Formal Languages ContextFree Languages Regular Languages 2 ContextFree

Formal Languages Context-Free Languages

Regular Languages 2

Context-Free Languages Regular Languages 3

Context-Free Languages Context-Free Grammars Pushdown Automata stack automaton 4

Context-Free Grammars 5

Grammars express languages Example: the English language 6

7

A derivation of “the dog walks”: 8

A derivation of “a cat runs”: 9

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” } 10

Notation Production Rules Variable Terminal 11

Another Example Grammar: Derivation of sentence : 12

Language? 13

Grammar: Derivation of sentence : 14

Other derivations: 15

Language of the grammar 16

More Notation Grammar Set of variables Set of terminal symbols Start variable Set of Production rules 17

Example Grammar : 18

More Notation Sentential Form: A sentence that contains variables and terminals Example: Sentential Forms sentence 19

We write: Instead of: 20

In general we write: If: 21

By default: 22

Example Grammar Derivations 23

Example Grammar Derivations 24

Another Grammar Example Grammar : Derivations: 25

Language? 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

Example A context-free grammar : A derivation: 31

A context-free grammar : Another derivation: 32

Describes parentheses: (((( )))) 33

Example A context-free grammar : A derivation: 34

Language? 35

A context-free grammar : Another derivation: 36

37

Example A context-free grammar : A derivation: 38

Language? 39

A context-free grammar : A derivation: 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 43

44

Definition: Context-Free Languages A language is context-free if and only if there is a context-free grammar with 45

Derivation Order Leftmost derivation: Rightmost derivation: 46

Language? 47

Leftmost derivation: Rightmost derivation: 48

Language? 49

Derivation Trees 50

51

52

53

54

Derivation Tree 55

Derivation Tree yield 56

Partial Derivation Trees Partial derivation tree 57

Partial derivation tree 58

sentential form Partial derivation tree yield 59

Sometimes, derivation order doesn’t matter Leftmost: Rightmost: Same derivation tree 60

Ambiguity 61

leftmost derivation 62

leftmost derivation 63

Two derivation trees 64

The grammar is ambiguous: string has two derivation trees 65

The grammar is ambiguous: string has two leftmost derivations 66

Definition: A context-free grammar if some string is ambiguous has: two or more derivation trees 67

In other words: A context-free grammar if some string is ambiguous has: two or more leftmost derivations (or rightmost) 68

Why do we care about ambiguity? take 69

70

71

Correct result: 72

• Ambiguity is bad for programming languages • We want to remove ambiguity 73

We fix the ambiguous grammar: New non-ambiguous grammar: 74

75

Unique derivation tree 76

The grammar : is non-ambiguous: Every string has a unique derivation tree 77

Another Ambiguous Grammar IF_STMT if EXPR then STMT else STMT 78

If expr 1 then if expr 2 then stmt 1 else stmt 2 IF_STMT if expr 1 if then expr 2 STMT then stmt 1 else STMT else stmt 2 IF_STMT if expr 1 if then expr 2 then stmt 1 79

Inherent Ambiguity Some context free languages have only ambiguous grammars Example: 80

The string has two derivation trees 81

Ambiguity in natural language? 82