PP (complexity class)

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 bounded) 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.

A language L is in PP if and only if there exists a probabilistic Turing machine M, such that

Alternatively, PP can be defined using only deterministic Turing machines. A language L is in PP if and only if there exists a polynomial p and deterministic Turing machine M, such that

In both definitions, "less than or equal" can be changed to "less than" (see below), and the threshold 1/2 can be replaced by any fixed rational number in (0,1), without changing the class.

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 > 1/2, such as 2/3 or 501/1000. If this is the case, then we can run the algorithm a 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.

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