The Sublogarithmic Alternating Space World

Maciej Liśkiewicz, Rüdiger Reischuk
1996 SIAM journal on computing (Print)  
This paper tries to fully characterize the properties and relationships of space classes defined by Turing machines that use less than logarithmic space -may they be deterministic, nondeterministic or alternating (DTM, NTM or ATM). 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 → 1 n+n! technique to alternating TMs. Let llog denote the logarithmic function log
more » ... twice, and Σ k Space(S), Π k 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 are both not empty. This implies that for each S ∈ Ω(llog ) ∩ o(log ) the classes 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 · 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 ∈ Ω(llog ) ∩ o(log ) and all k > 1 Σ k Space(S) and Π k Space(S) are not closed under complementation and concatenation. Moreover, Σ k Space(S) is not closed under intersection, and Π k 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 ≤ exp S we obtain the equality co-Σ k Space(S) = Π k 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 contextfree languages.
doi:10.1137/s0097539793252444 fatcat:csvph7zvnjc3pa2oiyazr4uyri