1 Context-Free Languages

2 Regular Languages

3 Context-Free Languages

4 Pushdown Automata Context-Free Grammars stack automaton

5 Context-Free Grammars

6 A context-free grammar G:

7 (((( ))))

8 A context-free grammar G:

9

10 A context-free grammar G:

11 () ((( ))) (( ))

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

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

14 Derivation Order Leftmost derivation: Rightmost derivation:

15 Derivation Trees

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20 Derivation Tree

21 yield sentence Derivation Tree

22 Partial Derivation Trees Partial derivation tree

23 Partial derivation tree sentential form yield

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

25 Ambiguity

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29 The grammar is ambiguous: string has two derivation trees

30 The grammar is ambiguous: stringhas two leftmost derivations

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

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

33 Why do we care about ambiguity? take

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36 The right solution:

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

38 We fix the ambiguous grammar: New grammar:

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40 Unique derivation tree

41 The grammar Is not ambiguous

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

43 The string has two derivation trees