Sublogarithmic Bounds on Space and Reversals

Abstract

The complexity measure under consideration is $\mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS}$ for Turing machines that are able to branch both existentially and universally. We show that, for any function $h(n)$ between $\log\log n$ and $\log n$, $\Pi_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$ is separated {}from $\Sigma_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$ as well as {}from $\mbox{\sf co}\Sigma_1 \mbox{\rm SPACE}\!\times\!\mbox{\rm REVERSALS} (h(n))$, for middle, accept, and weak modes of this complexity measure. This also separates determinism from the higher levels of the alternating hierarchy. For "well-behaved" functions h(n) between log log n and log n, almost all of the above separations can be obtained by using unary witness languages. In addition, the construction of separating languages contributes to the research on minimal resource requirements for computational devices capable of recognizing nonregular languages. For any (arbitrarily slow growing) unbounded monotone recursive function f(n), a nonregular unary language is presented that can be accepted by a \middle\ \mbox{$\Pi_1$ alternating} Turing machine in s(n) space and i(n) input head reversals, with $s(n)\cdot i(n)\in{\cal O}(\log\log n\cdot f(n))$. Thus, there is no exponential gap for the optimal lower bound on the product $s(n)\cdot i(n)$ between unary and general nonregular language acceptance---in sharp contrast with the one-way case.