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:

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the string by repeatedly replacing a non-terminal by the right-hand side of the production until all non-terminal have been replaced by terminal symbols.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

r.e. = a*   

Production rule for the Regular expression is as follows:

S → aS    rule 1
S → ε     rule 2

Now if we want to derive a string "aaaaaa", we can start with start symbols.

 S
aS 
aaS          rule 1
aaaS         rule 1
aaaaS        rule 1
aaaaaS       rule 1
aaaaaaS      rule 1
aaaaaaε      rule 2
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.

Example 2:

Construct a CFG for the regular expression (0+1)*

Solution:

The CFG can be given by,

Production rule (P):
S → 0S | 1S
S → ε

The rules are in the combination of 0's and 1's with the start symbol. Since (0+1)* indicates {ε, 0, 1, 01, 10, 00, 11, ....}. In this set, ε is a string, so in the rule, we can set the rule S → ε.

Example 3:

Construct a CFG for a language L = {wcwR | where w € (a, b)*}.

Solution:

The string that can be generated for a given language is {aacaa, bcb, abcba, bacab, abbcbba, ....}

The grammar could be:

S → aSa     rule 1
S → bSb     rule 2
S → c       rule 3

Now if we want to derive a string "abbcbba", we can start with start symbols.

S → aSa 
S → abSba       from rule 2
S → abbSbba     from rule 2
S → abbcbba     from rule 3

Thus any of this kind of string can be derived from the given production rules.

Example 4:

Construct a CFG for the language L = aⁿb²ⁿ where n>=1.

Solution:

The string that can be generated for a given language is {abb, aabbbb, aaabbbbbb....}.

The grammar could be:

S → aSbb | abb

Now if we want to derive a string "aabbbb", we can start with start symbols.

S → aSbb
S → aabbbb