PP (complexity)

From Academic Kids

In complexity theory, PP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of less than 1/2 for all instances. The abbreviation PP refers to Probabilistic, Polynomial time. The complexity class was defined by Gill in 1977.

If a decision problem is in PP, then there is an algorithm for it that is allowed to flip coins and make random decisions.It is guaranteed to run in polynomial time. If the answer is YES, the algorithm will answer YES with probability more than 1/2. If the answer is NO, the algorithm will answer YES with probability at most 1/2. In more practical terms, it is the class of problems that can be solved to any fixed degree of accuracy by running a randomized, polynomial-time algorithm a sufficient (but unbounded) number of times.



BPP is a subset of PP; it can be seen as the subset for which there are efficient probabilistic algorithms. The distinction is in the error probability that is allowed: in BPP, a YES instance must be accepted with probability exceeding some fixed constant c greater than 1/2, such as 2/3 or 501/1000. If this is the case, then we can run the algorithm a constant number of times and take a majority vote to achieve any desired probability of correctness less than 1, using the Chernoff bound. This number increases if c becomes closer to 1/2, but it does not depend on n.

The important thing is that this constant c is not allowed to depend on the input. A PP algorithm is permitted, on the other hand, to do something like the following:

  • On a YES instance, output YES with probability 1/2+1/2n, where n is the length of the input.
  • On a NO instance, output YES with probability 1/2.

Because these two probabilities are so close together, especially for large n, even if we run it a large number of times it is very difficult to tell whether we are operating on a YES instance or a NO instance. Attempting to achieve a fixed desired probability level using a majority vote and the Chernoff bound requires a number of repetitions that is exponential in n. This may be compared roughly to the problem of trying to figure out which side of a slightly-biased coin is more likely by flipping it many times.

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 that, 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 assignement 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 in NP-complete, and we can prefix any deterministic polynomial-time many-one reduction onto the PP algorithm, NP is contained in PP.

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.

A polynomial time Turing machine with a PP oracle is even more powerful (if P ≠ NP). Namely, PH is contained in PPP. This result was shown by Seinosuke Toda in 1989 and is known as Toda's theorem.

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.

Complete problems and other properties

Unlike BPP, PP is a syntantic, rather than semantic class. Any probabilistic machine recognizes some language in PP. In contrast, given a description of a probabilistic machine, it is hard to determine if it recognizes a language in BPP.

PP has natural complete problems, for example, MAJSAT. MAJSAT is a decision problem in which one is given a Boolean formula F. The answer must be YES if more than half of all assignments x1, x2, ..., xn make F true and NO otherwise.

PP is closed under union, intersection and complement.


  1. C. Papadimitriou. Computational Complexity, chapter 11. Addison-Wesley, 1994.

External Link

Important complexity classes (more)
P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NC | P-C

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