Michael Sipser Edition 3 Exercise 5 Question 34 (Page No. 241)

Question

Let $X = \{\langle M, w \rangle \mid  \text{M is a single-tape TM that never modifies the portion of the tape that contains the input $w$ } \}$

Is $X$ decidable? Prove your answer.

Answer 1

As we know, it is undecidable whether a string belong to the language accepted by a particular turing machine or not [Membership problem of Turing Machine]. So, How can it, in the first place, recognise whether some sub-string 'w' is a part of given input string 's'(say) or not. Only after it recognises whether substring 'w' is a part of string "s" which belongs to language accepted by it (i.e. whether it is present on tape or not) it can decide whether to write on that part or not.

So to conclude the answer to this question should be "it is undecidable"