PP Compared To Other Complexity Classes
PP contains BPP, since probabilistic algorithms described in the definition of BPP form a subset of those in the definition of PP.
PP also contains NP. To prove this, we show that the NP-complete satisfiability problem belongs to PP. Consider a probabilistic algorithm that, given a formula F(x1, x2, ..., xn) chooses an assignment x1, x2, ..., xn uniformly at random. Then, the algorithm checks if the assignment makes the formula F true. If yes, it outputs YES. Otherwise, it outputs YES with probability 1/2 and NO with probability 1/2.
If the formula is unsatisfiable, the algorithm will always output YES with probability 1/2. If there exists a satisfying assignment, it will output YES with probability more than 1/2 (exactly 1/2 if it picked an unsatisfying assignment and 1 if it picked a satisfying assignment, averaging to some number greater than 1/2). Thus, this algorithm puts satisfiability in PP. As SAT is NP-complete, and we can prefix any deterministic polynomial-time many-one reduction onto the PP algorithm, NP is contained in PP. Because PP is closed under complement, it also contains co-NP.
Furthermore, PP contains MA, which subsumes the previous two inclusions.
PP also contains BQP, the class of decision problems solvable by efficient polynomial time quantum computers. In fact, BQP is low for PP, meaning that a PP machine achieves no benefit from being able to solve BQP problems instantly. The class of polynomial time on quantum computers with postselection, PostBQP, is equal to PP (see #PostBQP below).
Furthermore, PP contains QMA, which subsumes inclusions of MA and BQP.
A polynomial time Turing machine with a PP oracle (PPP) can solve all problems in PH, the entire polynomial hierarchy. This result was shown by Seinosuke Toda in 1989 and is known as Toda's theorem. This is evidence of how hard it is to solve problems in PP. The class #P is in some sense about as hard, since P#P = PPPand therefore P#P contains PH as well.
PP strictly contains TC0, the class of constant-depth, unbounded-fan-in uniform boolean circuits with majority gates. (Allender 1996, as cited in Burtschick 1999).
PP is contained in PSPACE. This can be easily shown by exhibiting a polynomial-space algorithm for MAJSAT, defined below; simply try all assignments and count the number of satisfying ones.
PP is not contained in SIZE(nk) for any k (proof).
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