ContextFree Languages 1 Regular Languages 2 ContextFree Languages

Context-Free Languages 1

Regular Languages 2

Context-Free Languages Regular Languages 3

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

Context-Free Grammars 5

A context-free grammar G: 6

A context-free grammar G: 8

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A context-free grammar G: 10

Definition of Context-Free Grammars Any grammar G=(V, T, S, P) V: variables T: terminal symbols S: start variable P: production rules of the form : string of variables and terminals 12

Definition of Context-Free Languages A language L is context-free if and only if there is a grammar G with L = L(G) 13

Derivation Order Leftmost derivation: Rightmost derivation: 14

Derivation Trees 15

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Derivation Tree 20

sentence Derivation Tree yield 21

Partial Derivation Trees Partial derivation tree 22

sentential form Partial derivation tree yield 23

Often, derivation order doesn’t matter Same Derivation Tree 24

Ambiguity 25

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The grammar is ambiguous: string has two derivation trees 29

The grammar is ambiguous: string has two leftmost derivations 30

In general: A context-free grammar if some string is ambiguous has: two or more derivation trees 31

In other words: A context-free grammar if some string is ambiguous has: two or more leftmost derivations rightmost 32

Why do we care about ambiguity? take 33

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The right solution: 36

Therefore: Ambiguity is bad for programming languages We want to remove the ambiguity 37

We fix the ambiguous grammar: New grammar: 38

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Unique derivation tree 40

The grammar Is not ambiguous 41

Inherent Ambiguity Some context free languages have only ambiguous grammars Example: 42

The string has two derivation trees 43