1. Context-Free Languages Hinrich Schütze CIS, LMU, 2013-11-25 Slides based on RPI CSCI 2400 Thanks to Costas Busch

2. Take-away Definition context-free grammar Definition context-free language Derivation, sentential form, sentence Derivation trees Ambiguity Context-free grammars for natural language 2

3. Terminology For our purposes in this class: Context free grammar = Constituency grammar = Phrase structure grammar 3

4. 4 Grammars Grammars express languages Example: the English language

5. 5

6. 6 A derivation of “the dog walks”:

7. 7 A derivation of “a cat runs”:

8. 8 Language of the grammar: L = { “a cat runs”, “a cat walks”, “the cat runs”, “the cat walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

9. 9 Notation VariableTerminal Production Rules

10. 10 Another Example Grammar: Derivation of sentence :

11. 11 Language?

12. 12 Grammar: Derivation of sentence :

13. 13 Other derivations:

14. 14 Language of the grammar

15. 15 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

16. 16 Example Grammar :

17. 17 More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Formssentence

18. 18 We write: Instead of:

19. 19 In general we write: If:

20. 20 By default:

21. 21 Example Grammar Derivations

22. 22 Grammar Example Derivations

23. 23 Another Grammar Example Grammar :

24. 24 Language?

25. 25 Grammar : Derivations:

26. 26 More Derivations

27. 27 Language of a Grammar For a grammar with start variable : String of terminals

28. 28 Example For grammar : Since:

29. 29 A Convenient Notation

30. 30 Revisit first grammar A context-free grammar : A derivation:

31. 31 A context-free grammar : Another derivation:

32. 32 (((( )))) Describes parentheses:

33. 33 A context-free grammar : Example

34. 34 Language?

35. 35 A context-free grammar : Another derivation:

36. 36

37. 37 A context-free grammar : Example

38. 38 Language?

39. 39 A context-free grammar : Two derivations:

40. 40 Interpretation?

41. 41 () ((( ))) (( )) Describes matched parentheses:

42. 42 Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

43. 43

44. 44 Definition: Context-Free Languages A language is context-free if and only if there is a context-free grammar with

45. 45 Derivation Order Leftmost derivation: Rightmost derivation:

46. 46 Language?

47. 47 Leftmost derivation: Rightmost derivation:

48. 48 Language?

49. 49 Derivation Trees

50. 50

51. 51

52. 52

53. 53

54. 54 Derivation Tree

55. 55 yield Derivation Tree

56. 56 Partial Derivation Trees Partial derivation tree

57. 57 Partial derivation tree

58. 58 Partial derivation tree sentential form yield

59. 59 Same derivation tree Sometimes, derivation order doesn’t matter Leftmost: Rightmost:

60. 60 Ambiguity

61. 61 leftmost derivation

62. 62 leftmost derivation

63. 63 Two derivation trees

64. 64 The grammar is ambiguous: stringhas two derivation trees

65. 65 stringhas two leftmost derivations The grammar is ambiguous:

66. 66 Definition: A context-free grammar is ambiguous if some string has: two or more derivation trees

67. 67 In other words: A context-free grammar is ambiguous if some string has: two or more leftmost derivations (or rightmost)

68. 68 Why do we care about ambiguity? take

69. 69

70. 70

71. 71 Correct result:

72. 72 We want to remove ambiguity Ambiguity is bad for programming languages

73. 73 We fix the ambiguous grammar: New non-ambiguous grammar:

74. 74

75. 75 Unique derivation tree

76. 76 The grammar : is non-ambiguous: Every string has a unique derivation tree

77. 77 Another Ambiguous Grammar IF_STMTif EXPR then STMT if EXPR then STMT else STMT Ambiguity?

78. 78 If expr1 then if expr2 then stmt1 else stmt2 IF_STMT expr1then elseifexpr2then STMT stmt1 if IF_STMT expr1thenelse ifexpr2then STMTstmt2 if stmt1 stmt2

79. 79 Inherent Ambiguity Some context free languages have only ambiguous grammars Example:

80. 80 The string has two derivation trees

81. 81 Ambiguity in natural language?

82. Take-away Definition context-free grammar Definition context-free language Derivation, sentential form, sentence Derivation trees Ambiguity Context-free grammars for natural language 82