Formal Languages Simplifications of CFGs

Transcription

1 Forml Lnguges implifictions of CFGs

2 ubstitution Rule Equivlent grmmr b bc ubstitute b bc bbc b 2

3 ubstitution Rule b bc bbc ubstitute b bc bbc bc Equivlent grmmr 3

4 In generl: xz y 1 ubstitute y 1 xz xy1z equivlent grmmr 4

5 Lnguge? 5

6 Nullble Vribles production: Nullble Vrible: 6

7 Removing Nullble Vribles Exmple Grmmr: Mb M Mb M Nullble vrible 7

8 Finl Grmmr Mb M Mb M ubstitute M Mb b M Mb M b 8

9 Unit-Productions Unit Production: (single vribles on both sides) 9

10 Removing Unit Productions Observtion: Is removed immeditely 10

11 Exmple Grmmr: bb 11

12 ubstitute bb bb 12

13 Remove bb bb 13

14 ubstitute bb bb 14

15 Remove repeted productions Finl grmmr bb bb 15

16 Lnguge? 16

17 Useless Productions b Useless Production ome derivtions never terminte... 17

18 nother grmmr: b Useless Production Not rechble from 18

19 In generl: contins only terminls if xy w wl(g) then vrible is useful otherwise, vrible is useless 19

20 production x is useless if ny of its vribles is useless b Productions Vribles useless useless useless useless C useless useless C D useless 20

21 Removing Useless Productions Exmple Grmmr: C C Cb 21

22 Remove useless productions 22

23 First: find ll vribles tht cn produce strings with only terminls Round 1: C {, } C Cb Round 2: {,, } 23

24 Keep only the vribles tht produce terminl symbols: {,, } (other vribles re useless) C C Cb Remove useless productions 24

25 econd: Find ll vribles rechble from Use Dependency Grph not rechble 25

26 Keep only the vribles rechble from (the other vribles re useless) Finl Grmmr Remove useless productions 26

27 Removing ll tep 1: Remove Nullble Vribles tep 2: Remove Unit-Productions tep 3: Remove Useless Vribles 27

28 Norml Forms for Context-free Grmmrs 28

29 Chomsky Norml Form Ech production hs form: C or vrible vrible terminl 29

30 Exmples: b Chomsky Norml Form Not Chomsky Norml Form 30

31 Conversion to Chomsky Norml Form Exmple: b c Not Chomsky Norml Form 31

32 32 c b Introduce vribles for terminls: c T b T T T T T T T c b c b c b T T T,,

33 33 Introduce intermedite vrible: c T b T T T T T T T c b c b c T b T T T T T T T V V c b c b 1 1 V 1

34 34 Introduce intermedite vrible: c T b T T T T T V V T T V V c b c b V 2 c T b T T T T T T T V V c b c b 1 1

35 35 Finl grmmr in Chomsky Norml Form: c T b T T T T T V V T T V V c b c b c b Initil grmmr

36 In generl: From ny context-free grmmr (which doesn t produce ) not in Chomsky Norml Form we cn obtin: n equivlent grmmr in Chomsky Norml Form 36

37 The Procedure First remove: Nullble vribles Unit productions 37

38 Then, for every symbol : dd production T In productions: replce with T New vrible: T 38

39 Replce ny production C C 1 2 C n with C V 1 1 V 1 C V 2 2 V n2 C n1 C n New intermedite vribles: V1, V2,, V n 2 39

40 Theorem: For ny context-free grmmr (which doesn t produce ) there is n equivlent grmmr in Chomsky Norml Form 40

41 Observtions Chomsky norml forms re good for prsing nd proving theorems It is very esy to find the Chomsky norml form for ny context-free grmmr 41

42 exercise Find CNF for this grmmr: -> > C -> C -> epsilon (exercise 7.1.3, Hopcroft, Motwni, Ullmn) 42

43 Greibch Norml Form ll productions hve form: V V 1 2 V k k 0 terminl vribles 43

44 Exmples: c bb b b b Greibch Norml Form Not Greibch Norml Form 44

45 Conversion to Greibch Norml Form: T b T b bb T T T b b Greibch Norml Form 45

46 Theorem: For ny context-free grmmr (which doesn t produce ) there is n equivlent grmmr in Greibch Norml Form 46

47 Observtions Greibch norml forms re very good for prsing It is hrd to find the Greibch norml form of ny context-free grmmr 47

48 Try to compute Greibch Norml Form for grmmr in CNF exmple 48

49 Compilers 49

50 Progrm v = 5; if (v>5) x = 12 + v; while (x!=3) { x = x - 3; v = 10; }... Compiler Mchine Code dd v,v,0 cmp v,5 jmplt ELE THEN: dd x, 12,v ELE: WHILE: cmp x,

51 Compiler Lexicl nlyzer prser input progrm mchine code output 51

52 prser knows the grmmr of the progrmming lnguge 52

53 PROGRM TMT_LIT TMT Prser TMT_LIT TMT; TMT_LIT TMT; EXPR IF_TMT WHILE_TMT { TMT_LIT } EXPR IF_TMT WHILE_TMT EXPR + EXPR EXPR - EXPR INT if (EXPR) then TMT if (EXPR) then TMT else TMT while (EXPR) do TMT 53

54 The prser finds the derivtion of prticulr input input * 5 Prser E -> E + E E * E INT derivtion E => E + E => E + E * E => 10 + E*E => * E => * 5 54

55 derivtion derivtion tree E E => E + E => E + E * E => 10 + E*E => * E => * 5 E 10 + E E * E

56 derivtion tree E mchine code E + E mult, 2, 5 10 E * E dd b, 10,

57 Prsing 57

58 input string Prser grmmr derivtion 58

59 Exmple: input bb Prser b b derivtion? 59

60 Prsing lgorithm? 60

61 Exhustive erch b b Phse 1: Find derivtion of b bb b ll possible derivtions of length 1 61

62 bb b b 62

63 Phse 2 b b b bb Phse 1 b + 2 more b b b b bb b bb b b 63

64 Phse 2 b b b bb b b b bb Phse 3 b bb bb 64

65 Finl result of exhustive serch (top-down prsing) Prser input b bb b derivtion b bb bb 65

66 Is exhustive serch good prsing lgorithm? 66

67 Time complexity of exhustive serch uppose there re no productions of the form Number of phses for string : w 2 w 67

68 For grmmr with k rules Time for phse 1: k k possible derivtions 68

69 Time for phse 2: k 2 k 2 possible derivtions 69

70 Time for phse : 2 w k 2 w k 2 w possible derivtions 70

71 Totl time needed for string : w k k 2 2 w k phse 1 phse 2 phse 2 w Extremely bd!!! 71

72 There exist fster lgorithms for specilized grmmrs -grmmr: x symbol string of vribles Pir (, ) ppers once 72

73 -grmmr exmple: b c Ech string hs unique derivtion b bc bcc 73

74 For -grmmrs: In the exhustive serch prsing there is only one choice in ech phse Time for phse: 1 Totl time for prsing string w : w 74

75 For generl context-free grmmrs: There exists prsing lgorithm tht prses string in time 3 w w (we will show this in the next clss) 75

76 The CYK Prser 76

77 Input: The CYK Membership lgorithm Grmmr G in Chomsky Norml Form tring w Output: find if wl(g) 77

78 Input exmple: The lgorithm Grmmr G: b tring : w bbb 78

79 bbb b b b b bb bb b bb bbb bb bbb bbb 79

80 b b b b bb bb b b bb bbb bb bbb bbb 80

81 b b b b b bb, b bb bbb bb bb bbb bbb 81

82 b b, b, bb b bb bbb, b bb b bb bbb, bbb, 82

83 Therefore: bbb L(G) Time Complexity: 3 w Observtion: The CYK lgorithm cn be esily converted to prser (bottom up prser) 83

84 The following slides re courtesy of Professor Ppp, University of Debrecen. 84

85 1,7 1,1 7,7 b b b

86 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

87 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

88 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

89 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

90 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

91 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

92 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

93 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

94 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

95 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

96 Grmmr CD C C,C,D C b C DD C b D D, C, D D,C,C,C b b b

97 ,C Word is in lnguge if ppers here,d D, C, D D,C,C,C b b b

98 exercise Prse bb for this grmmr -> C -> -> CC b C -> (Hopcroft, Motwni, Ullmn, p301) 98