A note on parallel and alternating time

where equivalent under polynomial bounds. In other words, the complexity classes defined by parallel polynomial time, alternating polynomial time, and polynomial space are actually the same class. In contrast with the above, it was soon remarked that in the theory of complexity over the reals developed by Blum, Shub, and Smale [2], every decidable set can be decided using constant workspace [8]. This was at the expense of an exponential increase in the running time. Yet, the contrast with the classical situation prevailed. If we denote by PSPACER the class of sets of real vectors decidable in exponential time and using polynomial space, by PATR the class of sets decidable using polynomial alternating time, and by PARR the one of those decidable using parallel polynomial time, one can still show [4] that