Complexity Theory

From International Center for Computational Logic

Complexity Theory

Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2024

News

  • if you want to take an exam this semester, please register with your examination office (presumably done though Selma)
  • the preferred, yet not guaranteed, examination date is 11th February 2024; please contact Kati Domann for scheduling a date and time

Contents

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 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.

Examination Dates and Mode

The registration and scheduling for exams depend on the module they are for. All students need to contact the KBS secretary to arrange time slots for their oral exams, or the secretary of another chair in the case of another main examiner. The preferred date for oral exams is 11th February 2025'. Although we cannot generally accommodate wishes for specific dates, students with justified time constraints during the examination period can still let us know as early as possible.

Students taking Knowledge Graphs as part of any other module should contact the secretary of the main examiner to find a date. If the main examiner is Markus Krötzsch, the preferred date will also be 11th February 2025.

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

This page will publish all dates (see Dates & Materials above).

  • 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 and Motivation DS2, October 14, 2024 in APB E009 File 1 File 2
Lecture Turing Machines and Languages DS2, October 15, 2024 in APB E005 File 1 File 2
Lecture Undecidability (1) DS2, October 21, 2024 in APB E009 File 1 File 2
Lecture Undecidability (2) DS2, October 22, 2024 in APB E005 File 1 File 2
Exercise Mathematical Foundations, Decidability, and Recognisability DS5, October 22, 2024 in APB E005 File
Lecture Recursion DS2, October 28, 2024 in APB E009
Lecture Time Complexity and Polynomial Time DS2, October 29, 2024 in APB E005 File 1 File 2
Exercise Undecidability DS5, October 29, 2024 in APB E005 File
Lecture Nondeterministic Polynomial Time DS2, November 4, 2024 in APB E009 File 1 File 2
Lecture NP-Completeness DS2, November 5, 2024 in APB E005 File 1 File 2
Exercise Time Complexity DS5, November 5, 2024 in APB E005 File
Lecture NP-Complete Problems DS2, November 11, 2024 in APB E009 File 1 File 2
Exercise NP-Completeness DS2, November 12, 2024 in APB E005 File
Lecture Space Complexity DS2, November 18, 2024 in APB E009 File 1 File 2
Lecture Polynomial Space DS2, November 19, 2024 in APB E005 File 1 File 2
Consultation Time and Space Complexity DS5, November 19, 2024 in APB E005
Lecture Games/Logarithmic Space DS2, November 25, 2024 in APB E009 File 1 File 2
Lecture Hierarchy Theorems DS2, November 26, 2024 in APB E005 File 1 File 2
Exercise Space Complexity DS5, November 26, 2024 in APB E005 File
Lecture Space Hierarchy and Gaps DS2, December 2, 2024 in APB E009 File 1 File 2
Lecture P vs. NP: Ladner's Theorem DS2, December 3, 2024 in APB E005 File 1 File 2
Exercise Diagonalisation DS5, December 3, 2024 in APB E005 File
Lecture P vs. NP and Diagonalisation DS2, December 9, 2024 in APB E009 File 1 File 2
Lecture Alternation DS2, December 10, 2024 in APB E005 File 1 File 2
No session Lecture/Exercise Synchronization Break DS5, December 10, 2024 in APB E005
Lecture The Polynomial Hierarchy DS2, December 16, 2024 in APB E009 File 1 File 2
Lecture Questions and Answers DS2, December 17, 2024 in APB E005 File 1 File 2
Exercise Alternation DS5, December 17, 2024 in APB E005 File
Lecture Polynomial Hierarchy / Circuit Complexity DS2, January 6, 2025 in APB E009
Lecture Circuits and Parallel Computation DS2, January 7, 2025 in APB E005
Exercise Polynomial Hierarchy DS5, January 7, 2025 in APB E005 File
Lecture Probabilistic Turing Machines DS2, January 13, 2025 in APB E009
Lecture Probabilistic Complexity Classes (1) DS2, January 14, 2025 in APB E005
Exercise Circuit Complexity DS5, January 14, 2025 in APB E005
Lecture Probabilistic Complexity Classes (2) DS2, January 20, 2025 in APB E009
Lecture Quantum Computing (1) DS2, January 21, 2025 in APB E005
Exercise Randomised Computation DS5, January 21, 2025 in APB E005
Lecture Quantum Computing (2) DS2, January 27, 2025 in APB E009
Lecture Interactive Proof Systems (1) DS2, January 28, 2025 in APB E005
Exercise Quantum Computing (3) DS5, January 28, 2025 in APB E005
Lecture Interactive Proof Systems (2) DS2, February 3, 2025 in APB E009
Lecture Summary and Consultation DS2, February 4, 2025 in APB E005
Exercise Interactive Proof Systems (3) DS5, February 4, 2025 in APB E005


Calendar