Calculating Big-O

Question

I was asked to find the O-complexity of the algorithm accepting the language {0^(2^k) | k>=0} meaning the length of a string in the language will be of a power of two. (using a turing machine)

$ The algorithm is:

 1   With the head starting at the beginning of string, the headmoves right, marking every other 0.

 2     If there was a single 0, accept

 3     If there was more than a single 0 and the number of 0's was odd, reject

 4   Return head to left

 5   Repeat step one

$

So, for marking the 0's, I believe that is O(logn) because $$ n*(1/2)^x=1<=> n = 2^x <=> logn $$

where we start with n elements, cut in half x times until we have 1

I am unsure about when the cursor is moving to the start. At first, I thought that it was logn as well, but it being the same O-time as marking the 0's seems strange.

Also, I have been told: We observed that all context-free languages are decidable in polynomial time, i.e., CFL ⊆ P. Find a way to prove that in fact it is a proper subset.

Can I get a suggestion or help for this because I do not know how to.

================================================================= I believe that my question is different than those asked and answered because I am not analyzing an actual computer program; I am analyzing a turing machine, which, to me, seems more difficult because the algorithm says to traverse the beginning of the string, which is not considered when analyzing computer programs (at least I don't).

Possible duplicate of Is there a system behind the magic of algorithm analysis? — David Richerby, May 07 '16 at 20:09
One thing to bear in mind is that $n$ usually stands for the length of the input. It's a little confusing to describe the string as $0^{2^n}$ as, then, it's not clear whether you're being asked to measure the running time with respect to $n$ or to the length of the input. And, as you can see, it makes a big difference to the answer. — David Richerby, May 07 '16 at 20:10

Calculating Big-O

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