Complexity Theory

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Complexity Theory

Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2018

Dozent

Tutor

Umfang (SWS)

  • 4/2/0

Module

Leistungskontrolle

  • Mündliche Prüfung

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

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

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: There will be neither lectures nor exercises on the first week of the semester. Therefore, the first lecture of this course will be on Monday, 15th of October.
  • 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 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 DS2, 15. Oktober 2018 in APB E008 Download Download
Vorlesung Turing Machines and Languages DS2, 16. Oktober 2018 in APB E005 Download Download
Übung Mathematical Foundations, Decidability, and Recognisability DS3, 17. Oktober 2018 in APB E005 Download
Vorlesung Undecidability DS2, 22. Oktober 2018 in APB E008 Download Download
Vorlesung Undecidability and Recursion DS2, 23. Oktober 2018 in APB E005 Download Download
Übung Undecidability and Rice's Theorem DS3, 24. Oktober 2018 in APB E005 Download
Vorlesung Time Complexity and Polynomial Time DS2, 29. Oktober 2018 in APB E008 Download Download
Vorlesung NP DS2, 30. Oktober 2018 in APB E005 Download Download
Vorlesung NP-Completeness DS2, 5. November 2018 in APB E008 Download Download
Vorlesung NP-Complete Problems DS2, 6. November 2018 in APB E005 Download Download
Übung Time Complexity, PTime, and NP DS3, 7. November 2018 in APB E005 Download
Vorlesung Space Complexity DS2, 12. November 2018 in APB E008 Download Download
Vorlesung Polynomial Space DS2, 13. November 2018 in APB E005 Download Download
Übung NP-Completeness and Time Complexity DS3, 14. November 2018 in APB E005 Download
Vorlesung Games/Logarithmic Space DS2, 19. November 2018 in APB E008 Download Download
Vorlesung The Time Hierarchy Theorem DS2, 20. November 2018 in APB E005 Download Download
Vorlesung Space Hierarchy and Gaps DS2, 26. November 2018 in APB E008 Download Download
Vorlesung Space Hierarchy and Gaps (continued) DS2, 27. November 2018 in APB E005
Übung Space Complexity DS3, 28. November 2018 in APB E005 Download
Vorlesung P vs. NP: Ladner's Theorem DS2, 3. Dezember 2018 in APB E008 Download Download
Vorlesung P vs. NP and Diagonalisation DS2, 4. Dezember 2018 in APB E005 Download Download
Übung Diagonalisation DS3, 5. Dezember 2018 in APB E005 Download
Vorlesung Alternation DS2, 10. Dezember 2018 in APB E008 Download Download
Vorlesung The Polynomial Hierarchy DS2, 11. Dezember 2018 in APB E005 Download Download
Übung Diagonalisation and Alternation DS3, 12. Dezember 2018 in APB E005 Download
Vorlesung Circuit Complexity DS2, 17. Dezember 2018 in APB E008
Vorlesung Questions and Answers DS2, 18. Dezember 2018 in APB E005
Übung Alternation and the Polynomial Hierarchy DS3, 19. Dezember 2018 in APB E005
Vorlesung Circuits for Parallel Computation DS2, 7. Januar 2019 in APB E008
Vorlesung Probabilistic Turing Machines DS2, 8. Januar 2019 in APB E005
Übung Circuit Complexity DS3, 9. Januar 2019 in APB E005
Vorlesung Probabilistic Complexity Classes (1) DS2, 14. Januar 2019 in APB E008
Vorlesung Probabilistic Complexity Classes (2) DS2, 15. Januar 2019 in APB E005
Übung Probabilistic TMs and ComplexityClasses DS3, 16. Januar 2019 in APB E005
Vorlesung Quantum Computing (1) DS2, 21. Januar 2019 in APB E008
Vorlesung Quantum Computing (2) DS2, 22. Januar 2019 in APB E005
Übung Probabilistic Complexity Classes (3) DS3, 23. Januar 2019 in APB E005
Vorlesung TBA DS2, 28. Januar 2019 in APB E008
Vorlesung Summary, Outlook, Consultation DS2, 29. Januar 2019 in APB E005
Übung TBA DS3, 30. Januar 2019 in APB E005


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