1 Regular Grammars Generate Regular 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:

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13 Grammar NFA

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 strings Grammar looks like:

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23 ……

24 ……

25 ……

26 Since: We have: ……

27 Since: 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:

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 :

36 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

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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