Regular Grammars Generate Regular Languages

1. Regular Grammars GenerateRegular Languages

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

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. Grammar is right-linear Example:

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

7. Add edges for each production:

13. NFA Grammar

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

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

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

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

18. Resulting NFA looks like this:

19. Now, we need to show:

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

21. Grammar looks like: strings

23. ……

24. …… ……

25. …… …… …… ……

26. Since: • We have: …… …… …… ……

27. Since: • We have: • We have: and ……

28. The Case: Take We will show that in

29. Since • there is a path ……

30. Write: • There is a path …… …… …… ……

31. Since: • This derivation is possible …… …… …… ……

32. Since: • We have: • We have:

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

35. Construct right-linear grammar In : In :

36. In : In :

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

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

40. Convert to a right-linear grammar

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

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