Below linear-time : Dimensions versus time

Deterministic d-dimensional Turing machines are considered. We investigate the classes of languages acceptable by such devices with time bounds of the form id + r where r E o(id) is a sublinear function. It is shown that for any dimension d >= 1 there exist infinite time hierarchies of separated complexity classes in that range. Moreover, for the corresponding time bounds separated dimension hierarchies are proved. CR Subject Classification (1998): F.1.3, F.1.1, F.4.