Deterministic one-way Turing machines with sublinear space bounds

Deterministic one-way Turing machines with sublinear space bounds are systematically studied. We distinguish among the notions of strong, weak, and restricted space bounds. The latter is motivated by the study of P automata. The space available on the work tape depends on the number of input symbols read so far, instead of the entire input. The class of functions space constructible by such machines is investigated, and it is shown that every function f that is space constructible by a deterministic two-way Turing machine, is space constructible by a strongly f space-bounded deterministic one-way Turing machine as well. We prove that the restricted mode coincides with the strong mode for space constructible functions. It turns out that the strong mode is computationally less powerful than the weak mode, for any sublinear space constructible function. Moreover, the known infinite, dense, and strict hierarchy of strong space complexity classes is shown also for the weak mode by Kolmogorov complexity arguments. Finally, closure properties under Boolean operations are obtained for weak space bounds. These properties are different from the known properties of strong space bounds.