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- 1. Two Non Context-Free Languages (We will prove them later if we have time)
- 2. Regular languages Context-free languages Non Context-free languages
- 3. Positive Properties of Context-Free languages
- 4. Context-free languages are closed under: Union is context free is context free is context-free Union
- 5. Example Union Language Grammar
- 6. In general: The grammar of the union has new start variable and additional production For context-free languages with context-free grammars and start variables
- 7. Context-free languages are closed under: Concatenation is context free is context free is context-free Concatenation
- 8. In general: The grammar of the concatenation has new start variable and additional production For context-free languages with context-free grammars and start variables
- 9. Example Concatenation Language Grammar
- 10. Context-free languages are closed under: Star-operation is context free is context-free Star Operation
- 11. In general: The grammar of the star operation has new start variable and additional production For context-free language with context-free grammar and start variable
- 12. Example Language Grammar Star Operation
- 13. Context-free languages are closed under: reversal is context free is context-free Reversal
- 14. In general: The grammar of the reversal operation has production For context-free language with context-free grammar and every production
- 15. Example Language Grammar Reversal Operation
- 16. Negative Properties of Context-Free Languages
- 17. Context-free languages are not closed under: intersection is context free is context free not necessarily context-free Intersection
- 18. Example Context-free: Context-free: NOT context-free Intersection
- 19. Context-free languages are not closed under: complement is context free not necessarily context-free Complement
- 20. NOT context-free Example Context-free: Context-free: Complement
- 21. Is the complement of context free? Hw #1 p.219 For n a ( w ) > n b ( w ) : | cS 1 | S 1 c | cE | Ec This language is not context free (ref. Ex.8.8 p.217), but its complement is CF.
- 22. Intersection of Context-free languages and Regular Languages
- 23. The intersection of a context-free language and a regular language is a context-free language context free regular context-free
- 24. for for NPDA DFA Construct a new NPDA machine that accepts Machine Machine context-free regular simulates in parallel and i.e. whenever a symbol is read from input, M simultaneously executes the moves of M 1 and M 2
- 25. transition transition NPDA DFA transition NPDA
- 26. initial state initial state NPDA DFA Initial state NPDA
- 27. final state final states NPDA DFA final states NPDA
- 28. simulates in parallel and accepts string if and only if accepts string and accepts string
- 29. Therefore: (since is NPDA) is context-free is context-free
- 30. Applications of Regular Closure HW # 1, 7, 12, 17, 18, 21 ( 8.2 p.220)
- 31. The intersection of a context-free language and a regular language is a context-free language context free regular context-free Regular Closure
- 32. An Application of Regular Closure Prove that: is context-free
- 33. We know: is context-free
- 34. is regular is regular We also know:
- 35. regular context-free is context-free is context-free (regular closure)
- 36. Another Application of Regular Closure Prove that: is not context-free
- 37. context-free regular context-free If is context-free Then Impossible!!! (regular closure) Therefore, is not context free
- 38. Consider set difference: L 1 - L 2 Is it closed for context free languages? It is context free if L 2 is regular.

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