4.1.2 - Extended_Operators

And this is still only the beginning. Next Bowers' defines a whole new set of hyper-operators above and beyond the original set. He calls b{{3}}p "b power-expanded to p", b{{4}}p "b expando-tetrated to p", b{{5}}p "b expando-pentated to p", b{{6}}p "b expando-hexated to p", etc.

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

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

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

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

As you can see, Bowers' doesn't define these operations explicitly, but implicitly through following a pattern. He does however tell us that:

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

Again, Bowers' relies on his explicit definition of array notation to define the hyper-operators implicitly. Despite this brevity and informality, it can be gathered that the new series of operators follows the same definitions as the original series of hyper-operators. That is:

a{{2}}b = a{{1}}a{{1}} ... {{1}}a w/b a's

a{{3}}b = a{{2}}a{{2}} ... {{2}}a w/b a's

a{{4}}b = a{{3}}a{{3}} ... {{3}}a w/b a's

a{{5}}b = a{{4}}a{{4}} ... {{4}}a w/b a's

a{{6}}b = a{{5}}a{{5}} ... {{5}}a w/b a's

etc.

Consider the implication of the above and I think a SUPER YIKES, is in order!! Not only does this new operator sequence grow extremely rapidly, but it implies a way to continue even further. Just consider the strength of power-expansion:

3{{3}}3 = 3{{2}}3{{2}}3 = 3{{2}}3{{1}}3{{1}}3 = 3{{2}}3{{1}}3{3{3}3}3 =

3{{2}}3{{1}}3{27}3 = 3{{2}}3{3{3{ ... {3{3{3}3}3} ... }3}3}3 w/3{27}3 3's center out =

3{{1}}3{{1}}3{{1}}3{{1}}3{{1}} ... ... ... ... ... ... {{1}}3{{1}}3{{1}}3{{1}}3{{1}}3

w/3{3{3{ ... {3{3{3}3}3} ... }3}3}3 3's

w/3^^^^^^^^^^^^^^^^^^^^^^^^^3 3's center out

Now try to imagine expanding out the string of 3{{1}}3{{1}}3{{1}} ... {{1}}3 which itself has an unfathomable number of 3's. Now try to imagine a string of 3{{2}}3{{2}}3{{2}} ... {{2}}3 , a string of 3{{3}}3{{3}}3{{3}} ... {{3}}3, a string of 3{{4}}3{{4}}3{{4}} ... {{4}}3, etc.

It boggles the mind, and there is nothing I can really say to explain how big these numbers are. Even Graham's Number is lost in the sight of it. And Bowers' isn't even close to being done...

What next? Now he defines a new operator above all these: {{{1}}}, with the definition:

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

This is "b exploded to p". He also relates it to array notation:

a {{{c}}} b = {a,b,c,3} etc.

Oddly, the unusual behavior for 1{{1}}b, vanishes for 1{{{1}}}b :

1{{{1}}}1 = 1 [via Rule 3]

1{{{1}}}2 = 1{{1}}1 = 1 [via Rule 3]

1{{{1}}}3 = 1{{1{{1}}1}}1 = 1{{1}}1 = 1 [via Rule 3]

1{{{1}}}4 = 1{{1{{1{{1}}1}}1}}1 = 1{{1{{1}}1}}1 = 1{{1}}1 = 1 [via Rule 3]

etc.

Again we fail to get any growth when a=2:

2{{{1}}}1 = 2 [via Rule 3]

2{{{1}}}2 = 2{{2}}2 = 2{{1}}2 = 2{2}2 = 4

2{{{1}}}3 = 2{{2{{2}}2}}2 = 2{{4}}2 = 2{{3}}2 = 2{{2}}2 = 2{{1}}2 = 4

2{{{1}}}4 = 2{{2{{2{{2}}2}}2}}2 = 2{{4}}2 = ... = 2{{1}}2 = 4

2{{{1}}}5 = 2{{2{{{1}}}4}}2 = 2{{4}}2 = ... = 4

etc.

When a=3, we quickly find that there is no way to relate the numbers back to the ordinary hyper-operators:

3{{{1}}}1 = 3 [via Rule 3]

3{{{1}}}2 = 3{{3}}3 = 3{{2}}3{{2}}3 = 3{{2}}3{{1}}3{{1}}3 =

3{{2}}3{{1}}3{3{3}3}3 = 3{{2}}3{{1}}3{27}3 =

3{{2}}3{3{3{3{3{3{...{3{3{3}3}3}...}3}3}3}3}3}3 w/3{27}3 3's center out =

3{{1}}3{{1}}3{{1}} ... {{1}}3{{1}}3{{1}}3

w/3{3{3{ ... {3{3{3}3}3} ... }3}3}3 3s

w/3^^^^^^^^^^^^^^^^^^^^^^^^^3 3's center out

Already we can't really relate this back to hyper-operators in a natural way, because we have to resolve the long expression 3{{1}}3{{1}}3{{1}} ... {{1}}3{{1}}3. The next step however is totally mind boggling:

3{{{1}}}3 = 3{{3{{3}}3}}3 = 3{{3{{{1}}}2}}3 = 3{{N}}3

What this means is that 3{{{1}}}3 is equivalent to 3{{N}}3 where N is the previously insane number. The result would be that we would have to expand this to {{N-1}} operators, then {{N-2}} operators, ... all the way to {{1}}s before we could then expand this and relate it back to hyper-operators. Let's call operators of the form {c} the ordinary hyper-operators, and operators of the form {{c}} the expansion-operators. {{{1}}} is an operator that repeatedly plugs back into the expansion-operators, just like {{1}} repeatedly plugs back into the hyper-operators!!!

A word of caution as we move on ahead: Any attempt to comprehend this in relation to familiar operations should be abandoned immediately, because we can't really fully grasp it. There is nothing to compare these kinds of numbers to, and thus we can only understand them on their own terms, and no others! This is why it is so difficult to explain numbers like this to a laymen. It can't be explained in the familiar terms of decimal expressions and exponentiation. It really does take a few minutes of explanation, and a willingness on the part of the listener to "expand" their imagination in order to understand this. This is also why the most common rookie mistake is to take numbers in this range and -plex them. If you've read Section III, you should already know why that comes across as naive to the googologist. Plexing is far too slow at this point, and is no longer competitive!

At this point in the conversation it's usually a good idea to abandon lengthy expansions to familiar concepts and to just warp on ahead to the full pattern! You can probably already see what Bowers' is going for here. First, we can extend beyond the explosive-operator, and create a series of explosive-operators, that are defined as repetitions of the previous explosive-operators:

a{{{2}}}b = a{{{1}}}a{{{1}}}a{{{1}}} ... {{{1}}}a w/b a's

a{{{3}}}b = a{{{2}}}a{{{2}}}a{{{2}}} ... {{{2}}}a w/b a's

a{{{4}}}b = a{{{3}}}a{{{3}}}a{{{3}}} ... {{{3}}}a w/b a's

etc.

We follow this with a new operator, beyond all explosive-operators, that uses 1 additional layer of brackets: {{{{1}}}}. Bowers' calls this "a detonated to b" (continuing with his dynamite analogy):

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

At this point he doesn't even bother to tell us what this operator does. However, the astute reader will have already intuited the pattern established by the previous examples. Logically speaking a{{{{1}}}}b must be:

a{{{{1}}}}b = a{{{a{{{a{{{ ... {{{a{{{a{{{a}}}a}}}a}}} ... }}}a}}}a}}}a w/b a's center out.

Next we can create a series of detonative-operators, defined as repetitions of the previous detonative-operator:

a{{{{2}}}}b = a{{{{1}}}}a{{{{1}}}}a{{{{1}}}} ... {{{{1}}}}a w/b a's

a{{{{3}}}}b = a{{{{2}}}}a{{{{2}}}}a{{{{2}}}} ... {{{{2}}}}a w/b a's

a{{{{4}}}}b = a{{{{3}}}}a{{{{3}}}}a{{{{3}}}} ... {{{{3}}}}a w/b a's

etc.

Next up is something Bowers' calls "a petonated to b" (pent for 5 plus detonated):

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

Next logically would come a series of pentonative-operators. Again we create a new operator above all these, which would be {{{{{{1}}}}}}. That's 1 within 6 pairs of curly braces. Bowers' does not give this an official name, but again we can follow the pattern he establishes to continue. Since "pent" is a greek prefix, we can go further by using the greek prefixes:

a{{{{{{1}}}}}}b = a hexonated to b = {a,b,1,6}

a{{{{{{{1}}}}}}}b = a heptonated to b = {a,b,1,7}

a{{{{{{{{1}}}}}}}}b = a octonated to b = {a,b,1,8}

a{{{{{{{{{1}}}}}}}}}b = a ennonated to b = {a,b,1,9}

a{{{{{{{{{{1}}}}}}}}}}b = a deconated to b = {a,b,1,10}

etc.

Bowers' does not explicitly name any operators above pentonation, so the above isn't cannon, but it's an obvious enough continuation. It should be clear, that regardless of what we call these operators, we would be able to continue to an arbitrary number number of curly brackets. This is what I refer to as Bowers' "Bracket Notation". Bracket notation is as powerful as Conway's Chain arrow notation. In fact, it can be shown that the addition of each pair of curly brackets is equivalent to adding an extra argument to a Conway Chain. Bracket Notation however isn't the end of Bowers' Extended Operator Notation, ... far from it. Bracket notation does however represent the limit of "established" notation for the Extended Operators. Bowers' presents no official continuation, but he does give a description of notation that he uses on paper. We will be looking into this in the next heading. First however lets try to get stock of what we have just learned about Bracket Notation.

Bowers' does not provide a formal definition of the Extended Operators, even limited to the bracket notation. Instead he uses a litany of examples to give an informal "definition". I believe examples are very important. People learn best by doing, and in fact, I don't think there really is any other way to learn mathematics other than by doing it. The reason is, that its only by working out the consequences of definitions that their meaning becomes readily apparent. So I think worked out examples are an important when presenting googological functions. However, examples in and of themselves are not always enough. One should seek to simplify and uncover the general principle. The benefits of this are often that one gains far greater insight, and the keys to progressing further with far less confusion. In the case of Bracket Notation, it isn't too difficult to formalize the system and capture the general principle.

First, here is some terminology, directly from Bowers' writing:

I also have names for the various entries.

Consider {a,b,c,d,e,f}

a is the base

b is the extender

c is the operator

d is the level of operators

e is the level of level of operators

f is the level of level of level of operators.

Using this terminology, we can identify 4 rules to Bracket Notation:

1. When both the "level of operator" and "operator" are 1, add (base case)

2. If the extender = 1, and rule 1 does not apply, return the base.

3. When the operator=1 and "level of operator">1, repeatedly plug into the level of operator -1.

4. When the operator is greater than 1, repeat the lower operator

Next let's establish this convenient notation:

a{c}_db = a{{{...{{{c}}}...}}}b w/d bracket pairs

Formally we can now define bracket notation as follows:

Bower's Bracket Notation

Let a{c}_db be defined if a,b,c,d are positive integers as follows:

1. c,d=1 ; a{1}₁b = a+b

2. b=1 ; a{c}_d1 = a

3. c=1 ; a{1}_db = a{a{1}_d(b-1)}_d-1a

4. a{c}_db = a{c-1}_d(a{c}_d(b-1))

With this we now have a reliable definition to refer back to whenever we are in doubt about something. Note that this formal definition completely avoids the use of the ellipsis ( ... ). This is done to ensure that the definition is completely explicit and leaves nothing to interpretation.

Bowers' defines several numbers with his Bracket notation on his "Infinity Scrapers" page. The largest of these is the General, and Generalplex. The General is 10{{{{{{{{{{10}}}}}}}}}}10, that is 10{10}₁₀10. The Generalplex is 10{{{{ ... {{{{10}}}} ... }}}}10 where there are a General pairs of brackets, that is 10{10}_General10.

Bracket Notation is roughly equivalent to Ea###b in my Hyper-E Notation. That is to say:

10{10}_d10 < E10###(d+1)

ex.

10{10}10 < E10###2 = E10##10

10{{10}}10 < E10###3 = E10##10##10

10{{{10}}}10 < E10###4 = E10##10##10##10

etc.

With bracket notation understood, we have taken the first step towards understanding Bowers' system. But we are not yet even done with Extended Operator Notation...

Bowers' Extended Operator Notation

Part II : Beyond Bracket Notation

There are only 9 lines which describe the rest of Extended Operator Notation on Bowers' old site. Amazingly, even this is enough to reconstruct it, given it's close relation to array notation, which Bowers' does explicitly definite.

Bowers' first mentions an operator which goes beyond all the operators of bracket notation in these lines:

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

What exactly does all of this mean?! {a,b,1,1,2} refers to a new operator beyond all operators {c}_d. For now we will use "@" to represent this operator. What this line states is basically that:

a@b =

a{{...{{a}}...}}a w/a{{...{{a}}...}}a w/ ... ... ... ... ... ... ... ... w/a{{...{{a}}...}}a w/a }'s }'s ... ... }'s

[where there are b-1 a{{...{{a}}...}}a 's]

Still unclear? This next illustration should make it perfectly plain: