1. 1 Introduction to Computability Theory Lecture5: Context Free Languages Prof. Amos Israeli

2. On the last lecture we completed our study of regular languages. (There is still a lot to learn but our time is limited…). Introduction and Motivation 2

3. In our study of RL-s we Covered: 1.Motivation and definition of regular languages. 2.DFA-s, NFA-s, RE-s and their equivalence. 3.Non Regular Languages and the Pumping Lemma. Introduction and Motivation 3

4. In this lecture, we turn to Context Free Grammars and Context Free Languages. The class of Context Free Languages is an intermediate class between the class of regular languages and the class of Decidable Languages (To be defined). Introduction and Motivation 4

5. A Context Free Grammar is a “machine” that creates a language. A language created by a CF grammar is called A Context Free Language. (We will show that) The class of Context Free Languages Properly Contains the class of Regular Languages. Introduction and Motivation 5

6. Consider grammar : A CFL consists of substitution rules called Productions. The capital letters are the Variables. The other symbols are the Terminals. Context Free Grammar - Example 6

7. Consider grammar : The grammar generates the language called the language of, denoted by. Context Free Grammar - Example 7

8. Consider grammar : This is a Derivation of the word by : On each step, a single rule is activated. This mechanism is nondeterministic. Context Free Grammar - Example 8

9. This is A Parse Tree of the word by : Context Free Grammar - Example 9

10. Each internal node of the tree is associated with a single production. Context Free Grammar - Example 10

11. A Context Free Grammar is a 4-tupple where: 1. is a finite set called the variables. 2. is a finite set, disjoint from V called the terminals. 3. is a set of rules, where a rule is a variable and a string of variables and terminals, and 4. is the start variable. CF Grammar – A Formal Definition 11,

12. A word is a string of terminals. A derivation of a word w from a context Free Grammar is a sequence of strings, over, where: 1. is the start variable of G. 2. For each, is obtained by activating a single production (rule) of G on one of the variables of. A Derivation – A Formal Definition 12

13. A word w is in the Language of grammar G, denoted by, if there exists a derivation whose rightmost string is w. Thus, CF Grammar – A Formal Definition 13

14. Grammar : Rules: 1. 2. 3. Example2: Arithmetical EXPS 14

15. Derivation of by Grammar : input Example2: Arithmetical EXPS 15

16. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 16

17. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 17

18. Derivation of by Grammar : input Example2: Arithmetical EXPS 18

19. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 19

20. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 20

21. Derivation of by Grammar : input Example2: Arithmetical EXPS 21

22. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 22

23. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 23

24. Derivation of by Grammar : input Example2: Arithmetical EXPS 24

25. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 25

26. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 26

27. Derivation of by Grammar : input Example2: Arithmetical EXPS 27

28. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 28

29. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 29

30. Derivation of by Grammar : input Example2: Arithmetical EXPS 30

31. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 31

32. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 32

33. Derivation of by Grammar : input Example2: Arithmetical EXPS 33

34. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 34

35. Derivation of by Grammar : input output rule Example2: Arithmetical EXPS 35

36. Derivation of by Grammar : input Example2: Arithmetical EXPS 36

37. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 37

38. Derivation of by Grammar : output rule Example2: Arithmetical EXPS 38

39. Derivation of by Grammar : input Example2: Arithmetical EXPS 39

40. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 40

41. Derivation of by Grammar : output rule Example2: Arithmetical EXPS 41

42. Derivation of by Grammar : input Example2: Arithmetical EXPS 42

43. Derivation of by Grammar : input rule Example2: Arithmetical EXPS 43

44. Derivation of by Grammar : Example2: Arithmetical EXPS 44

45. Derivation of by Grammar : Example2: Arithmetical EXPS 45 Note: There is more than one derivation.

46. To be Demonstrated on the blackboard Example3: The Language of WF ( ) 46

47. We already saw that a word may have more then a single derivation from the same grammar. A Leftmost Derivation is a derivation in which rules are applied in order left to right. A grammar is ambiguous if it has Two parse trees. Ambiguity 47

48. Reminder: Two parse trees are equal if they are equal as trees and if all productions corresponding to inner nodes are also equal. Ambiguity 48

49. Grammar : Rules: Example4: Similar to Arith. EXPS 49,,

50. Example4: 1 st Parse Tree for______ 50

51. Example4: 2 nd Parse Tree for_____ 51

52. Note: Some ambiguous grammars may have an unambiguous equivalent grammar. But: There exist Inherently Ambiguous Grammars, i.e. an ambiguous grammar that does not have an equivalent unambiguous one. Ambiguity 52

53. Q: From a computational point of view, how strong are context free languages? A: Since the language is not regular and it is CF, we conclude that. Q: Can one prove ? A: Yes. Discussion 53

54. Q: A language is regular if it is recognized by a DFA (or NFA). Does there exist a type of machine that characterizes CFL? A: Yes, those are the Push-Down Automata (Next Lecture).. Q: Can one prove a language not to be CFL ? A: Yes, by the Pumping Lemma for CFL-s. For example: is not CFL. Discussion 54