Context-Free Languages Hinrich Sch ütze CIS, LMU, 2013-11-25 Slides based on RPI CSCI 2400

1. Context-Free Languages Hinrich Schütze CIS, LMU, 2013-11-25 Slides based on RPI CSCI 2400 Thanks to Costas Busch

2. Take-away • Definition context-free grammar • Definition context-free language • Derivation, sentential form, sentence • Derivation trees • Ambiguity • Context-free grammars for natural language

3. Terminology • For our purposes in this class: • Context free grammar • = • Constituency grammar • = • Phrase structure grammar

4. Grammars • Grammars express languages • Example: the English language

6. A derivation of “the dog walks”:

7. A derivation of “a cat runs”:

8. Language of the grammar: L = { “a cat runs”, “a cat walks”, “the cat runs”, “the cat walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

9. Notation Production Rules Variable Terminal

10. Another Example • Grammar: • Derivation of sentence :

11. Language?

12. Grammar: • Derivation of sentence :

13. Other derivations:

14. Language of the grammar

15. More Notation • Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

16. Example • Grammar :

17. More Notation • Sentential Form: • A sentence that contains • variables and terminals • Example: Sentential Forms sentence

18. We write: • Instead of:

19. In general we write: • If:

20. By default:

21. Example Grammar Derivations

22. Example Grammar Derivations

23. Another Grammar Example • Grammar :

24. Language?

25. Grammar : Derivations:

26. More Derivations

27. Language of a Grammar • For a grammar • with start variable : String of terminals

28. Example • For grammar : Since:

29. A Convenient Notation

30. Revisit first grammar A context-free grammar : A derivation:

31. A context-free grammar : Another derivation:

32. (((( )))) Describes parentheses:

33. Example A context-free grammar :

34. Language?

35. A context-free grammar : Another derivation:

37. Example A context-free grammar :

38. Language?

39. A context-free grammar : Two derivations:

40. Interpretation?

41. Describes matched parentheses: () ((( ))) (( ))

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

44. Definition: Context-Free Languages • A language is context-free • if and only if • there is a context-freegrammar • with

45. Leftmost derivation: Rightmost derivation: Derivation Order

46. Language?

47. Leftmost derivation: Rightmost derivation:

48. Language?

49. Derivation Trees