Is a Turing machine without the ability to write on blank cells less powerful than standard Turing?

Question

Is a Turing machine without the ability to write on blank cells less powerful than standard Turing?

I think the answer is yes but i'm unable to find a computation that standard Turing machine can do but this machine can't.

Any ideas?

Answer 1

The type of Turing machine you describe is a linear bounded automaton (it can only write on the parts of the tape containing the input). LBAs are the acceptors for context-sensitive languages so to find a specific example of a problem that can't be solved with this restriction but can be solved in general by a Turing machine, you just need a language that is decidable but not context-sensitive.

The example given on Wikipedia is:

An example of recursive language that is not context-sensitive is any recursive language whose decision is an EXPSPACE-hard problem, say, the set of pairs of equivalent regular expressions with exponentiation.

For more examples, see Is there an example of a recursive language which is not context sensitive?

Answer 2

A Turing Machine that cannot write on blanks is by the space version of the linear speedup theorem a linear bounded automaton. Therefore any decision problem outside $\textsf{DSPACE}(O(n))$ cannot be decided by it. Such problems do exist by the space hierarchy theorem.