Given any strings u and X (with u NOTEQUALTO e), define #u(x) to be the number o

Question

Given any strings u and X (with u NOTEQUALTO e), define #u(x) to be the number of times u occurs as a substring of x. For example, if u = 010 and z = 01010 then #u(x) = 2 because two occurrences are counted even though they overlap. Using the Myhill-Nerode technique (not the Pumping Lemma), prove that two the following languages are not regular and give a regular expression for the one that is regular. L_1 = {x element {0, 1}^*: # 00(x) > # 11(x)}. L_2 = {x element{0, 1}^*: # 01(x) > # 10(x)}. L_3 = {x0y: # 0 (x) = # 1(y)}.

Explanation / Answer

a) L = (aibi / i 0)

At first, we tend to assume that L is regular and n is that the variety of states.

Let w = anbn. so |w| = 2n n.

By pumping lemma, let w = xyz, wherever |xy| n.

Let x = ap, y = aq, and z = arbn, wherever p + letter of the alphabet + r = n, p 0, q 0, r 0. so |y| zero.

Let k = 2. Then xy2z = apa2qarbn.

Number of as = (p + 2q + r) = (p + letter of the alphabet + r) + letter of the alphabet = n + letter of the alphabet

Hence, xy2z = an+q bn. Since letter of the alphabet zero, xy2z isn't of the shape anbn.

Thus, xy2z isn't in L. thus L isn't regular.

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