# Answered Essay: 1.Prove the following grammar is ambiguous:

1.Prove the following grammar is ambiguous:

<assign> -> <id> = <expr>

<id> -> A | B | C

<expr> -> <expr> + <expr>

| <expr> * <expr>

| ( <expr>)

| <id>

2. Given the grammar:

<S> -> <A> a <B> b

<A> -> <A> b | b

<B> -> a <B>|a

Which of the following sentences are in the language generated by this grammar?

a. baab
b. bbbab
c. bbaaaaaS

d. bbaab

Problem 2

a) baab

Consider leftmost derivation : <S> => <A>a<B>b => ba<B>b => baab

d) bbaab

Consider leftmost derivation : <S> =><A>a<B>b => <A>ba<B>b => bba<B>b => bbaab

b) and c) are not part of this grammar

Problem 1)

A grammar is abmiguous if it produces atleast two different derivation of any language accepted by it.

Now, consider a language A = A*B+C

Consider Left Most Derivation

A = A*B+C

=> <id> = A*B+C

=> <id> = <id>*B+C

=> <id> = <expr> * B + C

=> <id> = <expr> * <id> +C

=> <id> = <expr> * <expr> + C

=> <id> = <expr> + C

=> <id> = <expr> + <id>

=> <id> = <expr> + <expr>

=> <id> = <expr>

=> <assign>

Now, consider rightmost derivation

A= A* B + C

=> A= A*B+ <id>

=>A=A*B+<expr>

=> A=A*<id>+<expr>

=> A = A * <expr> +<expr>

=> A= A* <expr>

=> A = <id> * <expr>

=> A = <expr> * <expr>

=> A = <expr>

=> <id> = <expr>

=> <assign>

Hence, grammar is ambiguous as atleast two different derivation exist for a language. Above example states that a expression 5*4+6, will be treated both as (5*4)+6 and 5*(4+6). Therefore grammar is ambiguous. Pages (550 words)
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