  It is well-known that one-tape Turing machines working in linear time are no
more powerful than finite automata, namely they recognize exactly the class of
regular languages. We prove that it is not decidable if a one-tape machine
works in linear time, even if it is deterministic and restricted to use only
the portion of the tape which initially contains the input. This motivates the
introduction of a constructive variant of one-tape machines, called
weight-reducing machine, and the investigation of its properties. We focus on
the deterministic case. In particular, we show that, paying a polynomial size
increase only, each weight-reducing machine can be turned into a halting one
that works in linear time. Furthermore each weight-reducing machine can be
converted into equivalent nondeterministic and deterministic finite automata by
paying exponential and doubly-exponential increase in size, respectively. These
costs cannot be reduced in general.
