1. Below is a tabular representation of an NFA. Its initial state is q0 and its lone accepting state is qf. Use the "subset construction" algorithm to compute a DFA that accepts the same language, and show it in the form of either a table or a transition graph.
| state | symbol | ||
|---|---|---|---|
| a | b | c | |
| q0 | q0,qa | q0,qb | q0,qc |
| qa | qf | qa | qa |
| qb | qb | qf | qb |
| qc | qc | qc | qf |
| qf | qf | qf | qf |
2. Below is a tabular representation of an NFA with ε-transitions. Its initial state of is q0 and its lone accepting state is qf. Use the algorithm described by Hopcroft & Ullman to compute an NFA having no such transitions, and show it in the form of either a table or a transition graph.
| state | symbol | ||
|---|---|---|---|
| a | b | ε | |
| q0 | q1,q3 | ||
| q1 | q1 | q2 | |
| q2 | q2 | q2 | qf |
| q3 | q4 | q3 | |
| q4 | q4 | q4 | qf |
| qf | |||
3. Below is a tabular representation of a DFA. Its initial state is q0 and its accepting states are qf1, qf2, and qf3. Minimize the DFA using either the algorithm described by Hopcroft & Ullman in the textbook or Hopcroft's algorithm (discussed in class after Phil had to leave). Show the result in the form of a transition graph.
| state | symbol | |
|---|---|---|
| a | b | |
| q0 | qa | qb |
| qa | qf1 | qab |
| qb | qab | qf2 |
| qab | qf3 | qf3 |
| qf1 | qf1 | qa |
| qf2 | qf3 | qf2 |
| qf3 | qf3 | qf2 |
4. Below is a tabular representation of a DFA. Its initial state is q0 and its accepting states are q3 and q7. As in the preceding problem, minimize the DFA.
| state | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| symbol | q0 | q1 | q2 | q3 | q4 | q5 | q6 | q7 | q8 | q9 | q10 |
| 0 | q1 | q2 | q3 | q4 | q4 | q0 | q6 | q6 | q4 | q8 | q9 |
| 1 | q2 | q1 | q3 | q4 | q7 | q5 | q5 | q6 | q9 | q10 | q10 |
5. Prove that the set { 0n | n is prime } is not regular.
6. Do several parts of Exercise 3.4 on page 72 of Hopcroft & Ullman. For those languages that are regular, good answers can be found in the form of DFA "constructions", as exemplified by the given solution for 3.4(b).