Context-Free Grammar (CFG)
CFG stands for context-free grammar. It is is a formal grammar which is used to generate all possible patterns of strings in a given formal language. Context-free grammar G can be defined by four tuples as:
1. G = (V, T, P, S)
Where,
G is the grammar, which consists of a set of the production rule. It is used to generate the string of a language.
T is the final set of a terminal symbol. It is denoted by lower case letters.
V is the final set of a non-terminal symbol. It is denoted by capital letters.
P is a set of production rules, which is used for replacing non-terminals symbols(on the left side of the production) in a string with other terminal or non-terminal symbols(on the right side of the production).
S is the start symbol which is used to derive the string. We can derive the string by repeatedly replacing a non-terminal by the right-hand side of the production until all non-terminal have been replaced by terminal symbols.
Example 1: Construct the CFG for the language having any number of a's over the set ∑= {a}. Solution: As we know the regular expression for the above language is 1. r.e. = a*
Production rule for the Regular expression is as follows:
1. S → aS rule 1
2. S → ε rule 2
Now if we want to derive a string "aaaaaa", we can start with start symbols.
1. S
2. aS
3. aaS rule 1
4. aaaS rule 1
5. aaaaS rule 1
6. aaaaaS rule 1
7. aaaaaaS rule 1
8. aaaaaaε rule 2
9. aaaaaa
The r.e. = a* can generate a set of string {ε, a, aa, aaa,.....}. We can have a null string because S is a start symbol and rule 2 gives S → ε
