1 Context-Free Languages

2 Regular Languages

3 Context-Free Languages

4 Pushdown Automata Context-Free Grammars stack automaton

5 Context-Free Grammars

6 Example A context-free grammar : A derivation:

7 A context-free grammar : Another derivation:

8 (((( ))))

9 A context-free grammar : A derivation: Example

10 A context-free grammar : Another derivation:

11

12 A context-free grammar : A derivation: Example

13 A context-free grammar : A derivation:

14 () ((( ))) (( ))

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

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

17 Derivation Order Leftmost derivation: Rightmost derivation:

18 Leftmost derivation: Rightmost derivation:

19 Derivation Trees

20

21

22

23

24 Derivation Tree

25 yield Derivation Tree

26 Partial Derivation Trees Partial derivation tree

27 Partial derivation tree

28 Partial derivation tree sentential form yield

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

30 Ambiguity

31 leftmost derivation

32 leftmost derivation

33 Two derivation trees

34 The grammar is ambiguous: stringhas two derivation trees

35 stringhas two leftmost derivations The grammar is ambiguous:

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

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

38 Why do we care about ambiguity? take

39

40

41 Correct result:

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

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

44

45 Unique derivation tree

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

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

48 The string has two derivation trees