FABER Formal Languages, Automata and Models of Computation

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1 DVA 325 FABR Forml Lnguges, Automt nd Models of Computtion Lecture 7 chool of Innovtion, Design nd ngineering Mälrdlen University 27 November

2 Content ü Regulr vs. Non-regulr Lnguges ü Context-Free Lnguges ü Context-Free Grmmrs ü Derivtion Trees. Ambiguity ü Applictions ü Push-Down Automt (PDA) 2

3 Time to tke the next step: beyond Regulr Lnguges n l n l { b c : n, l 0} { n! : n 0} Non-regulr Context-Free lnguges Lnguges n n R { b } { ww } Regulr Lnguges 3

4 Automt theory: forml lnguges nd forml grmmrs Grmmr Lnguges Automton Type-0 Type-1 Type-2 Recursively enumerble Contextsensitive Context-free Turing mchine Liner-bounded nondeterministic Turing mchine Non-deterministic pushdown utomton Type-3 Regulr Finite stte utomton Production rules No restrictions nd 4

5 Context-Free Lnguges Bsed on C Busch, RPI, Models of Computtion 5

6 Context-Free Lnguges Context-Free Grmmrs Pushdown Automt 6

7 Context-Free Grmmrs 7

8 V T P : : : : Grmmr Forml Definition G = ( V, T,, P) et of vribles et of terminl symbols trt vrible et of production rules 8

9 Repetition: Regulr Grmmrs Grmmr G = ( V, T,, P) Vribles Terminl symbols trt vribles Right or Left Liner Grmmrs. Productions of the form: A xb A Bx or C x x is string of terminls 9

10 Definition: Context-Free Grmmrs Grmmr G = ( V, T,, P) Vribles Terminl symbols trt vribles Productions of the form: A x x is string of vribles nd terminls 10

11 Regulr vs. Context-free Grmmr A regulr grmmr is either right or left liner, wheres context free* grmmr is ny combintion of terminls nd non-terminls. Hence regulr grmmrs re subset of context-free grmmrs. Grmmr generting plindromes is not regulr: A B ABA something something *The nme context-free grmmr is explined by the property of productions tht re independent of the surrounding symbols. There re lso context-sensitive grmmrs where productions depend on the context (symbols tht surround vribles). 11

12 xmple 1: A context-free grmmr G b A derivtion λ b bb bb 12

13 A context-free grmmr G b λ Another derivtion b bb bbb bbb 13

14 b λ n L (G) = { b : n 0} n!( ( ( ( ) ) ) )! 14

15 xmple 2: A context-free grmmr G bb λ A derivtion bb bb 15

16 A context-free grmmr G bb λ Another derivtion bb bb 16

17 bb λ R L( G) = { ww : w {, b}*} 17

18 xmple 3: A context-free grmmr G b λ A derivtion b b b 18

19 A context-free grmmr G b λ A derivtion b b bb bb 19

20 20 λ b } prefix in ny ) ( ) ( nd ), ( ) ( : { ) ( v v n v n w n w n w G L b b = =! )! ) ( ( ) ) ) ( ( )( (

21 xmple 4: Lnguge L = { b :n m} For the 1 A For the B 1 A, A. cse n n b λ, m cse n > m : B, 1 bb b. b λ, < m : is context - free.

22 bb b B A A b λ B A bb b. B b λ B, m n < :, A. A b λ, A m n >, : is: the lnguge for grmmr The m} :n b { L m n =

23 Definition: Context-Free Grmmrs Grmmr G = ( V, T,, P) Vribles Terminl symbols trt vribles Productions of the form: A x x is string of vribles nd terminls 23

24 Definition: Context-Free Lnguges A lnguge L is context-free if nd only if there is context-free grmmr L = L(G) G with 24

25 A A B B AB A λ Bb λ Derivtion Order Leftmost derivtion AB AB BBb b

26 A A B B λ λ Derivtion Order AB A Bb Rightmost derivtion 1 4 AB 5 ABb 2 3 AbAbb 26

27 A AB bbb B A λ Leftmost derivtion AB bbbb babb bbbbbb bbbbb bbbb 27

28 A AB bbb B A λ Rightmost derivtion AB A bbb bab bbbbb bbbb 28

29 Derivtion Trees 29

30 Derivtion cn be represented in tree form AB A A λ B Bb λ AB A B 30

31 AB A A λ B Bb λ AB AB A B A 31

32 AB A A λ B Bb λ AB AB ABb A B A B b 32

33 AB A A λ B Bb λ AB AB ABb Bb A B A B b λ 33

34 AB A A λ B Bb λ AB AB ABb Bb b Derivtion Tree A B A B b λ λ 34

35 AB A A λ B Bb λ AB AB ABb Bb b Derivtion Tree A B yield A B b λλ b = b λ λ 35

36 Prtil Derivtion Trees AB A A λ B Bb λ AB Prtil derivtion tree A B 36

37 AB AB Prtil derivtion tree A B A 37

38 AB AB sententil form Prtil derivtion tree yield A B AB A 38

39 ometimes, derivtion order doesn t mtter Leftmost: AB AB B Bb b Rightmost: AB ABb Ab Ab b The sme derivtion tree A B A B b λ λ 39

40 Ambiguity 40

41 41 ) ( leftmost derivtion derivtion (* denotes multipliction)

42 42 ) ( derivtion leftmost derivtion

43 ( ) 43

44 ( ) Two derivtion trees 44

45 45 The grmmr ) ( is mbiguous! tring hs two derivtion trees

46 46 hs two leftmost derivtions: * The grmmr ) ( is mbiguous s the string

47 Definition A context-free grmmr G is mbiguous if some string w L(G) hs two or more derivtion trees (two or more leftmost/rightmost derivtions). 47

48 Why do we cre bout mbiguity? = 2 48

49 Why do we cre bout mbiguity?

50 2 2 Why do we cre bout mbiguity? = = 8 50

51 Correct result: =

52 Ambiguity is bd for progrmming lnguges We wnt to remove mbiguity! 52

53 53 We fix the mbiguous grmmr ) ( by introducing prentheses () to indicte grouping, (precedence) F F F T F T T T T ) ( Non-mbiguous grmmr

54 54 F F F T F T T T T ) ( F F F F T T T F T T T T F T F F T

55 Unique derivtion tree T T T F F F 55

56 G The grmmr : T T T T F T F F ( ) is non-mbiguous. very string derivtion tree. w L(G) F hs unique 56

57 57 Inherent Ambiguity ome context free lnguges hve only mbiguous grmmrs! xmple: } { } { m m n m n n c b c b L = λ 1 1 Ab A A c λ 2 2 bbc B B 1 2

58 The string n b n c hs two derivtion trees n c 2 58

59 n l n l { b c : n, l 0} { n! : n 0} Non-regulr Context-Free lnguges Lnguges n n R { b } { ww } Regulr Lnguges 59

60 Applictions: Compilers 60

61 Progrm v = 5; if (v>5) x = 12 v; while (x!=3) { x = x - 3; v = 10; }... Compiler Mchine Code Add v,v,0 cmp v,5 jmplt L THN: dd x, 12,v L: WHIL: cmp x,

62 Compiler Lexicl nlyzer prser input output progrm mchine code 62

63 A prser knows the grmmr of the progrmming lnguge 63

64 PROGRAM TMT_LIT TMT Prser TMT_LIT TMT; TMT_LIT TMT; XPR IF_TMT WHIL_TMT { TMT_LIT } XPR IF_TMT WHIL_TMT XPR XPR XPR - XPR ID if (XPR) then TMT if (XPR) then TMT else TMT while (XPR) do TMT 64

65 The prser finds the derivtion of prticulr input input 10 2 * 5 Prser * INT derivtion * 10 * 10 2 * 10 2 * 5 65

66 derivtion derivtion tree * 10 * 10 2 * 10 2 * 5 10 *

67 derivtion tree mchine code mult, 2, 5 10 * dd b, 10,

68 Prsing exmples 68

69 input string Prser grmmr derivtion 69

70 xmple: input bb Prser b b derivtion? λ 70

71 xhustive erch b b λ Phse 1: Find derivtion of b bb b λ All possible derivtions of length 1 71

72 bb b b λ 72

73 73 Phse 2 b bb b b Phse 1 b b bb b bb b b b λ b b

74 Phse 2 b b λ b bb b b b bb Phse 3 b bb bb 74

75 Finl result of exhustive serch (top-down prsing) Prser input b bb b λ derivtion b bb bb 75

76 Another use of context free grmmrs: Context Free Art 76

77 Context Free Art 77

78 Context-Free Lnguges Context-Free Grmmrs Pushdown Automt stck utomton 78

79 Pushdown Automt PDAs 79

80 Pushdown Automton - PDA Input tring tck ttes 80

81 The tck A PDA cn write symbols on stck nd red them lter on. POP reding symbol PUH writing symbol All ccess to the stck only on the top! (tck top is written leftmost in the string, e.g. yxz) A stck is vluble s it cn hold n unlimited mount of informtion. The stck llows pushdown utomt to recognize some non-regulr lnguges.! y x z 81

82 The ttes Input symbol Pop old reding stck symbol Push new writing stck symbol, b / c q 1 q 2 82

83 , b / c q 1 q 2 input!!!! stck b top Replce c e h $ e h $ (An lterntive is to strt nd finish with empty stck) 83

84 ,λ / c q 1 q 2 input!!!! stck c b top Push b e h $ e h $ 84

85 input,b / λ q 1 q 2!!!! stck b top Pop e h $ e h $ 85

86 , λ / λ q 1 q 2 input!!!! stck b top No Chnge b e h $ e h $ 86

87 Forml Definition Pushdown Automton is defined s 7-tuple M = ( Q, Σ, Γ, δ, q0, z, F) ttes Finl sttes Input lphbet tck lphbet Trnsition function strt stte tck strt symbol 87

88 xmple 3.7 lling: A PDA for simple nested prenthesis strings Time 0 ( ( ( ) ) ) ε Input tck (, ε / ( ), ( / ε strt s ),(/ε q end 88

89 xmple 3.7 Input ( ( ( ) ) ) ( ε Time 1 tck (, ε / ( ), ( / ε strt s ),(/ε q end 89

90 xmple 3.7 Time 2 Input ( ( ( ) ) ) ( ( ε (, ε / ( ), ( / ε tck strt s ),(/ε end q 90

91 xmple 3.7 Input ( ( ( ) ) ) ( ( Time 3 ( ε (, ε / ( ), ( / ε tck strt s ),(/ε q end 91

92 xmple 3.7 Input ( ( ( ) ) ) ( ( Time 4 (, ε / ( ), ( / ε ( ε tck strt s ),(/ε q end 92

93 xmple 3.7 Input ( ( ( ) ) ) (, ε / ( ), ( / ε ( ( ε tck Time 5 strt s ),(/ ε q end 93

94 xmple 3.7 Input ( ( ( ) ) ) ( ε Time 6 (, ε / ( ), ( / ε tck strt s ),(/ε q end 94

95 xmple 3.7 Input ( ( ( ) ) ) ε Time 7 tck (, ε / ( ), ( / ε strt s ),(/ε q end 95

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge