Numbers: non-negative integers, represented by lower case letters.
Strings: Collections of numbers and separators, represented by capital letters
 Non-Empty String: A string that contains at least one number and zero or more separators.
 Main String: A string that is not a substring nor enclosed by square brackets.
Separators: Numbers enclosed by square brackets []
Substrings: Strings that are parts of another string
Argument: A number at the end of the main string, the letter n is reserved for it.
Syntax Rules
A number ends the main string
Except in the main string, Separators must always be inbetween two numbers
Strings can be zero, or they can be empty, except this would create an illegal string (check above rule)
If the main string contains more than 1 separator, solve the substring consisting only of the argument and the separator adjacent to it.
Symbol Glossary

x- Reduce x

() Group

x > y x is greater than y

x^y copy x y times

Note: Reducing and Copying applies to grouped things as a whole. If grouping symbols are nested, Reducing and Copying is done from the inside out.

Examples of teminology use
In this string: Sa , the substring S must end with a separator, because it precedes a number. So S could look like P[5]a, where its substring P must end with a number, because it precedes a separator. Finally, in this string aSb, S be at least one separator, but it can also look like [p]Q[r]
Illegal string example: in the string aSb, the substring S can't be empty because it would leave two numbers unseparated.

 [0]n = n^n

 [S]n = [S-]^nn

Defining -
A string of this form Reduces to this
m+1S S may be an empty string mS
(0[0])^{m+1}p+1 (n[0])^{m+1}p
(0[0])^{m+1}0 (n[0])^{m}n
(0S)^{m+1}T Where S => All separators in T (n(S-n)^nS)^{m+1}T-
0Sm+1 n(S-n)^nSm
0(S0)^{m+1} n(S-n)^n(Sn(S-n)^n)^m
RST Where S => All separators in R, > All separators in T R-ST
Ordering of separators
[S] > [0]
[m+1] > [m]
important edit
took out multiseparators and nesting out of level 1 csbn, the reason is that while writing down the ordering rules i considered them too complex for a basic notation ,so this edit brings the power down from

\varphi_w(0) to a more modest \omega^{\omega^\omega}

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