Tips for creating “Context Free Grammar”

How to write CFG with example a^mbⁿ

L = {a^m bⁿ | m >= n}.

Language description: a^m bⁿ consist of a followed by b where number of a are equal or more then number of b.

some example strings: {^, a, aa, aab, aabb, aaaab, ab......}

So there is always one a for one b but extra a are possible. infect string can be consist of a only. Also notice ^ null is a member of language because in ^ NumberOf(a) = NumberOf(b) = 0

How to write a grammar that accepts the language formed by strings a^m bⁿ?

In the grammar, there should be rules such that if you add a b symbol you also add a a symbol.

and this can be done with something like:

   S --> aSb 

But this is incomplete because we need a rule to generate extra as:

   A --> aA | a

Combine two production rules into a single grammar CFG.

   S --> aSb | A
   A --> aA  | a

So you can generate any string that consist of a also a and b in (a^m bⁿ) pattern.

But in above grammar there is no way to generate ^ string.

So, change this grammar like this:

   S --> B   | ^
   B --> aBb | A
   A --> aA  | a

this grammar can generate {a^m bⁿ | m >= n} language.

Note: to generate ^ null string, I added an extra first step in grammar by adding S--> B | ^, So you can either add ^ or your string of symbol a and b. (now B plays role of S from previous grammar to generate equal numbers of a and b)

Edit: Thanks to @Andy Hayden
You can also write equivalent grammar for same language {a^m bⁿ | m >= n}:

   S --> aSb | A
   A --> aA | ^

notice: here A --> aA | ^ can generate zero or any number of a. And that should be preferable to my grammar because it generates a smaller parse tree for the same string.
(smaller in height preferable because of efficient parsing)

The following tips may be helpful to write Grammar for a formal language:

• You are to be clear about language that what it describes (meaning/pattern).
• You can remember solutions for some basic problems(the idea being that you can write new grammars).
• You can write rules for fundamental languages like I have written for RE in this example to write Right-Linear-Grammmar. The rules will help you to write Grammar for New Languages.
• One different approach is to first draw automata, then convert automata to Grammar. We have predefined techniques to write grammar from automata from any class of formal language.
• Like a Good Programmer who learns by reading the code of others, similarly one can learn to write grammars for formal languages.

_{Also the grammar you have written is wrong.}