Two Design Problems.

1. Given an FA, describe its language precisely.
2. Given a definition of a regular language, design an FA that recognizes it.

Solving Problem 1 is an art. No general principles are known.

To solve Problem 2, work with state diagrams and follow these steps.

1. Get all the machine states. Each state memorizes one unique machine condition.
2. Pinpoint the accept states.
3. Draw the edges.

Assume Σ = {a, b}. Design machines to recognize strings that

1. starts with a
2. ends with a
3. ends with bb
4. starts with a and ends with b
5. starts with a or ends with a
6. if it starts with a, then it doesn’t end with a
7. does not start or end with a but a occurs somewhere
8. contains (at least) an a

Assume Σ = {a, b}. Design machines to recognize strings that

9. contains exactly one a
10. contains (at least) two a’s
11. contains exactly two a’s
12. contains ab
13. contains ab and ba
14. contains ab or ba
15. contains neither aa nor bb
16. negate each of the above

We can always get the complement of a language of a DFA simply by exchanging the role of the accepting and the nonaccepting states. Does this technique work for NFAs?

The Product Construction can be used to obtain the language of binary set operations (like union, intersection, set difference, symmetric difference, etc) of two regular languages.