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Simplifications of Context-Free Grammars

Learn about simplifications of context-free grammars by Dr. Sandip Mal, including useless productions, nullable variables, unit productions, Chomsky normal form, and Greibach normal form. Discover conversion techniques and the importance of different grammar forms.

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Simplifications of Context-Free Grammars

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  1. Simplifications of Context-Free Grammars Dr. Sandip Mal

  2. Dr. Sandip Mal

  3. Useless Production Some derivations never terminate... Useless Productions Dr. Sandip Mal

  4. Useless Production Another grammar: Not reachable from S Dr. Sandip Mal

  5. Nullable Variables Nullable Variable: Example: Nullable variable Dr. Sandip Mal

  6. Unit-Productions Unit Production: (a single variable in both sides) Example: Unit Productions Dr. Sandip Mal

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  15. Normal FormsforContext-free Grammars Dr. Sandip Mal

  16. Chomsky Normal Form Each productions has form: or variable variable terminal Dr. Sandip Mal

  17. Examples: Chomsky Normal Form Not Chomsky Normal Form Dr. Sandip Mal

  18. Conversion to Chomsky Normal Form • Example: Not Chomsky Normal Form We will convert it to Chomsky Normal Form Dr. Sandip Mal

  19. Introduce new variables for the terminals: Dr. Sandip Mal

  20. Introduce new intermediate variable to break first production: Dr. Sandip Mal

  21. Introduce intermediate variable: Dr. Sandip Mal

  22. Final grammar in Chomsky Normal Form: Initial grammar Dr. Sandip Mal

  23. In general: From any context-free grammar (which doesn’t produce ) not in Chomsky Normal Form we can obtain: an equivalent grammar in Chomsky Normal Form Dr. Sandip Mal

  24. The Procedure • First remove: • Nullable variables • Unit productions (Useless variables optional) Dr. Sandip Mal

  25. Then, for every symbol : New variable: Add production In productions with length at least 2 replace with Productions of form do not need to change! Dr. Sandip Mal

  26. Replace any production with New intermediate variables: Dr. Sandip Mal

  27. Observations • Chomsky normal forms are good • for parsing and proving theorems • It is easy to find the Chomsky normal • form for any context-free grammar Dr. Sandip Mal

  28. Greinbach Normal Form All productions have form: symbol variables Dr. Sandip Mal

  29. Examples: Greinbach Normal Form Not Greinbach Normal Form Dr. Sandip Mal

  30. Conversion to Greinbach Normal Form: Greinbach Normal Form Dr. Sandip Mal

  31. Observations • Greinbach normal forms are very good • for parsing strings (better than Chomsky Normal Forms) • However, it is difficult to find the • Greinbach normal of a grammar Dr. Sandip Mal

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