Alternating space is closed under complement and other simulations for sublogarithmic space

Abstract

We present some new simulations for ASpace(s(n)), the class of languages accepted by alternating Turing machines with O(s(n)) space, with absolutely no assumptions on s(n). These simulations provide the following inclusions:(a) ASpace(s(n))⊆DTime(n⋅2O(s(n))). This extends, to sublogarithmic space bounds, the classic result stating that ASpace(s(n))⊆DTime(2O(s(n))), proved under the assumption s(n)≥log⁡n.(b) ASpace(s(n))⊆NTimeSpace(n⋅2O(s(n)),2O(s(n))), a simulation by nondeterministic machines with simultaneous bounds on time and space. This improves the known inclusion, stating that ASpace(s(n))⊆NSpace(2O(s(n))), proved under the assumption s(n)≥log⁡log⁡n.(c) ASpace(s(n))=co-ASpace(s(n)), i.e., the alternating space is closed under complement, independently of whether s(n) is above log⁡n and of whether the original machine can get into an infinite loop. This solves a long-standing open problem. Quite surprisingly, this complementary simulation does not eliminate infinite loops—the new machine itself goes to infinite loops along some computation paths.