Separating the lower levels of the sublogarithmic space hierarchy

Abstract

For S(n) ≥ log n it is well known that the complexity classes NSPACE(S) are closed under complementation. Furthermore, the corresponding alternating space hierarchy collapses to the first level. Till now, it is an open problem if these results hold for space complexity bounds between log log n and log n, too. In this paper we give some partial answer to this question. We show that for each S between log log n and log n, Σ2SPACE(S) and Σ3SPACE(S) are not closed under complement. This implies the hierarchy (formula presented). We also compare the power of weak and strong sublogarithmic space bounded ATMs.