Complexity Theory

From International Center for Computational Logic

Complexity Theory

Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2021

This course covers the fundamental concepts as well as advanced topics of complexity theory.

Key topics are:

  • Turing Machines (revision): Definition of Turing Machines; Variants; Computational Equivalence; Decidability and Recognizability; Enumeration
  • Undecidability: Examples of Undecidable Problems; Mapping Reductions; Rice’s Theorem (both for characterizing Decidability and Recognizability); Recursion Theorem; Outlook into Decidability in Logic
  • Time Complexity: Measuring Time Complexity; Many-One Reductions; Cook-Levin Theorem; Time Complexity Classes (P, NP, ExpTime); NP-completeness; pseudo-NP-complete problems
  • Space Complexity: Space Complexity Classes (PSpace, L, NL); Savitch’s Theorem; PSpace-completeness; NL-completeness; NL = coNL
  • Diagonalization: Hierarchy Theorems (det. Time, non-det. Time, Space); Gap Theorem; Ladner’s Theorem; Relativization; Baker-Gill-Solovay Theorem
  • Alternation: Alternating Turing Machines; APTime = PSpace; APSpace = ExpTime; Polynomial Hierarchy
  • Circuit Complexity: Boolean Circuits; Alternative Proof of Cook-Levin Theorem; Parallel Computation (NC); P-completeness; P/poly; (Karp-Lipton Theorem, Meyer’s Theorem)
  • Probabilistic Computation: Randomized Complexity Classes (RP, PP, BPP, ZPP); Sipser-Gács-Lautemann Theorem
  • Quantum Computing: Quantum circuits, BQP, some basic results

Mode of Teaching and Registration

The sessions are taught in presence, using a large enough room at TU Dresden. Starting from 22 Nov 2022, the lecture is also streamed live via Zoom, so that students can join while in Covid quarantine or if unable to attend due to the general pandemic situation. The link can be found in Opal under "Links" (see below) and can also be found (or requested anew) in the Matrix chat group.

The course generally does not require a special registration and there is no limit for participants. However, students in programmes that use the Selma system (esp. students in CMS Master) will need to register there to obtain credits. Additionally, we kindly ask you to enroll in the OPAL course so we can contact you, should the need arise. Most of the materials will be freely available world-wide.

Contact

Besides the regular meetings in the lectures and exercise classes, you can also contact the teachers and other students in the public discussion channel on Matrix shown on the side.

Acknowledgements

The slides for some of the foundational lectures of this course are based on slides used by Markus Krötzsch for the course Complexity Theory at the University of Oxford, which were adopted from slides created by Stefan Kreutzer and Ian Horrocks for that course.

Further material has been prepared first by Daniel Borchmann during his time at TU Dresden.

Schedule and Location

All dates will be published on this page (see Dates & Materials above).

  • The weekly lecture sessions will take place on Mondays DS2 (9.20 - 10.50) and Tuesdays DS2 (9.20 - 10.50).
  • The weekly exercise session will take place on Wednesdays DS3 (11.10 - 12.40).
  • All lecture and exercise sessions will take place in room APB/E005.
  • Important: Stay informed about current covid-19 regulations of TU Dresden. In particular, participants need to provide a certificate of vaccination or recovery, or a current negative test.
  • Michael Sipser: Introduction to the Theory of Computation, International Edition; 3rd Edition; Cengage Learning 2013
Introductory text that covers all basic topics in this lecture.
Free lecture notes with a general overview of main results; more detailed than Sipser on oracles and alternation; main reference for randomized computation
  • John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation; Addison Wesley Publishing Company 1979
The Cinderella Book; contains a lot of information not contained in most other books; the hierarchy of undecidable problems as well as Rice' characterization of recognizable properties of recognizable languages are from here.
  • Christos H. Papadimitriou: Computational Complexity; 1995 Addison-Wesley Publishing Company, Inc
Standard reference text for many advanced aspects on complexity theory; the proofs of the Linear Speedup Theorem, the Gap Theorem, and Ladner's Theorem as given in the lecture are from here
  • Sanjeev Arora and Boaz Barak: Computational Complexity: A Modern Approach; Cambridge University Press 2009
Extensive book covering the state of the art of Complexity Theory
  • Michael R. Garey and David S. Johnson: Computers and Intractability; Bell Telephone Laboratories, Inc. 1979
The classical book on Complexity Theory; contains a long list of problems with their complexities

Subscribe to events of this course (icalendar)

Lecture Introduction DS2, October 11, 2021 in APB E008 File
Lecture Turing Machines and Languages DS2, October 12, 2021 in APB E005 File
Lecture Undecidability DS2, October 18, 2021 in APB E005 File
Lecture Undecidability (continued) DS2, October 19, 2021 in APB E005
Exercise Mathematical Foundations, Decidability, and Recognisability DS3, October 20, 2021 in APB E005 File
Lecture Undecidability and Recursion DS2, October 25, 2021 in APB E005 File
Lecture Time Complexity and Polynomial Time DS2, October 26, 2021 in APB E005 File
Exercise Undecidability DS3, October 27, 2021 in APB E005 File
Lecture Time Complexity and Polynomial Time (continued) DS2, November 1, 2021 in APB E005
Lecture NP DS2, November 2, 2021 in APB E005 File
Exercise Rice's Theorem and PTime DS3, November 3, 2021 in APB E005 File
Lecture NP-Completeness DS2, November 8, 2021 in APB E005 File
Lecture NP-Complete Problems DS2, November 9, 2021 in APB E005 File
Exercise NP and NP-Completeness DS3, November 10, 2021 in APB E005 File
Lecture Space Complexity DS2, November 15, 2021 in APB E005 File
Lecture Polynomial Space DS2, November 16, 2021 in APB E005 File
No session Day of Prayer and Repentance (Public Holiday) DS3, November 17, 2021 in APB E005
Lecture Games/Logarithmic Space DS2, November 22, 2021 in APB E005 File
Lecture Games/Logarithmic Space (continued) DS2, November 23, 2021 in APB E005
Exercise NP-Completeness and Time Complexity DS3, November 24, 2021 in APB E005 File
Lecture The Time Hierarchy Theorem DS2, November 29, 2021 in APB E005 File
Exercise Space Complexity DS2, November 30, 2021 in APB E005 File
Lecture Space Hierarchy and Gaps DS2, December 1, 2021 in APB E005 File
Lecture P vs. NP: Ladner's Theorem DS3, December 6, 2021 in APB E005 File
Lecture P vs. NP and Diagonalisation DS2, December 7, 2021 in APB E005 File
Lecture P vs. NP and Diagonalisation (continued) DS3, December 8, 2021 in APB E005
Lecture Alternation DS2, December 13, 2021 in APB E005 File
Lecture The Polynomial Hierarchy DS2, December 14, 2021 in APB E005 File
Exercise Diagonalisation and Alternation DS3, December 15, 2021 in Video conference File
Lecture The Polynomial Hierarchy / Circuit Complexity DS2, December 20, 2021 in APB E005 File
Lecture Questions and Answers DS2, December 21, 2021 in APB E005 File
Exercise Alternation DS3, December 22, 2021 in Video conference File
Lecture Circuits for Parallel Computation DS2, January 10, 2022 in APB E005 File
Lecture Probabilistic Turing Machines DS2, January 11, 2022 in APB E005 File
Exercise Polynomial Hierarchy DS3, January 12, 2022 in APB E005 File
Lecture Probabilistic Complexity Classes (1) DS2, January 17, 2022 in APB E005 File
Lecture Probabilistic Complexity Classes (2) DS2, January 18, 2022 in APB E005 File
Exercise Circuit Complexity DS3, January 19, 2022 in APB E005 File
Lecture Quantum Computing (1) DS2, January 24, 2022 in APB E005
Lecture Quantum Computing (2) DS2, January 25, 2022 in APB E005
Exercise Probabilistic TMs and Complexity Classes DS3, January 26, 2022 in APB E005 File
Lecture Interactive Proof Systems DS2, January 31, 2022 in APB E005
Lecture Summary and Consultation DS2, February 1, 2022 in APB E005
Exercise Randomised Computation and Quantum Computing DS3, February 2, 2022 in APB E005 File


Calendar