PP (complexity)

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 less than or equal to 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.

An alternative characterization of PP is the set of problems that can be solved by a nondeterministic Turing machine in polynomial time where the acceptance condition is that a majority (more than half) of computation paths accept. Because of this some authors have suggested the alternative name Majority-P.


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, an algorithm must give correct answer (YES or NO) 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 of repeats increases if c becomes closer to 1/2, but it does not depend on the input size n.

The important thing is that this constant c is not allowed to depend on the input. On the other hand, a PP algorithm is permitted 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 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.

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.

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 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.

Complete problems and other properties

Unlike BPP, PP is a syntactic, rather than semantic class. Any polynomial-time probabilistic machine recognizes some language in PP. In contrast, given a description of a polynomial-time probabilistic machine, it is undecidable in general 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 complement and symmetric difference, and also under union and intersection. The proof of the latter two closures is significantly more difficult than the former two, and was an open problem for 14 years.


  • C. Papadimitriou. Computational Complexity, chapter 11. Addison-Wesley, 1994.
  • E. Allender. A note on uniform circuit lower bounds for the counting hierarchy. In Proceedings 2nd International Computing and Combinatorics Conference (COCOON), volume 1090 of Springer Lecture Notes in Computer Science, pages 127-135, 1996.
  • Burtschick, Hans-Jörg; Heribert Vollmer (1999). "Lindström Quantifiers and Leaf Language Definability". Electronic Colloquium on Computational Complexity (TR96-005): Retrieved on 2006-11-20.

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