Reversible Turing machines with a working tape and a one-way or two-way read-only input tape are considered. We investigate the classes of languages acceptable by such devices with small time bounds in the range between real time and linear time,
, time bounds of the form
is a sublinear function. It is shown that there exist infinite time hierarchies of separated complexity classes in that range. We then turn to the question of whether reversible Turing machines in the range of interest are weaker than general ones or not. This is answered in the affirmative by proving that there are languages accepted by irreversible one-way Turing machines in real time that cannot be accepted by any reversible one-way machine in less than linear time.
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