1. Chapter 1 Regular Language
What is a computer?
Let’s begin with the simplest computational model : the finite state machine or finite automaton

2. 1.1 Finite Automata 1.1.1 An Example : Automatic Door State diagram
State transition table

3. 1.1.2 Another Example : M1 Alphabet : { 0 , 1 } States : q1 , q2 , q3
Start state : q1
Accept state : q3
Transitions

4. 1.1.3 Formal Definition of Finite Automata

5. we can describe M1 formally now

6. 1.1.4 Formal Definition of computation
A language is called a regular language if some finite automaton recognize it
If A is the set of all strings that machine M accepts, we say that M recognizes A or M accepts A

7. 1.1.5 The Regular Operations
Let A and B be language, let’s define the following three regular operations :
Union :
Concatenation :
Star :

8. Theorem 1.25 : The class of regular languages is closed under the union operation
PROOF :

9. Theorem 1.26 : The class of regular languages is closed under the concatenation
Proof idea : Break the string into two pieces such that the first one can be accepted by the first machine, and the other one can be accepted by the second machine.
But, where to break?

10. 1.2 Nondeterminism Deterministic VS. nondeterministic
Deterministic computation : Determined next move
Nondeterministic computation : Several choices may exist for the next move
“ fork a process “ on every possible choice / computational path

12. 1.2.1 Formal Definition of NFAs

13. Example 1.38
NFA N1

14. 1.2.2 Equivalence of NFAs and DFAs
Theorem 1.39 : Every NFA has an equivalent DFA
Proof idea : Convert NFA N into an equivalent DFA M that simulates the NFA N ( L(N) = L(M) )
Corollary 1.40 : A language is regular if and only if some nondeterministic finite automaton recognizes it

15. Proof by construction :

16. Example 1.41
NFA N4
Construct a DFA M

17. Example 1.41

18. 1.2.3 Closure under the Regular Operations
Theorem 1.45 : The class of regular language is closed under the union operation

19. Theorem 1.47 : The class of regular language is closed under the concatenation operation

20. Theorem 1.49 : The class of regular language is closed under the star operation