The correct answer is Option 1: b* a b * a b * a b *.

Key Points

• The regular expression b* a b * a b * a b * represents the language accepted by the given finite automata.
• This regular expression can be broken down as follows:
  • b* matches zero or more occurrences of the character 'b'.
  • a matches a single occurrence of the character 'a'.
  • Repeating the pattern b* a three more times ensures that the sequence 'a' appears exactly three times, each possibly preceded by zero or more 'b's.
• This pattern matches any string of 'b's followed by 'a', and this sequence is repeated three times.

Additional Information

• The finite automata can be visualized as having states that transition between each other upon reading specific inputs (characters).
• Understanding the structure of the automata and the transitions helps in deriving the correct regular expression that represents the language it accepts.
• Finite automata are used in various fields such as text processing, compiler design, and network protocols to recognize patterns and validate input sequences.
• Regular expressions are a powerful tool for describing patterns in strings and are widely used in search algorithms, text processing, and input validation.