(b.) → for proving B to be not regular,
by Myhill – Nerode Theorem,
As, B = { 1^ky | y belongs to (0+1)* and y consists of at max k 1’s }
Consider, S = { 1, 11, 111, … 1^k , 1^k+1 , .. }
Now, take two pairs of set S to prove that they are distinguishable i.e, → 1^k , 1^k+1
and, z = { all strings consisting at most k 1’s \\ k >=1}
--- As , (1^k+1)(z) → { 1^k+1.( all strings consisting k 1’s ) \\ k >=1 }
{ which is accepted by (B) }
and, (1^k)(z) → { 1^k.( all strings consisting k 1’s ) \\ k >=1 )
VIOLATING CONDITION – ( As, by def. of B, string y can’t have more than k 1’s)
{ which is not accepted by (B) for every string of z }
As, by def. of Myhill- Nerode Theorem,
there are infinite distinguishable equivalence strings { corresponding to elements in S }
results in NOT REGULAR nature of B.
