Grammars Costas Busch RPI 1 Grammars express languages

Grammars Costas Busch - RPI 1

Grammars express languages Example: the English language Costas Busch - RPI 2

Costas Busch - RPI 3

A derivation of “the dog walks”: Costas Busch - RPI 4

A derivation of “a cat runs”: Costas Busch - RPI 5

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” } Costas Busch - RPI 6

Notation Production Rules Variable Terminal Costas Busch - RPI 7

Another Example Grammar: Derivation of sentence Costas Busch - RPI : 8

Grammar: Derivation of sentence Costas Busch - RPI : 9

Other derivations: Costas Busch - RPI 10

Language of the grammar Costas Busch - RPI 11

More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules Costas Busch - RPI 12

Example Grammar : Costas Busch - RPI 13

More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Forms Costas Busch - RPI sentence 14

We write: Instead of: Costas Busch - RPI 15

In general we write: If: Costas Busch - RPI 16

By default: Costas Busch - RPI 17

Example Grammar Derivations Costas Busch - RPI 18

Example Grammar Derivations Costas Busch - RPI 19

Another Grammar Example Grammar : Derivations: Costas Busch - RPI 20

More Derivations Costas Busch - RPI 21

Language of a Grammar For a grammar with start variable : String of terminals Costas Busch - RPI 22

Example For grammar : Since: Costas Busch - RPI 23

A Convenient Notation Costas Busch - RPI 24

Linear Grammars Costas Busch - RPI 25

Linear Grammars with at most one variable at the right side of a production Examples: Costas Busch - RPI 26

A Non-Linear Grammar Number of : in string Costas Busch - RPI 27

Another Linear Grammar : Costas Busch - RPI 28

Right-Linear Grammars All productions have form: or Example: string of terminals Costas Busch - RPI 29

Left-Linear Grammars All productions have form: or Example: string of terminals Costas Busch - RPI 30

Regular Grammars Costas Busch - RPI 31

Regular Grammars A regular grammar is any right-linear or left-linear grammar Examples: Costas Busch - RPI 32

Observation Regular grammars generate regular languages Examples: Costas Busch - RPI 33

Regular Grammars Generate Regular Languages Costas Busch - RPI 34

Theorem Languages Generated by Regular Grammars Costas Busch - RPI Regular Languages 35

Theorem - Part 1 Languages Generated by Regular Grammars Regular Languages Any regular grammar generates a regular language Costas Busch - RPI 36

Theorem - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by a regular grammar Costas Busch - RPI 37

Proof – Part 1 Languages Generated by Regular Grammars Regular Languages The language generated by any regular grammar is regular Costas Busch - RPI 38

The case of Right-Linear Grammars Let be a right-linear grammar We will prove: Proof idea: is regular We will construct NFA with Costas Busch - RPI 39

Grammar is right-linear Example: Costas Busch - RPI 40

Construct NFA such that every state is a grammar variable: special final state Costas Busch - RPI 41

Add edges for each production: Costas Busch - RPI 42

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NFA Grammar Costas Busch - RPI 48

In General A right-linear grammar has variables: and productions: or Costas Busch - RPI 49

We construct the NFA each variable such that: corresponds to a node: special final state Costas Busch - RPI 50

For each production: we add transitions and intermediate nodes ……… Costas Busch - RPI 51

For each production: we add transitions and intermediate nodes ……… Costas Busch - RPI 52

Resulting NFA looks like this: It holds that: Costas Busch - RPI 53

The case of Left-Linear Grammars Let be a left-linear grammar We will prove: is regular Proof idea: We will construct a right-linear grammar with Costas Busch - RPI 54

Since is left-linear grammar the productions look like: Costas Busch - RPI 55

Construct right-linear grammar Left linear Right linear Costas Busch - RPI 56

Construct right-linear grammar Left linear Right linear Costas Busch - RPI 57

It is easy to see that: Since Regular Language is right-linear, we have: Regular Language Costas Busch - RPI Regular Language 58

Proof - Part 2 Languages Generated by Regular Grammars Regular Languages Any regular language is generated by some regular grammar Costas Busch - RPI 59

Any regular language is generated by some regular grammar Proof idea: Let be the NFA with Construct from such that . a regular grammar Costas Busch - RPI 60