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**The power of the middle bit of a \(\#\)P function.**
*(English)*
Zbl 0849.68036

Summary: This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from \(a\neq\)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 \(\text{Mod}_k\)P, \(k\geq 2\), 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 S. Toda [SIAM J. Comput. 20, No. 5, 865-877 (1991; Zbl 0733.68034)]. Consequences of these results for circuit complexity are obtained using the concept of a MidBit gate, which is defined to take binary inputs \(x_1,\dots, x_w\) and output the \(\lfloor \log_2(w)/2\rfloor\)th bit in the binary representation of the number \(\sum^w_{i= 1} x_i\). Every language in ACC can be computed by a family of depth-2 deterministic circuits of size \(2^{(\log n)^{O(1)}}\) with a MidBit gate at the root and AND-gates of fan-in \((\log n)^{O(1)}\) at the leaves. This result improves the known upper bounds for the class ACC.

### MSC:

68Q15 | Complexity classes (hierarchies, relations among complexity classes, etc.) |