COSC 3340: Introduction to Theory of Computation

1. COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 8 UofH - COSC 3340 - Dr. Verma

2. Pumping lemma applications. • Proving L = {anbn | n  0 } is not regular. Proof: Assume L is regular. Certainly L is infinite and therefore the pumping lemma applies to L. Let p be the constant for L (of the pumping lemma). UofH - COSC 3340 - Dr. Verma

3. Pumping lemma applications (contd.) To show there exist a string wL of length at least p such that Q where Q is the rest of the statement of pumping lemma. Let w =apbp such that |w|  p write apbp= xyz But according to pumping lemma, UofH - COSC 3340 - Dr. Verma

4. Pumping lemma applications (contd.) PL statement (i)  |xy|  p Therefore, a…aa…ab…b p p x y z y = am m > 0 xyz = apbp UofH - COSC 3340 - Dr. Verma

5. Pumping lemma applications (contd.) PL statement (ii) xyiz L i = 0,1,2,3,… Therefore, xy2z  L xy2z = xyyz = ak+mbk  L But, L ={anbn | n 0 } which means ap+mbp  L since m > 0 CONTRADICTION !! UofH - COSC 3340 - Dr. Verma

6. Pumping lemma applications (contd.) Therefore our assumption that L ={anbn | n 0 } is a regular language cannot be true. UofH - COSC 3340 - Dr. Verma

7. Using Pumping Lemma -- Very Important points • Above example is the typical application of pumping lemma, to show that a language is not regular. • You must choose string w so that w in Land |w| is at least the pumping length. • Example: choosing w = aaabbb is wrong since we do not know the exact value of p. • You must consider all possibilities for x, y and z such that w = xyz and |xy| p. • The pumping lemma CANNOT be used to show that a language is regular, since it assumes that L is regular. UofH - COSC 3340 - Dr. Verma