
Next construct the incredible, ungraspable number v2; = 3^...3 {^#v1;}
Now continue this process vd1; = 3^...3 {^#vd;} until step d+1 = 63
That is Graham's number.

Postscript

Since Graham and Rothschild wrote their "Ramsey's Theorem for n-Parameter Sets" in 1971, Graham's upper bound didn't change. The lower bound for a solution – which experts think is a small number – increased from 6 to 11 (Exoo 2003) and recently to 13 (Jerome Barkley 2008).

Note that the Dutch mathematician Van der Waerden, working in this field, proved a theorem in 1927 on arithmetical progressions of colours, for which he defined bounds that grow like an Ackermann-type function. So we can't be absolutely sure if Graham came up with the all-time record bound in Ramsey theory. But because Graham's number involves a statement about squares inside hypercubes we believe it will always remain the most elegant.

§5.1.2. How to express a world record

In the same fashion as David Wells does in the quote above, we'll devise an operatorial function Ôg that expresses Graham's number exactly as Ôg(3,3,4,63)
We'll graft Ôg on superpowers – this is the dominant rule for all functions Og.

  • Ôg(a,b,c) = I(a,b,c1) = a^...b {^#c}
  • Og(a,b,c,1) = Og(a,b,Og(a,b,c))
  • Og(a,b,c,d1) = Og(a,b,Og(a,b,c,d))
  • Og(R,y,z1) = Og(R,Og(R,y,z))

Our circumflexed Ô functions start with multiplication instead of addition of 2 parameters. This is a matter of initialisation and of minor importance. The main axiom above applies to Ôg too.
Ôg travels not nearly as fast as bigE or our champion bigI, and it doesn't leap to the next function subclass. The progression of Graham's Og is comparable to our baby Om.
The fact that Og applies single substitution does not reduce function speed that much. Because multiple substitution is futile both Og and Om belong to the same class of recursive functions as On, which hardly increases any faster towards the end of the row.

The substitute expression in Og (with addition at c=0) is the same as that of Ono = On' {h=0}
By analogy define a new function Of which reduces Of(a,b,c,d1) = Of(a,b,Of(c,c,c,d)) with a similar substitution as Om {h=0} and check that Of and Og roughly keep pace.

It's trivial that Ôg has an edge over function On. Substitution of the rightmost parameter is dominant and Ôg has a head start, so On will perpetually lag behind Ôg, provided the original parameter values are not too extreme.
Despite its handicaps (single substitution, dropping the outer iterator) Og apparently has the same expressive power as any other Ackermann class {K.2.0} function.

On(x,x,y,1) = I(I(x,x,y),I(x,x,y),I(x,x,y)) <
              I(x,x,I(x,x,y1)1) = Ôg(x,x,Ôg(x,x,y)) = Ôg(x,x,y,1)
Og(R) < On(R) < Ôg(R)

Graham's number Ôg(3,3,-2,Ôg(4,3,1)) can be approximated by chained arrows or by bigE. Here operatorial bigE represents typed majority arrows, and by translation from chained arrows the above expression in Ôg of Graham's number rolls out.
Imagine the fathomless space in between such relatively close Big numbers!

   m→n→64→2 <
E(3,65,2,1) < Ôg(3,3,4,63) < E(4,65,2,1) < 2→3→65→2

§5.1.3. Og hops back

With good reason we found that the functions {Omh,Onh,Og,..} reside in the same Ackermann class {K.2.0}. Then we take a step back to measure the ur-functions Nn and Mm, which remain primitive recursive, although they apply multiple substitution to a whole row of parameters.

Similarly we'll define the ur-function Ôg0, which applies the rules of Graham's Og from the start, only to find that the Ackermann-Conway level {K.2} is never reached. Og0's first row of parameters stays on the primitive recursive level – just before the jump to the class {K.2.0} of Ackermann functions.
From this we may conclude that the rules of Graham's Og are essentially slower than the main superpower rule, even though functions Ogn typically operate on a higher class of numbers. It is the grafting that gets them there.
To get the approximation for Og0 just replace arrows ^.. {#c} by stars *.. {#c}

  • Ôg0(a,b) = a*b
  • Og0(R,y,z1) = Og0(R,Og0(R,y,z))
Ôg0(a,b,1) = Ôg0(a,Ôg0(a,b)) = a*a*b
Ôg0(a,b,c) = a^c1*b ~ a^c2 {a~b}
Ôg0(a,b,c,1) = a^(a^c*a*b)*a*b
Ôg0(a,b,c,d) = (a^..c.*a*b) {#d1#} ~ abc^^d2
Ôg0(a,b,c,d,1) ~ abc^^abc^^d2
Ôg0(a,b,c,d,e) ~ abc^^^e1\^^d2 ~ abcd^^^e2
Ôg0(ai,...z) {ai,#n1} ~ a^..z2 {^#n}

These functions Og0 are clearly primitive recursive, belonging to the Grzegorczyk hierarchy {K.1.c}. It takes a lengthy row of parameters to obtain a high value for c. This means the ur-graft Og0 is covered by the 3rd parameter c of chained arrows.
Then we draw the conclusion that Og1 (Graham's Og), having the same rules as Og0 past its 3 parameter stock, will behave the same, in that it won't make another Ackermann jump and with its first row Og1 will cover the initial Ackermann-Conway class {K.2.0} completely.

We'll prepare Ôg0 for the job of initializing the sequence of recursive grafts on Qy.
The following equations can be worked out as an exercise.

Qy(a,b) = a→b = a*b = Ôg(a,b) = Ôg0(a,b)
Qy(a,1,c) = Qy(a,Qy(a,a,c-),c-) = Ôg0(a,...Ôg0(a,...)) {a,#c a#c1}
Qy(a,b,c) = Qy(a,Qy(a,b-,c),c-) = Ôg0(a,...,b) {a#c1}

§5.1.4. Og is jumped over

What lies beyond {K.1} class recursion? And who makes the second jump and the next, counting an arbitrary number of Ackermann jumps? We've suggested before that Conway's chained arrows fill the bill. Now it's shown how the first row of Ôg can be covered by the single parameter d of a special successor function Qy of chained arrows.
We'll first define the operatorial Qy given Conway's chained arrows, and then express the row of Graham's Ôg in terms of Qy. All subrows R = ti,... {ti#r r>1} have at least two parameters.

  • a→b→c = Qy(a,b,c) = a^...b {^#c} [same base]
  • R→y1→1 < Qy(R,y1,1) = Qy(R,Qy(R,y,1)) [drops z=0]
  • Qy(R,1,z1) = Qy(R,Qy(R,0,z1),z) = Qy(R,Qy(R,Qy(R,-,z1),z),z)
               = Qy(R,Qy(R,Qy(R),z),z) [O: nests 2 extra iterations,
    compare :→]
      R→3→z1 = R→(R→(R→1→z1)→z)→z = R→(R→(R)→z)→z
  • Qy(R,y1,z1) = Qy(R,Qy(R,y,z1),z) [same main rule]

This tells us chained arrows R→x→y1→z1 are slower than Qy(R,x,y,z) given 4 or more parameters. And both expressions are predecessors of R→x→y→z2 which increases notably faster (its outer iterator dominates).

  • a→b→c→1 = Qy(a,b,c) = Ôg(a,b,c) = a^...b {^#c}
  • Qy(a,b,1,1) = Qy(a,b,Qy(a,b,a*b)) = Ôg(a,b,a*b,1)
  • Qy(a,b,c,1) = Qy(a,b,Qy(a,b,c-,1)) = Qy(a,b,..a*b.) {#c1#}
                = Ôg(a,b,a*b,c)
  • Qy(a,b,1,2) = Qy(a,b,Qy(a,b,a*b,1),1)
                = Ôg(a,b,a*b,Ôg(a,b,a*b,a*b)) = Ôg(a,b,a*b,a*b,1)
  • Qy(a,b,2,2) = Qy(a,b,Qy(a,b,1,2),1)
                = Ôg(a,b,a*b,Ôg(a,b,a*b,a*b,1)) = Ôg(a,b,a*b,a*b,2)
  • Qy(a,b,c,2) = Qy(a,b,Qy(a,b,c-,2),1)
                = Ôg(a,b,a*b,a*b,c)
  • Qy(a,b,1,d) = Qy(a,b,Qy(a,b,a*b,d-),d-)
                = Ôg(a,b,a*b,...,Ôg(a,b,a*b,...)) {a*b#d- a*b#d}
                = Ôg(a,b,a*b,...,1) {a*b#d}
  • a→b→c1→d1 < Qy(a,b,c,d) = Ôg(a,b,a*b,...,c) {a*b#d}

The first row of Graham's Ôg creates a new superpower function past the Ackermann jump, and Conway's chained arrows cover this whole subclass {K.2.0} with their 4th parameter d.
The length of Og's row keeps increasing as fast as chained arrows' final parameter value, so it's definitely the 4th chained arrow which jumps to the initial subclass on the Ackermann-Conway row.

§5.1.5. Recursive grafting on Og

A row of Ôg0 is equal to 3 parameters of chained arrows or Qy, and a row of Ôg1 takes 4 parameters of Qy.
Suppose we'd graft a similar function Ôg2 on Qy(a,b,c,d) instead of superpowers, then Ôg2 expresses the next recursive class {K.2.1} with its first row of parameters. But how much of the value of the 5th chained arrow parameter does that take? And ultimately – do we need a whole row of chained arrows to express all recursively grafted Ôgn completely or will a limited number of chained arrows be sufficient?

Find the key to enumerate the whole level of countable Ackermann jumps with a single coefficient.
Start with the two axioms of Ôg2 and then write out the necessary calculations like we did before.

  • Ôg2(a,b,c,d) = Qy(a,b,c,d)
  • Og2(R,y,z1) = Og2(R,Og2(R,y,z))
  • Qy(a,b,c,1,1) = Qy(a,b,c,Qy(a,b,c,Qy(a,b,c)))
                  = Ôg2(a,b,c,Ôg2(a,b,c),1)
  • Qy(a,b,c,d,1) = Qy(a,b,c,Qy(a,b,c,d-,1))
                  = Ôg2(a,b,c,Ôg2(a,b,c),d)
  • Qy(a,b,c,1,2) = Qy(a,b,c,Qy(a,b,c,Qy(a,b,c),1),1)
                  = Ôg2(a,b,c,Ôg2(a,b,c),Ôg2(a,b,c),1)
  • Qy(a,b,c,d,2) = Qy(a,b,c,Qy(a,b,c,d-,2),1)
                  = Ôg2(a,b,c,Ôg2(a,b,c),Ôg2(a,b,c),d)
  • Qy(a,b,c,d,e) = Ôg2(a,b,c,v,...,d) {v#e v:Ôg2(a,b,c)}

The family of functions indexed as Ôgn, share the main rule of Og and are recursively grafted on the operatorial Qy (increasing slightly faster than Conway's chained arrows) with parameter row length n+2. With each next grafting n>0 function Ôgn jumps to a new subclass on the Ackermann-Conway level.
Give the general formula of recursive grafting on superpowers, which translates expressions of Ôgn into Qy. First the stock, then the shoot. The first shoot is the front row of Ôg0 superpowers.

  • Ôgn(a,ti;,..) {ti;#n1} = Qy(a,ti;,..) {ti;#n1}
  • Qy(a,ti;,..y,z) {ti;,#n} = Ôgn(a,ti;,..v,..y) {ti;,#n v,#z}
                   where v = Ôgn(a,ti;,..) {ti;#n}

So ultimately yes, the row of chained arrows covers all recursive functions Ôgn grafted on Ackermann jumps – both occupy the Ackermann-Conway class {K.2} of functions.
That primitive recursion {K.1} should supply the initial functions here is not evident (addition is a likely candidate at the root). But expressing Ôg in Qy we cannot graft any lower than on Ôg0(a,b) = a*b because the early one parameter stock Ôg-(a) = a leads to contradiction in this system.

A row of Og = Og1 acts as a 2nd row, appended after the superpower front row of Og0, which is the stock Og is grafted on. The following row of Og2 is like a 3d row in the square of Og0 or a 2nd row in the square of Og.
From this follows that a row of chained arrow parameters of length n3 is about as fast as an n row square of Og, and that a row of chained arrow parameters of length ng2 generally increases as fast as an n row square of Ogg.
To put it bluntly – a square array of Og is equivalent to a row of chained arrows – the superpower grafting square.

Summarizing our proof for the superpower grafting square

We knew by analogy with Og0 that Og1 should cover the next superpower-type subclass with its first row of parameters. We defined the operatorial Qy (an altered form of chained arrows) to show that 4 parameter Qy and a specific expression over the row of Og1 are equally fast.
In this chapter we extrapolated the method of grafting so it could be used recursively, taking n2 parameter Qy as a stock for the first n2 parameters of the next function Ogn covering the n1th recursive subclass.
Because Qy is exactly expressible by functions Ogn and because Qy uses similar rules as Conway's chained arrows (and does not increase significantly faster), it follows that both methods cover the Ackermann-Conway level, where every n2th chained arrow signifies the nth Ackermann jump.

# 5.2. Front row bigE

§5.2.1. Home of subscripted arrows

The first row of parameters of the operatorial function bigE continues the basic definitions for 3 parameters from chapter 4, according to the plan introduced by the general formula for subscripted majority arrows in chapter 3.
Because bigE is our majority arrow function, the first 4 parameters a,b,c,d in E() are equal to the operations on item a with iterator b, where c counts the arrowhead operators ^ typed by a single subscript d.
The fabulous fourth parameter d takes us beyond superpowers and is explained below.

  • E(a,1,c,d) = a
  • E(a,b,1,1) = E(a,b,b) = a^1;b = a^...b {^#b}
  • E(a,b,1,d1) = E(a,b,b,d) = a^d1;b = a^d;...b {^#b}
  • E(a,b1,c1,d) = E(a,E(a,b,c1,d),c,d) = a^d;...b1 {^d;#c1}
                == E(a,..a.,c,d) {#b#} = a^d;.. ... {^d;#c a#b1}

The definition of the first row of parameters of bigE follows the same pattern.

  • E(a,b,1,...) {1#k k>1} = E(a,b,...) {b#k}
  • E(a,b,1,...,n1,R) {1#k k>0} = E(a,b,...,n,R) {b#k1}
  • E(a,1,c1,R) = E(a,1,c,R) == E(a,1) = a
  • E(a,b1,c1,R) = E(a,E(a,b,c1,R),c,R) = a^R;...b1 {^R;#c1}
                == E(a,..a.,c,R) {#b#} = a^R;.. ... {^R;#c a#b1}

Recursive scheme Nº2 for bigE with a single row of parameters of arbitrary length.

  1. 1.1. Root recursion: multiplication [in basic scheme]
  2. 2.1. Upload example: substitution of a reduced row <= 2.2
  3. 2.2. Partial upload: upward substitution of a range of parameters
  4. 2.3. Parameter destruction: row iterator collapse => 1.3
  5. 2.4. Main row iteration: super-chained(?) expansion => 1.4

What's so great about the expansion of bigE is that its reduction is such a lengthy undertaking. Before the final parameter z is counted down a lot of water has flowed under the bridge and the basic iterator b is fat and juicy, spawning its value high up to replace the penultimate parameter y=1.
This makes numbers grow very big indeed, much bigger than those defined by Conway's chained arrow notation.

The rule that a lower positioned iterator (left parameter) substitutes for an iterator in higher position (right parameter), can be taken as the defining rule of the operatorial functions of which bigE and bigA are prominent examples.
This so called uploading was deployed in the same way by the ^R; arrow operator subscripts for bigE. The multiple substitution of parameters in this type of uploading is less important.

§5.2.2. Race against chained arrows

We've explored the widening gap between bigE's subscripted arrows and Conway's chained arrows in an example in chapter 3. Subscripted arrows win the race, that's not the point, but how fast are both functions exactly?
The definition for the first row of chained arrows is repeated here.

  • a→b = a^b
  • a→b→c = E(a,b,c)
  • R→1 = R
  • R→1→z = R
  • R→y1→z1 = R→(R→y→z1)→z

Compare or approximate chained arrows with bigE past 3 parameters, like we expressed bigE in bigI.
Using the following equations we positioned Graham's number between two expressions of bigE.

  • a→b→c→1 = E(a,b,c)
  • a→b→2→2 = a→b→(a→b) = E(a,b,a^b) <~ E(a^b,a^b,1,1)
  • a→b→3→2 = E(a,b,E(a,b,a^b)) <~ E(a^b,3,2,1)
  • a→b→c1→2 = a→b→(..a→b.) {#c#} = E(a,b,..a^b.) {#c#}
     <~ E(a^b,c1,2,1) = E(a^b,v,v) {v:E(a^b,c,2,1)}

Because the final parameter tends to dominate the result, we inferred that the terms left and right of the predecessor sign <~ are close enough (approximate) within this context. But for small extreme values where a^b < 5 these equations do not offer proper (algorithmic) estimates.

  • a→b→2→3 = a→b→(a→b)→2 <~ E(a^b,a^b,2,1)
  • a→b→3→3 <~ E(a^b,E(a^b,a^b,2,1),2,1) = E(a^b,3,3,1)
  • a→b→c1→3 = a→b→(..a→b.)→2 {#c#} <~ E(a^b,c1,3,1)
  • a→b→c1→d1 = a→b→(..a→b.)→d {#c#} <~ E(a^b,c1,d1,1)

The question is how bigE's 4th parameter increases, while chained arrows expand to higher parameters.
For extra examples, click here!

  • a→b→c→2→2 = a→b→c→(a→b→c) <~ E(a^b,c,E(a,b,c),1)
     <~ E(a^b,E(a,b,c),1,2)
    <~ E(E(a,b,c),2,2,2)
  • a→b→c→3→2 <~ E(a^b,c,a→b→c→2→2,1) <~ E(E(a,b,c),3,2,2)
  • a→b→c→d→2 <~ E(E(a,b,c),d,2,2)
  • a→b→c→2→e1 <~ E(E(a,b,c),a→b→c,e,2)
  • a→b→c→d→e <~ E(E(a,b,c),d,e,2)
  • a→b→c→d→2→2 <~ E(E(a,b,c),d,E(a^b,c,d,1),2)
     <~ E(E(a,b,c),E(a^b,c,d,1),1,3) <~
    E(E(a^b,c,d,1),2,2,3)
  • a→b→c→d→e→f <~ E(E(a^b,c,d,1),e,f,3)
  • a→b→c→d→e→f→g <~ E(E(E(a,b,c),d,e,2),f,g,4)
  • a→b→c→d→e→f→g→h <~ E(E(E(a^b,c,d,1),e,f,3),g,h,5)

It has become obvious that bigE covers the first row of chained arrows with its first 4 parameters. Interestingly we've seen this phenomenon with minority stars versus majority arrows in chapter 3.1 – providing testimony that bigI is as fast as a whole row of chained arrows, merely by appending a unit value I(a,b,c,1)

ai→..y→z {ai→#n} <~ E(ai→..,y,z,n-) {ai#n}
                 <~ E(ai^..,y,z,n) or I(ai*..,yz,n,1) {ai#n}

When n=0 in the above formula we have E(0,b,c,-) < b→c and from this we conjecture that E(a,b,c,-) = E(b,c) and by inverse construction that a negative final parameter value of -*z let drop a number z of leftmost parameters starting with item a.

# 5.3. Front row bigI with waitor

Indian number 2^2*3^3*4^4*5^5*6^6*7^7*8^8*9^9*10^10 engraved in ceramic foot

A thousand heads has Purusha,
a thousand eyes, a thousand feet.

On every side pervading earth
he fills a space ten fingers wide.

This Purusha is all that yet has been
and all that is to be...

Rig Veda 10:90

Under 2011 Construction

§5.3.1. Home of subscripted stars

An operatorial bigI which covers mixed minority stars in a single row of parameters is harder to achieve. With one type-subscript d stars pass by a whole row of arrow subscripts or parameters d,..,z of bigE. It seems the function concept itself is too slow to contain bigI's operator expansion!
Let's see what actually happens when mixed stars reduce and mimic that behaviour in an augmented form of bigI.

a*1;*1;*1;b {c=3} = a*1;*1;*..b {*#b}
               = a*1;*1;*..(a*1;*1;*..b-) {*#b- *#b}
              == a*1;*1;*.. ... {*#b- a#b}
I(a,b,c1,1) = I(w,b,1,1) {w:I[a,1,c,1]} = I(w,b,b)
           == I(w,..a.,b-) {#b-#}

This new functional construct w is called a waitor and is inserted in an expression O(w,Z) in the format O[X] with special waitor brackets, meaning that it must wait before certain rules can be invoked for its evaluation.
Normally inner expressions can (or must) be reduced to natural number first, but not waitors, which have a lower functional precedence depending on the current situation (reduction phase) of its outer expression.

When in an operatorial bigI I(a,b,c1,R) = I(w,b,1,R) a waitor w = I[a,1,c,R] is substituted for the item parameter a (in 1st outer position), this item waitor is not a function, but it is iterated over and waits until all parameters except the initial iterator (in 2nd outer position) are resolved. With just two parameters left the next reduction step will substitute the inner value b=1 in w for the outer accumulated value b' so that I(w,b') becomes a normal operatorial I(a,b',c,R) and the process can repeat itself from the start.
But in all other outer positions (except the 1st) an iterator value is necessary to drive the reduction train, and the offspring iterator parameters are evaluated first as I[a,1,R] = a (as usual).

With the help of a train of item waitors w, the rules for the first row of operatorial bigI can thus be formulated.
This continues the basic definition for bigI up to 3 parameters.

  • I(a,b,1,...,n1,R) {1#k k>0} = I(a,b,...,n,R) {b#k1}
  • I(a,1,c1,R) = I(a,1,c,R) == I(a,1,1)
  • I(a,b,c1,R) = I(w,b,1,R) {w:I[a,1,c,R]}
                = a*R;..b {*R;#c1} = (a*R;..)*-R;..b {*R;#c *-R;#b}
  • I(w,b) {w:I[a,1,R]} := I(a,b,R)

Below we'll show how this construction with waitors covers superpowers too, provided that multiplication is reduced to repeated addition by a separate rule – the repetition step of multiplication – to precede the above definition. Consequently the higher rules for I(a,b,c) {c>1} were justly marked as provisional before.
Because waitor items can't just be multiplied and have to be substituted after a single repetition step, both step and the result of multiple steps – namely multiplication – are individual constructs in the evaluation of expressions of bigI. This means that addition keeps its status as root axiom and that a primary extra axiom must be inserted here.
Without multiplication rule I(a,b,1) would explode via I(I[a,1],b,1) to irreducibility!

  • I(a,b1,1) = I(a,I(a,b,1))
             == I(a,..a.) {#b#} = a... {a#b1 0*} = a*b1
  • I(a,b1,c1) = I(w,b1,1) {w:I[a,1,c]} = I(w,I(w,b,1))
              := I(a,I(w,b,1),c) == I(a,..I(w,1,1).,c) {#b#}
               = I(a,..a.,c) {#b#} = a*..b1 {*#c1}

Recursive scheme Nº2 for bigI with a single row of parameters of arbitrary length.

  1. 1.1. Root axiom: addition [axiom in basic scheme]
  2. 1.3. Primary axiom: repetition step [case in basic scheme]
  3. 2.1. Partial upload: upward substitution of a range of parameters
  4. 2.2. Parameter destruction: row iterator collapse
  5. 2.3. Waitor introduction: row item substitution => 1.4
  6. 2.4. Waitor elimination: row iterator expansion => 1.5

Under 2011 Construction

# 5.4. Front row bigA

§5.4.1. Comparing pluses with Fa via Fb

We've seen in chapter 4.2 that bigA's basic 3 parameters are equal to the superpower function Fb(a,b,c), but the issue still stands whether bigA equals Fb for its expansion over the whole first row. We'll deal with the issue here, and show how this depends on your choice of axioms.
Recapitulate the rules we've established for unary pluses in the definition of bigA's front row.

  • A(a,1) = a+ = a1
  • A(a,b1) = a+.. {+#b1} = a1+.. {+#b} = ab1
  • A(a,1,1R) = a+1R; = a+R;.. {+R;#a} = A(a,a,R)
  • A(a,1,...) {1#k1} = A(a,1,..,a1) {1#k>0}
             == A(a,...) {a#k1}
  • A(a,1,...,1R) {1#+k1} = A(a,1,..,a1,R) {1#k>0}
             == A(a,..,R) {a#k2} (see +R; def. list upload rule)
  • A(a,b1,R) = a+R;.. {+R;#b1} = (a+R;)+R;.. {+R;#b}
              = A(A(a,1,R),b,R)

The unbracketed unmixed right minority function Fb is to be derived from Fa again. We'll begin by defining the first row of bracketed unmixed subscripted right minority operators for function Fa.
As with the unmixed subscripted majority arrows for bigE, we'll use a notation where operator counter c is prefixed to the row of subscripts, so that ×c,R; = ×R;.. R;#c}
Presume the generic axioms, and that addition applies to the 2 parameter cases Fa(a,b) = Fb(a,b) = ab
The first rule in the list covers the case Fa(a,1,1) and the main axiom covers Fa(a,b,1) = a*b1

  • Fa(a,1,c1,R) = a×c1,R;1 = a×c,R;a = Fa(a,a,c,R)
  • Fa(a,b1,c1,R) = a×c1,R;b1 = a×c,R;(a×c1,R;b) == a×c,R;.. {a#b11}
                  = Fa(a,Fa(a,b,c1,R),c,R)
  • Fa(a,b,1,..,1R) {1#k k>0} = Fa(a,b,..,R) {b#k1}

In this row definition of Fa we've copied the last rule straight from the partial upload axiom for bigE.
Again Fb follows from unbracketing (0 the main rule of Fa in combination with minority precedence. Keep the other two axioms the same for Fa and Fb, we won't repeat those here, only the main rule for first row Fb is shown.

  • Fb(a,b1,c1,R) = a×c1,R;b1 = (a×c,R;a)×c1,R;b
                  = Fb(Fb(a,a,c,R),b,c1,R)

Again we can check that the main rule of A(a,b1,c1,R) and Fb is the same for c>1.
But case c=1 is different, because the upload axiom of bigA is more powerful.

A(a,b1,1,1R) = A(A(a,1,1,1R),b,1,1R) = A(A(a,a,a,R),b,1,1R)
     Fb(a,b1,1,1R) = Fb(a,b1,b1,R) = Fb(Fb(a,a,b,R),b,b1,R)

Ψ.6. Dimensions

Under 2011 Construction

# 6.1. Bowers' extended arrays

Images of airplane in front of WTC building, twin tower hit at the top, issuing clouds of black smoke

The ancients have used pillars to prop up the teachings since time immemorial, and Zen practitioners through the years have made them a nesting place. Linji wants to shake the tree and loosen the nest, so he points to a pillar and asks, "Is this ordinary or sacred?"
If you understand it the instant it is uttered you are free and unhindered. But if you hesitate, you have fallen into the pit of brambles... Do you understand?

When affirmation and negation have merged, there is no way to speak of it.
The truth of ordinary and sacred, wherever it is encountered, is, after all, in your hands alone.

– Loori's True Dharma Eye 300.

§6.1.1. Front row BEAF

Thinking of operatorial dimensions Bowers' Exploding Array Function comes to mind. But what does Jonathan Bowers actually do? And how fast does BEAF increase?
In this subchapter you'll learn how the array notation from the Hedrondude website works out up to multiple dimensions.

This is followed in chapter 7.2 by a rigorous theorem of the dimensional grafting cycle Bowers well-nigh envisions, when he rewrites a box of parameters of n dimensions to its abstract form X^n of exponentiation – the very brick he started building from!
Thereupon he suggests we should call his master function BEAF and that its rules, "..can be continued as long as the structures [dimension boxes] can be defined."

We won't keep the typical curly braces {X} that Bowers applies in his latest versions of the BEAF (he used angled brackets <X> in the early days). Instead in the rules and calculations in this chapter all expressions of BEAF are rendered in minipop-format with `X` database brackets. In this format we can opt to leave away the outer brackets.
Actually the BEAF is just a function, albeit a dimensional one, and in a function comma separated sequences of parameters are usually delimited by round brackets (X), so those would make a most intelligible choice.

On the first row BEAF(a,b,c,d) runs parallel with Bowers' extended operators a.{.c}..b {#d#} so we can just skip this single entry operator story. Remarkable (but unpractical) is that both superpower operators and dimension separators in BEAF can be quantified by brackets instead of (subscripted) coefficients.

We start with the five axioms that define Bowers' front row of array entries. The simpler rule for the reduction of a,b is listed first, despite that the 2nd rule covers the initial case `a,1` = a and is arguably more fundamental. The same priority issue concerns rules 4 and 5, but by convention the dominant main rule is listed last.
Note that in an earlier version of his array notation Bowers evaluated `a,b` as addition, and that he later changed this to exponentiation as a strict application of the 2nd rule.

  • BEAF {}
  • a,b1 = `a,b`.. {#a 0*} == `a,1`.. {#a**b 0*}
         = a.. {#a**b 0*} = a^b1
  • R,1 = R
  • a,1,R = a
  • a,b1,1,..,11R {1#k} = a,..,`a,b,1,..,11R`,1R {a#k1 1#k}
                       == `a,..,.a.,1R`.. {a#k1 #b#}
  • a,b1,c1,R {b>0 c>0} = a,`a,b,c1,R`,c,R
                       == `a,.a.,c,R`.. {#b#}

BEAF starts off very fast, and speeds away from Conway's chained arrows in the 4th parameter.
Characteristic is BEAF's upload rule (4th axiom above). It transports the accumulated value in b to the high end of the parameter row, where the rightmost consecutive entry with value 1 is replaced by a direct predecessor (minimally counted down copy) of the original expression. Iterator b is still there, contained in the inner expression, whose evaluation will magnify b zillionfold before it takes its position as higher iterator.

Our own upload rule for bigE applies a similarly elevated substitution, but the top substitutable parameter is simply replaced by the value of b sent straight up the row (not as a subexpression).
BEAF's upload gives it a head start with respect to the value of its dominant rightmost array entries. This makes the countdown of its first row very slow – resulting in Bowers' proverbial infinity scrapers.

And what are infinity scrapers?
They are numbers which are so big, that even though they are finite, they seem to be reaching for infinity, like a sky scraper is a building that reaches for the sky.

J. Bowers (Hedrondude) on Big numbers

Bowers had the bright idea to upload the lowest array entries a,b to the higher entries when those are counted down fully. This supersedes the mechanism of a,2,c1 = a,a,c with super-squares.
But in one sweep he replaces the accumulated value of the iterator b by the small item a and that's not optimal. Generally multiple substitution doesn't significantly increase function speed, but when Bowers substitutes each parameter in the left outer row for item a, this seems to jam the motor of the iteration train. A safer approach offers the upload method of bigE, where the accumulated value of b replaces all parameters 1 in series.

To count down E(a,b1,2,..) by bigE's main rule creates the same nesting as BEAF(a,b,2,..) counted down with the single nest of its elevated subexpression. Consider that most b are Big numbers, then the effect of adding 1 is negligible. But the rewards from multiple substitution shouldn't be overrated too.
So at this stage it's far from clear how bigE translates to BEAF. In the next section we'll get our hands dirty and make the necessary calculations.

§6.1.2. Company on the Bowery

Now we will approximate Bowers' extended array function BEAF with Conway's chained arrows →before, and our operatorial bigE (after) on the first row. This is tough, but we've expressed Conway's chained arrows in bigE back in chapter 5.2, that helps.
To show extra examples in between stations click here!

  • a→b = a,b = E(a,b,1) = a^b
  • a→b→c = a,b,c = E(a,b,c) = a^..b {^#c}
  • a→a→2→2 = a→a→(a→a) <~
      a,3,1,2 = a,a,`a,2,1,2`,1 = a,a,`a,a,a` <~
        E(a,3,2,1) = E(a,E(a,a,a),E(a,a,a))
  • a→a→b→2 = a→a→(a→a→b-→2) == a→a→(..1.) {#b#} <~
      a,b1,1,2 = a,a,`a,b,1,2` == `a,a,..a.` {#b#} <~
        E(a,b1,2,1) = E(a,v,v) {v:E(a,b,2,1)}
              == E(a,vi,..a.) {vi:E(a,b-i,2,1) #b# i≥0}
  • a→a→b→3 == a→a→(..1.)→2 {#b#} <~
      a,b1,2,2 == `a,..a.,1,2` {#b#} <~
        E(a,b1,3,1) == E(a,..a.,2,1) {#b#}
  • a→a→b→c1 == a→a→(..1.)→c {#b#} <~
      a,b1,c,2 == `a,..a.,c-,2` {#b#} <~
        E(a,b1,c1,1) == E(a,..a.,c,1) {#b#}
  • a→a→a→a <~ a,2,1,3 = a,a,a,2
            <~ E(a1,2,2,2) = E(a1,a1,a1,1)

While chained arrows exhaust their 4th parameter in full, BEAF requires just a single value d=2. So the latter already increases much faster, a trend which continues for the rest of Conway's row – each appended chained arrow best approximated by the next value of entry d in BEAF. On Bowers' website the proof that BEAF with its 5th entry far surpasses a whole row of chained arrows is attributed to Chris Bird.
What follows must be similar to Bird's proof – we translate a row of chained arrows to successors in BEAF and bigE. Iterated reductions == are notated with an ordinary equals = sign from now on. For more examples click here!

  • a→a→a→2→2 = a→a→a→(a→a→a) <~
      a,3,1,3 = a,a,`a,a,a,2`,2 <~
        E(a1,3,2,2) = E(a1,v,v,1) {v:E(a1,2,2,2)}
  • a→a→a→b→2 = a→a→a→(..1.) {#b#} <~
      a,b1,1,3 = `a,a,..a.,2` {#b#} <~
        E(a1,b1,2,2) = E(a1,..a1.,1,2) {#b#}
  • a→a→a→b→c1 = a→a→a→(..1.)→c {#b#} <~
      a,b1,c,3 = `a,a,..a.,c-,3` {#b#} <~
        E(a1,b1,c1,2) = E(a1,..a1.,c,2) {#b#}
  • a→...→b→2 {a#d} = a→...→(..1.) {a#d #b#} <~
      a,b1,1,d = `a,a,..a.,d-` {#b#} <~
        E(a1,b1,2,d-) = E(a1,..a1.,1,d-) {#b#}
  • a→...→b→c1 {a#d} = a→...→(..1.)→c {a#d #b#} <~
      a,b1,c,d = `a,a,..a.,c-,d` {#b#} <~
        E(a1,b1,c1,d-) = E(a1,..a1.,c,d-) {#b#}
  • a→... {a#a2} <~ a,2,1,1,2 = a,a,a,a
                 <~ E(a,2,2,1,1) = E(a,a,a,a)

Because bigE too uses parameter d to cover the full row of chained arrows, we think BEAF will approximately match bigE in velocity over the whole front row. To prove this we need only compare expressions where these two operatorials differ significantly, that is for cases where the upload rule determines the result.
Below are two of those – for examples of earlier cases click here!

  • a,3,1,1,2 = a,a,a,`a,a,a,a` <~
      E(a,3,2,1,1) = E(a,v,v,v) {v:E(a,a,a,a)}
  • a,b1,1,1,2 = `a,a,a,..a.` {#b#} <~
      E(a,b1,2,1,1) = E(a,v,v,v) {v:E(a,b,2,1,1)}
              = E(a,vi,vi,..a.) {vi:E(a,b-i,2,1,1) #b# i≥0}
  • a,b1,c,1,2 {c>1} = `a,..a.,c-,1,2` {#b#} <~
      E(a,b1,c1,1,1) = E(a,..a.,c,1,1) {#b#}
  • a,2,1,2,2 = a,a,a,1,2 <~
      E(a1,2,2,2,1) = E(a1,a1,a1,1,1)
  • a,b1,1,d1,e1 {d>0} = `a,a,..a.,d,e1` {#b#} <~
      E(a1,b1,2,d1,e) = E(a1,v,v,d,e) {v:E(a1,b,2,d1,e)}
         = E(a1,vi,..a1.,d,e) {vi:E(a1,b-i,2,d1,e) #b# i≥0}
  • a,b1,1,..,2 {1#k} = a,..,`a,b,1,..,2` {a#k1 1#k}
                      = `a,..,..a.` {a#k1 #b#}
    <~
      E(a,b1,2,1,..) {1#k} = E(a,v,..) {v#k1 v:E(a,b,2,1,..) 1#k}
         = E(a,vi,..,:.a.) {vi:E(a,b-i,2,1,..) 1#k vi#k #b# i≥0}

The equations to compare BEAF and bigE along the first row have finally reached their general and exact form. Provided that value a is sufficiently large, both operatorial functions increase at roughly the same speed.
At least a>2 because accumulated values v are supposed to be very Big, as the length of the reduction cascade indicates. We feel the rule for single upload in BEAF is less pretty than multiple upload in bigE. Whether first rows starting with a=2 must increase explosively or extinguish after many ages to a mere 4 is a matter of taste.

`2,b,c,R` == `2,v,1,2` = `2,2,v` = 4
E(2,b,c,R) {b>2} == E(2,v,v) >> 4 (much larger than 4)

By analysing their respective upload rules we could have concluded directly that BEAF and bigE are about equal. Let's try and see what happens in the BEAF expression if we add 1 to the huge accumulated value v after it is uploaded (as a thought experiment, the reduction rules don't cause this).
The seemingly insignificant 1 spawns a series of parameters xi that dwarf the v1 subexpression, because each contains nest upon nest of v in its elevated position. Such is the power of a high iterator.

`a,..,v+1,z` {a#k1} == `a,x,1,..,v+1,z` {1#k-}
                      = `a,..,x,v,z` {a#k x>v1}
                     == `a,xi,..,v,z` {xi#k xi>xi1>v1}
      ~> E(a,v,..,z-) {v#k1 z≥1}

So there is no significant difference between the rules of BEAF and bigE from the perspective of the length of a row. Neither walks away from the other, except in the details (still there are major gaps between numbers that Big).
The general comparison for first rows of BEAF and bigE follows. Note that cases c=1 get represented on both lines.

  • `a,b,c,1,...,2` {1#k} <~ E(a,b,c1,1,...) {1#k1}
  • E(a,b,c1,R) {R:≠1,..} <~ `a,b,c,R1` <~ E(a1,b,c1,R)

An equal final value z gives bigE the edge, secondarily the value of the 3d entry c favours BEAF, and if all else is comparable Bowers' base entry a has the upper hand. These differences are not so significant for the whole of the first row, and certainly have no import in higher dimensions (plane, cube, etc.) or what lies beyond…

§6.1.3. Folding dimension boxes

Proud young man with beard in blue shirt
Jabe Bowers
Tyler TX 1969

The way Jonathan Bowers explains his extending array notation past the first row is rather chaotic, supernumerary yet incomplete. We must prop up his list of rules to cover every possible expression in BEAF – see our rule 3 which drops a lonely dimension entry. Later we'll take advantage of the omission as we try to give BEAF a significant boost by re-expanding dimensions before they are counted down completely.

Bowers' principal explosive device is a standard dimension box D[a,b,c] with base items a filling all entries, sizes set equal to the accumulated value of prime iterator b and dimensions symmetrically nested and counted by an inline number c which functions at a deeper level.
Follows a recursive definition with two simple rules to construct dimension boxes. For examples click here!

  • D[a,b,1] := a,.. {a#b} (initial row)
  • D[a,b,c1] = D[a,b,c][c]... {D[a,b,c]#b} (subdimension)
  • example  D[5,3,2] := 5,5,5[1]5,5,5[1]5,5,5
         < D[4,2,3] := 4,4[1]4,4[2]4,4[1]4,4

On Bowers' website bracketed numbers (n) are adopted as separators between nth dimensional subarrays. We will write [n] instead, which is commonly restricted with n>0 to positive integer values (in BEAF), and adopt the convention that [0] or [] stands in for ,0; or , the single comma (uncountable by default).
We can also notate a dimension box with a dual repetition a,...., with an extra , to append its closing ,.. {,#c1} separator series. To show this alternative definition, click here!

  • D[a,b,0] = a  and [0] : ,
  • I[A,b,c1] := A[c].. {A#b} (subdivision)
  • D[a,b,c1] = I[D[a,b,c],b,c1]
             == I[.a.,b,i].. {#c1#} (dimension box)
  • D[a,b,c][c] := a,...., {,..#c #b} (closed dimension box)
    example  D[3,3,2][2] := 3,3,3,,3,3,3,,3,3,3,,,

Written in the format F[X] of a parametrized wildcard or word function, dimension boxes D[a,b,n] are introduced in BEAF by the axioms 6 & 7 shown below. Those subsequences are not pre-evaluated, but substituted as such [by : words, not as = values) for the reduced first row a,b in the outer expression.
Bowers' five axioms for the front row can simply be extended to multiple dimensions. This foundation is supplemented by two rules which are meant to explode the counted down base a,b by substituting it for a dimension box.
Only at first a series of boxes is issued when multiple [mi].. are reduced.

Follows the definition list of dimensional BEAF. To select the rules to introduce dimension boxes click twice!

  • BEAF {} (preceding rules take precedence,
                              word Z contains any allowed characters,
    where pZ := p[m]Z {p>0 m≥0})
  • a,b = a^b (initial rule)
  • a,1[m]Z {m≥0} = a (collapse if 2nd entry on 1st row is 1)
  • A1[mi]..1[n]Z {mi≥0 n>0} = A1[n]Z (chop off any trailing entry 1)
        so  A1,..[n]Z {1#k n>0} = A1[n]Z (drop last row entries 1)
           
    A[mi]..1 = A (when Z is empty, given that  A[ni].. = A )
            a[ni]..Z {ni>0} = a,1[ni]..Z = a (reverse rule)
  • a,b1,1,...,2Z {1#k} = a,...,c,1Z {a#k1} (upload on 1st row)
            with  c = `a,b,1,...,2Z` {1#k}
  • a,b1,2Z = a,c,1Z (main rule)
            with  c = `a,b,2Z`
  • a,b1[ni]..1,..,p1Z {[ni]#s 1#k1 p>0} =
        D[a,b1,ni][ni]..a,..,c,pZ {#s a#k} (explosive dimensional upload)
            with  c = `a,b[ni]..1,...,p1Z` {[ni]#s 1#k1}
  • a,b[ni]..p1Z {[ni]#s} = D[a,b,ni][ni]..pZ {#s} (explosion)
            each  D[a,b,ni] as defined above

Note that the rules in a definition list apply in the order given. For example in the last two axioms, if s=1 the condition that n1>0 need not be stated explicitly, because the 4th and 5th axiom evaluate expressions where n=0 first.
Also a character sequence like 1Z or pZ {p>0} should be read as p[m]Z which can simply refer to p,1Z or generally expand to p[mj]....1Z (separator series are discussed below), or reduce to p in case Z is empty.

Bowers' inline notation [ni].. opens up a new level, where a maximum number n cannot increase by evaluation, although predecessor subexpressions are nested on the right and dimension boxes m≤n inserted left.
Inline separators [n1] repeatedly generate boxes with many [n] in the evaluation of BEAF, so there's no need for a rule to count down separator coefficients. In rule 3 we assume that Bowers drops them flat when their time has come, that is when the array on the right side of the separator crashes to 1.

Airplane analogy

We think of p {p>1} as the outer iterator, because it is to be counted down in the outer expression. Bowers calls this entry the pilot, when it's the first iterable value right after prime entry b.
He also invented the concept of a co-pilot, which is the outer receptor directly left of the pilot. The co-pilot c is most important as a substitute in the constructive earlier rules. But in the last rule above there's no parameter fit to be co-pilot, because the pilot entry is seated in first position on the row.
All other entries in this airplane analogy of Bowers are called passengers. But the heavy ones on the right of the pilot have to wait a long time – stranded on the airport sequence Z – before their flight arrives!

§6.1.4. Diacritical remarks

In this section we want to make the transition to the operatorial format lighter for readers who are more familiar with Bowers' system. His extended arrays use a different notation than our Big operatorials and the construction of dimension boxes normally isn't our foremost objective, but overall their habitat is the same.
We translate the inline bracketed separator [n] to a subscripted comma ,n; with depth delimiter, so that the number of separated array dimensions n is equal, and the function of the support characters , for [ to open and ; for ] to close the dimension enumerator is the same. Because subscripts can be understood without ; we'll usually omit the closing semicolon, but under the hood it is still there (at least if we allow single , separators).

In BEAF a series of inline separators is torn apart at the very first reduction step – typically rule 7 – that addresses it, sprinkling individual separators [ni] in between the dimensional arrays each one of them helps define.
The short-lived inline separators [ni].. .:. {ni>ni1} that Bowers mentions are at best a precision notation for expressing various dimensional subsequences. His mixing apparatus is not robust enough to create significantly Bigger numbers, but it does offer a better resolution (is more precise) than our simplified (initially additive) commas.

Bowers' idea to mix numbers of separators feels natural. Our own translation covers all dimensions of BEAF with countable and mixable separator commas. These can either be notated as a sequence ,.. of length c1 or enumerated ,c; with a single coefficient. Here our separator system starts with a type of comma mixing that is similar to counting, simply adding the initial coefficients as defined below…..

,ci;.. .:. {,ci;#ki ,ci;..#m 0*} = ,.. {,#n}
                             = ,n-; := [n-]
where  n = (ci1*ki)... {#m 0*}

example  a,b,2;,2;p1 = a,b,,,,,,p1 = a,b,5p1
                  := a,b[5]p1 = D[a,b,5]p
       a,b[2][2]p1 = D[a,b,2]D[a,b,2]p

Take for example a sequence of [2].. in Bowers' system. When we count a single dimension higher with ,3; or ,,,, this easily surpasses all physically expressible BEAF made up of lower type than [3] separator series.
Compare a few expressions (using various separator styles) as an exercise.

`a,2[2]q1` = a,2,2;q1 = a,a,1;a,a,2;q
           = a,a,,a,a,,,q {1<q{a,2[2]..p} [2]#s} <
`a,2[2]..p1` {[2]#s} = a,a,,a,a,2;...p {#s p>1} <
  `a,2[3]p1` = a,a,,a,a,2;a,a,,a,a,3;p {s{a,2[3]p}}
             = a,2,3;p1 = a,..,..,..,p {#2 #2 #2}
                      
= a,....,p {,..#3 #2}

The dimension box a,...., {,..#n #b} substituting for a,b,.. {,#n1} is rock solid. It contains b^n parameter positions that first have to be counted down from a and next from increasingly Big b' refilling those passenger entries each time 1 is reached. Bowers uses that b^n as a name to classify dimension boxes.
If we'd write a,i;..:.,n; {,i;..#n #b i≥0 0+} to define such boxes by counting dimensions with a coefficient, we fall down to the seemingly absurd context 0+ where subscripted commas do not add but relate by zeration. For to make the dual repetition work, the rightmost of a series of commas ,p;,q; = ,q; must be left over.

However, we expect a plain repetition as for example a,a,.. .., {,#n #b^b} to increase faster, because almost b^b more pockets a can be refilled, given that b outgrows n quickly. And b^b on such a scale means next to nothing – adding an extra 1 to c in an earlier expression frankly already has Bigger consequences.
So dimension boxes are beautiful and characteristic for BEAF, but to erect the whole structure with all the lower dimensions in place is less convincing than a simple but oversized array of penultimate dimensions. The higher dimensions spread out to the lower anyway…

What matters most is the main engine of the first row. And only when the "main row" is eventually count down to two parameters, the accumulated value of prime iterator b explodes into a new oversized dimension box. One by one every subdimension is then counted down from the left until it is discarded.
Here we've met with a shortcoming – apart from the so called prime block where explosions to multiple dimensions of size b take place – dimension sizes can only be counted down in BEAF, not expanded directly.

# Boosting BEAF

Instead of just collapsing counted down dimensions, a better deal would give a rule that upgrades the size of existing dimensions. Subarrays can freely be increased when they are positioned left of a higher pilot (outer iterator) as it is counted down in a long recursive loop. This means that (except for the unique final row) BEAF should keep a single entry on rows ending on 1 to be able to refill them and expand their size by Bigger b.

First we limit our old axiom to a simpler version that leaves room for the resurrection of counted down rows.
Then the new axiom follows, which performs three actions: On the left it replaces a,b by a prime block in Bowers' grand style. In the middle it reinflates further dimensions consisting of single parameters 1 with our method of repeating penultimate dimensions. On the right its "co-pilot" value b may look small and simplistic, but we've upheld that its strength is comparable to Bowers' inner expression in section 6.1.1 before.
Thy mighty triple formula… (Brahma the creator, Vishnu the preserver and Shiva on the outer side :3)

  • A1,1,,Z = A1,,Z
  • a,b,m;1,nj1;...,p;2Z {1#k1 nj>0 p>0} =
      a,....,a,nj;..,nj1;.:.,nk1;b,p;1Z {,..#m #b a#b a,nj;..#k}

We don't define series a,b[mi].. here like Bowers tends to do. This obscure and evanescent construct in his notation should reduce to the same s subsequent dimension boxes as by rule 6 & 7 of BEAF.
Note that at the illipsis .:. two new comma series (typed by nj and nj1) are inserted at each step j to k.

Exploding "orphan passengers" 1 to a series of a within predecessor nj dimensions of size b delivers a significant boost to BEAF, we hope. It protects arrays from being eaten away slowly from the left, as long as "crashed pilots" can be brought back to life and restart their career.

Under 2011 Construction

§ The Biggest number

So what's the Biggest number?
That depends on the kind of system you'd like to represent your numbers in and the resources (time, money) converted to that system.

Suppose you'd only allow a number of fingers stuck up in the air, counted off verbally by a person walking by. Already a lot of motivation and organisation is required to move past the number 1000, for which more than a hundred foolish people would have to hold up both hands for up to an hour.
Dependent on the verbal skills of the person counting and the perseverance of the last batch of participants, we estimate the Guinness record of physical finger counting at no more than ~ 10^5.

On the other hand you might approve of decimal or hexadecimal numbers being printed out on sheets of paper. Then let me tell you, there have been print outs of the digits of the number Pi running past the million. And if you're prepared to fell some more trees – with 5000 signs on an A4 sheet, double sided, 500 in a package, you'd need 1661 packages to machine-write a hexadecimal number of size ~ 10^10^10.
We'd advise against it. Why not allow digital storage instead? On your 1 TB hard disk about 2^48 bits are available to write a number as large as ~ 10^10^14.
For the same reason we expect the blue-ray DVD won't turn out to be a commercial success, we estimate the digital capacity eventually used by the human race not to exceed 2^128 bits – with shared computing this implies a practical maximum binary number of at most ~ 10^10^38.

Broadly speaking, the all time random number record will have to lie below 10^^4, for all the information our universe will ever create is far smaller Yet with mathematical means it's easy to build Bigger numbers. That's what this article is about.
Knuth already made it so much easier for you with his up-arrows. And then comes Graham's Og, Conway's chained arrows, bigE and bigI in the first row, dimensions with countable separator commas, subscriptable commas in bigI style, the Big type of functions, countable Big function types in a cycle, operable cycles, higher cycles, essential characters, meta-characters, enumerable meta-levels, etc.
Mathematical methodology seems endless, but each time a new axiom comes into play we have to be able to cope with that. The human mind does have its limitations…

Attrition "The Voice Of God" record sleeve - inverse flip

Also the number of well-defined axioms for any operatorial number system has its history (as in the book) and therefore its physical limits, and can't be standardized and automated (or can it?o)
A lot more water has to flow under the bridge before we might be able to hint at a practical mathematical limit for the largest number. Logically there can be no limit, every operatorial system is extendible and every format of extension can be enumerated with a counter that is iterable in a larger system.
One thing we do know: working exclusively with integers, or discrete units 1 and -, neither infinity nor the continuum of real numbers is reached. The unit of infinity ω is bigger than all natural numbers.

Within operatorial context the first infinity ω1; becomes countable by the same methods we counted 1. Infinitely many next ωn were expressed by the Operatorial Ω(X). Then we suggested the higher infinities ψ and its subscript functionality Ψ to lie beyond. Greek letter counting was to be our next hobby, etc... Infinities know no end.
The concept of 0=1 would be a candidate for the largest sort of infinity, where it not that the system which produces it is at once in an axiomatic state of demise. Perhaps physically the collapse of a Big or infinite system isn't that straightforward – it may take a while before we can witness all of its constituent parts fallen to the ground. In the mean time 0=1 may sort of exist and even become extended, who knows?
Clever mathematicians may think of new ways to define new concepts. The future is definitely uncertain.

But why should we make the effort to describe an extremely Big number with mathematical rigour, when hardly anybody appreciates it? A grandmaster chess player isn't necessarily best qualified to explain youngsters how to play the game.
Worse than that – when our systems become too large, we tend to lose oversight. One day a mathematical computer may become so proficient at building operatorials, that nobody understands the results any more.
It's here where the metaphysical definitions come into their own right, whose only boundary is set by intelligent imagination. This profoundly fantastic aspect makes them conspicuous, but their lack of scientific basis is compensated by ease of understanding. Man has always appreciated the hugeness of What can be like this.

# St. Peter's Subway

We'll propose a metaphysical definition of St. Peter's Subway as the Biggest number ever conceived by man. It speculates on the potential growth of the axiomatic power (and what have you) of mathematics – this thing that mathematicians do. And it's description is not absolute, but recursive.
St. Peter's Subway starts off as a parable.

Adam, the first man, comes at the gate of heaven and is asked for the Biggest number. He holds up his hand and St. Peter counts the fingers. "Five is not much", he says, but following the rules Adam is now given 5 new lives to come up with a higher number.
"Had he pointed at his missing rib, he'd been walking above the clouds instead of below", St. Peter tells his boss, who shakes her head 5 times.

After his lives are over, Adam returns at the gate and is asked again, "What is your Biggest number?"
Adam's last life had been in India, so now he knows decimal numbers, and he starts writing out his solution on the patch of sand below St. Peter's feet. It is a one followed by a hundred zeros.
"Googol is not too many", St. Peter observes, but still Adam is awarded a googol number of lives. God just shakes her head as St. Peter tells her, "Had he drawn a plain circle he could have stepped through."

After a long time Adam returns and he looks almost a Buddha – his head shaven, prayer beads moving round his fingers. St. Peter stands firmly at the gate of heaven and asks him the same question, concerning the Biggest number. Thereupon Adam Buddha writes down a wonderful mathematical formula in the style of bigI, which expresses a number far beyond human imagination.
"What's that?", St. Peter asks.
"It's an extension of the operatorial bigI", Adam explains, "I found it on the internet".
"The internet?", St. Peter sighs.
"Yes, I was busy reaching enlightenment, you see, I'm fed up with returning time and time again."
"I see…"
And so Adam has to spend an unimaginable aeon of lives down in the worldly realm.
God gets a little itchy as St. Peter tells her the news, "Had he pointed to the lack of his empty head, the man from nothing should have gone free".

Before Adam returns, St. Peter passes away, as there is a limit to each man's years, even if that man is a divine being. In turn St. Peter has to try NOT to enter the Greek hell, known as Hades, which is just one big solid mist without a scream.
Outside of Hades the Hell Hound is waiting, and the animal barks a question at the approaching Saint.
"What is the Biggest number of bones?"
As St. Peter repeats the last answer he has heard from Adam in heaven, he feels lucky the Hound is kept on chains or else he would have been torn apart by the sharp teeth of this ferocious Beast.
"Why?", growls the multi-headed dog named Kerberos, "This time I will answer you!"

"If a host of mathematicians as huge as the number you just told me were to search for the Biggest number and send an envoy to me after their civilisation had perished to hand me their solution, I would instantly raise a new mathematical army to the last given number to search for a way to formulate an even Bigger number. And I would continue sending out mathematical souls and receiving the envoy until one would come back to me whose answer was about the same as last time. Then I would decide to repeat this process of sending and receiving an extra number of times as Big as the final answer, just to make sure the solution remains in essence the same. Only then would I accept that final number to be the Biggest number conceivable in mathematics."
"It is that many of my dog boned lives that you, Peter, must suffer the sulphur fumes in the metro system of Hades", spoke the mythical Cerberus.

"Two questions before I go", said Peter, "How long does your dogship usually live, and what if these mathematicians keep returning with entirely different functions of numbers?"
"If so my former method would become the initial event of a successive process to which the same rules of formation and termination and eventually succession apply", refereed the Dog of dogs.
"And regarding the first question you've asked, I live exactly for as long as you can stay, exactly."
"Wu!"

The Saint, a Dog, who can tell what follows next?
These days nobody stands at a loss of words, nor sits totally perplexed.
Such a painful process 00 the bitter end...endarkenment.
I beg you a smaller question please

Σ Reference

Σ.1. Dictionary of Signs

Index of mathematical symbols – with a name and a short explanation. The name of the symbol links to a passage in the article which declares its use or elaborates on it.

0 Zero the empty natural number   reduced by removing the sign.
1 One the number 1 or the unit 1 that composes the natural numbers 1..
2 Two the natural number 11 as a character, substitute with X2Y {2:11}
3 Three the natural number 111 as a character sign.
4 Four the natural number 1111 as a character.
5 Five number 11111, etc. 6,7,8,9 – but then 10 is an old-style ten.
ω Omega the infinite unit ω > 1.. counts the set of all natural numbers.
- Inverse unit generates negative numbers -*n and inverse functions.
- Minus the old-style operator of subtraction 1-10=-9 is not countable.
+ Binary plus the old-style operator of addition 1+9=10 is never counted.
+ Unary plus the left associative subscriptable operator a++..;.. of bigA.
* Star operator single * for multiplication, multiple **.. superpowers.
^ Arrow operator single ^ of exponentiation, multiple ^.. superpowers.
× Abstract operator to illustrate locally defined operations in an example.
/ Division operator for division a/b, roots a//b and super-roots ///..
% Remainders super-modulo %.. of the above super-division operation.
£ Logarithmic operator for £.. super-logarithms, e.g. 2££65536 = 4
\ Backslash insert a guest operation into a host series for more precision.
E Decimal power scientific notation, example 6.8E9 ~ 6793605222
Dagger signifies that the previous calculation is cut short in bigO style.
Infinity the utopia ↑∞ of a higher number or infinity not composed of units.
Limit up approach a number from below, or lim n↑ω towards infinity.
Limit down let a variable approach a limit from above, usually towards zero.
Unchained arrow countable operator to construct the operatorial bigY.
Chained arrow parameter separator in Conway's chained arrow notation.
! Factorial unary operator of the super-factorial and fastorial functions.
? Choice random operator n? picks an integer in the range from 0 to n
$ Sequality function $(R) returns 1 if all its arguments are equal, else 0
x Mean algorithmic mean value, the common average xi;../n {xi;#n}
, Separator sign between two parameters or ,.. array dimensions.
; Delimiter marks the end of a row of type subscripts a,0;b,d,e;c
@ Generic char stand-in for any sign possible in the expression context.
. Dots select a subsequence in an expression. Use a brown . for decimals.
R Row row of parameters or subscripts of arbitrary length, also S,T.
Κ Kappa signal a set of functions of similar signature {K.n.m}
= Equality in a=b expression a equals the b that results from its reduction.
Equivalence when two or more different forms of reduction seem possible.
~ Approximation estimate two numbers a~b as about equal in some context.
Not equal comparison a≠b where number a does not equal b
< Less than comparison a<b where number a is smaller than b
> Greater than comparison a>b where number a is larger than b
Less or equal comparison a≤b where number a is not larger than b
Greater or equal comparison a≥b where number a is not smaller than b
& And the logical expression a&b means both a and b are true.
| Or the logical expression a|b means either a or b or both are true.
: Substitution replaces a variable by an expression with equal := value.
# Meta-operator to repeat a sequence of signs within a preceding expression.
Clubs shorthand for a dimension box :: where value = sizes = dimensions.
10 Radix generic number base, counting with fingers is based on 10 ten.
TB Terabyte data capacity of 2^40 byte = 2^48 bit priced at €50,-
:. Illipsis or .:. marks a sequence with a changing meta-variable.
:: Dimension box dual repetition of item a along b sizes and b dimensions.
() Brackets subdivide an expression or function for ordered evaluation.
`` Mini-brackets in many evaluation trains one bracket sign will be sufficient.
[] Waitor brackets special brackets for waitors complement mini-brackets.
{} Set brackets denote the class level of functions or a set of unique elements.
{} Meta-brackets contain meta-statements that denote an expression.
(n Temporary brackets are removed after an n number of reduction steps.
0* Star space the default sign context adds adjoined units and variables.
0^ Arrow space the classic sign context that multiplies adjoined variables.
=: Double equal reduction which does not apply to nested waitor expressions.
-> Abstract step a locally defined single step from a left to a right expression.
=> Right implication the left expression L=>R implies or creates the right.
<= Left implication the right expression implies or causes the left.
<~ Predecessor two algorithmically close expressions, the right comes before.
~> Successor two expressions are algorithmically close, the right comes next.
->> Abstract iteration multiple steps from the left to the right expression.
== Equal by iteration iterating onwards the two expressions will be equal.
<=> Double implication both expressions imply the other, so both are equal.

The precedence rules for most signs are illustrated in chapter 2.5.
The purpose of the four list markers for function definition lists is discussed in chapter 4.0.

Variables, wildcards, subscripts and colour conventions are explained in chapter 1.1.
Mathematical constants can have their signs π ê î capped to avoid conflict with our standard variables, and to celebrate that they are generated by the exponential ^ functions.

# Standard variables

The letters of the alphabet are used as variable names that come in groups.

  • abcde operands or parameters at the left side of a function array.
  • fgh coefficients added to a function or specific parameter types.
  • ij íì incrementors add n times up from 1 or down from n
  • lo Take care, lookalikes of 10
  • kmnpq natural numbers, 0 inclusive, e.g. to count indices.
  • rst denote the size of a dimensional or higher construct.
  • uvw typify substitutes in the evaluation of an operatorial.
  • xyz the parameters at the right end of a function array.

Σ.2. Pedigree of Functions

Explains the context in which a signed function is constructed.

F Function example function, as is G, enumerable with a coefficient Fc.
Γ Gamma Euler's factorial function over the real and complex numbers.
φ Phi lower recursive function, which is iterable by a higher function.
ψ Psi higher recursive function, which iterates over a lower function.
A( bigA operatorial function based on the +R unary plus operator.
E( bigE operatorial function based on the ^R arrow operator.
I( bigI operatorial function based on the *R star operator.
O( bigO an umbrella function to express generic properties of operatorials.
O[ Waitor bigO inner expression, waiting for special evaluation rules to apply.
U( bigU operatorial function derived from the factorial operator.
Y( bigY operatorial function based on unchained arrows.
Fa Special function denoted by two or more characters, like Grz or Ack_ψ'
int Integer part discrete function that returns the integer part of a number.
log Logarithm inverse power log(10^m) = m where radix 10 is generic.
BEAF BEAF Bowers' Exploding Array Function, a famous Big number algorithm.

Σ.3. Glossary of Concepts

Ackermann function
iterate over the enumeration of a family of primitive recursive functions to make the first Ackermann jump on the Ackermann-Conway level {K.2} entering class {K.2.0} of Ackermann functions.
Algorithm
a set of instructions to construct or reduce an input sequence step by step and to eventually output a richer construction or a simpler but larger sequence such as 1.. or to never halt.
Chained arrows
John H. Conway's recursive extension ai→..→y→z of Superpowers, that covers the whole Ackermann-Conway level of functions, and jumps beyond in the form of super-chained arrows →→..
Dimension box
is a box or multidimensional array or discrete hypercube, where standard boxes are iteratively stashed inside standard boxes, until the number of dimensions and all dimension sizes are equal.
Dominant rule
rule in the definition of an operatorial function, dominating the evaluation of natural expressions thereof, because it contributes most to the size of the resulting numbers (function speed).
Dual repetition
two repetitions of characters AB.... {B..#m #n} that work in tandem to create a series of open repetitions, which is structurally comparable to exponentiation.
Enumeration
externally keep count of some construct (such as a function) with a natural number coefficient. A so called predecessor was previously enumerated, next comes the successor.
Feedback
feeding the result of an expression back into one or more variables in the same formula. Higher recursions are usually first found by enumeration of iterated feedback.
General recursion
any function or construct, which grows by substituting a predecessor for a parameter or for an inner subordinate construct. Usually only called general past the level of primitive recursion.
Graft on a stock
when the left part of a function array (the parameter stock) is defined with different rules than the remaining function space (the expanding row shoot), the function is said to be grafted.
Operatorial
array functions bigO that we construct following the natural expansion of some operator. All operatorials count down final iterators to z=0 and then apply O(X,) = O(X) as a generic axiom.
Power tower
an expression with powers a^bi... {^bi#n n>1} of n1 storeys or n exponents, evaluated from right to left. Small power towers can be used to express astronomical improbabilities.
Primitive recursion
the class of functions starting from addition of 1 and closed under iterated substitution of earlier primitive recursive functions for any number of parameters.
Structure
a structure in its ground state is a pure construction, composed of smaller patterns, with characters devoid of arithmetical meaning. It can be filled and grafted on and interpreted on different levels.
Superpowers
operations higher than exponentiation a^b that cover primitive recursion. Also known as super-exponentiation. Specially the subclasses n of multiple stars *.. {*#n1} or arrows ^.. {^#n}
Upload
substitute the counted down value of higher iterators (eventually y=1) for the accumulated value of a lower left parameter (notably b). Rules for uploading give the biggest boost to our operatorials.
Waitor
an inner expression within waitor brackets that waits in place after it is introduced, until its outer expression has been reduced maximally. Then the waitor is applied to restart the evaluation train.

Φ Big Family

Φ.2. Artwork

Our house photographer is Tess Jungblut.
Photography and politics weblog Tessonome (leave a message after the beep).

Under 2011 Construction

Comes a table with artist's info (copyright), reason to insert it in the book (context), and thumbnails zoomable to large pictures.

Φ.3. Commentary

Emails, visitor comments, answers and questions, FAQ, Guestbook.

Under 2011 Construction