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Simplifications of Context-Free Grammars

Simplifications of Context-Free Grammars. A Substitution Rule. equivalent grammar. substitute B. In general:. substitute B. equivalent grammar. Useless Productions. never terminates. Useless Production. Another grammar:. Not reachable from S. Useless Production. In general:. If.

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Simplifications of Context-Free Grammars

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  1. Simplifications of Context-Free Grammars

  2. A Substitution Rule equivalent grammar substitute B

  3. In general: substitute B equivalent grammar

  4. Useless Productions

  5. never terminates...

  6. Useless Production

  7. Another grammar:

  8. Not reachable from S Useless Production

  9. In general: If Then is useful Otherwise is useless

  10. Removing Useless Productions Example Grammar

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

  12. Keep only the variables that produce terminal symbols

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

  14. Keep only the variables reachable from S Final Grammar

  15. Nullable Variables Nullable Variable:

  16. Removing Nullable Variables Example Grammar: Nullable variable

  17. Substitute Final Grammar

  18. Unit-Productions Unit Production:

  19. Removing Unit Productions Observation: Is removed immediately

  20. Example Grammar:

  21. substitute

  22. substitute

  23. Remove repeated productions Final grammar

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

  25. Normal FormsforContext-free Grammars

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

  27. Examples: Chomsky Normal Form NOT Chomsky Normal Form

  28. Convertion to Chomsky Normal Form • Example: NOT Chomsky Normal Form

  29. Introduce variables for terminals:

  30. Introduce intermediate variables:

  31. Final grammar in Chomsky Normal Form: Initial grammar

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

  33. The Procedure: • Remove • nullable variables • unit productions

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

  35. Replace any production: with: New variables:

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

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

  38. Examples: NOT Greinbach Normal Form Greinbach Normal Form

  39. Conversion to Greinbach Normal Form Greinbach Normal Form

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

  41. An ApplicationofChomsky Normal Forms

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

  43. Algorithm: Input Examples Grammar : String :

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

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