# BPP

In complexity theory, BPP is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1/3 for all instances. The abbreviation BPP refers to Bounded-error, Probabilistic, Polynomial time.

BPP algorithm (1 run)
Correct
YES NO
YES ≥2/3 ≤1/3
NO ≤1/3 ≥2/3
BPP algorithm (n runs)
Correct
YES NO
YES > 1-e-n/18 < e-n/18
NO < e-n/18 > 1-e-n/18

If a problem is in BPP, 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. On any given run of the algorithm, it has a probability of at most 1/3 of giving the wrong answer. That is true, whether the answer is YES or NO.

The choice of 1/3 in the definition is arbitrary. It can be any constant between 0 and 1/2 (exclusive) and the set BPP will be unchanged; however, this constant must be independent of the input. The idea is that there is a probability of error, but if the algorithm is run many times, the chance that the majority of the runs are wrong drops off exponentially as a consequence of the Chernoff bound [1] (http://www.cs.sfu.ca/~kabanets/cmpt710/lec16.pdf). This makes it possible to create a highly accurate algorithm by merely running the algorithm several times and taking a "majority vote" of the answers.

BPP is one of the largest practical classes of problems, meaning most problems of interest in BPP have efficient probabilistic algorithms that can be run quickly on real modern machines, by the method described above. For this reason it is of great practical interest which problems and classes of problems are inside BPP.

It is known that BPP is closed under complement; that is, BPP=Co-BPP. It is an open question whether BPP is a subset of NP. It is also an open question whether NP is a subset of BPP; if it is, then NP=RP (many consider this unlikely, since it would imply practical solutions for a range of difficult NP-complete problems). It is known that RP is a subset of BPP, and BPP is a subset of PP. It is not known whether those two are strict subsets. BPP is contained in PH.

The existence of certain strong pseudorandom number generators is conjectured by most experts of the field. This conjecture implies that randomness does not give additional computational power to polynomial time computation, that is, P=RP=BPP. We also have P = BPP if EXPTIME collapses to MA, 1 if the exponential-time hierarchy collapses to E = DTIME(2O(n)), 1 or if E has exponential circuit complexity.2

For a long time, one of the most famous problems that was known to be in BPP but not in P was the problem of determining whether a given number is a prime. However, in the 2002 paper PRIMES in P, Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena found a deterministic polynomial-time algorithm for this problem, thus showing that it is in P.

This class is defined for an ordinary Turing machine plus a source of randomness. The corresponding class for a quantum computer is BQP.

## References

László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307–318. 1993.
Russell Impagliazzo and Avi Wigderson. P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma.
Valentine Kabanets. "CMPT 710 - Complexity Theory: Lecture 16". Simon Fraser University. 2003.

## Footnotes

 Important complexity classes (more) P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NC | P-C PSPACE | PSPACE-C | EXPTIME | EXPSPACE | BQP | BPP | RP | ZPP | PCP | IP | PH
de:BPP (Komplexitätsklasse)

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