The Sublogarithmic Alternating Space World

Abstract

This paper tries to fully characterize the properties and relationships of space classes defined by Turing machines (TMs) that use less than logarithmic space—be they deterministic, nondeterministic, or alternating (DTMs, NTMs, or ATMs). We provide several examples of specific languages and show that such machines are unable to accept these languages. The basic proof method is a nontrivial extension of the $1^n \mapsto 1^{n + n!} $ technique to alternating TMs. Let 1log denote the logarithmic function log iterated twice, and let $\Sigma _k {\textit{Space}}(S) $ and $\prod _k {\textit{Space}}(S)$ be the complexity classes defined by S-space-bounded ATMs that alternate at most $k - 1$ times and start in an existential (resp., universal) state. Our first result shows that for each $k > 1$, the sets \[ \begin{gathered} \hfill \Sigma _k {\textit{Space}}(1\log )\backslash \Pi _k Space(o(\log )) \quad {\text{and}} \\ \hfill \Pi _k {\textit{Space}}(1\log )\backslash \Sigma _k {\textit{Space}}(o(\log )) \\ \end{gathered} \] are both not empty. This implies that for each $S \in \Omega (1\log ) \cap o(\log )$, the classes \[ \begin{gathered} \Sigma _1 {\textit{Space}}(S) \subset \Sigma _2 {\textit{Space}}(S) \subset \Sigma _3 {\textit{Space}}(S) \subset \cdots \\ \subset \sum\nolimits_k {Space(S) \subset } \sum\nolimits_{k + 1} {Space(S) \subset } \cdots \\ \end{gathered} \] form an infinite hierarchy. Furthermore, this separation is extended to space classes defined by ATMs with a nonconstant alternation bound A provided that the product $A \cdot S$ grows sublogarithmically. These lower bounds can also be used to show that basic closure properties do not hold for such classes. We obtain that for any $S \in \Omega (1\log ) \cap o(\log )$ and all $k > 1$, $\Sigma _k {Space(S)} $ and $\prod _k {\textit{Space}}(S)$ are not closed under complementation and concatenation. Moreover, $\Sigma _k {{\textit{Space}}(S)} $ is not closed under intersection and $\prod _k {\textit{Space}}(S)$ is not closed under union. It is also shown that ATMs recognizing bounded languages can always be guaranteed to halt. For the class of Z-bounded languages with $Z \leqslant \exp S$, we obtain the equality co-$\Sigma _k {{\textit{Space}}(S)} = \Pi _k {\textit{Space}}(S)$. Finally, for sublogarithmic bounded ATMs, we give a separation between the weak and strong space measure and prove a logarithmic lower space bound for the recognition of nonregular context-free languages.