1 Context-Free Languages

2 Regular Languages

3 Context-Free Languages

4 Pushdown Automata Context-Free Grammars stack automaton

5 Context-Free Grammars

6 Grammars Grammars express languages Example: the English language

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8 A derivation of “the dog walks”:

9 A derivation of “a cat runs”:

10 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” }

11 Notation VariableTerminal Production Rules

12 Another Example Grammar: Derivation of sentence :

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 Formssentence

19 We write: Instead of:

20 In general we write: If:

21 By default:

22 Example Grammar Derivations

23 Grammar Example Derivations

24 Another Grammar Example Grammar : Derivations:

25 More Derivations

26 Language of a Grammar For a grammar with start variable : String of terminals

27 Example For grammar : Since:

28 A Convenient Notation

29 Example A context-free grammar : A derivation:

30 A context-free grammar : Another derivation:

31 (((( )))) Describes parentheses:

32 A context-free grammar : A derivation: Example

33 A context-free grammar : Another derivation:

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35 A context-free grammar : A derivation: Example

36 A context-free grammar : A derivation:

37 () ((( ))) (( )) Describes matched parentheses:

38 Definition: Context-Free Grammars Grammar Productions of the form: String of variables and terminals VariablesTerminal symbols Start variable Variable

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

40 Derivation Order Leftmost derivation: Rightmost derivation:

41 Leftmost derivation: Rightmost derivation:

42 Derivation Trees

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47 Derivation Tree

48 yield Derivation Tree

49 Partial Derivation Trees Partial derivation tree

50 Partial derivation tree

51 Partial derivation tree sentential form yield

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

53 Ambiguity

54 leftmost derivation

55 Rightmost derivation

56 Two derivation trees

57 The grammar is ambiguous: stringhas two derivation trees

58 stringhas two leftmost derivations The grammar is ambiguous:

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

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

61 Why do we care about ambiguity? take

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64 Correct result:

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