TITLE: reversible Turing machines
QUESTION [12 upvotes]: Hello,
Let T be a Turing machine such that
1) it operates on the alphabet {0,1}, 
2) its set of states is A
3) the language it accepts is $L$ . 
Does there exists a Turing machine S which also operates on the alphabet {0,1} and such that the language it accepts is L (the set of states might be different though) and such that, crucially, S is reversible? 
By reversible I mean "the computational paths of S are disjoint". More precisely, the transition table of S gives rise to a map $K_S: \text{Tapes}\times B \to \text{Tapes} \times B$, where Tapes is the subset of the infinite product $\{0,1\}^Z$ consisting of those sequences which have a finite number of 1's, and B is the set of states of S. S is reversible iff, by definition, $K_S$ is injective on the set
$$
 \bigcup_{i=0}^\infty K_S^{i}(\text{Tapes}\times \{Initial \} ),
$$
where $Initial\in B$ is the initial state of S.
If the answer to the above question is "no" then what if we allow S to operate on an alphabet which is larger then {0,1}?

REPLY [5 votes]: Well there are universal reversible Turing machines (RTMs), for example [1].
So you can simulate T with a universal RTM.
The resulting machine would be reversible and accept L.

[1] Kenichi Morita and Yoshikazu Yamaguchi, "A Universal Reversible Turing Machine" MCU2007