Sublogarithmic-space turing machines, nonuniform space complexity, and closure properties

Abstract

We show that some of the fundamental closure properties (such as concatenation) that hold for Turing machines (TMs) operating in space above logn, do not hold for TMs operating in space below logn. We also compare the powers of TMs andsweeping TMs operating in space below logn. While the proof that the powers of TMs and sweeping TMs are the same is trivial for space greater than or equal to logn, it is not obvious when the space is sublogarithmic. To explore the nature of sublogarithmic space computations further, we introduce a nonuniform space complexity measure and study some of its fundamental properties (such as closure, hierarchy, and gap) in the sublogarithmic range.