1. CSCI 3130: Formal languages and automata theory Tutorial 2 Chin

2. Reminder Homework 1 is due at 23:59, today! Subscribe Moodle for tutorial information My office hour: 11:00 – 12:00, Tuesday

3. Outline Regular expression Closure properties In this (only this) tutorial. Assume  = {0, 1}.

4. Regular Expressions The symbols  and  are regular expressions Every a in  is a regular expression If R and S are regular expressions, so are R+S, RS and R* Remember for R*, * could be 0

5. Regular Expressions What are their regular expressions? 1. L 1 = {w : w is a string of even length} ((0 + 1)(0 + 1))* 2. L 2 = {w : w begins with 0 or ends with 0} 0(0 + 1)* + (0 + 1)*0 3. L 3 = {w : w contains the pattern 01 at least once} (0 + 1)* 01 (0 + 1)*

6. Regular Expressions What are their regular expressions? 1. L 1 = {w : w is a string of odd length} 2. L 2 = {w : w begins with 0 or ends with 0 but not both} 3. L 3 = {w : w contains the pattern 01 at least twice} 4. L 4 = {w : w contains less 1s than 0s and at most two 0s}

7. Regular Expressions 1. L 1 = {w : w is a string of odd length} odd means of length 2n + 1 string of length 2n = ((0 + 1)(0 + 1))* (0 + 1)((0 + 1)(0 + 1))*

8. Regular Expressions 2. L 2 = {w : w begins with 0 or ends with 0 but not both} w must be of length at least 2 Either starts with 0 and ends with 1, or starts with 1 and ends with 0 0(0 + 1)*1 + 1(0 + 1)*0

9. Regular Expressions 3. L 3 = {w : w contains the pattern 01 at least twice} w must contain 01 twice in the expression can be anything before, after, or between the two 01s (0 + 1)* 01 (0 + 1)* 01 (0 + 1)*

10. Regular Expressions 4. L 4 = {w : w contains less 1s than 0s and at most two 0s} There can be at most two 0s Enumerate all possible cases: 0, 00, 001, 010, 100 OR them using + 0 + 00 + 001 + 010 + 100

11. Regular Expressions What do these regular expressions represent? 1. 0(0 + 1)*0 + 1(0 + 1)*1 + 0 + 1 2. 1*(011*)* 3. (0*10*10*)* 4. ((0 + 1)1)*

12. Regular Expression 1. 0(0 + 1)*0 + 1(0 + 1)*1 + 0 + 1 First alphabet must be equal to the last alphabet

13. Regular Expression 2. 1*(011*)* 011* = start with 0 then followed by at least one 1 (011*)* = start with 0 and every 0 must be followed by at least one 1 1* at the beginning = possibly start with 1s Every 0 is followed by at least one 1

14. Regular Expression 3. (0*10*10*)* There are two 1s inside the brackets 0*10*10* = there can be 0 before, after, or between the two 1s = there are two 1s (0*10*10*)* = there are 2n 1s All strings that contain even number of 1s

15. Regular Expression 4. ((0 + 1)1)* (0 + 1)1 = X1, X = 0/1 ((0 + 1)1)* = strings of the form X1X1…X1, X = 0/1 Strings of even length and every even position is 1

16. Closure Properties Suppose L and L’ are both regular 1. L is also regular 2. L R is also regular 3. L ∪ L’ is also regular 4. L ∩ L’ is also regular

17. Closure Properties Suppose L and L’ are both regular. Then L and L’ are also regular L ∪ L’ is also regular (L ∪ L’) is also regular L ∩ L’ = (L ∩ L’) by De Morgan’s law 4. Hence L ∩ L’ is also regular The construction idea in lecture is also important.

18. Closure Properties Let L be a language comprising all strings w such that 1.w contains an even number of 1s, 2.an odd number of 0s, and 3.no occurrences of the substring 10. Is L regular?

19. Closure Properties To show a language L is regular, 1.represent L using a DFA / NFA / RE, or 2.use closure properties

20. Closure Properties 1. L 1 = {w : w contains an even number of 1s} regular 2. L 2 = {w : w contains an odd number of 0s} regular 3. L 3 = {w : w contains the substring 10} regular

21. Closure Properties Let L be a language comprising all strings w such that 1.w contains an even number of 1s, 2.an odd number of 0s 3.no occurrences of the substring 10. Is L regular? L = L 1 ∩L 2 ∩L 3. Hence it is regular.

22. Regular Expression (Extra) L = {1 k y : y ∈ {0, 1}* and y contains at least k 1s, k ≥ 1} What is the regular expression of L?

23. Regular Expression (Extra) L = {1 k y : y ∈ {0, 1}* and y contains at least k 1s, k ≥ 1} e.g. 111 0001111 (k = 3) is in L. 1k1k y

24. Regular Expression (Extra) L = {1 k y : y ∈ {0, 1}* and y contains at least k 1s, k ≥ 1} What about 111 00011? Same as 1 1100011. Hence is in L! L = Strings that start with 1 and contain at least two 1s 1k1k y

25. Non-regular languages Homework 1 Problem 4 What happens if we allow nested parans?

26. Non-regular languages Consider the case ((((( ))))) same as L = {( n ) n : n ≥ 0} L’ = {0 n 1 n : n ≥ 0} is NOT regular (lecture) More on non-regular languages next week.

27. The End Questions?