Foundations of Complexity Theory
Foundations of Complexity Theory
Course with SWS 2/2/0 (lecture/exercise/practical) in WS 2020
- 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
- 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
Schedule and Location
Because of the ongoing COVID-19 pandemic, we are offering this lecture as an online course. Here are all the details:
- We will host the tutorials as "live sessions" on Tuesdays from 14:50 to 16:20. See the schedule of these sessions under the "Dates and Materials" tab for more information.
- These "live sessions" will take place in a dedicated BigBlueButton room. To access this room, click on this link 10 minutes before the beginning of each session.
- Exercise sheets preparing for the tutorials and the final exams will be uploaded at least one week before the tutorial takes place.
- Every week on Tuesday, we will publish either one video (if there is a tutorial happening on that week) or two videos (if there is none) with the weekly lectures. These videos will be posted on this webpage under the "Dates and Materials" tab.
Simliar courses have been taught at TU Dresden by Prof. Dr. Markus Krötzsch in previous years:
Note that the lecture this year is a bit more compact than the ones offered in 2018 and 2019.
AcknowledgementsThe 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 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||DS7, October 27, 2020 in||Download 1, Download 2, Download 3|
|Lecture||Mathematical Foundations, Decidability, and Recognisability||DS8, October 27, 2020 in||Download 1, Download 2, Download 3|
|Exercise||Mathematical Foundations, Decidability, and Recognisability||DS5, November 3, 2020 in||Download|
|Lecture||Time Complexity and Polynomial Time||DS8, November 3, 2020 in|
|Lecture||NP||DS7, November 10, 2020 in|
|Lecture||NP-Completeness||DS8, November 10, 2020 in|
|Exercise||Time Complexity, PTime, and NP||DS5, November 17, 2020 in|
|Lecture||NP-Complete Problems||DS8, November 17, 2020 in|
|Exercise||NP-Completeness and Time Complexity||DS5, November 24, 2020 in|
|Lecture||Space Complexity||DS8, November 24, 2020 in|
|Lecture||Polynomial Space||DS7, December 1, 2020 in|
|Lecture||Games/Logarithmic Space||DS8, December 1, 2020 in|
|Exercise||Space Complexity||DS5, December 8, 2020 in|
|Lecture||The Time Hierarchy Theorem||DS8, December 8, 2020 in|
|Exercise||Diagonalisation||DS5, December 15, 2020 in|
|Lecture||Space Hierarchy and Gaps||DS7, December 15, 2020 in|
|Lecture||P vs. NP: Ladner's Theorem||DS7, January 5, 2021 in|
|Lecture||P vs. NP and Diagonalisation||DS8, January 5, 2021 in|
|Exercise||Diagonalisation and Alternation||DS5, January 12, 2021 in|
|Lecture||Alternation||DS7, January 12, 2021 in|
|Exercise||Alternation and the Polynomial Hierarchy||DS5, January 19, 2021 in|
|Lecture||The Polynomial Hierarchy||DS7, January 19, 2021 in|
|Exercise||Circuit Complexity||DS5, January 26, 2021 in|
|Lecture||Circuit Complexity||DS7, January 26, 2021 in|
|Exercise||Probabilistic Turing Machines and Complexity Classes||DS5, February 2, 2021 in|
|Lecture||Probabilistic Turing Machines and Complexity Classes||DS7, February 2, 2021 in|