Can the following sets of strings be accepted by finite auto

Can the following sets of strings be accepted by finite automata? Justify your answers! {1^n | n is a prime number} {0^2n 1^2m | n and m are integers} {x | x is a binary power of two} {x | the center symbol of x is a 1}

Solution

a)No because anything power of 1 is 1 itself.There will be no more states than just 1.

b)No because anything power of 0 is 0.There will be no more states than 0.

c)Yes it will be accepted because x is a binary power of 2 .This implies it can have 2 states 0 and 1(x belongs to 0,1).

d)Yes since the center of symbol of x is 1.X can have various values and many states like a,b,c.This will be accept.