 
Summary: For Completeness, Sublogarithmic Space is No Space
Manindra Agrawal
Department of Computer Science
Indian Institute of Technology, Kanpur
Kanpur 208016, India
email: manindra@iitk.ac.in
Abstract
It is shown that for any class C closed under lineartime reductions, the complete sets for
C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are
isomorphic under firstorder reductions.
Keywords: Isomorphisms; Sublogarithmic reductions; Computational Complexity.
1 Introduction
Logarithmic space is a critical bound in space complexity. For the class DLOG (= DSPACE(log n))
we do not have, till now, any nontrivial upper bound, while, on the other hand, it is not too
difficult to exhibit languages in DLOG  DSPACE(o(log n)) [16]. The reason for this is that
the TMs working within sublogarithmic space cannot even record the the length of the input,
and thus can be `fooled' easily. In fact, when the space bound of a DTM is o(log log n), the
TM cannot recognize any nonregular language [20, 12]. So, there exists a gap between the
classes DSPACE(log log n) and DSPACE(1) (the class of regular sets) in the sense that an in
termediate space bound does not yield a different class (this result has been generalized to even
