Alternating on-line turing machines with only universal states and small space bounds

Abstract

Let L [AONTM( L( n))] be the class of sets accepted by L( n) space bounded alternating on-line Turing machines, and L [UONTM( L( n))] be the class of sets accepted by L( n) space bounded alternating on-line Turing machines with only universal states. This note first shows that, for any L( n) such that L( n) ⩾ log log n and lim n → ∞ [ L( n)/log n] = 0, (i) L [UONTM( L( n))] ⊋ L [AONTM( L( n))], (ii) L [UONTM( L( n))] is not closed under complementation, and (iii) L [UONTM( L( n))] is properly contained in the class of sets accepted by L( n) space bounded alternating Turing machines with only universal states. We then show that there exists an infinite hierarchy among L [UONTM( L( n))]'s with log log n ⩽ L( n) ⩽ log n.