1. 1 Grammars

2. 2 Grammars express languages Example: the English language

3. 3

4. 4 A derivation of “the boy walks”:

5. 5 A derivation of “a dog runs”:

6. 6 Language of the grammar: L = { “a boy runs”, “a boy walks”, “the boy runs”, “the boy walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

7. 7 Notation Variable or Non-terminal Terminal Production rule

8. 8 Another Example Grammar: Derivation of sentence :

9. 9

10. 10 Other derivations:

11. 11 Language of the grammar

12. 12 More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules

13. 13 Example Grammar :

14. 14 More Notation Sentential Form: A sentence that contains Variables and terminals Example: Sentential Formssentence

15. 15 We write: Instead of:

16. 16 In general we write: If:

17. 17 By default:

18. 18 Examples Grammar:

19. 19 Another Example Grammar G: Derivations:

20. 20

21. 21 Language of a Grammar For a grammar with start variable

22. 22 Example For grammar G: Since:

23. 23 A Convenient Notation

24. 24 Linear Grammars

25. 25 A Linear Grammar Grammars with at most one variable on the right side of a production Examples:

26. 26 A Non-Linear Grammar Grammar

27. 27 Another Linear Grammar Grammar

28. 28 Right-Linear Grammars All productions have the form: Example: or

29. 29 Left-Linear Grammars All productions have form: Example: or

30. 30 Regular Grammars

31. 31 Regular Grammars Definition: A regular grammar is any right-linear or left-linear grammar Examples:

32. 32 Observation Regular grammars generate regular languages Examples: Grammar G:

33. 33 Theorem A Language is regular if and only if there is a grammar such that