On Some Most Probable Separations of Complexity Classes

This thesis is a study of separations of some complexity classes which take place in almost all relativized worlds. We achieve probability one separations of PSPACE from the Polynomial-time Hierarchy PH. Also we separate with probability one all levels of the Boolean Hierarchy BH. The study on the Boolean Hierarchy is a continuation of the work by Bennet and Gill in [BG81] and the joint work in [CH86], where we introduced the "sawing" argument. This "sawing" technique is adapted here to yield probability one separation. The study on PSPACE versus the Polynomial-time Hierarchy is more intriguing. Several novel techniques are employed here. The connection with Boolean circuit is exploited to reduce the problem to a Boolean circuit computation problem. The fixed depth unbounded fan-in Boolean circuit model is considered in connection with the parity function. We show that with an exponential bound of the form $exp(n^{\lambda}$ on the size of the circuits, they make asymptomatically 50% error on all possible inputs, uniformly. Certain probabilistic and game theoretic methods are applied extensively to conclude the result