1 Normal Forms for Context-free Grammars

2 Chomsky Normal Form All productions have form: variable and terminal

3 Examples: Not Chomsky Normal Form Chomsky Normal Form

4 Convertion to Chomsky Normal Form Example: Not Chomsky Normal Form

5 Introduce variables for terminals:

6 Introduce intermediate variable:

7

8 Final grammar in Chomsky Normal Form: Initial grammar

9 From any context-free grammar not in Chomsky Normal Form we can obtain: An equivalent grammar in Chomsky Normal Form In general:

10 The Procedure First remove: Nullable variables Unit productions

11 For every symbol : In productions: replace with Add production New variable:

12 Replace any production with New intermediate variables:

13 Theorem: For any context-free grammar there is an equivalent grammar in Chomsky Normal Form

14 Observations Chomsky normal forms are good for parsing and proving theorems It is very easy to find the Chomsky normal form of any context-free grammar

15 Greinbach Normal Form All productions have form: symbolvariables

16 Examples: Greinbach Normal Form Not Greinbach Normal Form

17 Conversion to Greinbach Normal Form: Greinbach Normal Form

18 Theorem: For any context-free grammar there is an equivalent grammar in Greinbach Normal Form

19 Observations Greinbach normal forms are very good for parsing It is hard to find the Greinbach normal form of any context-free grammar

20 An Application of Chomsky Normal Forms

21 The CYK Membership Algorithm Input: Grammar in Chomsky Normal Form String Output: find if

22 The Algorithm Grammar : String : Input example:

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27 Therefore: Time Complexity: The CYK algorithm can be easily converted to a parser Observation:

28 Pushdown Automata PDAs

29 Pushdown Automaton -- PDA Input String Stack States

30 Initial Stack Symbol Stack bottom special symbol

31 The States Input symbol Pop symbol Push symbol

32 top input stack Replace

33 Push top input stack

34 Pop top input stack

35 No Change top input stack

36 Non-Determinism These are allowed transitions in a Non-deterministic PDA (NPDA)

37 NPDA: Non-Deterministic PDA Example:

38 Execution Example: Input current state Time 0 Stack

39 Input Time 1 Stack

40 Input Stack Time 2

41 Input Stack Time 3

42 Input Stack Time 4

43 Input Stack Time 5

44 Input Stack Time 6

45 Input Stack Time 7

46 Input Time 8 accept Stack

47 A string is accepted if there is a computation such that: All the input is consumed The last state is a final state At the end of the computation, we do not care about the stack contents

48 The input string is accepted by the NPDA:

49 is the language accepted by the NPDA: In general,

50 Another NPDA example NPDA

51 Execution Example: Input Time 0 Stack

52 Input Time 1 Stack

53 Input Time 2 Stack

54 Input Time 3 Stack Guess the middle of string

55 Input Time 4 Stack

56 Input Time 5 Stack

57 Input Time 6 Stack accept

58 Rejection Example: Input Time 0 Stack

59 Input Time 1 Stack

60 Input Time 2 Stack

61 Input Time 3 Stack Guess the middle of string

62 Input Time 4 Stack

63 Input Time 5 Stack There is no possible transition. Input is not consumed

64 Another computation on same string: Input Time 0 Stack

65 Input Time 1 Stack

66 Input Time 2 Stack

67 Input Time 3 Stack

68 Input Time 4 Stack

69 Input Time 5 Stack No final state is reached

70 There is no computation that accepts string