PP (complexity)
In complexity theory, PP, or PPT 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.
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.
[2] This characterization of Turing machines does not require a bounded error probability.
Hence, PP is the complexity class containing all problems solvable by a PPT machine with an error probability of less than 1/2.
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.[3] 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 this definition, the string y corresponds to the output of the random coin flips that the probabilistic Turing machine would have made.
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.
On the other hand, a PP algorithm is permitted to do something like the following: Because these two probabilities are exponentially close together, even if we run it for a polynomial 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. PP includes BPP, since probabilistic algorithms described in the definition of BPP form a subset of those in the definition of PP.
To prove this, we show that the NP-complete satisfiability problem belongs to PP.
PP also includes BQP, the class of decision problems solvable by efficient polynomial time quantum computers.
Furthermore, PP includes QMA, which subsumes inclusions of MA and BQP.
[6] PP strictly includes uniform TC0, the class of constant-depth, unbounded-fan-in boolean circuits with majority gates that are uniform (generated by a polynomial-time algorithm).
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.
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.
[1] 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.
denote the complement of L. By the definition of PP there is a polynomial-time probabilistic algorithm A with the property that We claim that without loss of generality, the latter inequality is always strict; the theorem can be deduced from this claim: let
be the polynomial upper bound on the running time of A on input x.
Then and This justifies the assumption (since A′ is still a polynomial-time probabilistic algorithm) and completes the proof.
David Russo proved in his 1985 Ph.D. thesis[8] that PP is closed under symmetric difference.
It was an open problem for 14 years whether PP was closed under union and intersection; this was settled in the affirmative by Beigel, Reingold, and Spielman.
By adding postselection, a larger class called PostBQP is obtained.
Informally, postselection gives the computer the following power: whenever some event (such as measuring a qubit in a certain state) has nonzero probability, you are allowed to assume that it takes place.
PP is also equal to another quantum complexity class known as PQP, which is the unbounded error analog of BQP.
It denotes the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of less than 1/2 for all instances.
Even if all amplitudes used for PQP-computation are drawn from algebraic numbers, still PQP coincides with PP.
