Relationship between grammar and language in Theory of Computation

In the Theory of Computation, grammar and language are fundamental concepts used to define and describe computational problems. A grammar is a set of production rules that generate a language, while a language is a collection of strings that conform to these rules. Understanding their relationship is crucial for designing programming languages, parsers, and automata.

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.

Example Questions

Que-1: 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:

(A) All palindromes
(B) All odd-length palindromes
(C) Strings that begin and end with the same symbol
(D) All even-length palindromes

Solution:
Using S -> a and S -> b, we can generate 'a' and 'b'. Similarly, using S => aSa => aba, the string 'aba' is generated. Other strings generated from the grammar include:
{ a, b, aba, bab, aaa, bbb, ababa, ... }
Thus, the correct answer is (B) All odd-length palindromes.

Que-2: Consider the following context-free grammars: (GATE-CS-2016)

Which one of the following pairs of languages is generated by G1 and G2, respectively?

Solution:
Consider the grammar G1: Using S=>B=>b, b can be generated. Using S=>B=>bB, bb can be generated. Using S=>aS=>aB=>ab can be generated. Using S=>aS=>aB=>abB=>abb can be generated. As we can see, number of a’s can be zero or more but number of b is always greater than zero. Therefore,

L(G1) = {a^mbⁿ| m>=0 and n>0}

Consider the grammar G2: Using S=>aA=>a, a can be generated. Using S=>bB=>b, b can be generated. Using S=>aA=>aaA=>aa can be generated. Using S=>bB=>bbB=>bb can be generated. Using S=>aA=>aB=>abB=>abb can be generated. As we can see, either a or b must be greater than 0. Therefore,

L(G2) = {a^mbⁿ| m>0 or n>0}

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:

Que-3.

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

Solution:

The given language L contains the strings :

{a, b, aa, bb, aaa, bbb, aab, abb…}

It means either the string must contain one or more number of a OR one or more number of b OR a followed by b having unequal number of a and b. If we consider grammar in option (A), it can generate ab as:

S=>AC=>aAC=>aC=>ab

However, ab can’t be generated by language L. Therefore, grammar in option (A) is not correct. Similarly, grammar in option (B) can generate ab as:

S=>aS=>ab

However, ab can’t be generated by language L. Therefore, grammar in option (B) is not correct. Similarly, grammar in option (C) can generate ab as:

S=>AC=>C=>aCb=>ab

However, ab can’t be generated by language L. Therefore, grammar in option (C) is not correct. Therefore, using method of elimination, option (D) is correct.

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Article Tags :

• Misc
• GATE CS
• Theory of Computation