Solution

The correct answer is : option 4

Comprehension Recap: The machine accepts all strings over Σ = {a, b} that contain the substring "abb", and can start and end with any combination of 'a' and 'b'.

Key Points

• A regular expression defines a language using a pattern over an alphabet (here, Σ = {a, b}).
• We are looking for a regular expression that:
  • Allows any combination of a and b before abb
  • Must contain abb somewhere in the string
  • Allows any combination of a and b after abb
• In regular expression terms, this translates to: (a + b)* abb (a + b)*

Option Analysis:

Option 1: (a + b)* aab

• This accepts all strings that end with aab.
• But it does not ensure the occurrence of 'abb' — hence, incorrect.
• ❌ Rejected

Option 2: aba(a + b)*

• Matches strings that start with aba and are followed by any characters.
• But abb is not mandatory in this structure.
• ❌ Rejected

Option 3: b(a + b)* b(a + b)* a(a + b)*

• Complicated structure, but does not guarantee the abb sequence.
• More importantly, it's disorganized and doesn't represent the core substring abb.
• ❌ Rejected

Option 4: (a + b)* abb (a + b)*

• This is the canonical regular expression for strings that:
  • Can have any characters before abb
  • Must contain abb
  • Can have any characters after it
• This is a classic regular expression format for "string containing a pattern".
• ✅ Correct Answer

 Final Answer: Option 4 — (a + b)* abb (a + b)*

Explanation: This expression ensures the presence of the required substring abb while allowing flexibility for the string to have any content before or after it, aligning with the machine's definition in the passage.