Relations among simultaneous complexity classes of nondeterministic and alternating Turing machines

Ruzzo [Tree-size bounded alternation, J. Comput. Syst. Sci. 21] introduced the notion of tree-size for alternating Turing machines (ATMs) and showed that it is a reasonable measure for classification of complexity classes. We establish in this paper that computations by tree-size and space simultaneously bounded ATMs roughly correspond to computations by time and space simultaneously bounded nondeterministic TMs (NTMs). We also show that not every polynomial time bounded and sublinear space simultaneously bounded NTM can be simulated by any deterministic TM with a slightly increased time bound and a slightly decreased space bound simultaneously.