1. 1 Simplifications of Context-Free Grammars

2. 2 A Substitution Rule substitute B equivalent grammar

3. 3 In general: substitute B equivalent grammar

4. 4 Useless Productions

5. 5 never terminates...

6. 6 Useless Production

7. 7 Another grammar:

8. 8 Not reachable from S Useless Production

9. 9 In general: If Then is uselessOtherwise is useful

10. 10 Removing Useless Productions Example Grammar

11. 11 First:find all variables that produce strings with only terminals Round 1: Round 2:

12. 12 Keep only the variables that produce terminal symbols

13. 13 Second: Find all variables reachable from S Dependency Graph not reachable

14. 14 Keep only the variables reachable from S Final Grammar

15. 15 Nullable Variables Nullable Variable:

16. 16 Removing Nullable Variables Example Grammar: Nullable variable

17. 17 Substitute Final Grammar

18. 18 Unit-Productions Unit Production:

19. 19 Removing Unit Productions Observation: Is removed immediately

20. 20 Example Grammar:

21. 21 substitute

22. 22

23. 23 substitute

24. 24 Remove repeated productions Final grammar

25. 25 Removing All 1. Remove Nullable Variables 2. Remove Unit-Productions 3. Remove Useless Variables

26. 26 Normal Forms for Context-free Grammars

27. 27 Chomsky Normal Form All productions have form: variable OR terminal

28. 28 Examples: Chomsky Normal Form NOTChomsky Normal Form

29. 29 Convertion to Chomsky Normal Form Example: Chomsky Normal Form NOT

30. 30 Introduce variables for terminals:

31. 31 Introduce intermediate variables:

32. 32 Final grammar in Chomsky Normal Form: Initial grammar

33. 33 From any context-free grammar Not in Chomsky Normal Form We can obtain: An equivalent grammar in Chomsky Normal Form In general:

34. 34 The Procedure: Remove nullable variables unit productions

35. 35 For every symbol : Add new variable In productions, replace with Add production

36. 36 Replace any production: with: New variables:

37. 37 Theorem: For any context-free grammar there is an equivalent grammar in Chomsky Normal Form

38. 38 Greinbach Normal Form All rules have form: symbol variables

39. 39 Examples: Greinbach Normal Form Greinbach Normal Form NOT

40. 40 Conversion to Greinbach Normal Form Greinbach Normal Form

41. 41 Theorem: For any context-free grammar there is an equivalent grammar in Greinbach Normal Form

42. 42 An Application of Chomsky Normal Forms

43. 43 The CYK Membership Algorithm Inputs: Grammar in Chomsky Normal Form String w Output: find if

44. 44 Algorithm: Input Examples Grammar : String :

45. 45

46. 46

47. 47

48. 48

49. 49 Therefore: Time Complexity: It can be easily converted to a parser