Today we shoot past the practical physical world with the operation of tetration, which a man uses to raise power towers of variable height. From there on the other superpowers are developed.
Arrows ^^{..}
for comparison
§2.1
In 1976 Donald Knuth invented his arrow notation to express with operators the numbers that result from superexponential functions (constructible by primitive recursion, classified in the Grzegorczyk hierarchy – that some functions increase faster was proven by Wilhelm Ackermann in 1928).
We use arrowheads
a^^{..}b
in comparisons, and shown below how to
evaluate them.^{ }
Knuth continued the rules for
+ * ^
by^{ }reducing a superpower with
c+1
arrows
to a tower of b-1
times the operation
with c
^{ }arrows.
- a*(b+1) = a+a*b == a+..a*1 {:b}^{ }= a+..a {:b}
- a^(b+1) = a*a^b == a*..a^1 {:b}^{ }
- a^^{..}1 {c1} = a^^{..}1 {c} == a
- a^^{..}b1 {c1} = a^^{..}a^^{..}b {c c1} ^{ } == a^^{..}..a {c :b}
The 2nd repetition :b
applies to the word from the left start of the Exp
up to the 2nd marker dots ..
(words in the Exp correspond to factors in the Rep by position).
Same level arrow operations are always associated from right to left.
The thing with arrowheads is that they are ruled by
majority precedence, so for their evaluation brackets are needed, soon as
3^^^3
= 3^^(3^3^3)
for example.
And if we choose stars ruled by minority precedence (is maximal!),
again these cannot be reduced stepwise without brackets,
with 3**3 = 3*(3**2)
for example.
So to be strict about the characters your algorithm employs,
best^{ }discard the precedence issue
and have operators +^{..}
associate from right to left in all cases!^{ }
Apply
a+^{..}b {+:n2}
= a*^{..}b {*:n1}
= a^^{..}b {^:n}
to transcribe intermediate reduction forms of these operations to
1..
and +^{..}
signs
(binary, without bracket).^{ }
Further simplify these expressions to a triple format
a+_{c+}b
{c=n2}
or to a file format with a single fixed a
plus lots of double recursions
,b,c
on top._{ }
Even further, at the expense_{ }of operator compactness,
compression of the reduction forms of superpowers reaches its binary low in a
basic row
function._{ }
Tetration with higher exponents a,b,c
§2.2
We have set up an elementary
system ♥
for redoubling and extend its rules slowly.
- 2.0 a,1,1 = a,a1 _{1.}= a.. {:2^a1} = a*2^(1+a)
- 1.3 a,b1,1 = aa,b,1 == a..,1,1 {:2^b} = a*2^(1+b+a*2^b)
Three examples how you can reduce such expressions completely.
We'll topple the googolduplex 10^10^10^100
here, a number that escapes our physical universe.
- ♥(6,1,1) _{2.}= ♥(6,7) = 6*2^7 = 768
- ♥(2,5,1) _{1.}= ♥(4,4,1) == ♥(2^6,1,1) _{2.}= ♥(64,65) = 64*2^65 = 2^71
- ♥(2,1,5) _{2.}= ♥(2,3,4) _{1.}= ♥(4,2,4) _{1.}= ♥(8,1,4) _{2.}= ♥(8,9,3) _{1.}== ♥(8*2^8,1,3) _{2.}= ♥(2^11,2049,2) _{1.}== ♥(2^(11+2048),1,2) _{ }== ♥(2^(2059+2^2059),1,1) ~> ♥(2^2^2059,1,1) _{ }~> 2^2^2^2059 ~> 2^^4\^11 ~> 2^^6
The last reduction applies the following rules that
result in a base 2 tetration.
The approximation given will be worked out generally in
section 5.
Assign helper variables u,v,w,..
to store numbers.
- 2.1 a,1,c1 = a,a1,c _{1.}== a..,1,c {:2^a}
- 1.4 a,b1,c = aa,b,c == a..,1,c {:2^b} _{2.}= u,u1,c- {u=a*2^b} ~> 2^^c\^(a*2^b)
So the three parameters are translated to a tetration
2^^c
with a postponed power operation
\^P
on top, marked by the backslash.
This is by convention worked out to a power tower
2^..(2^P)
{:c-}
to get more detail from P
.
Backslashed arrow notation is a convenient tool
to compare superexponential functions.
However, the few extra steps backslashed \
on top
can usually be neglected, given that the direct size of the stairway
of a higher superpower is so dominant.
Our superexponential row §2.3
Let wildcards A,X,Y,Z
in a rule substitute
for any allowed word (sign sequence) in the expression.
We can add some restrictions to the words covered
by a wildcard with Regular Expressions,
and call such a standard predefined wildcard a wordcard.
Wildcards are allowed to remain empty, with no characters at all.
This in contrast to variables n>0
inbetween separators which do not count down to zero.
The exception is when an end parameter is
deleted
(it is thought not to hold a value
0
anymore).
By default the wordcard R
will stand in
(R: RegExp)
for (part_{ }of) an array row, made up of parameters
p_{i}
and single separators ,
but without begin or end separator.
The regular expression for R
shows (when you click it!)_{ }it can also
be an empty array_{ }.
Complete rules for the first row of parameters in my
system ♥
with multiple parameters and single comma in between.
The new upload rule does look beautiful!
- 0.1 A, = A (comma elimination)
- 1.0 a,1 = aa, (initial doubling)
- 1.5 a,1bZ = aa,bZ (redoubling motor) == a..,1Z {a:2^b}
- 2.2 a,1..R {,1:s1} = a,1..a,R {,1:s} (uploads) == a,1a,..R {a,:s} _{1.}= aa,..R {a,:s1}
Note that technicalities of greedy vs. lazy selection change the recipe for a wildcard:
Selections are greedy (in Regular Expressions),
which means that all subsequent instances of a word to match
are to be included, so that none follow right after.
When in the declaration of a rule a wildcard X
is placed after a repetition ..
of variable size,
the greedy match on the left forces X
not to start with the repeated token.
Because we've defined a variable as a repetition of ones
1.. {b}
the wildcard in rule ♥.1.5
above
(is Z)
is not supposed to start on 1
(although the rule would still work).
If the wildcard R
had not been predefined as a (partial) row,
in rule ♥.2.2
the greediness of the left series of ,1
would have required us to adapt R
to start on another token.
As soon as a wildcard is declared other repetitions that follow
can be lazy again – making it alright for
R
to start with word a,
in the continuation of rule
♥.2.2
above.
This being said, a clean policy will avoid the greedy issue
by using well defined wildcards.
After reaching t_{1}
= 256
in blog
turf 1
it is now time to create the superexponential record t_{2}
with ten characters up to single ,
comma between numbers 1..
- t_{2} = 11,1,1,1,1 = ♥(2,1,1,1,1) _{ } _{ 2.}= ♥(2,1,1,3) == ♥(4,2,2,2) _{ } _{ 1.}= ♥(8,1,2,2) _{2.}= ♥(8,9,1,2) _{ } _{ 1.}= ♥(2^11,1,1,2) _{ } _{ 2.}= ♥(2^12,2^11,2^11,1) _{ } _{1.4} ~> ♥(v,1,1,1) {v=2^^(2^11)} _{ } _{ 2.} ~> ♥(v,v,v) ~> 2^^2^^(2^11) _{ } _{ } ~> t_{1}^^^3 ~> 2^^^4\+1
The number t_{2}
is already unimaginably large,
a full tetration step larger_{ }than the ancient
unspeakable tetration
10^^(10^(5*2^120))
that mahayana buddhism aspired to._{ }
Clean rules and the basic row §2.4
In our clean systems:
- You don't repeat variables – only words consisting of fixed units are repeated.
- Try to do without brackets – in the primary rules do not nest subexpressions.
- Although a lower number rule comes before a higher rule in the list
(we usually match
♥.1
before♥.2
), avoid dependence on rule preference. - Avoid greedy issues with repetitions while using predefined wildcards.
- Don't rely on Regular Expressions to give back characters (possessive).
- Don't allow value zero in parameters, but keep final countdowns smooth.
Knuth's arrow operation a^^{..}b
can be projected on a basic row of parameters,
for the evaluation of superpowers in a system
♦
similar to ours.
- 0. a,b = b
- 1. a,R,1 = a,R
- 2. a,b,1R = a,f,R {f=ab R≠0}
- 3. a,b.,1..R {:s1} = a.,1..,b1,R {:s} = a,a.,1..,b,R {:s-} == a,a,a.,a-..,b-,R {:s--}
- ♦ a,b.,1..,2 {:s} = a,a.,1..,b {:s-} = a+^{..}..a {s :b-} = a*^{..}b {s}
An input expression runs through many reduction forms
until a number output is reached.
But in upload rule ♦.3
the smallest steps, like
a^b = a*a^(b-1)
,
are skipped and Knuth's substitution directly applies, so that
a^(b+1) = a*..a
{:b}
.
In ♦.2
the function motor f
adds constant a
to the beast b
, which is slower
than the motor rule ♥.1
doubling its animal a
.
Also ♥
gains a parameter
for it lacks a constant
and it iterates down to zero instead of 1
.
Perhaps more significant is that small a
instead of big b
is substituting the left over entries 1
in the upload series.
We expect ♥
to outpace ♦
but not in the long run!
The higher upload dominates and both upload rules let
their best(ial) value substitute for the rightmost entry 1
in the series.
Less powerful rules in the beginning don't matter so much in the end.
In section 5
below we will run down the calculation
to compare both systems and prove this for the row.
Operators turned superators §2.5
Vice^{ }versa system ♥
has an operator format ♥
that offers compactness of expression.
But instead of counting operators
♥^{..}
we'd rather put their number in a superscript,
so you might call them superoperators or
superators then.^{ }
In both formats the hearts are left associative
and we can park a heart wildcard
^{♥}
on the right to hide
some advanced operations from view.^{ }
Translate
♥(2,1,1,1) = ♥(2,1,3) = ..
^{ }
to come to grips with the definition.^{ }
2♥♥♥1 = 2^{3}1 = 2^{2}2^{3}0 = 2^{2}3 = 2♥♥3 = 2^{1}2^{2}2 = 2^{0}2^{1}1^{2}2 = 4^{1}1^{2}2 = 8^{1}0^{2}2 = 8^{2}2 = 8^{1}8^{2}1 = 2^3^{1}8^{♥} == 2^11^{1}0^{♥} = 2^11^{2}1 = 2^11^{1}2^11^{2}0 = 2^11^{1}2^11+1 = 2^12♥2^11 == 2^12*2^2^11^{1}0 = 2^2060
For comparison the system ♦
with right associative superators
^{s}
counting stars
*^{..}
is shown right.
We stub diamonds ^{♦}
left, but heart wildcards ^{♥}
on either side.
Dare try to wrap your head
around these alien^{ }abacuses!
a^{0}b^{♥} = ab^{♥} ^{ } ^{ } ^{♦}a^{0}b = ^{♦}ab a^{s}0^{♥} = a^{♥} {s<2 or ^{♥}≠0} ^{ } ^{♦}a^{s}1 = ^{♦}a ^{♥}a^{s1}0 = ^{♥}a1 {s>0} a^{s}b^{♥} = a^{s-}a^{s}b-^{♥} ^{♦}a^{s}b1 = ^{♦}a^{s-}a^{s}b ^{ } ^{ } == a.a^{í}..b-^{♥} {:s} == ^{♦}.a^{s-}..a {:b}
An incrementor í
starts at 1
and is increased by 1
in each next step of a repetition :s
.
For example
1*í..
{*í:n}
defines the factorial,
but in that case order doesn't matter so you can write
n! = 1*i..
{*i:n n≥0}
as well.
With these special operator notations we like
to^{ }demonstrate that superexponential numbers
can be expressed in a simpler way.^{ }
Pity the party people on planet^{ }earth
have hijacked superscripts for their petty exponents
^{♦}a^{2}b
,
but for a machine superscripts and subscripts need to be delimited anyhow,
in order to be read from the line.^{ }
Generally there are two options for indexing operators and/or separator signs:
You can expand them directly a[c]b
between_{ }brackets – or almost directly
after the initial ,
like Bowers did in his Beaf
(Jabe is wasting a comma sign there)._{ }
Or you can first introduce them ,
and then start
to_{ }count ,.. {c}
and append an enumeration ,d;..
or an array ,;d,e,..;
(like we did before)
or ,`d,e,..`
(like we do on this journey), so that a mono-bracket type expansion
is eventually reached._{ }
The main difference between the two systems –
the redoubling row versus the basic row system –
appears to be that
♥
operations are in effect reduced by
minority precedence. This is certain to result in larger numbers
(example 5*2**3
= 1000
)
than the majority precedence
(where 5*2^3
= 40
)
which rules system ♦
operators.
But how much larger are those numbers exactly?
Evaluating the first row §2.6
Devise a formula to approximate the first row of
system ♥
with arrow
^^{..}
operations.
The usual reduction order is ♥:1:2:0
cascading from the left until rules wear out.
- a,b1,R = u,1,R {u=a*2^b ~ 2^(b+log(a))}
- a,b1,c1^{♥} = u,1,c1^{♥} = u,u1,c^{♥} = u*2^u,1,c^{♥} ~> 2^u,2^u,c-^{♥} ~> 2^..u,1,1 {:c} = w,1,1^{♥} {w=2^^c\^u}
- a,b1,c1,d^{♥} = w,1,1,d^{♥} ~> 2^^w,1,1,d-^{♥} == x,1,1,1^{♥} {x=2^^^d\^^c\^u} = x,x1,x ~> 2^^x,1,1 ~> 2^^x^{ } = 2^^^(d+1)\^^c\^u^{ }
- a.,r_{í}.. {:n} ~> 2^^{..}r_{n}1\.^^{..}r_{ì}\..log(a)- {n ì :n-}
In this general approximation formula we've used
both incrementor í
(as explained above)
and decrementor ì
meta-variables.
A decrementor starts at s
and is decreased by 1
each step of a repetition :s
until finally index 1
is reached.
Now we can compare some Big numbers between systems
and find that our ♥
is two parameters faster
– or one if you don't count the
♦
constant
2
item.
a,1.. {,1:s1} ♥= a,1..,a1 {,1:s-} = a^{s1}1 = a^{s}a1 ~> 2^^{..}(a+1) {^:s a≥1} ♦= 2,a1,1..,2 {,1:s1} = ^{♦}2^{s1}a1
Of course these differences in function speed lose significance once you learn to express row length in the very Big numbers created on the row.
Can there be a final generalization in the
evolution of a rule (or rule principle)?
Ponder a peculiar candidate that may answer this
fundamental question affirmatively.
The doubling rule ♥.1.0
from
day one
has been extended a few times to cover further iterations of redoubling
on the row.
Now we propose a universal motor rule ♥.1.6
with a restriction on the wildcard
(is Z)
And meet the dire consequences if we allow any
()
wildcard Z
as an opportunity to create new numbers:
fractions, roots, tetration roots, etc…
- 1.6 a,1Z = aa,Z
- => aa,,1 = a,1,1 = aa,a
- aa,,11Z = a,1,11Z = aa,a,1Z
- aa,,1,..1Z {1,:s} = aa,..Z {a,:s2}
You can choose rule ♥.1.6
as an alternative where usually an upload rule
♥.2
applies.
To ponder an example, derive
1,,1,1
= ♥(1,½,½)
a mystifying iteration indeed!
We won't go into the field of fractional parameters from odd a,,Z
because we are out to build Big numbers in higher dimensions.
There we evaluate strictly to whole numbers.