1. Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

2. Example Grammar :

3. Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

4. Leftmost derivation: Rightmost derivation:

9. Derivation Tree

10. yield Derivation Tree

11. Simplifications of Context-Free Grammars

12. Example

13. Normal Forms for Context-free Grammars

14. Chomsky Normal Form Each productions has form: variable or terminal

15. Introduce variables for terminals:

16. Introduce intermediate variable:

18. Final grammar in Chomsky Normal Form: Initial grammar

19. Then, for every symbol : In productions: replace with Add production New variable:

20. Replace any production with New intermediate variables:

21. Greinbach Normal Form All productions have form: symbolvariables

22. Pushdown Automata PDAs

23. Pushdown Automaton -- PDA Input String Stack States

24. The States Input symbol Pop symbol Push symbol

25. top input stack Replace

26. Push top input stack

27. Pop top input stack

28. No Change top input stack

29. Pop top input stack Empty Stack empty The automaton HALTS No possible transition after

30. Pop top input stack A Possible Transition

31. Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions

32. Formalities for PDAs

33. Transition function:

35. Formal Definition Pushdown Automaton (PDA) States Input alphabet Stack alphabet Transition function Final states Stack start symbol Initial state

36. Instantaneous Description Current state Remaining input Current stack contents

37. A computation:

38. Deterministic PDAs - DPDAs

39. Deterministic PDA: DPDA Allowed transitions: (deterministic choices)

40. Allowed transitions: (deterministic choices)

41. Not allowed: (non deterministic choices)

42. PDAs Have More Power than DPDAs

43. Positive Properties of Context-Free languages

44. Context-free languages are closed under: Union is context free is context-free Union

45. Context-free languages are closed under: Concatenation is context free is context-free Concatenation

46. Context-free languages are closed under: Star-operation is context freeis context-free Star Operation

47. Negative Properties of Context-Free Languages

48. Context-free languages are not closed under: intersection is context free not necessarily context-free Intersection

49. Context-free languages are not closed under: complement is context freenot necessarily context-free Complement

50. Intersection of Context-free languages and Regular Languages

51. The intersection of a context-free language and a regular language is a context-free language context free regular context-free

52. An Application of Regular Closure Prove that: is context-free

53. Decidable Properties of Context-Free Languages

54. Membership Question: for context-free grammar find if string Membership Algorithms: Parsers Exhaustive search parser CYK parsing algorithm

55. Empty Language Question: for context-free grammar find if Algorithm: 1.Remove useless variables 2.Check if start variable is useless

56. Infinite Language Question: for context-free grammar find if is infinite Algorithm: 1. Remove useless variables 3. Create dependency graph for variables 4. If there is a loop in the dependency graph then the language is infinite 2. Remove unit and productions