Friday, 21 December 2012

Big Arrows Compass

Painting of labyrinth of arrows on concentric circles

Today is selling archery gear to the Maya gods at a discount. We create a natural extension of Knuth-Conway style arrows, for comparison against various array functions for Big numbers.

Knuth's up .. arrows  §3.1

There are two popular types of arrows available for the definition of Big numbers.
In this chapter we use the countable up-arrow as operators for superpowers. Next chapter we study the right-arrow parameter separator for the row of Conway's chained arrows.
All our arrow structures are painted blue and resolved from right to left (right associatively).

We prefer to speed up multiplication (the first recursion of repeated addition) by substituting a*b = a.. {b} directly into an expression. Applied as in rule *. below which goes straight to the result.
To start more rigorously we can define addition as the shifting of 1 between variables, and multiplication by repeated addition of the same number. Though Karatsuba multiplication for large numbers is the faster algorithm, we like to keep it simple. Adding step by step while avoiding brackets is achieved with the plus + sign in between operands.

  • *.0 X+a1 = X1+a == Xa+1 = Xa1
  • *,1 a*1 = a
  • *,2 a*b1 = a+a*b = a(a*b) == a+..a*1 {:b} = a..(a*1) {b} = a+..a {:b} = a.. {b1}

We use RepExp notation in the definition of rules to repeat substrings in the expression.
A wildcard X stands for an arbitrary substring (which honours the context), and X may be void of characters. But parameters are never empty, and our variables p are greedy for number signs (eat all 1..). So that rules that start on a variable (above rule *,2 and rules A, below) can be thought to have an inactive (possibly empty) left part X with an operator sign on the right end (not ending on 1 or ω etc.).

  • *. Xy*z {X†?} = Xy.. {z}

A virtual left part X is indicated by a comma , instead of a dot . in the rule number. To make this optional X explicit we let it follow by a question mark {X?} in the repertory. The dagger character is a generic operator sign, for instance * number repetition.

Follows the rule list for the evaluation of .. operations (in the active right part of an expression) in the context of up-arrows. Knuth counted his arrows up from the initial A,2 power and so on == by direct substitution, but we prefer a stepwise = approach.

  • A,1 a↑..1 {c} = a => a↑1 = a
  • A,2 a↑b1 = a*a↑b == a*..a↑1 {:b} = a*..a {:b}
  • A,3 a↑..b1 {c1} = a↑..a↑..b {c c1} == a↑....a {c :b}

The superpowers ..{c≥2} expressible now are already far greater than any realistic quantity you can ever hope to imagine. The fantastic world record number, the unspeakable tetration 10^^(10^(5*2^120)) from Buddhist scripture, is smaller than 4↑↑↑3 its neighbour pentation. 

Because the counter starts c=1 at a power (not multiplication), we have a nice shortcut.

  • 1*1=1 => X↑1↑..z {↑:c>0} = X↑1 = X

At chained arrows rule C.3 we get the same total right drop at the penultimate operator. So we'd like to see this feature return as a consequence of other arrow systems. But in the extended system B. below, we can only drop the t→ is 1→ from in between.

Backslashed operations on top of up-arrows are helpful to compare the growth of various functions in the range of superpowers (to assess the Grzegorczyk hierarchy).
A simpler extension with one right arrow on top fulfils a similar purpose (and heralds the continuation of chained arrows to higher dimensions). This refined up-top system B. is defined here, and illustrated by expressions with the old \^ backslashes.

  • B,1 a1 = a where =.. {k>0} or =
  • B,2 a↑b1 = a*a↑b == a*..a {:b} = a^(b+1)
  • B,3 a↑..t→b1 {c} = a↑..a↑..t→b {c c} == a↑....t→1 {c :b1} = a↑....t {c :b1} = a^..(b+1)\^..t {c+1 c}
  • B,4 a↑..b1 {c1} = a↑..a→b {c} = a^..b\^..a {c+1 c}

Multiplication * is the first recursive rule, which repeats the addition of a number. The series of up-arrows .. repeat earlier operations of repetition. 
First come the a↑b powers, next the power towers a↑a↑t up until a↑..t {:b} which equals a↑t→b and in turn a↑a→b equals a↑↑b1 tetration. Then generalize this process to superpowers expressed by multiple a↑..b {:c} arrows, in total a double recursion. 

Superpowers have no counterpart in physical reality – their numbers are just too big to be of practical use. Consider the size of 3↑↑4 evaluated both in the system A. of up-arrows and in the extended system B. for example.

  • 3↑↑4 A.3= 3↑3↑↑3 = 3↑3↑3↑↑2 = 3↑3↑3↑3↑↑1 A.1= 3↑3↑3↑3 A.2= 3↑3↑3*3↑2 = 3↑3↑3*3*3↑1 A.1= 3↑3↑3*3*3 *.= 3↑3↑27 == 3↑7625597484987 ~ 1E3638334640024
  • 3↑↑4 B.4= 3↑3→3 B.3= 3↑3↑3→2 = 3↑3↑3↑3→1 B.1= 3↑3↑3↑3 B.2= 3↑3↑3*3*3 *.= 3↑3↑27 ~ 1511 GB

Though this tetration still fits on a 2 TB harddisk, larger superpowers can't be digitalised or written physically in number base. All of our 1E120 bit of quantum-information space fails to store 4↑↑4 in binary. Working out the recipe would run the cosmos to ruin.

Conway's chained arrows  §3.2

In 1996 John H. Conway published his chained arrow notation in “The Book of Numbers”. We think his arrow operators should become the benchmark of choice to further classify functions in the fast-growing hierarchy.

Define a self-sufficient list of rules for Conway's chained arrows.

  • C.1 X→1 = X
  • C.2 a→b1 = a*a→b = a↑b1 == a*..a→1 {:b} = a*..a {:b}
  • C.3 X→1→z = X <=> X→1→Z = X
  • C.4 X→y1→z1 = X→(X→y→z1)→z == X→(..X→1→z1.)→z {:y:} = X→(..X.)→z {:y:}

Conway directly substituted the three entries a→b→c = a↑..b {c} but we can reduce the three chained arrow entries to multiple up-arrows step by step. This proves that the system C. can express every superpower and reduce it to multiplications. 

  • a→b1→1 = a→b1 = a↑b1
  • a→b1→2 = a→(a→b→2) = a↑(a→b→2) == a↑..(a→1→2) {:b} = a↑..a {:b} = a↑↑b1
  • a→b1→3 = a→(a→b→3)→2 = a↑↑(a→b→3) == a↑↑..a {:b} = a↑↑↑b1
  • a→b1→c1 = a↑..(a→b→c1) {c} == a↑..b1 {c1}

The main rule C.4 in Conway's system relies on bracket characters to nest expressions. Now we transform it to a chained-up arrow system D. where brackets are not required, because in X→y→z the operations starting on take over as word delimiters.

In this context we allow multiple up .. and single right arrows, with temporary up-right arrows ..{z>0} as auxiliary constructs for stepwise operator change. In wildcards all contextual signs are valid, so in X the X may end on .. arrows.

Parts of a rule that are painted grey remain unchanged in its evaluation. A part denoted by a question mark {X?} may be lacking (if void the expression still matches the rule).
To demand that string X is part of Y we write XY, and not to allow X to match in Y we write XY, in the Rep (repertory) for the Exp (expression).

Our rigorous list of rules for alternative chained-up arrow algorithm D. It expands the up-arrow context of system A, but is incompatible with system B, up-top. (Can you make it compatible by adapting rules D.2.3?)

  • D.1 X11 = X1 where =.. {k>0} or =
  • D.2 Wx→y1 {†=or* W†?} = Wx*x→y == W.x*..x {:y}
  • D.3 X1↑y = X1→y
  • D.4a X→y→z1 = X↑→y→z == X↑..→y→1 {z} .4b = X↑..→y {z} = X..y {z1}
  • D.5 W↑1X↑↑y1 {W↑? ↑1X} = W↑1X1X↑↑y == W↑.1X..1X {:y}

Rule D.3 handles both the referral of x↑y↑z powers to rule D.2 and the removal of (virtual) brackets from x→y↑z chained arrows. Rule D.4 can be done step by step (strictly a dual rule), or substituted straight away in grand Conway style.

Rule D.5 reduces .. {c1z1} up-arrows by repetition of 1X chains pi..pn1.. {:n c}  where 1X is delimited by (but not contains) lone .. up-arrows.
This clause of matching the repeated string 1X from the right down to (the left end or) ↑1 (or generally until q↑‡p is met) replaces the nesting brackets of rule C.4. 

Take rule D.4 and D.5 together to meet the conclusion of Conway's original system.

  • X→y1→z1 D.4= X..y1 {z1} D.5= X..X..y {z z1} C.4≡ X→(X→y→z1)→z =D.5= X....X..1 {z :y z1} C.4≡ X→(..X→1→z1.)→z {:y:} D.1= X....X {z :y} C.3≡ X→(..X.)→z {:y:}

From this format it is clear that chained arrows repeat the double recursion of up-arrows, taking two entries from the end of the row. By rule D.3 the result is appended to the row again as a new chained entry, so each double recursion consumes a single entry. Then over the length of the row we count the 3d recursive algorithm.

We'll reduce two chained arrows expressions 3→4→2 (the example 3↑↑4 in last chapter) and 3→3→3→3 (introducing the →↑↑ row creator), so you can see the rules at work.

  • 3→4→2 D.4= 3↑→4→1 = 3↑→4 = 3↑↑4 D.5= 3↑3↑↑3 = 3↑3↑3↑↑2 = 3↑3↑3↑3↑↑1 D.1= 3↑3↑3↑3 D.3= 3↑3↑3→3 =D.2= 3↑3↑3*3*3 =*.= 3↑3↑27 =D.3.2.*.= 3*..3 {:7625597484986}
  • 3→↑↑2 E.7= 3→↑3→2 E.6= 3→3→↑3→1 E.1= 3→3→↑3 E.7= 3→3→3→3 = 3→3↑→3→2 = 3→3↑↑→3→1 = 3→3↑↑→3 D.4= 3→3↑↑↑3 C.4= 3→3→(3→3→2→3)→2 D.5≡ 3→3↑↑3→3↑↑↑2 C.4= 3→3→(3→3→(3→3→1→3)→2)→2 D.5≡ 3→3↑↑3→3↑↑3→3↑↑↑1 C.3= 3→3→(3→3→(3→3)→2)→2 D.1≡ 3→3↑↑3→3↑↑3→3 C.2.*.= 3→3→(3→3→27→2)→2 D.2≡ 3→3↑↑3→3↑↑3*3*3
  • a→b→c1→2 ≡ a→b↑↑c1 a→b→(a→b→c→2)→1 ≡ a→ba→b↑↑c a→b→(a→b→c→2) ≡ a→b→(a→b↑↑c) a→b→(..a→b→1→2.) {:c:} ≡ a→b..↑1 {:c1} a→b→(..a→b.) {:c:} ≡ a→b..a→b {:c}
  • #Garham = 3→3→(..4.) {:64:} < 3→3→65→2

Watch the tetration 3↑3↑3↑3 being reduced to mere trillions of multiplications. Compare the chained arrows 3→3→3→3 which is even larger than Gardner's Graham's number.

And this is hardly where it all ends… Long after the sky has fallen down a tiny typist will travel the universe of Big numbers on the back of an arrow.

Friday, 20 January 2012

Growing Numbers in all Dimensions

Trajectory of a cloud of gas near the supermassive black hole

Multidimensional array functions for Big numbers.

Past ,, into the second row  §3.4

After the first row of function arguments in our system a second row is appended.
We employ a double comma ,, to separate rows within a multidimensional array.
In a definition list we mark a rule m.♥ if it was covered by some rule m.n before.
We approximate ~> expressions by preceding (just smaller) chained arrows.

  • 3.0 a,1,,1 = a,1.. {,1:a1} ~> 2→a1→a
  • 1.♥ a,b1,,1 = aa,b,,1 == a*2^b,1,,1 ~> 2→a*2^ba*2^b

When entry a has accumulated its first row so that all but entry b=1 is counted down, its job becomes to inflate the size of the row again.
Work out three cases. The approximation as calculated for the first row applies.

  • ♥(5,1,,1) 3.= ♥(5,1,1,1,1,1,1) 2.= ♥(10,5,5,5,5,5) ~> 2^^^^^6\^^^^5 ~> 2→8→5
  • ♥(2,4,,1) 1.= ♥(16,1,,1) 3.=2.= ♥(32.,16..) {:16} ~> 2^..17\^..16 {16 15} ~> 2→19→16
  • ♥(2,1,2,,1) 2.= ♥(2,3,1,,1) 1.= ♥(8,1,1,,1) 2.= ♥(8,9,,1) 1.= ♥(u,1,,1) {u:2^11} ~> 2^..u1\^..u {u u-} ~> 2→u3→u ~> 2→11→2→2

By default a partial row R either starts on 1 or is empty, which implicates we have to follow up with comma elimination A,=A in rules ♥.3.1 and ♥.2.5 below.
Wildcard Z (or in calculations) fits any part to the right of the expression, but because the preceding repetition is greedy, Z and later R won't start with ,1 in the rules ♥.2 respectively.  This can also be read from the matching Z/RegExp descriptions. 

  • 1.♥ a,1bZ = aa,bZ == a*2^b,1Z (redoubling motor)
  • 2.3 a,1..Z {,1:n1 Z/(,(,1+)*)?} = a,1..aZ {,1:n} (upload to end of first row) == aa,a..Z {,a:n}
  • 2.4 a,1..Z {,1:n1 Z/1+(,1+)*,?(,1+)*} = a,1..a,Z {,1:n} (upload within first row) == a,1a,..Z {a,:n} 1.= aa,..Z {a,:n1}
  • 3.1 a,1,,1R = a,1..,,R {,1:a1} (upgrade first row size) 2.= aa,a..,,R {,a:a} ~> 2→a1→a,1,,R
  • 2.5 a,1,,1..R {,1:n1} (uploads to second row entry) = a,1,,1..,a1,R {,1:n-}

We apply the old rules for the first row until the primer pattern a,1,, occurs and then the new dimensional rules kick in. Generally a lower level structure is treated as if it is a subexpression of the higher level waiting on the right.
Only the rightmost structures cannot be inflated and are just counted down and off (this alone guarantees reducibility of the expression).

Two further examples on the second row.

  • ♥(2,1,,4) 3.= ♥(2,1,1,1,,3) 2.= ♥(4,2,2,,3) == ♥(2^11,1,1,,3) = ♥(2^12,2048,,3) ~> ♥(v,1,,3) {v:2^2059} 3.= ♥(v,1..,,2) {,1:v1} ~> ♥(w,1,,2) {w:2→2059→2→2} ~> ♥(x,1,,1) {x:2→2059→3→2} ~> 2→2059→4→2
  • ♥(2,1,,1,2) 2.= ♥(2,1,,3,1) 3.= ♥(2,1,1,1,,2,1) ~> ♥(w,1,,1,1) ~> ♥(w,1,,w) {w:2→2059→2→2} ~> 2→2059→w→2 ~> 2→2059→2→3

With this last example we have passed Graham's number(s) – the largest integer ever used in a serious mathematical article (in 1971, for an upper bound in Ramsey theory).

Formulate the approximation of two rows in system compared to chained arrows.
From the expressions that were reduced to natural number before we've learned that:

  • a,R = a,rí.. {,rí:n} ~ 2^..rn3,1 {^:n} ~> vr,1 {vr:2→rn→n}
  • a,R,,1b ~> 2→vr→vr,1,,b ~ 2→vr→2→2,1,,b ~> 2→@→1b→2,1,,1 {@aR}
  • ri,..,b {ri,:n} ~> 2→@→b1→2 {@ri..}

A meta-mix shortcut @ over a set of variables r1,..rn will ideally derive some lifted mean from this set, but in practice a maximal parameter rm is a safe choice.
Because the numbers are already so Big and there's no need to split any hairs, the @ mix-maximized value is just to make sure that, when rm has a superiorly (or infinitely) larger value than the other parameters our approximation formula still holds. 
It is convenient to write the set of variables assigned to @ without commas and to append new next entries at an existing max-mixed set without further ado. 

  • a,R,,b,1c ~> 2,1,,2→@→b→2,c ~> 2→@→(2→@→b→2)→2,1,,1,c ~> 2→@b→2→3,1,,1,c ~> 2→@b→(2→@b→2→3)→2,1,,1,-c ~> 2→@b→1c→3,1,,1,1
  • a,R,,b,c,1d ~> 2,1,,1,2→@b→c→3,d ~> 2→@b→(2→@b→c→3)→3,1,,1,1,d ~> 2→@bc→2→4,1,,1,1,d ~> 2→@bc→1d→4,1,,1,1,1
  • Rm,,Rn,z ~> 2,1,,1,..2→@,-z {1,:n- @RmRn} ~> 2→@→2→n2,1,,1,..-z {1,:n} ~> 2→@→z→n2,1,,1,..1 {1,:n} ~> 2→@→z1→n2,1,,1
  • ri,1,..,rj,2,..z {ri,1,:m rj,2,:n} ~> 2→@→z1→n2 {@RmRn}

Let the subscripts under a partial row sign R keep count of its number of parameters.
In system one row with length n1 and final entry z can be approximated by Rn→z→n using 3 parameters of chained arrows. Two rows in the system require 4 parameters of chained arrows. You can guess where we are heading at… 

Square array  §3.5

Just like every parameter holds some variable number 1.. every row in what's called a square will have different lengths. In the course of a reduction cascade the sizes in a multidimensional function array behave extremely erratic.
Parameters are counted down to value 1 and likewise row size is reduced until a leftmost entry 1 remains and then the upgrade rule appends a series of ,1 to the row.

First we sketch an overview of the reduction cascade for a system square array.
The rule correspondences (rules rendered m.n= green) could then be sought out.

  • a,b,c,,d,e,,f,g,,h :: a,1,c 2.7= a,a1,c- : a,1,1 2.7= a,a1 :: a,1,,d,e 3.2= a,1..,,d-,e {,1:a1} 2.6=2.7=1.5 aa,a..,,d-,e {,a:a} : a,1,,1,e 2.7= a,1,,a1,e- : a,1,,1,1 2.6= a,1,,a1 :: a,1,,1,,f,g 3.2= a,1,,1..,,f-,g {,1:a1} : a,1,,1,,1,g 2.7= a,1,,1,,a1,g- : a,1,,1,,a1 :: a,1,,1,,1,,h 3.2= a,1,,1,,1..,,h- {,1:a1} : a,1,,1,,1,,1 3.2= a,1,,1,,1.. {,1:a1} 2.6= .

Reuse the rules ♥.0.1,1.5 in the definition of the second dimension or square.
Use a wordcard S/(1+(,1+)*(,(,1+)+)*)? for a partial square (possibly void).

  • 2.6 a,1,,1..,1..Z {,,1:n- ,1:m Z/(,(,1+)+)*} = a,1,,1..,1..aZ {,,1:n- ,1:m-} (upload to end of row, for n=1 of first row, for n>1 a higher row)
  • 2.7 a,1,,1..,1..S {,,1:n- ,1:m} = a,1,,1..,1..a,S {,,1:n- ,1:m-} (upload within a row, for n=1 in first row, for n>1 a higher row)
  • 3.2 a,1,,1..S {,,1:n} (upgrade row size) = a,1,,1..,1..,,S {,,1:n- ,1:a}

From an earlier reduction of ♥(4,2,2,,3) to ♥(w,1,,2) take over the value of w in the course of the following example.

  • ♥(2,1,,1,,1) = ♥(2,1,,1,1,1) = ♥(2,1,,1,3) = ♥(2,1,,3,2) = ♥(2,1,1,1,,2,2) = ♥(4,2,2,,2,2) ~> ♥(w,1,,1,2) ~> ♥(w,1,,w,1) {w:2→2059→2→2} ~> ♥(y,1,,y) {y:2→2059→w→2 ~> 2→2059→2^2059→2} ~> 2→2059→y→2 ~> 2→2059→3→3

To name the size of this number, among Bowers' infinity scrapers it comes just after the Corporalplex, and yet it is among the smallest members of a very large family.

Build a formula for approximation of a square upon the two rows already calculated. 
A full row Rn has n entries ri,..rn {:n-} and the sign @ is a dynamic mean, best represented by the largest of all entries previously included in the reduction. 

  • Rm,,Rn,,b ~> 2→@→rn→n1,1,,1,,b {@RmRn} ~> @,1,,1,..@→@→@,,-b {1,:2→@→rn→n1} ~> @→@→@→2→2,1,,1,,-b ~> @→@→@→3→2,1,,1,,--b ~> @→@→@→b→2,1,,1,,1
  • Rm,,Rn,,b,c,d ~> @,1,,1,,@→@→@→b→2,-c,d ~> @→@→@→2→3,1,,1,,1,-c,d ~> @→@→@→c→3,1,,1,,1,1,d ~> @,1,,1,,1,@→@→@→c→3,-d ~> @→@→@→2→4,1,,1,,1,1,-d ~> @→@→@→d→4,1,,1,,1,1,1
  • Rm,,Rn,,Rp ~> @→@→@→rp→p1
  • Rm,,Rn,,Rp,,Rq ~> @→@→@→@→rq→q1
  • Rni,,..Rns {:s-} ~> @→..rns→ns1 {:s}

Let S in the system be a two dimensional array with s rows, where the last row has a length z with a final entry y and let S overall have a maximal parameter value x, then S is approximately larger than a series x→..y→z1 {:s} of Conway's chained arrows – or larger than z1→↑s2 when represented by superchained arrows.

We conclude the square of and the row of chained arrows increase equally fast. Given a large enough size s our spouses won't notice the difference (just 2 parameters).
Now we go one dimension higher.

Cubic arrays  §3.6

The new rules for the cube are an extension of those for the square in the section above, and form a special case in the definition of dimensional arrays in the next section.
For the moment work out an example of a cubic array 11,1,,,111 (in binary notation).

  • ♥(2,1,,,3) = ♥(2,1,,1,,1,,1,,,2) = ♥(2,1,,1,,1,1,1,,,2) = ♥(2,1,,1,,3,2,,,2) = ♥(4,2,2,,2,2,,2,2,,,2) = ♥(2^v0,1,,2,2,,2,2,,,2) {v0:2059} ~> ♥(v1,1,,1,,2,2,,,2) {v1:2→v0→3→3} ~> ♥(v1,1,,1,..v1,,1,2,,,2) {1,:v1} ~> ♥(v2,1,,1,,1,2,,,2) {v2:2→v1→v1→v1 ~ 2→v0→v0→2→2} ~> ♥(v2,1,,1,..v2,,v2,1,,,2) {1,:v2} ~> ♥(v3,1,,1,,v2-,1,,,2) {v3:2→v2→v2→v2 ~ 2→v0→v0→3→2} ~> ♥(v4,1,,1,,v4,,,2) {v4:2→v0→v0→v2→2 ~ 2→v0→v0→2→3} ~> ♥(v5,1,,,2) {v5:2→v0→v0→v4→2 ~ 2→v0→v0→3→3} ~> ♥(v5,..,,..7,,,1) {v5,..:v5 v5,..,,:v5} ~> ♥(v6,1,,,1) {v6:v5→..2 v5→..:v5 ~ v5→↑v5} ~> ♥(v6,..,,..7) {v6,..:v6 v6,..,,:v6} ~> v6→..2 {v6→..:v6} ~> v6→↑v6 ~> v5→↑(v5→↑v5) = v5→↑3→2 ~> v0→↑(v0→↑(v0→v0→v0→3→3)) ~> v0→↑(v0→↑(v0→↑4)) ~> 4→↑4→2

Notice the cube is upgraded with squares when it is reduced to a primer a,1,,,
To compare this number with another notation – in Beaf the next larger would be {3,4,1,1,2} a small Pentadecal.

Now we approximate the cube starting from the earlier results on the square.
A member square Sn holds n rows R and in @ we store a past largest entry. 

  • Sn,,,b ~> @→↑n2,1,,,b {@Sn}

This wraps up the cube.

Multiple dimensions  §3.7

A structure is anything that is systematically countable. A bare structure is counted down and either waiting to be upgraded or to be counted off.
Iterator variables 1.. are the smallest structures in this respect and get an upload from the accumulator value a in rule ♥.4 (more precisely: the bare variable 1 rightmost from the primer has the substructure unit 1 added a times).

We've arranged the primer in the system to be a,1 so the countdown of variable b and the first row stops there if a higher structure is waiting.
Philosophically the accumulator a is not part of any structure and all structures are to be counted down to 1 (except final variables z in expressions of which are counted down to zero, leaving a piece of the structural skeleton to drop off from the right).

To address separators in general we introduce a generic shorthand $ which signifies any type of variable separator.  This translation should be consistent, meaning that similar types are intended at similar positions on both sides of an equation.
Then a dimensional series p$i.. {:n} with various commas $i is ,.. {,:ki} can be written in the shorthand p$.. {:n} in a rule list as below .

  • 0.2 A$ = A
  • 2.? a,1$..1Z {1$:n} = a,1$..a1$Z {1$:n-}

After countable commas ,.. to separate higher dimensions, extradimensional types of separators ,Xi;.. or rather ,`Xi`.. will follow suit.