Relationship Between Grammar & Language - Theory of Computation - Computer

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:

➤ 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: