1. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b List all unordered pairs of distinct states (p, q). Make a sequence of passes through these pairs as follows. On the first pass, mark each pair (p, q) for which exactly one of the two states p and q is in A. On each subsequent pass, examine each unmarked pair (r, s); if there is a symbol σ ∈ such that δ(r, σ) = p, δ(s, σ) = q, and the pair (p, q) has already been marked, then mark (r, s). After a pass in which no new pairs are marked, stop. At that point, the marked pairs (p, q) are precisely those for which p q.

2. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 Pair (2, 1) is marked on the first pass because 2 is an accepting state and 1 is not.

3. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 Pair (6, 1) is marked on the first pass because 6 is an accepting state and 1 is not. 1

4. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (3, 2) is marked on the first pass because 2 is an accepting state and 3 is not. 1 1 1

5. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (4, 2) is marked on the first pass because 2 is an accepting state and 4 is not. 1 1 1 1

6. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (5, 2) is marked on the first pass because 2 is an accepting state and 5 is not. 1 1 1 1 1

7. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (7, 2) is marked on the first pass because 2 is an accepting state and 7 is not. 1 1 1 1 1 1

8. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (6, 3) is marked on the first pass because 6 is an accepting state and 3 is not. 1 1 1 1 1 1 1

9. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (6, 4) is marked on the first pass because 6 is an accepting state and 4 is not. 1 1 1 1 1 1 1 1

10. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (6, 5) is marked on the first pass because 6 is an accepting state and 5 is not. 1 1 1 1 1 1 11 1

11. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (7, 6) is marked on the first pass because 6 is an accepting state and 7 is not. 1 1 1 1 1 1 11 1 1

12. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b Pair (4, 1) is marked on the second pass because δ(4, a) = 4 and δ(1, a) = 2. (4, 2) was marked in pass one. 2 1 1 1 1 1 1 111 1 a a

13. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 111 1 2 Pair (4, 3) is marked on the second pass because δ(4, a) = 4 and δ(3, a) = 6. (6, 4) was marked in pass one. a a

14. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 11 1 Pair (5, 4) is marked on the second pass because δ(5, a) = 6 and δ(4, a) = 4. (6, 4) was marked in pass one. a a

15. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 Pair (7, 4) is marked on the second pass because δ(7, a) = 6 and δ(4, a) = 4. (6, 4) was marked in pass one. a a

16. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 Using the above information we can now determine the states in the minimum-state FA.

17. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 We will combine states 1, 3, 5, and 7.

18. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 We will also combine states 6 and 2.

19. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 State 4 will be a new state. 212

20. Find a minimum-state FA recognizing the same language as the given FA. 1 3 7 2 5 4 6 b b a a a a a a a b b b b 123456 2 3 4 5 6 7 b 1 2 1 1 1 1 1 2 1 2 1 2 1 1 Below is our minimum-state FA as outlined before: 1,3, 5, 7 6,2 4 b b b a a a