Question

] Derive (a101 + b1) * (a1 + b) using leftmost derivation and right most derivation. Is this grammar ambiguous? If so, prove it or construct a non-ambiguous grammar.

E⇒ I
E→ E+E
E⇒ E* E
E→ (E)
I a
I ⇒ b
I → Ia
I → Ib
I⇒ 10
I→ I1 ] Convert the given CFG to GNF, where V = {S,A,B}, T = {a,b}, and P is, S→ ABA A✈ aA | ε BbB | ε

Answer 1

The given context-free grammar (CFG) is ambiguous. To demonstrate, consider the string "a1 + b1 * a1." This string has two distinct leftmost derivations: E → E + E → I + E → a1 + E → a1 + E * E → a1 + I * E → a1 + b1 * a1 and E → E * E → E * I → E * b → I * b → a1 * b1 * a1. Therefore, the grammar is ambiguous.

To resolve this ambiguity, a non-ambiguous grammar can be constructed. One possible solution is to introduce separate non-terminals for addition and multiplication expressions. The modified grammar is as follows:

R

E → T + E | T

T → F * T | F

F → (E) | I

I → a | b | 10 | I1

This grammar ensures a unique parse for any given string, eliminating the ambiguity present in the original grammar.

Step-by-step explanation:

The given CFG is ambiguous because there exists at least one string with multiple leftmost derivations, indicating different parsing possibilities. In this case, "a1 + b1 a1" has two distinct leftmost derivations, leading to ambiguity.

To resolve this, a non-ambiguous grammar is constructed by introducing separate non-terminals for addition (E → T + E | T) and multiplication (T → F T | F) expressions. This modification ensures that each string has a unique parse, eliminating the ambiguity present in the original grammar.

The new grammar follows the guidelines for Greibach Normal Form (GNF), where each production has the form A → aα, and α can be empty or contain terminals but not non-terminals. This transformation simplifies the grammar, making it more manageable for parsing and analysis.