Computer Sciences Dept.

CS 710 - Complexity Theory

Spring 2015

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 mapping of inputs to 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, quantum effects, and nonuniformity. 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, 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.


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.


MWF 2:30-3:45pm in 1325 CS&S. On average only 2 of the 3 weekly slots will be used.

Course Work

  • Homework (40%). There will be 2 assignments. You should work on the assignments in pairs.

  • Paper Presentation (60%). Read a recent paper on computational complexity and present it to the instructor.


Dieter van Melkebeek. Email: dieter@cs.wisc.edu. Office hours: F 4-5pm and by appointment.

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