Complexity Theory
From International Center for Computational Logic
Complexity Theory
Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2025
Lecturer
SWS
- 4/2/0
Modules
Examination method
- Oral exam
Lecture series
Contents
This course covers both fundamental concepts and advanced topics in 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 course generally does not require a special registration, and there is no participant limit. However, students in programmes that use the Selma system (esp., students in CMS Master) will need to register there to obtain credits. Most of the materials will be freely available worldwide.
Schedule and Location
This page will publish all dates (see Dates & Materials above).
- The weekly lecture sessions will take place on Mondays DS4 (13.00 - 14.30) in APB E005 and Tuesdays DS3 (11.10 - 12.40) in MER 0E23.
- The weekly exercise session will take place on Tuesday DS5 (14.50 - 16.20), also in APB E005.
Contact
Besides the regular meetings in 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.- 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.
- Erich Grädel: Complexity Theory; Lecture Notes, Winter Term 2009/10. Available online at https://logic.rwth-aachen.de/Teaching/KTQC-WS09/index.html.en
- 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 and Motivation | DS4, October 13, 2025 in APB E005 | File 1, File 2 |
| Lecture | Turing Machines and Languages | DS3, October 14, 2025 in MER 0E23 | File 1, File 2 |
| Lecture | Undecidability | DS4, October 20, 2025 in APB E005 | File 1, File 2 |
| Lecture | Undecidability and Recursion | DS3, October 21, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | Mathematical Foundations, Decidability, and Recognisability | DS5, October 21, 2025 in APB E005 | File |
| Lecture | Time Complexity and Polynomial Time | DS4, October 27, 2025 in APB E005 | File 1, File 2 |
| Lecture | Polynomial-Time Reductions and Nondeterministic Polynomial Time | DS3, October 28, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | Undecidability | DS5, October 28, 2025 in APB E005 | File |
| Lecture | NP-Completeness | DS4, November 3, 2025 in APB E005 | File 1, File 2 |
| Lecture | NP-Complete Problems | DS3, November 4, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | Time Complexity | DS5, November 4, 2025 in APB E005 | File |
| Lecture | Space Complexity and PSPACE | DS4, November 10, 2025 in APB E005 | File 1, File 2 |
| Lecture | Polynomial Space and Games | DS3, November 11, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | NP-Completeness | DS5, November 11, 2025 in APB E005 | File |
| Lecture | Logarithmic Space | DS4, November 17, 2025 in APB E005 | File 1, File 2 |
| Lecture | Hierarchy Theorems | DS3, November 18, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | More NP-Completeness and P-Completeness | DS5, November 18, 2025 in APB E005 | |
| Lecture | Space Hierarchy and Gaps | DS4, November 24, 2025 in APB E005 | File 1, File 2 |
| Lecture | P vs. NP: Ladner's Theorem | DS3, November 25, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | Space Complexity | DS5, November 25, 2025 in APB E005 | File |
| Lecture | P vs. NP and Diagonalisation | DS4, December 1, 2025 in APB E005 | File 1, File 2 |
| Lecture | Alternation | DS3, December 2, 2025 in MER 0E23 | File 1, File 2 |
| Exercise | Time and Space Complexity | DS5, December 2, 2025 in APB E005 | File |
| Lecture | The Polynomial Hierarchy | DS4, December 8, 2025 in APB E005 | |
| Lecture | Polynomial Hierarchy / Circuit Complexity | DS3, December 9, 2025 in MER 0E23 | |
| Exercise | Alternation | DS5, December 9, 2025 in APB E005 | |
| Lecture | Circuits and Parallel Computation | DS4, December 15, 2025 in APB E005 | |
| Lecture | Questions and Answers (before Christmas) | DS3, December 16, 2025 in MER 0E23 | |
| No session | Christmas Break | DS5, December 16, 2025 in APB E005 | |
| Lecture | Probabilistic Turing Machines | DS4, January 5, 2026 in APB E005 | |
| Lecture | Probabilistic Complexity Classes (1) | DS3, January 6, 2026 in MER 0E23 | |
| Exercise | Polynomial Hierarchy | DS5, January 6, 2026 in APB E005 | |
| Lecture | Probabilistic Complexity Classes (2) | DS4, January 12, 2026 in APB E005 | |
| Lecture | Quantum Computing (1) | DS3, January 13, 2026 in MER 0E23 | |
| Exercise | Circuit Complexity | DS5, January 13, 2026 in APB E005 | |
| Lecture | Quantum Computing (2) | DS4, January 19, 2026 in APB E005 | |
| Lecture | Interactive Proof Systems (1) | DS3, January 20, 2026 in MER 0E23 | |
| Exercise | Randomised Computation | DS5, January 20, 2026 in APB E005 | |
| Lecture | TBA | DS4, January 26, 2026 in APB E005 | |
| Lecture | TBA | DS3, January 27, 2026 in MER 0E23 | |
| Exercise | Quantum Computing | DS5, January 27, 2026 in APB E005 | |
| Lecture | TBA | DS4, February 2, 2026 in APB E005 | |
| Lecture | Summary and Consultation | DS3, February 3, 2026 in MER 0E23 | |
| Exercise | TBA | DS5, February 3, 2026 in APB E005 |
Calendar
