The speed of copying on one-tape off-line Turing machines

Abstract

It is shown that one-tape off-line Turing machines (TMs), i.e., Turing machines with one work tape and a two-way input tape, can copy a string of length s across d cells on the work tape in O(d+ sd log(min{n,d})) steps, where n denotes the length of the input. This observation is used to show that such TMs can simulate f( n)-time bounded multi-tape TMs in O( f(n) 2 log(n)) steps. This is faster by a factor of log n than the straightforward simulations. Further it is shown (by a Kolmogorov complexity argument) that often this upper bound for copying is optimal: one-tape off-line TMs need ω( sd log(min{n,d})) steps for copying a string of lenght s across d cells, if d⩾ s⩾ log n.