New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.

One of the major concerns of theoretical computer science is the classifi cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac~that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function.

By virtue of the close relationship between logic and relational databases, it turns out that complexity has important applications to databases such as analyzing the parallel time needed to compute a query, and the analysis of nondeterministic classes. This book is a relatively self-contained introduction to the subject, which includes the necessary background material, as well as numerous examples and exercises.

Property Testing is the study of super-fast (randomized) algorithms for approximate decision making. These algorithms are given direct access to items of a huge data set, and determine, whether this data set has some predetermined (global) property or is far from having this property. Remarkably, this approximate decision is made by accessing a small portion of the data set. This state-of-the-art survey presents a collection of extended abstracts and surveys of leading researchers in property testing and related areas; it reflects the program of a mini-workshop on property testing that took place in January 2010 at the Institute for Computer Science (ITCS), Tsinghua University, Beijing, China. The volume contains two editor's introductions, 10 survey papers and 18 extended abstracts.