Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Komplexität und Kryptografie

The Power of the Middle Bit of a #P Function

Frederic Green, Johannes Köbler, Kenneth W. Regan, Thomas Schwentick, and Jacobo Toran


This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function f. The middle bit of f(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hierarchy and the classes ModkP are shown to be low for MP. They are also low for a class we call AmpMP which is defined by abstracting the "amplification" methods of Toda (SIAM J. Comput. 20, 865--877, 1991).
Consequences of these results for circuit complexity are obtained using the concept of a MidBit gate. Every language in ACC can be computed by a family of depth-2 deterministic circuits of quasi-polynomial size with a MidBit gate at the root and AND-gates of poly-log fan-in at the leaves. This result improves the known upper bounds for the class ACC.

Ps-File: The Power of the Middle Bit of a #P Function