User blog:Chronolegends/CSBN - Chronolegend's Square Bracket Notation Level 1

The notation consists of expressions called strings, formed by at least one pair of square brackets containing and being followed by a number. Such as [6]25. A string enclosed by [] is called a separator, as it is used to separate what is inside from the numbers or numbers from eachother.

A number n superscripted above a symbol n copies of that symbol, unless it is a group-closing symbol in which case it indicates copies of that entire group. This is done for purposes of shorthand, and doesnt actually appear in the notation. The use is to avoid overly long descriptions involving ellipses. For example: (s....s with 50 copies of s)...repeat 25 times becomes $$(s^{50})^{25}$$

Lower case letters represent non-negative integers, except z, which represents variable non-negative integers

Example $$a(+b)^3=2+5+5+5, (b+z)^3=(8+2)(8+7)(8+9) $$

Upper case letters enclosed by square brackets represent separators, currently the only valid form for separators is :

$$[X] = [a]$$

$$ [X]n = {([X]--)}^nn $$
 * General form.

Base Case $$ [0]-- = n* $$
 * Defining [S]--.

$$ [a+1([B]z)^{d}]-- = [a([B]z)^{d}] $$
 * Plus One Case.

d can be zero, in which case the rule is simply $$ [a+1]-- = [a] $$

$$ [0[B]a+1] = [n([B]--n)^n[B]a] $$
 * Multiplier Case.

$$ [0[B]0] = [n([B]--n)^n] $$
 * Power Case.

$$ [0([0]0)^{a+1}]-- = [n([0]n)^a] $$ a can be zero, in which case the rule is simply $$[0[0]0]-- = [n] $$
 * All-zero Case.

$$ [(z[Z])^{d+1}z[B]z([Z]z)^{e+1}]--, [B] > [Z] = [[(z[Z])^{d+1}z]--[B]z([Z]z)^{e+1}] $$
 * Chains, all different separators.

Groups and reduce everything which is to the left of the greatest separator.

$$ [0[B]a([Z]z)^{c+1}]--, [B] > [Z] = [n([B]--n)^n[B][a([Z]z)^{c+1}]--] $$
 * Chains, all different separators, leftmost zero.

Groups and reduce everything which is to the right of the greatest separator.

$$ [0([B]0)^{a+1}]-- = [n([B]--n)^n([B]n([B]--n)^n)^a] $$
 * Chains, equal separators, all-zero elements.

Eats one separator and replaces zeroes with $$n([B]--n)^n$$

$$ [(z[Z])^az[B](z[Z])^cz[B](z[Z])^dz]--, [B] > [Z] = [[(z[Z])^az[B](z[Z])^cz]--[B](z[Z])^dz]  $$
 * Chains, equal separators

Groups and reduces everything which is to the left and right of the leftmost greatest separator.


 * Ends the definition of []-- For part 1

Level comparison of the notation:

$$ [0[m]0]n > f_{\omega^{\omega^m}}(n) $$

$$ [a[b]c[b]d]e > f_{({(\omega^{\omega^b}})^2)d+(\omega^{\omega^b})c+a}(e) $$

ToDo list (hooks for later blog posts)

PART 2 Forms [X] = [a[B]c] (epsilon level)

PART 3 Forms [X] = [A][B] (phi omega level)

PART 4 Forms [X] = [A[B]] (gamma zero level)

PART 5 Forms [X] = a[]b (unknown???)