The Complexity of Finding Middle Elements

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 define 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). 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 differences between these classes and completely determine the inclusion structure between these classes and some other well-known classes of functions like Valiant’s # P and Kobler, Schoning, and Toran’s span-P, that hold under general accepted complexity theoretic assumptions such as the counting hierarchy does not 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 PPNP.