Grammars 1 Grammars express languages Example the English

Grammars 1

Grammars express languages Example: the English language 2

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A derivation of “the boy walks”: 4

A derivation of “a dog runs”: 5

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” } 6

Notation Variable or Non-terminal Production rule Terminal 7

Another Example Grammar: Derivation of sentence : 8

Derivation of sentence : 9

Other derivations: 10

Language of the grammar 11

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

Example Grammar : 13

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

We write: Instead of: 15

In general we write: If: 16

By default: 17

Examples Grammar: 18

Another Example Grammar G: Derivations: 19

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Language of a Grammar For a grammar with start variable 21

Example For grammar G: Since: 22

A Convenient Notation 23

Linear Grammars 24

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

A Non-Linear Grammar 26

Another Linear Grammar 27

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

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

Regular Grammars 30

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

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

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