Let's say we have a Turing machine which head can only write 1 or blank to the tape (although it can read all symbols from any input alphabet correctly). Can we operate with it on any input?
My approach was that we can rewrite our input tape with unary coding and symbol-codings separated with blanks. If we had, for example, an input
we could try to encode 0 as
1, 1 as
11 and so on. We would know when we reached the end of our input word when we read two blanks in a row. However, rewriting the example string from above would cause some problems. When we code a single symbol with multiple ones, we overwrite another symbols and need to remember them in state. In the example above, after encoding 0 our tape would be like
And in our state we'd remember that we need to encode 1 now. Hovever, writing the encoding would overwrite 2, 3 and a blank, so we would need to remember those ones as well in the state. Such an approach leads to creating a lot of states: for n-letter alphabet and unary encoding I proposed it would be $(n+1)^n$ (all possible combinations of symbols on n+1 positions on the tape) "basic states", and for each of those we'd need a few more so that we could write an adequate number of "1" finished by a blank. I am not sure if this is acceptable at all.
Is my approach correct? What other approaches could I take?