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

UofH - COSC Dr. Verma 2 Closure properties of Regular Languages Regular languages are closed under: – Union Notation:  – Intersection Notation:  L 1  L 2 is regular if L 1 and L 2 are regular. L 1  L 2 is regular if L 1 and L 2 are regular.

Lecture 3UofH - COSC Dr. Verma 3 Examples Let  = {a,b}. Let L 1 = { w in  * | w has even number of a's}. – Is L 1 regular? L 2 = { w in  * | w has odd number of b's}. – Is L 2 regular? L 1  L 2 = ? – Ans: {w in  * | w has even a's or odd b's}. L 1  L 2 = ? – Ans: {w in  * | w has even a's and odd b's}. By closure properties, both these are regular.

Lecture 3UofH - COSC Dr. Verma 4 DFA of L 1 = {w in {a,b}* | w has even number of a's}.

Lecture 3UofH - COSC Dr. Verma 5 DFA of L 2 = { w in {a,b}* | w has odd number of b's}.

Lecture 3UofH - COSC Dr. Verma 6 DFA of L 1  L 2 = {w in {a,b}* | w has even a's or odd b's}.

Lecture 3UofH - COSC Dr. Verma 7 DFA of L 1  L 2 = {w in {a,b}* | w has even a's and odd b's}.

Lecture 3UofH - COSC Dr. Verma 8 DFA of L 1  L 2 = {w in {a,b}* | w has even a's or odd b's}. aba

Lecture 3UofH - COSC Dr. Verma 9 DFA of L 1  L 2 = {w in {a,b}* | w has even a's or odd b's}. aba

Lecture 3UofH - COSC Dr. Verma 10 DFA of L 1  L 2 = {w in {a,b}* | w has even a's or odd b's}. abaaba

Lecture 3UofH - COSC Dr. Verma 11 DFA of L 1  L 2 = {w in {a,b}* | w has even a's or odd b's}. aba

Lecture 3UofH - COSC Dr. Verma 12 DFA of L 1  L 2 = {w in {a,b}* | w has even a's and odd b's}. aba

Lecture 3UofH - COSC Dr. Verma 13 DFA of L 1  L 2 = {w in {a,b}* | w has even a's and odd b's}. aba

Lecture 3UofH - COSC Dr. Verma 14 DFA of L 1  L 2 = {w in {a,b}* | w has even a's and odd b's}. abaaba

Lecture 3UofH - COSC Dr. Verma 15 DFA of L 1  L 2 = {w in {a,b}* | w has even a's and odd b's}. aba

Lecture 3UofH - 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 3UofH - COSC Dr. Verma 17 General Construction for  and  Idea: Simulate two DFA's in parallel. Let M 1 = (Q 1, ,  1, s 1, F 1 ) and M 2 = (Q 2, ,  2, s 2, F 2 ) M = (Q, , , s, F) where: Q = Q 1 X Q 2 s = (s 1, s 2 )  ((q 1, q 2 ),  ) = (  1 (q 1,  ),  2 (q 2,  )) For Union, F = ? – Ans: (Q 1 X F 2 ) U (F 1 X Q 2 ) For Intersection, F = ? – Ans: F 1 X F 2

Lecture 3UofH - 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 3UofH - COSC Dr. Verma 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 3UofH - COSC Dr. Verma 20 Example: {w in {0,1}* | the second last symbol of w is a 1}

Lecture 3UofH - COSC Dr. Verma 21 NFA A = {w in {0,1}* | the second last symbol of w is a 1} 110

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

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

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

Lecture 3UofH - 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.