Definitions

# 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 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.

## Relationship to other complexity classes

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 and PH $subseteq$ BPP() (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. As a result, P=NP leads to BPP=P since PH collapses to P in this case.

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. Note that ordinary generators are not sufficient to show this result; any probabilistic algorithm implemented using a typical random number generator will always produce incorrect results on certain inputs irrespective of the seed (though these inputs might be rare). We also have P = BPP if the exponential-time hierarchy collapses to E = $textrm\left\{DTIME\right\} left\left(2^\left\{O\left(n\right)\right\} right\right)$ (Babai et al.), or if E has exponential circuit complexity (Impagliazzo and Wigderson). Also BPP is contained in i.o.-SUBEXP = $bigcap_\left\{varepsilon>0\right\}hbox\left\{i.o.-DTIME\right\}\left(2^\left\{n^varepsilon\right\}\right)$ if EXPTIME does not collapse to MA (Babai et al.).

A Monte Carlo algorithm is a randomized algorithm which is likely to be correct. Problems in the class BPP have Monte Carlo algorithms with polynomial bounded runtimes. This is compared to a Las Vegas algorithm which is a randomized algorithm which is likely to be correct and is never incorrect, the alternative being to state failure. Las Vegas algorithms with polynomial bound runtimes are used to define the class ZPP.

BPP is contained in P/poly; in fact, as a consequence of the proof of this fact, every BPP algorithm operating on inputs of bounded length can be derandomized into a deterministic algorithm using a fixed string of random bits. Finding this string may be expensive, however.

## Other properties

For a long time, one of the most famous problems that was known to be in BPP but not known to be in P was the problem of determining whether a given number is prime. However, in the 2002 paper , 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.

An important example of a problem in BPP (in fact in co-RP) still not known to be in P is polynomial identity testing, the problem of determining whether a polynomial is identically equal to the zero polynomial. In other words, is there an assignment of variables such that when the polynomial is evaluated the result is nonzero? It suffices to choose each variable's value uniformly at random from a finite subset of at least d values to achieve bounded error probability, where d is the total degree of the polynomial.

BPP is low for itself, meaning that a BPP machine with the power to solve BPP problems instantly (a BPP oracle machine) is not any more powerful than the machine without this extra power.

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

Membership in any language in BPP can be determined by a polynomial-size boolean circuit (see circuit complexity).

## References

• László Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson (1993). "BPP has subexponential time simulations unless EXPTIME has publishable proofs". Computational Complexity, 3:307–318.
• Russell Impagliazzo and Avi Wigderson (1997). "P=BPP if E requires exponential circuits: Derandomizing the XOR Lemma". Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pp. 220–229.
• Valentine Kabanets (2003). "CMPT 710 – Complexity Theory: Lecture 16". Simon Fraser University.
• Christos Papadimitriou (1993). Computational Complexity. 1st edition, Addison Wesley. ISBN 0-201-53082-1. Pages 257–259 of section 11.3: Random Sources. Pages 269–271 of section 11.4: Circuit complexity.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Section 10.2.1: The class BPP, pp.336–339.

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