Deterministic. Finite Automata. And Regular Languages. Fall 2018 Costas Busch - RPI 1

Transcription

1 Deterministic Finite Automt And Regulr Lnguges Fll 2018 Costs Busch - RPI 1

2 Deterministic Finite Automton (DFA) Input Tpe String Finite Automton Output Accept or Reject Fll 2018 Costs Busch - RPI 2

3 Trnsition Grph q3 q4 initil stte stte trnsition ccepting stte Fll 2018 Costs Busch - RPI 3

4 Alphet {, } q3 q4 For every stte, there is trnsition for every symol in the lphet Fll 2018 Costs Busch - RPI 4

5 hed Initil Configurtion Input Tpe Input String q 0 q3 q4 Initil stte Fll 2018 Costs Busch - RPI 5

6 Scnning the Input q3 q4 Fll 2018 Costs Busch - RPI 6

7 q3 q4 Fll 2018 Costs Busch - RPI 7

8 q3 q4 Fll 2018 Costs Busch - RPI 8

9 Input finished q3 q4 ccept Fll 2018 Costs Busch - RPI 9

10 A Rejection Cse Input String q 0 q3 q4 Fll 2018 Costs Busch - RPI 10

11 q3 q4 Fll 2018 Costs Busch - RPI 11

12 q3 q4 Fll 2018 Costs Busch - RPI 12

13 Input finished q3 q4 reject Fll 2018 Costs Busch - RPI 13

14 Another Rejection Cse Tpe is empty () Input Finished q 0 q3 q4 reject Fll 2018 Costs Busch - RPI 14

15 Lnguge Accepted: L q3 q4 Fll 2018 Costs Busch - RPI 15

16 To ccept string: ll the input string is scnned nd the lst stte is ccepting To reject string: ll the input string is scnned nd the lst stte is non-ccepting Fll 2018 Costs Busch - RPI 16

17 Another Exmple L,, q3 q4 Fll 2018 Costs Busch - RPI 17 Accept Accept Accept stte stte stte

18 Empty Tpe () Input Finished q3 q4 ccept Fll 2018 Costs Busch - RPI 18

19 Another Exmple, Accept stte trp stte Fll 2018 Costs Busch - RPI 19

20 Input String, Fll 2018 Costs Busch - RPI 20

21 , Fll 2018 Costs Busch - RPI 21

22 , Fll 2018 Costs Busch - RPI 22

23 Input finished ccept, Fll 2018 Costs Busch - RPI 23

24 A rejection cse Input String, Fll 2018 Costs Busch - RPI 24

25 , Fll 2018 Costs Busch - RPI 25

26 , Fll 2018 Costs Busch - RPI 26

27 Input finished, reject Fll 2018 Costs Busch - RPI 27

28 Lnguge Accepted: L { : n 0} n, Fll 2018 Costs Busch - RPI 28

29 Another Exmple Alphet: {1} Lnguge Accepted: 1 1 EVEN { x : x * nd x is even} {, 11, 1111, , } Fll 2018 Costs Busch - RPI 29

30 Forml Definition Deterministic Finite Automton (DFA) M Q,,,, F Q : set of sttes : input lphet q 0 F : trnsition function : initil stte : set of ccepting sttes Fll 2018 Costs Busch - RPI 30

31 Set of Sttes Q Exmple Q q 0 1, 2, 3, 4,, q q q q q 5 q3 q4 Fll 2018 Costs Busch - RPI 31

32 Input Alphet :the input lphet never contins Exmple q3 q4 Fll 2018 Costs Busch - RPI 32

33 Initil Stte q 0 Exmple q 0 q3 q4 Fll 2018 Costs Busch - RPI 33

34 Set of Accepting Sttes F Q Exmple F q 4 q3 q 4 Fll 2018 Costs Busch - RPI 34

35 Trnsition Function : Q Q ( q, x ) q q x q Descries the result of trnsition from stte q with symol x Fll 2018 Costs Busch - RPI 35

36 Exmple:, q 1 q3 q4 Fll 2018 Costs Busch - RPI 36

37 , q5 q 0 q3 q4 Fll 2018 Costs Busch - RPI 37

38 3 2, q q q3 q4 Fll 2018 Costs Busch - RPI 38

39 Trnsition Tle for sttes q q 0 q 1 q 2 q 3 symols 5 q5 q q5 3 q q4 5 q 4 q3 q4 Fll 2018 Costs Busch - RPI 39

40 Extended Trnsition Function * : Q * Q * ( q, w ) q Descries the resulting stte fter scnning string w from stte q Fll 2018 Costs Busch - RPI 40

41 Exmple: * q, q 0 2 q 2 q3 q4 Fll 2018 Costs Busch - RPI 41

42 * q, q 0 5 q 0 q3 q4 Fll 2018 Costs Busch - RPI 42

43 * q, q 1 4 q3 q4 Fll 2018 Costs Busch - RPI 43

44 Specil cse: for ny stte q * q, q Fll 2018 Costs Busch - RPI 44

45 In generl: * q, w q implies tht there is wlk of trnsitions w 1 2 k q 1 2 k q sttes my e repeted q w q Fll 2018 Costs Busch - RPI 45

46 Lnguge Accepted y DFA Lnguge of DFA : M it is denoted s L ll the strings ccepted y M nd contins M We sy tht lnguge is ccepted (or recognized) y DFA M if L LM L Fll 2018 Costs Busch - RPI 46

47 For DFA M Q,,, q, 0 F M Lnguge ccepted y : L M w * * q, w : 0 F w q q F Fll 2018 Costs Busch - RPI 47

48 M Lnguge rejected y : L M w * * q, w : 0 F w q q F Fll 2018 Costs Busch - RPI 48

49 More DFA Exmples {, } q 0 q 0 L( M) { } Empty lnguge L( M) All strings * Fll 2018 Costs Busch - RPI 49

50 {, }, q 0 L( M) { } Lnguge of the empty string Fll 2018 Costs Busch - RPI 50

51 {, } L M = { ll strings with prefix } Communiction strings Preceding 01 or suffix 110 ccept q 3 Fll 2018 Costs Busch - RPI 51

52 L M = Wht is the lnguge??? { } , Fll 2018 Costs Busch - RPI 52

53 L M = { ll inry strings contining sustring 001} security codes 1 0 0, Fll 2018 Costs Busch - RPI 53

54 L M = Wht is the lnguge {??} ,1 Fll 2018 Costs Busch - RPI 54

55 L M = { ll inry strings without sustring 001 } ,1 Fll 2018 Costs Busch - RPI 55

56 L (M) = {??} q3 q 4 Fll 2018 Costs Busch - RPI 56

57 L( M) w : w, * q3 q 4 Fll 2018 Costs Busch - RPI 57

58 Definition: A lnguge M L Regulr Lnguges is regulr if there is DFA tht ccepts it ( L( M) L ) The lnguges ccepted y ll DFAs form the fmily of regulr lnguges Fll 2018 Costs Busch - RPI 58

59 Exmple regulr lnguges:,, { n : n 0} { ll strings in {}* with prefix } { ll inry strings without sustring 001} { x : x { } {} {1} * nd {, There exist utomt tht ccept these lnguges (see previous slides). x * } w Fll 2018 Costs Busch - RPI 59 : w, is even} *

60 There exist lnguges which re not Regulr: L n n { : n 0} ADDITION { x y z : x 1 n, y 1 m,z 1 k, n m k } There is no DFA tht ccepts these lnguges (we will prove this in lter clss) Fll 2018 Costs Busch - RPI 60

61 Exmple Build nd utomt which ccepts the regulr lnguge L = { 2 2 W, W ε {} * } Fll 2018 Costs Busch - RPI 61