ContextFree Languages 1 Regular Languages 2 ContextFree Languages

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Context-Free Languages 1

Context-Free Languages 1

Regular Languages 2

Regular Languages 2

Context-Free Languages Regular Languages 3

Context-Free Languages Regular Languages 3

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

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

Context-Free Grammars 5

Context-Free Grammars 5

Example A context-free grammar : A derivation: 6

Example A context-free grammar : A derivation: 6

A context-free grammar : Another derivation: 7

A context-free grammar : Another derivation: 7

Example A context-free grammar : A derivation: 9

Example A context-free grammar : A derivation: 9

A context-free grammar : Another derivation: 10

A context-free grammar : Another derivation: 10

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Example A context-free grammar : A derivation: 12

Example A context-free grammar : A derivation: 12

A context-free grammar : A derivation: 13

A context-free grammar : A derivation: 13

Definition: Context-Free Grammars Grammar Variables Terminal symbols Start variables Productions of the form: is

Definition: Context-Free Grammars Grammar Variables Terminal symbols Start variables Productions of the form: is string of variables and terminals 15

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

Definition: Context-Free Languages A language is context-free if and only if there is a grammar with 16

Derivation Order Leftmost derivation: Rightmost derivation: 17

Derivation Order Leftmost derivation: Rightmost derivation: 17

Leftmost derivation: Rightmost derivation: 18

Leftmost derivation: Rightmost derivation: 18

Derivation Trees 19

Derivation Trees 19

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Derivation Tree 24

Derivation Tree 24

Derivation Tree yield 25

Derivation Tree yield 25

Partial Derivation Trees Partial derivation tree 26

Partial Derivation Trees Partial derivation tree 26

Partial derivation tree 27

Partial derivation tree 27

sentential form Partial derivation tree yield 28

sentential form Partial derivation tree yield 28

Sometimes, derivation order doesn’t matter Leftmost: Rightmost: Same derivation tree 29

Sometimes, derivation order doesn’t matter Leftmost: Rightmost: Same derivation tree 29

Ambiguity 30

Ambiguity 30

leftmost derivation 31

leftmost derivation 31

leftmost derivation 32

leftmost derivation 32

Two derivation trees 33

Two derivation trees 33

The grammar is ambiguous: string has two derivation trees 34

The grammar is ambiguous: string has two derivation trees 34

The grammar is ambiguous: string has two leftmost derivations 35

The grammar is ambiguous: string has two leftmost derivations 35

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

Definition: A context-free grammar if some string is ambiguous has: two or more derivation trees 36

In other words: A context-free grammar if some string is ambiguous has: two or

In other words: A context-free grammar if some string is ambiguous has: two or more leftmost derivations (or rightmost) 37

Why do we care about ambiguity? take 38

Why do we care about ambiguity? take 38

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Correct result: 41

Correct result: 41

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

• Ambiguity is bad for programming languages • We want to remove ambiguity 42

We fix the ambiguous grammar: New non-ambiguous grammar: 43

We fix the ambiguous grammar: New non-ambiguous grammar: 43

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Unique derivation tree 45

Unique derivation tree 45

The grammar : is non-ambiguous: Every string has a unique derivation tree 46

The grammar : is non-ambiguous: Every string has a unique derivation tree 46

Inherent Ambiguity Some context free languages have only ambiguous grammars Example: 47

Inherent Ambiguity Some context free languages have only ambiguous grammars Example: 47

The string has two derivation trees 48

The string has two derivation trees 48