1 Grammars

2 Grammars express languages Example: the English language

3

4 A derivation of “the boy walks”:

5 A derivation of “a dog runs”:

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 Notation Variable or Non-terminal Terminal Production rule

8 Another Example Grammar: Derivation of sentence :

9

10 Other derivations:

11 Language of the grammar

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

13 Example Grammar :

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

15 We write: Instead of:

16 In general we write: If:

17 By default:

18 Examples Grammar:

19 Another Example Grammar G: Derivations:

20

21 Language of a Grammar For a grammar with start variable

22 Example For grammar G: Since:

23 A Convenient Notation

24 Linear Grammars

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

26 A Non-Linear Grammar Grammar

27 Another Linear Grammar Grammar

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

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

30 Regular Grammars

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

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

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