The Computational Complexity Column Recent Advances towards Proving P = Bpp

Are there too many complexity classes? Merely trying to understand one aspect of computation, such as the power of randomness, leads to a whole range of complexity classes, such as ZPP, RP, and BPP, to name but a few. Do we really need all of these classes? One of the most exciting developments in complexity theory in the past few years is the growing body of evidence that all of the aforementioned classes are merely pseudonyms for P. Our guest column this issue gives an overview of this area. Abstract Two independent techniques have been developed recently that yield suucient conditions for P = BPP in terms of worst-case circuit complexity of functions computable in exponential time. Andreev, Clementi and Rolim proved that P = BPP provided that a sparse \eeciently enumerable" language exists of suuciently high circuit complexity. This result has been subsequently improved by Impagliazzo and Wigderson by showing that either P = BPP or all the decision problems solvable in time 2 O(n) are solvable by circuits of size 2 o(n). In this column we discuss these results and their relation with previously known suucient conditions for P = BPP.