Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2021
- Oral exam
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
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).
- Monday lecture sessions will take place in room APB/E008. All other lecture and exercise sessions will take place in room APB/E005.
- Important: Stay informed about current covid-19 regulations of 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
|Lecture||Introduction||DS2, October 11, 2021 in APB E008|
|Lecture||Turing Machines and Languages||DS2, October 12, 2021 in APB E005|
|Lecture||Undecidability||DS2, October 18, 2021 in APB E008|
|Lecture||Undecidability and Recursion||DS2, October 19, 2021 in APB E005|
|Exercise||Mathematical Foundations, Decidability, and Recognisability||DS3, October 20, 2021 in APB E005|
|Lecture||Time Complexity and Polynomial Time||DS2, October 25, 2021 in APB E008|
|Lecture||NP||DS2, October 26, 2021 in APB E005|
|Exercise||Undecidability and Rice's Theorem||DS3, October 27, 2021 in APB E005|
|Lecture||NP-Completeness||DS2, November 1, 2021 in APB E008|
|Lecture||NP-Complete Problems||DS2, November 2, 2021 in APB E005|
|Exercise||Time Complexity, PTime, and NP||DS3, November 3, 2021 in APB E005|
|Lecture||Space Complexity||DS2, November 8, 2021 in APB E008|
|Lecture||Polynomial Space||DS2, November 9, 2021 in APB E005|
|Exercise||NP-Completeness and Time Complexity||DS3, November 10, 2021 in APB E005|
|Lecture||Games/Logarithmic Space||DS2, November 15, 2021 in APB E008|
|Lecture||The Time Hierarchy Theorem||DS2, November 16, 2021 in APB E005|
|No session||Day of Prayer and Repentance (Public Holiday)||DS3, November 17, 2021 in APB E005|
|Lecture||Space Hierarchy and Gaps||DS2, November 22, 2021 in APB E008|
|Lecture||P vs. NP: Ladner's Theorem||DS2, November 23, 2021 in APB E005|
|Exercise||Space Complexity||DS3, November 24, 2021 in APB E005|
|Lecture||P vs. NP and Diagonalisation||DS2, November 29, 2021 in APB E008|
|Lecture||Alternation||DS2, November 30, 2021 in APB E005|
|Exercise||Diagonalisation||DS3, December 1, 2021 in APB E005|
|Lecture||The Polynomial Hierarchy||DS2, December 6, 2021 in APB E008|
|Lecture||Circuit Complexity||DS2, December 7, 2021 in APB E005|
|Exercise||Diagonalisation and Alternation||DS3, December 8, 2021 in APB E005|
|Lecture||Circuits for Parallel Computation||DS2, December 13, 2021 in APB E008|
|Lecture||Probabilistic Turing Machines||DS2, December 14, 2021 in APB E005|
|Exercise||Alternation and the Polynomial Hierarchy||DS3, December 15, 2021 in APB E005|
|Lecture||Probabilistic Complexity Classes (1)||DS2, December 20, 2021 in APB E008|
|Lecture||Questions and Answers||DS2, December 21, 2021 in APB E005|
|Exercise||Circuit Complexity||DS3, December 22, 2021 in APB E005|
|Lecture||Probabilistic Complexity Classes (2)||DS2, January 10, 2022 in APB E008|
|Lecture||Quantum Computing (1)||DS2, January 11, 2022 in APB E005|
|Exercise||Probabilistic TMs and Complexity Classes||DS3, January 12, 2022 in APB E005|
|Lecture||Quantum Computing (2)||DS2, January 17, 2022 in APB E008|
|Lecture||Interactive Proof Systems||DS2, January 18, 2022 in APB E005|
|Exercise||Probabilistic Complexity Classes (3)||DS3, January 19, 2022 in APB E005|