Computer Sciences Dept.

CS 710 - Complexity Theory

Fall 2017

Course Description

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 relation between inputs and outputs in a time- and space-efficient way. We develop models of computation that represent the various capabilities of digital computing devices, including parallelism, randomness, and quantum effects. 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 relations. The meat of this part of the course consists of intricate connections between these models, as well as some separation results.

In the second part, depending on the interest of the students, we may study other computational processes that arise in diverse areas of computer science, each with their own relevant efficiency measures. Specific possible topics include:

  • proof complexity, interactive proofs, and probabilistically checkable proofs -- motivated by verification,
  • pseudorandomness and zero-knowledge -- motivated by cryptography and security,
  • computational learning theory -- motivated by artificial intelligence,
  • communication complexity -- motivated by distributed computing,
  • query complexity -- motivated by databases.
All of these topics have grown into substantial research areas in their own right. We cover the main concepts and some of the key results, as time permits.


Complexity theory at the level of CS 520. We will start the course with a quick review of the relevant CS 520 material. If you haven't taken CS 520 or an equivalent course before but are mathematically mature, you should be able to pick up the prerequisite material from the review.


There is no required text. Lecture notes will be made available from the course web page. Relevant references are the text books Complexity Theory: A Modern Approach by Sanjeev Arora and Boaz Barak, and Computational Complexity: A Conceptual Perspective by Oded Goldreich.


TR 2:30-3:45pm in 2540 Engineering Hall.

Course Work

  • Homework (75%). There will be 3 assignments. You can work on the assignments in pairs.

  • Scribes (25%). Write lecture notes for one of the lectures with new material. Detailed instructions will be provided in due time.


Dieter van Melkebeek <dieter@cs.wisc.edu>.
Office hours: T 4-5pm and by appointment.

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