Solution

The correct answer is : option 3

Important Points

Question Objective:

We are asked to identify the grammar that correctly represents the language accepted by the machine. From the comprehension, the machine:

• Accepts all strings over Σ = {a, b}
• Strings can start and end with any combination
• Must contain the substring "abb"

Desired Grammar Structure:

We can divide the string into 3 parts to construct the grammar:

• Prefix: Any combination of 'a' and 'b' — this can be represented using recursion (A → aA | bA | ε)
• Middle: The fixed substring 'abb'
• Suffix: Again, any combination of 'a' and 'b' — again can be represented by A

Thus, an ideal grammar looks like:

S → AabbA  
A → aA | bA | ε

Option-wise Evaluation:

✅ Option 3:

S → AabbA
A → aA | bA | ε

• This grammar allows any string with 'abb' in the middle
• A before and after 'abb' ensures prefix and suffix can be anything

✅ Correct – matches the requirement precisely

❌ Option 1:

S → AabbB  
A → aA | ε  
B → bB | ε

• Introduces a separate non-terminal B after 'abb' which is not necessary
• Creates a possibility that "abb" may not be present if A and B generate empty string

Incorrect – structure deviates from strict placement of 'abb'

❌ Option 2:

S → abbA  
A → aA | ε | bA

• Prefix flexibility is missing
• Only strings starting with "abb" will be accepted

Incorrect – does not accept strings with prefix before "abb"

❌ Option 4:

S → Aabb  
A → aA | bA | ε

• No suffix A after "abb"
• Only strings ending in "abb" are accepted

Incorrect – fails to allow suffix characters after "abb"

Final Answer: Correct Option: Option 3

This grammar ensures that "abb" is included in the middle of the string and is surrounded by any number of valid symbols from Σ = {a, b}.