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

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

41 Derivation Order Leftmost derivation: Rightmost derivation:

42 Leftmost derivation: Rightmost derivation:

43 Derivation Trees

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48 Derivation Tree

49 yield Derivation Tree

50 Ambiguity

51 leftmost derivation

52 leftmost derivation

53 Two derivation trees

54 The grammar is ambiguous: stringhas two derivation trees

55 stringhas two leftmost derivations The grammar is ambiguous:

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

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

58 Why do we care about ambiguity? take

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61 Correct result:

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

63 Another Ambiguous Grammar IF_STMTif EXPR then STMT if EXPR then STMT else STMT

64 If expr1 then if expr2 then stmt1 else stmt2 IF_STMT expr1then elseifexpr2then STMT stmt1 if IF_STMT expr1thenelse ifexpr2then STMTstmt2 if stmt1 stmt2

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

66 The string has two derivation trees