1. Formal Language & Automata Theory
Shyamanta M Hazarika
Computer Sc. & Engineering
Tezpur University

2. Regular Languages Regular languages are closed under: Union
Concatenation
Star operation

3. Regular Languages Namely, for regular languages and : Union
Concatenation
Star operation
Regular
Languages

4. We will prove Regular languages are closed under: Complement
Intersection

5. We will prove Namely, for regular languages and : Complement Regular
Intersection
Regular
Languages

6. Complement
Theorem:
For regular language
the complement is regular

7. Complement Theorem: For regular language the complement is regular
Proof:
Take DFA that accepts and make
nonfinal states final
final states nonfinal
Resulting DFA accepts

8. Example

9. Intersection Theorem: For regular languages and
the intersection is regular

10. Intersection Theorem: For regular languages and
the intersection is regular
Proof:
Apply DeMorgan’s Law:

11. Intersection
regular
regular
regular
regular
regular

12. Standard Representations of Regular Languages
DFAs
Regular
Grammars
NFAs
Regular
Expressions