I was thinking about TM (Turing Machine) as a computation model, and I came up with the following question :
Is it possible to make a TM that answers binary search (tell wether $x$ belong to a sorted array $A$) in $\log$ time ? That is, is it possible to simulate random access on a TM ?
My thoughts so far are that it is not possible, because the memory of a TM has to be accessed sequentially, e.g. in binary search, reading the element at $n/2$ in the array (first comparison for binary search) takes at least $n/2$ steps.
If it is indeed impossible, does the same hold for structures like binary trees ?