Grammar generated by language
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The document discusses different types of grammars and languages: - Type 3 grammars generate regular languages using rules of the form A → aB or A → a. - Type 2 grammars generate context-free languages using rules of the form A → α. - Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|. - Type 0 grammars generate recursively enumerable languages using rules of the form α → β. The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's, and c's.
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Grammar generated by language
- 1. Prof.Neeraj Bhargava Abhishek Kumar Department of Computer Science School of Engineering & System Sciences, MDS, University Ajmer, Rajasthan, India 1
- 2. A grammar express languages with the help of four tuples:- (Vn , ∑ , R , S) • V is a finite set of variables/non-terminals • Σ is a finite set of terminals • S ∈ V is the start symbol • R ⊆ (Σ∪V )∗×(Σ∪V )∗ is a finite set of rules/productions. We say γ1 α γ2 ⇒G γ1 β γ2 , iff (α → β) ∈ R. And L(G) = {w ∈ Σ∗|S ∗ ⇒G w} 2
- 3. Grammar Rules Languages Type 3 A → aB or A → a Regular Type 2 A → α Context Free Type 1 α → β with |α|≤|β Context Sensitive Type 0 α → β Recursively Enumerable Figure 2: In the above table, α,β ∈ (Σ∪V )∗, A,B ∈ V and a ∈ Σ∪{} 3
- 4. Example 2. Consider the grammar G with Σ = {a} with S → $Ca# | a | ∈ Ca → aaC $D → $C C# → D# | E a D → Da aE → Ea $E → ∈ The following are derivations in this grammar S ⇒ $Ca# ⇒ $aaC# ⇒ $aaE ⇒ $aEa ⇒ $Eaa ⇒ aa S ⇒ $Ca# ⇒ $aaC# ⇒ $aaD# ⇒ $aDa# ⇒ $Daa# ⇒ $Caa# ⇒ $aaCa# ⇒ $aaaaC# ⇒ $aaaaE ⇒ $aaaEa ⇒ $aaEaa ⇒ $aEaaa ⇒ $Eaaaa ⇒ aaaa L(G) = {ai |i is a power of 2} 4
- 5. Example 8. Consider the grammar G with Σ = {a,b,c}, V = {S,B,C,H} and S → aSBC | aBC CB → HB HB → HC HC → BC aB → ab bB → bb bC → bc cC → cc L(G) = {anbncn |n ≥ 0} 5
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