1. 1 Regular Grammars Generate Regular Languages

2. 2 Theorem Regular grammars generate exactly the class of regular languages: If is a regular grammar then is a regular language If is a regular language then there is a regular grammar with

3. 3 Proof First we prove: If is a regular grammar then is a regular language can be: Right-linear grammar or Left-linear grammar

4. 4 The case of Right-Linear Grammars Let be a right-linear grammar We will show: is regular The proof: We will construct an NFA with

5. 5 Grammar is right-linear Example:

6. 6 Construct NFA such that every state is a variable: special final state

7. 7 Add edges for each production:

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13. 13 Grammar NFA

14. 14 In General A right-linear grammar has variables: and productions: or

15. 15 We construct the NFA such that: each variable corresponds to a node:

16. 16 For each production: we add transitions and intermediate nodes ………

17. 17 For each production: we add transitions and intermediate nodes ………

18. 18 Resulting NFA looks like this:

19. 19 Now, we need to show:

20. 20 The Case: Take We will show: there is a path with label in from state to state …………

21. 21 strings Grammar looks like:

22. 22

23. 23 ……

24. 24 ……

25. 25 ……

26. 26 Since: We have: ……

27. 27 Since: We have: …… and

28. 28 The Case: Take We will show that in

29. 29 Since there is a path ……

30. 30 Write: There is a path ……

31. 31 Since: This derivation is possible ……

32. 32 Since: We have:

33. 33 The Case of Left-Linear Grammars Let be a left-linear grammar We will show: is regular The proof: We will construct a right-linear grammar with

34. 34 Since is left-linear grammar the productions look like:

35. 35 Construct right-linear grammar In :

36. 36 In :

37. 37 It is easy to see that: Since is right-linear, we have: is regular language is regular language (homework) is regular language

38. 38 Proof - Part 2 Now we will prove: If is a regular language then there is a regular grammar with Proof outline: we will take an NFA for and convert it to a regular grammar

39. 39 Since is regular There is an NFA such that Example:

40. 40 Convert to a right-linear grammar

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43. 43 We can generalize this process for any regular language : For any regular language we obtain an right-linear grammar with

44. 44 Since is right-linear grammar is also a regular grammar with