Complexity Theory

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Complexity Theory

Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2017

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 are published on this page (see Dates & Materials above)

  • The weekly lecture sessions will take place on Tuesdays DS3 (11:10 to 12:40), and Wednesdays DS6 (16:40 to 18:10).
  • The weekly exercise session will take place on Tuesdays DS5 (14:50 to 16:20).
The first exercise will take place in the second week, i.e., on 17 Oct 2017
  • All sessions will take place in room APB/E005.
  • Important: There will be no lectures or exercises in the third week (24th and 25th Oct 2017)
  • 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 DS3, 10. Oktober 2017 in APB E005 Download Download
Vorlesung Turing Machines and Languages DS6, 11. Oktober 2017 in APB E005 Download Download
Vorlesung Undecidability DS3, 17. Oktober 2017 in APB E005 Download Download
Übung Mathematical Foundations, Decidability, and Recognisability DS5, 17. Oktober 2017 in APB E005 Download
Vorlesung Undecidability and Recursion DS6, 18. Oktober 2017 in APB E005 Download Download
Vorlesung Time Complexity and Polynomial Time DS6, 1. November 2017 in APB E005 Download Download
Vorlesung NP DS3, 7. November 2017 in APB E005 Download Download
Übung Undecidability and Rice's Theorem DS5, 7. November 2017 in APB E005 Download
Vorlesung NP-Completeness DS6, 8. November 2017 in APB E005 Download Download
Vorlesung NP-Complete Problems DS3, 14. November 2017 in APB E005 Download Download
Übung Time Complexity, PTime, and NP DS5, 14. November 2017 in APB E005 Download
Vorlesung Space Complexity DS6, 15. November 2017 in APB E005 Download Download
Vorlesung Polynomial Space DS3, 21. November 2017 in APB E005 Download Download
Übung NP-Completeness and Time Complexity DS5, 21. November 2017 in APB E005 Download
Vorlesung Games/Logarithmic Space DS3, 28. November 2017 in APB E005 Download Download
Übung Space Complexity DS5, 28. November 2017 in APB E005 Download
Vorlesung The Time Hierarchy Theorem DS6, 29. November 2017 in APB E005 Download Download
Vorlesung Space Hierarchy and Gaps DS3, 5. Dezember 2017 in APB E005 Download Download
Übung Log-space Complexity and Diagonalization DS5, 5. Dezember 2017 in APB E005 Download
Vorlesung P vs. NP: Ladner's Theorem DS6, 6. Dezember 2017 in APB E005 Download Download
Vorlesung P vs. NP and Diagonalisation DS3, 12. Dezember 2017 in APB E005 Download Download
Übung Diagonalisation DS5, 12. Dezember 2017 in APB E005 Download
Vorlesung Alternation DS6, 13. Dezember 2017 in APB E005 Download Download
Vorlesung The Polynomial Hierarchy DS3, 19. Dezember 2017 in APB E005 Download Download
Übung Diagonalisation and Alternation DS5, 19. Dezember 2017 in APB E005 Download
Vorlesung Questions and Answers DS6, 20. Dezember 2017 in APB E005 Download Download
Vorlesung Circuit Complexity DS3, 9. Januar 2018 in APB E005 Download Download
Übung Alternation and the Polynomial Hierarchy DS5, 9. Januar 2018 in APB E005 Download
Vorlesung Circuits for Parallel Computation DS6, 10. Januar 2018 in APB E005 Download Download
Vorlesung Probabilistic Turing Machines DS3, 16. Januar 2018 in APB E005 Download Download
Übung Circuit Complexity DS5, 16. Januar 2018 in APB E005 Download
Vorlesung Probabilistic Complexity Classes (1) DS6, 17. Januar 2018 in APB E005 Download Download
Vorlesung Probabilistic Complexity Classes (2) DS3, 23. Januar 2018 in APB E005 Download Download
Übung Probabilistic TMs and ComplexityClasses DS5, 23. Januar 2018 in APB E005 Download
Vorlesung Quantum Computing (1) DS6, 24. Januar 2018 in APB E005 Download Download
Vorlesung Quantum Computing (2) DS3, 30. Januar 2018 in APB E005 Download Download
Übung Probabilistic Complexity Classes (3) DS5, 30. Januar 2018 in APB E005 Download
Vorlesung Summary, Outlook, Consultation DS6, 31. Januar 2018 in APB E005


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