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hw7-solutions - Com S 381 Homework 7 Solutions 1 L wwR |...

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Com S 381 Homework 7 Solutions 1. L : { ww R | w (a+b)* } is not a regular set. Suppose L is regular. Then there exists a constant n such that for every string w in L such that | w | n , we can break w into three strings w = xyz such that: y ε |xy| n For all k 0, the string xy k z is also in L. Let w be strings of the form a n bba n . Because | xy | n , the substring xy must be in the form a m , we can pump y up and let k = 2, then the w becomes a n + |y| bba n . This is not in the language L, therefore we proved L not regular by the pumping lemma. 2. We can express this language as C 1 C 2 where, C 1 : {a m b m c n d n | n 1, m 1} Grammar: S aMbcNd M aMb | ε N cNd | ε C 2 : {a*b n c n d* | n 1} Grammar: S aSd | bBc B bBc | ε Because C 1 forces a and b, c and d to be the same length, and C 2 forces b and c to be the same length, the intersection of the two language has a, b, c, d all be the same length.
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