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# Relationship Between Grammar & Language Notes | EduRev

## Computer Science Engineering (CSE) : Relationship Between Grammar & Language Notes | EduRev

The document Relationship Between Grammar & Language Notes | EduRev is a part of the Computer Science Engineering (CSE) Course Theory of Computation.
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Relationship between grammar and language in Theory of Computation

A grammar is a set of production rules which are used to generate strings of a language. In this article, we have discussed how to find the language generated by a grammar and vice versa as well.

➤ Language generated by a grammar –
Given a grammar G, its corresponding language L(G) represents the set of all strings generated from G. Consider the following grammar,
G: S-> aSb|ε
In this grammar, using S-> ε, we can generate ε. Therefore, ε is part of L(G). Similarly, using S=>aSb=>ab, ab is generated. Similarly, aabb can also be generated.
Therefore,
L(G) = {anbn, n>=0}
In language L(G) discussed above, the condition n = 0 is taken to accept ε.

➤ Key Points –

• For a given grammar G, its corresponding language L(G) is unique.
• The language L(G) corresponding to grammar G must contain all strings which can be generated from G.
• The language L(G) corresponding to grammar G must not contain any string which can not be generated from G.

Let us discuss questions based on this:

Try yourself:Consider the grammar: (GATE-CS-2009)
S -> aSa|bSb|a|b
The language generated by the above grammar over the alphabet {a,b} is the set of:

Try yourself:Consider the following context-free grammars: (GATE-CS-2016)
G1: S→aS/B, B→b/bB
G2: S→aA|bB, A→aA|B|ε, B→bB|ε
Which one of the following pairs of languages is generated by G1 and G2, respectively?

➤ Grammar generating a given language –

Given a language L(G), its corresponding grammar G represents the production rules which produces L(G). Consider the language L(G):
L(G) = {anbn, n>=0}
The language L(G) is set of strings ε, ab, aabb, aaabbb….

For ε string in L(G), the production rule can be S->ε.

For other strings in L(G), the production rule can be S->aSb|ε.

Therefore, grammar G corresponding to L(G) is:
S->aSb| ε
Key Points –

• For a given language L(G), there can be more than one grammar which can produce L(G).
• The grammar G corresponding to language L(G) must generate all possible strings of L(G).
• The grammar G corresponding to language L(G) must not generate any string which is not part of L(G).

Let us discuss questions based on this:

Try yourself:Which one of the following grammar generates the language L = {ai bj | i≠j}? (GATE-CS-2006)

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## Theory of Computation

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