CS 810 : Complexity Theory 1 / 22 / 2007 Lecture 1 : Introduction

This course provides a graduate-level introduction to computational complexity theory, the study of the power and limitations of efficient computation. In the first part of the course we focus on the standard setting, in which one tries to realize a given mapping of inputs to outputs in a timeand space-efficient way. We develop models of computation that represent the various capabilities of digital computing devices, including parallelism, randomness, quantum effects, and non-uniformity. We also introduce models based on the notions of nondeterminism, alternation, and counting, which precisely capture the power needed to efficiently compute important types of mappings. The meat of this part of the course consists of intricate relationships between these models, as well as some separation results. In the second part we study other computational processes that arise in diverse areas of computer science, each with their own relevant efficiency measures. Specific topics include: