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This is a question from the lecture about Toda's theorem:

http://www.cs.princeton.edu/courses/archive/spring01/cs522/lecnotes/lec8.ps

The lecture uses theorem 1 and theorem 2 but not include proof. My question is how to proof?

Theorem 2:

$L \in \oplus P$ iff $L$ can be computed by a DC uniform circuit $\{D_n\}$ that

use $AND$, $OR$, $NOT$, $XOR$ gates.

has size $2^{n^{O(1)}}$ and constant depth

$XOR$ gates can have unbounded (exponential) fanin, but $AND$, $OR$ gates have fanin at most $n^{O(1)}$

$NOT$ gates can appear anywhere in the circuit.

This question may not be a research level question, but I really want to know the answer. If someone can give me a detailed reference, I will appreciate that.

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The circuit characterization are by Kannan, Venkateswaran, Vinay, and Yao given in the paper:

A Circuit-Based Proof of Toda′s Theorem

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