The Complexity of Finding Middle Elementsheribert

Seinosuke Toda introduced the class Mid P of functions that yield the middle element in the set of output values over all paths of nondeterministic polynomial time Turing machines. We deene two related classes: Med P consists of those functions that yield the middle element in the ordered sequence of output values of nondeterministic polynomial time Turing machines (i.e. we take into account that elements may occur with multiplicities greater than one). Med P consists of those functions that yield the middle element of all accepting paths (in some resonable encoding) of nondeterministic polynomial time Turing machines. We exhibit similarities and diierences between these classes and completely determine the inclusion structure between these classes and some well-known other classes of functions like Valiant's # P and KK obler, Schh oning, and Torr an's span-P, that holds under general accepted complexity theoretic assumptions such as the counting hierarchy doesn't collapse. Our results help in clarifying the status of Toda's very important class Mid P in showing that it is closely related to the class PP NP .