Introduction to strong array notation

Strong array notation is a googological notation developed by Hyp cos. It is one of the most powerful recursion based large number functions defined. This page will serve as a step by step walk through of how the notation works.

Linear array notation

3-entry Arrays

Three entry arrays in this notation are exactly equivalent to those in BEAF. s(a,b,c) is equal to a^^^...^^^b with c arrows. However the definitions are not identical.

s(a,b) is a^b, and s(a,b,c) = s(a,s(a,b-1,c),c-1). However, s(a,1,c) is not directly defined as a. Instead we have to solve it using the process. Since the third entry is the first non-1 entry, it decreases by 1, the 1 before it stays the same (since it is the second entry), and the first entry also stays the same. In fact this is a general property of the notation: s(a,1,c#) = s(a,1,1#).

Here is an example:

s(3,2,4) = s(3,s(3,1,4),3) = s(3,s(3,1,1),3) = s(3,s(3,1),3) = s(3,3,3) = s(3,s(3,2,3),2) = s(3,s(3,s(3,1,3),2),2) = s(3,s(3,3,2),2) = ...

The resulting number is \(3\uparrow\uparrow\uparrow3\).

4-entry Arrays

Adding a fourth argument is relatively easy. When the third entry is more than 1, it works very similar to 3-entry arrays. But when the third entry is 1, things change. As we need to start the process, and the third entry is 1, we jump to the fourth entry. Since it's more than 1, we decrease it by 1, change the third entry into the original second entry's value, and replace the first two entries into the base value.

So:

s(a,b,1,d) = s(a,a,b,d-1)

Note how this works even if b is 1. This is one major difference between this notation and BEAF.

Some examples should help you understand this:

s(3,2,2,2) = s(3,s(3,1,2,2),1,2) = s(3,s(3,1,1,2),1,2) = s(3,s(3,3,1,1),1,2) = s(3,s(3,3),1,2) = s(3,3,s(3,3)) = s(3,3,27)

The result is:

\(3\uparrow\uparrow\cdots\uparrow\uparrow3\) with 27 up arrows.

Or in chained arrow notation: \(3\rightarrow3\rightarrow2\rightarrow2\)

s(3,1,1,4) = s(3,3,1,3) = s(3,3,3,2) = s(3,s(3,2,3,2),2,2) = s(3,s(3,s(3,1,3,2),2,2),2,2) = s(3,s(3,27,2,2),2,2) = ...

The resulting number is exactly equal to \(3\rightarrow3\rightarrow3\rightarrow3\) in chained arrow notation.

s(4,2,2,3) = s(4,s(4,1,2,3),1,3) = s(4,s(4,1,1,3),1,3) = s(4,s(4,4,1,2)1,3) = s(4,4,s(4,4,4),2) = s(4,4,4^^^^4,2)

...

This notation so far reaches \(\omega^2\) in the FGH.

5+ entry Arrays

Arrays with 5 or more entries behave almost like 4 entry arrays except the length of chain a:

• s(a,b,1,1,n) = s(a,a,a,b,n-1)
• s(a,b,1,1,1,n) = s(a,a,a,a,b,n-1)
• s(a,b,1,1...1,1,n) = s(a,a...,a,b,n-1)
• ...

An example is s(3,3,3,3,3,3):

s(3,3,3,3,3,3) = s(3,s(3,2,3,3,3,3),2,3,3,3) = s(3,s(3,s(3,1,3,3,3,3),2,3,3,3),2,3,3,3) =  s(3,s(3,s(3,1,2,3,3,3),2,3,3,3),2,3,3,3) = s(3,s(3,s(3,1,1,3,3,3),2,3,3,3),2,3,3,3) =s(3,s(3,s(3,3,1,2,3,3),2,3,3,3),2,3,3,3) = s(3,s(3,s(3,3,3,1,3,3),2,3,3,3),2,3,3,3)

This notation so far reaches \(\omega^{\omega}\) in the FGH.

Extended array notation

Dimensional arrays

Next we add dimensional seprators. {1} is comma and {2} is a row separator.

When the entry is at the beginning of a row, replace 1{a}b with 1{a}2{a}b-1, and jump inside the first {a}.

If you are inside a dimensional separator, replace the {a} with {a-1}1{a-1}...{a-1}1{a-1} with b-1 1's, where b is the second entry of the main array.

Note s(a,b{2}2) is not equal to s(a,b,1{2}2) or s(a,b,1,1{2}2) because we do not remove a 1 from the end of a row that is not the last. They are equal to s(a,b,1,1...1,2) with b-1, b and b+1 respectively 1's

An example is s(3,3{2}3,3)

Because the third entry is at the beginning of a row, it becomes s(3,3{2}2{2}2,3) and we jump inside the first {2}

Because the entry is inside a dimensional separator, we replace it with n commas separated by 1's\

=s(3,3,1,1,2{2}2,3)

= s(3,3,3,3,1{2}2,3)

We do not remove the 1 from this array

This notation reaches \(\omega^{\omega^{\omega}}\) in FGH 

Nested arrays

We can now have arrays instead of just numbers in dimensional seperators, and when you jump inside a dimensional separator, it behaves just like an array, with a and b referring to the first and second entries of the main array.

We need to revise a rule. The new rule is

If the entry follows a left brace, replace it with b copies of it with the first entry subtracted by 1

An example is s(3,3{1{2}3}2)

=s(3,3{1{2}3}2{1{2}3}1)

=s(3,3{1{2}3}2)

=s(3,3{1{2}2{2}2}2)

=s(3,3{1,1,1,2{2}2}2)

=s(3,3{3,3,3,1{2}2}2)

This has level \(\varepsilon_0\) in the FGH.