A Note on Square Rooting of Time Functions of Turing Machines

Let f and g be functions from N to N. We write f g to denote that there exists a constant c > 0 such that for all but finitely many n it holds that f (n) ≤ c(gcn) + cn. We say that f and g are equivalent if f g and g f . In [SBR] Sapir et al. conjecture that for all nondeterministic time functions T that are (n2) the square root of T is equivalent to the time function of a nondeterministic Turing machine. In this note we study this conjecture. We show that the deterministic version of this conjecture is false. For every rational number α, 0 < α < 1, there is a deterministic time function T = (n2) such that T α is not equivalent to any time function of a deterministic Turing machine.