Question

Which one of the following regular expressions is equivalent to the language accepted by the DFA given below? \begin{center} \includegraphics[width=6cm]{22.png} \end{center}

• A. \(0^*1(0 + 10^*1)^*\)
• B. \(0^*(10^*11)^*0^*\)
• C. \(0^*1(010^*1)^*0^*\)
• D. \(0(1 + 0^*10^*1)^*0^*\)

Hint:

To find the regular expression for a DFA, analyze transitions for each state and consider all possible loops and paths to the accepting state(s).

Answer 1

The Correct Option is A

The DFA has three states:
\(q_0\): Initial state.
\(q_1\): Intermediate state.
\(q_2\): Accepting state. Analysis:
From \(q_0\), a \(1\) transitions to \(q_1\).
From \(q_1\), a \(0\) transitions to \(q_2\), and another \(1\) transitions back to \(q_0\).
From \(q_2\), \(0\) loops back to \(q_2\). This DFA accepts strings that:
Start with \(0^*\) (any number of 0s at the beginning).
Have at least one \(1\) to reach \(q_1\).
Follow by \(0 + 10^*1\) repeatedly (to loop between \(q_1\) and \(q_0\)).
Optionally end in \(0^*\) (to stay in \(q_2\)). The equivalent regular expression is: \[ 0^*1(0 + 10^*1)^* \] Final Answer: \[ \boxed{0^*1(0 + 10^*1)^*} \]