Complexity Theory

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Complexity Theory

Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2024

Dozent

Tutor

Umfang (SWS)

  • 4/2/0

Module

Leistungskontrolle

  • Mündliche Prüfung

Matrix-Kanal

Vorlesungsreihe


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

Veranstaltungskalender abonnieren (icalendar)

Vorlesung Introduction and Motivation DS2, 14. Oktober 2024 in APB E009 Datei 1 Datei 2
Vorlesung Turing Machines and Languages DS2, 15. Oktober 2024 in APB E005 Datei 1 Datei 2
Vorlesung Undecidability (1) DS2, 21. Oktober 2024 in APB E009 Datei 1 Datei 2
Vorlesung Undecidability (2) DS2, 22. Oktober 2024 in APB E005 Datei 1 Datei 2
Übung Mathematical Foundations, Decidability, and Recognisability DS5, 22. Oktober 2024 in APB E005 Datei
Vorlesung Recursion DS2, 28. Oktober 2024 in APB E009
Vorlesung Time Complexity and Polynomial Time DS2, 29. Oktober 2024 in APB E005 Datei 1 Datei 2
Übung Undecidability DS5, 29. Oktober 2024 in APB E005 Datei
Vorlesung Nondeterministic Polynomial Time DS2, 4. November 2024 in APB E009 Datei 1 Datei 2
Vorlesung NP-Completeness DS2, 5. November 2024 in APB E005 Datei 1 Datei 2
Übung Time Complexity DS5, 5. November 2024 in APB E005 Datei
Vorlesung NP-Complete Problems DS2, 11. November 2024 in APB E009 Datei 1 Datei 2
Übung NP-Completeness DS2, 12. November 2024 in APB E005 Datei
Vorlesung Space Complexity DS2, 18. November 2024 in APB E009 Datei 1 Datei 2
Vorlesung Polynomial Space DS2, 19. November 2024 in APB E005 Datei 1 Datei 2
Konsultation Time and Space Complexity DS5, 19. November 2024 in APB E005
Vorlesung Games/Logarithmic Space DS2, 25. November 2024 in APB E009 Datei 1 Datei 2
Vorlesung Hierarchy Theorems DS2, 26. November 2024 in APB E005 Datei 1 Datei 2
Übung Space Complexity DS5, 26. November 2024 in APB E005 Datei
Vorlesung Space Hierarchy and Gaps DS2, 2. Dezember 2024 in APB E009 Datei 1 Datei 2
Vorlesung P vs. NP: Ladner's Theorem DS2, 3. Dezember 2024 in APB E005 Datei 1 Datei 2
Übung Diagonalisation DS5, 3. Dezember 2024 in APB E005 Datei
Vorlesung P vs. NP and Diagonalisation DS2, 9. Dezember 2024 in APB E009 Datei 1 Datei 2
Vorlesung Alternation DS2, 10. Dezember 2024 in APB E005 Datei 1 Datei 2
Entfällt Lecture/Exercise Synchronization Break DS5, 10. Dezember 2024 in APB E005
Vorlesung The Polynomial Hierarchy DS2, 16. Dezember 2024 in APB E009 Datei 1 Datei 2
Vorlesung Questions and Answers DS2, 17. Dezember 2024 in APB E005 Datei 1 Datei 2
Übung Alternation DS5, 17. Dezember 2024 in APB E005 Datei
Vorlesung Polynomial Hierarchy / Circuit Complexity DS2, 6. Januar 2025 in APB E009
Vorlesung Circuits and Parallel Computation DS2, 7. Januar 2025 in APB E005
Übung Polynomial Hierarchy DS5, 7. Januar 2025 in APB E005 Datei
Vorlesung Probabilistic Turing Machines DS2, 13. Januar 2025 in APB E009
Vorlesung Probabilistic Complexity Classes (1) DS2, 14. Januar 2025 in APB E005
Übung Circuit Complexity DS5, 14. Januar 2025 in APB E005
Vorlesung Probabilistic Complexity Classes (2) DS2, 20. Januar 2025 in APB E009
Vorlesung Quantum Computing (1) DS2, 21. Januar 2025 in APB E005
Übung Randomised Computation DS5, 21. Januar 2025 in APB E005
Vorlesung Quantum Computing (2) DS2, 27. Januar 2025 in APB E009
Vorlesung Interactive Proof Systems (1) DS2, 28. Januar 2025 in APB E005
Übung Quantum Computing (3) DS5, 28. Januar 2025 in APB E005
Vorlesung Interactive Proof Systems (2) DS2, 3. Februar 2025 in APB E009
Vorlesung Summary and Consultation DS2, 4. Februar 2025 in APB E005
Übung Interactive Proof Systems (3) DS5, 4. Februar 2025 in APB E005


Kalender