1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 3
Lecture 3
UofH - COSC Dr. Verma

2. Closure properties of Regular Languages
Regular languages are closed under:
Union
Notation: 
Intersection
Notation: 
L1  L2 is regular if L1 and L2 are regular.
L1  L2 is regular if L1 and L2 are regular.
Lecture 3
UofH - COSC Dr. Verma

3. Examples
Let  = {a,b}.
Let L1 = { w in * | w has even number of a's}.
Is L1 regular?
L2 = { w in * | w has odd number of b's}.
Is L2 regular?
L1  L2 = ?
Ans: {w in * | w has even a's or odd b's}.
L1  L2 = ?
Ans: {w in * | w has even a's and odd b's}.
By closure properties, both these are regular.
Lecture 3
UofH - COSC Dr. Verma

4. DFA of L1 = {w in {a,b}* | w has even number of a's}.
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5. DFA of L2 = { w in {a,b}* | w has odd number of b's}.
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6. DFA of L1  L2 = {w in {a,b}* | w has even a's or odd b's}.
Lecture 3
UofH - COSC Dr. Verma

7. DFA of L1  L2 = {w in {a,b}* | w has even a's and odd b's}.
Lecture 3
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8. DFA of L1  L2 = {w in {a,b}* | w has even a's or odd b's}.
aba
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UofH - COSC Dr. Verma

9. DFA of L1  L2 = {w in {a,b}* | w has even a's or odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

10. DFA of L1  L2 = {w in {a,b}* | w has even a's or odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

11. DFA of L1  L2 = {w in {a,b}* | w has even a's or odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

12. DFA of L1  L2 = {w in {a,b}* | w has even a's and odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

13. DFA of L1  L2 = {w in {a,b}* | w has even a's and odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

14. DFA of L1  L2 = {w in {a,b}* | w has even a's and odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

15. DFA of L1  L2 = {w in {a,b}* | w has even a's and odd b's}.
aba
Lecture 3
UofH - COSC Dr. Verma

16. DFA of A = {w | w contains at least one 1 and an even number of 0s follow the last 1}
Lecture 3
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17. General Construction for  and 
Idea: Simulate two DFA's in parallel.
Let M1 = (Q1, , 1, s1, F1) and
M2 = (Q2, , 2, s2, F2)
M = (Q, , , s, F) where:
Q = Q1 X Q2
s = (s1, s2)
((q1, q2), ) = (1(q1, ), 2(q2, ))
For Union, F = ?
Ans: (Q1 X F2) U (F1 X Q2)
For Intersection, F = ?
Ans: F1 X F2
Lecture 3
UofH - COSC Dr. Verma

18. Nondeterministic Finite Automaton (NFA)
Generalization of DFA. Allows:
0 or more next states for the same (q, ).
Guessing
Transitions labeled by the empty string.
Changing state without reading input
Motivation: Flexibility.
Easier to prove many closure properties.
Lecture 3
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19. How does an NFA work?
w is accepted by an NFA provided there is a sequence of guesses that leads to a final state.
Language accepted by NFA is the set of all strings accepted by it.
Lecture 3
UofH - COSC Dr. Verma

20. Example: {w in {0,1}* | the second last symbol of w is a 1}
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21. NFA A = {w in {0,1}* | the second last symbol of w is a 1}
110
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UofH - COSC Dr. Verma

22. NFA A = {w in {0,1}* | the second last symbol of w is a 1}
110
Lecture 3
UofH - COSC Dr. Verma

23. NFA A = {w in {0,1}* | the second last symbol of w is a 1}
110
Lecture 3
UofH - COSC Dr. Verma

24. NFA A = {w in {0,1}* | the second last symbol of w is a 1}
110
Lecture 3
UofH - COSC Dr. Verma

25. Formal definition of NFA
Notation: e =  U {e}.
NFA M = (Q, , , s, F) where:
Q - finite set of states
 - input alphabet
s - initial state
F  Q - set of final states
 is a subset of Q X e X Q.
If (p, u, q) in , then NFA in state p can read u and go to q.
Lecture 3
UofH - COSC Dr. Verma