Construct nondeterministic finite automata for the following regular expressions using Algorithm 3.3. Show the sequence of moves made by each in processing the input string ababbab.
The Correct Answer and Explanation is :
To construct nondeterministic finite automata (NFA) for the regular expressions and process the input string ( ababbab ), we use Algorithm 3.3. The key steps include constructing an NFA for each regular expression and tracing the sequence of moves for the input string. Below is the solution:
Step 1: Regular Expressions
The given regular expressions are:
- ( a^* )
- ( (a \cup b)^* )
- ( ab^*a )
Step 2: Constructing NFAs
1. NFA for ( a^* ):
- States: ( q_0, q_1 )
- Transitions:
- ( q_0 \xrightarrow{\epsilon} q_1 ) (loop back to ( q_0 ))
- ( q_0 \xrightarrow{\epsilon} \text{Accept})
- This automaton accepts any string with zero or more ( a )’s.
2. NFA for ( (a \cup b)^* ):
- States: ( q_0, q_1 )
- Transitions:
- ( q_0 \xrightarrow{\epsilon} q_1 ) for both ( a ) and ( b ).
- ( q_1 \xrightarrow{\epsilon} q_0 ).
- This automaton accepts any string of ( a )’s and ( b )’s.
3. NFA for ( ab^*a ):
- States: ( q_0, q_1, q_2, q_3 )
- Transitions:
- ( q_0 \xrightarrow{a} q_1 )
- ( q_1 \xrightarrow{b} q_1 ) (loop for ( b ))
- ( q_1 \xrightarrow{a} q_2 )
- ( q_2 \rightarrow \text{Accept}).
Step 3: Sequence of Moves for Input ( ababbab )
1. ( a^* ):
- Start at ( q_0 ):
- ( a: q_1 \rightarrow q_0 ) (loop).
- ( b ): Reject (halt).
2. ( (a \cup b)^* ):
- Start at ( q_0 ):
- Traverse ( a, b, a, b, b, a, b ) via ( q_1 ) loops (accept).
3. ( ab^*a ):
- Start at ( q_0 ):
- ( a: q_0 \rightarrow q_1 ).
- ( b: q_1 \rightarrow q_1 ) (loop).
- ( a: q_1 \rightarrow q_2 ) (accept ( ab^*a )).
Explanation
- NFAs handle nondeterminism using epsilon (( \epsilon )) transitions and multiple paths.
- The sequence of moves is determined by tracking all valid paths for the input.
- Each NFA structure matches its respective regular expression:
- ( a^* ): Accepts repeated ( a )’s.
- ( (a \cup b)^* ): Accepts any combination of ( a ) and ( b ).
- ( ab^*a ): Accepts strings starting and ending with ( a ), with ( b )’s in between.