The proofs presented in this appendix were not included in the main text for a variety of reasons (e.g., they were deemed too technical and/or out of pace for the corresponding location). On the other hand, since our presentation of them is sufficiently different from the original and/or standard presentation, we see a benefit in including them in the current book.

Summary: This appendix contains proofs of the following results:

1. PH is reducible to #P (and in fact to ⊕P) via randomized Karpreductions. The proof follows the underlying ideas of Toda's original proof, but the actual presentation is quite different.

2. For any integral function f that satisfies f(n) ∈ {2, …, poly(n)}, it holds that IP(f) ⊆ AM(O(f)) and AM(O(f)) ⊆ AM(f). The proofs differ from the original proofs (provided in and, respectively) only in the secondary details, but these details seem significant.