Introduction
Have you ever wondered if you have a much easier language to use than a regular language and can also somehow help in the programming language?

Then your search is over! Context-Free Language is the answer to your problem. We will learn all about this language in this blog, so let's proceed with our topic without wasting further time.
You can also read about - Moore Machine
What is Context-Free Grammar (CFG)?
A Context-Free Grammar (CFG) is a formal system used to define the syntactical structure of a language. It consists of a set of production rules, where each rule specifies how a symbol or group of symbols can be replaced by other symbols. CFG is widely used in computer science for programming language design and parsing, as it provides a way to describe the hierarchical structure of sentences in natural or programming languages. CFG ensures that a language can be generated by recursive patterns, making it easier to analyze and process syntactically.
G = (V, T, P, S)
Also read , Arden's theorem
Explanation
- G in the above expression stands for grammar. It consists of the set of all the production rules with the help of which we can generate the string of a language.
- T stands for the final set of the terminal symbol. We have to use the lower case alphabet to show them.
- V and T are somewhat opposite as V stands for the set of all non-terminal symbols, and we have to use capital letters to denote it.
- P is the collection of all the production rules used to replace all non-terminals symbols of a string with terminal or other non-terminal symbols.
- S stands for the start symbol, used for deriving string.
Examples
Example 1
Construct the CFG for having any number of b’s over the set ∑= {b}.
Solution:
The regular expression for the above language is
r.e. = b*
Production rules are as given below:
S → bS rule 1
S → ε rule 2
Now, we will derive the string “bbbbbb.”
S
bS
bbS rule 1
bbbS rule 1
bbbbS rule 1
bbbbbS rule 1
bbbbbbS rule 1
bbbbbbε rule 2
bbbbbb
The r.e. = b* can generate a set of string {ε, b, bb, bbb,.....}
Example 2
Construct a CFG for the expression (0+1)*
Solution:
Production rule (P):
S → 0S | 1S
S → ε
This will result in the set {ε, 0, 1, 01, 10, 00, 11, ....}.
Example 3
Construct a CFG for language L = {wcwR | where w € (a, b)*}.
Solution:
The production rules can be
S → aSa rule 1
S → bSb rule 2
S → c rule 3.
Now, we can use it to derive different strings. Let’s take an example of the string “abbcbba”
S → aSa
S → abSba from rule 2
S → abbSbba from rule 2
S → abbcbba from rule 3
Example 4
Construct a CFG for the language L=anb2n
Solution:
This can generate the following output {abb, aabbbb, aaabbbbbb....}.
The production rule is
S → aSbb | abb.
If we want to derive the string “aabbbb,” we will proceed like this.
S → aSbb
S → aabbbb
Also see, Turing Machine in TOC.