1 NFAs accept the Regular Languages

2 Equivalence of Machines Definition: Machine is equivalent to machine if

3 Example of equivalent machines NFA FA

4 We will prove: Languages accepted by NFAs Regular Languages NFAs and FAs have the same computation power Languages accepted by FAs

5 Languages accepted by NFAs Regular Languages accepted by NFAs Regular Languages We will show:

6 Languages accepted by NFAs Regular Languages Proof-Step 1 Proof: Every FA is trivially an NFA Any language accepted by a FA is also accepted by an NFA

7 Languages accepted by NFAs Regular Languages Proof-Step 2 Proof: Any NFA can be converted to an equivalent FA Any language accepted by an NFA is also accepted by a FA

8 Convert NFA to FA NFA FA

9 Convert NFA to FA NFA FA

10 Convert NFA to FA NFA FA

11 Convert NFA to FA NFA FA

12 Convert NFA to FA NFA FA

13 Convert NFA to FA NFA FA

14 Convert NFA to FA NFA FA

15 NFA to FA: Remarks We are given an NFA We want to convert it to an equivalent FA With

16 If the NFA has states the FA has states in the powerset

17 Procedure NFA to FA 1. Initial state of NFA: Initial state of FA:

18 Example NFA FA

19 Procedure NFA to FA 2. For every FA’s state Compute in the NFA Add transition to FA

20 Exampe NFA FA

21 Procedure NFA to FA Repeat Step 2 for all letters in alphabet, until no more transitions can be added.

22 Example NFA FA

23 Procedure NFA to FA 3. For any FA state If is accepting state in NFA Then, is accepting state in FA

24 Example NFA FA

25 Theorem Take NFA Apply procedure to obtain FA Then and are equivalent :

26 Proof AND

27 First we show: Take arbitrary: We will prove:

28

29 denotes

30 We will show that if then

31 More generally, we will show that if in : (arbitrary string) then

32 Proof by induction on Induction Basis: Is true by construction of

33 Induction hypothesis:

34 Induction Step:

35 Induction Step:

36 Therefore if then

37 We have shown: We also need to show: (proof is similar)

38 Single Accepting State for NFAs

39 Any NFA can be converted to an equivalent NFA with a single accepting state

40 NFA Equivalent NFA Example

41 NFA In General Equivalent NFA Single accepting state

42 Extreme Case NFA without accepting state Add an accepting state without transitions

43 Properties of Regular Languages

44 Concatenation: Star: Union: Are regular Languages For regular languages and we will prove that: Complement: Intersection: Reversal:

45 We say: Regular languages are closed under Concatenation: Star: Union: Complement: Intersection: Reversal:

46 Regular language Single accepting state NFA Single aceepting state Regular language NFA

47 Example

48 Union NFA for

49 Example NFA for

50 Concatenation NFA for

51 Example NFA for

52 Star Operation NFA for

53 Example NFA for

54 Reverse NFA for 1. Reverse all transitions 2. Make initial state accepting state and vice versa

55 Example

56 Complement 1. Take the FA that accepts 2. Make final states non-final, and vice-versa

57 Example

58 Intersection regular We show regular

59 DeMorgan’s Law: regular

60 Example regular

61 for FA Construct a new FA that accepts Machine simulates in parallel and Another Proof for Intersection Closure

62 States in State in

63 transition FA transition FA

64 initial state Initial state FA

65 accept state accept states FA Both constituents must be accepting states

66 Example:

67 Automaton for intersection

68 simulates in parallel and accepts stringif and only if accepts string and accepts string