BPP - Definition and Overview

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.

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. The idea is that there is a small probability of error, but if the algorithm is run many times, the chance that the majority of the runs are wrong drops off exponentially.

It is known that 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. 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(2^O(n)), ¹ or if E has exponential circuit complexity.²

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

External links

• Princeton CS 597E: Derandomization paper list (http://www.cs.princeton.edu/courses/archive/fall03/cs597E/)

References