Session-18.pdf - Procedure to prove a Language is not...

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Procedure to prove a Language is not regular: 1) Assume the Language L is Regular 2) a FA with 'n' states 3) Take w 4) 5) i z L 6) Our assumption is false, so L is not regular. Problems: 1. Prove that L = {a i b i Sol: Assume that L is regular then a FA 'M' with 'n' states accepting L. Let w = a n b n & |w| = 2n > n. = a n-1 ab n = xyz, where x=a n-1 , y=a & z=b n Consider xy i z for i=2, xy 2 z = a n-1 a 2 b n = a n+1 b n L xy i z L, for i=2. Our assumption is false, so L is not regular. 2. Prove that L = { 0 p / p is prime number} is not regular. Assume that L is regular then a with 'n' states accepting L. Let m be the next immediate prime number after n. Let w = 0 m = n 0 m-n n and z=0 m-n Clearly, , |xy|=| n | = n( ) Consider xy i z for i=m+1,
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xy i z = n ) i 0 m-n = 0 n(m+1) 0 (m-n) | xy i z| = n(m+1)+m-n = mn + n + m n =m(n+1), a composite number xy i z L, for i=m+1. Our assumption is false, so L is not regular. 3. Prove that L = { 0 i ^ 2 / p is prime number} is not regular. Assume that L is regular then Here ----- (1) Let w = 0 n ^ 2 & |w|=n 2 > n = 0 0 0 (n^2 - 2) = xyz, where x=0, y=0 and z=0 (n^2 -2) Clearly, |xy|=| 00 | = 2( (from (1)) |y| Consider xy i z for i=3, n 2
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  • Fall '14
  • rani
  • Prime number, xyiz

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