Complexity Theory

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Complexity Theory

Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2015

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

Legacy

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.
  • 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 DS4, 14. Oktober 2015 in APB E005 Download
Vorlesung Turing Machines and Languages DS4, 16. Oktober 2015 in APB E005 Download
Vorlesung Undecidability DS2, 20. Oktober 2015 in APB E005
Vorlesung Undecidability DS4, 21. Oktober 2015 in APB E005
Übung Mathematical Foundations DS4, 23. Oktober 2015 in APB E005
Vorlesung Rice's Theorem for Recognizability DS2, 27. Oktober 2015 in APB E005
Vorlesung A Hierarchy of Undecidable Problems DS4, 28. Oktober 2015 in APB E005
Übung Turing Machines and Languages DS4, 30. Oktober 2015 in APB E005
Vorlesung The Recursion theorem DS2, 3. November 2015 in APB E005
Vorlesung Decidability and Logic DS4, 4. November 2015 in APB E005
Übung Undecidability and Rice's Theorem DS4, 6. November 2015 in APB E005
Vorlesung Time Complexity: Polynomial Time DS2, 10. November 2015 in APB E005 Download
Vorlesung Time Complexity: NP DS4, 11. November 2015 in APB E005 Download
Übung The Recursion Theorem and Decidability in Logic DS4, 13. November 2015 in APB E005
Vorlesung Time Complexity: NP Completeness DS2, 17. November 2015 in APB E005 Download Download
Übung Time Complexity DS4, 20. November 2015 in APB E005
Vorlesung NP-Complete Problems DS2, 24. November 2015 in APB E005 Download Download
Vorlesung Space Complexity DS4, 25. November 2015 in APB E005 Download Download
Übung Time Complexity DS4, 27. November 2015 in APB E005
Vorlesung Polynomial Space DS2, 1. Dezember 2015 in APB E005 Download Download
Vorlesung Games/Logarithmic Space DS4, 2. Dezember 2015 in APB E005 Download Download
Übung Space Complexity DS4, 4. Dezember 2015 in APB E005
Vorlesung The Time Hierarchy Theorem DS2, 8. Dezember 2015 in APB E005
Vorlesung More on Hierarchy Theorems, the Gap Theorem DS4, 9. Dezember 2015 in APB E005
Übung Space Complexity DS4, 11. Dezember 2015 in APB E005
Vorlesung Ladner's Theorem DS2, 15. Dezember 2015 in APB E005
Vorlesung The Baker-Gill-Solovay Theorem DS4, 16. Dezember 2015 in APB E005
Übung Mid-Term Consultation DS4, 18. Dezember 2015 in APB E005
Vorlesung Alternating Turing Machines DS2, 5. Januar 2016 in APB E005 Download Download
Vorlesung Alternating Complexity Classes DS4, 6. Januar 2016 in APB E005 Download Download
Übung Diagonalization DS4, 8. Januar 2016 in APB E005
Vorlesung The Polynomial Hierarchy DS2, 12. Januar 2016 in APB E005 Download Download
Vorlesung Circuit Complexity DS4, 13. Januar 2016 in APB E005 Download Download
Übung Alternation DS4, 15. Januar 2016 in APB E005
Vorlesung Circuits for Parallel Computation DS2, 19. Januar 2016 in APB E005 Download Download
Vorlesung Randomized Computation DS4, 20. Januar 2016 in APB E005
Übung Circuit Complexity DS4, 22. Januar 2016 in APB E005
Vorlesung Randomized Computation DS2, 26. Januar 2016 in APB E005
Vorlesung Randomized Computation DS4, 27. Januar 2016 in APB E005
Übung Randomized Computation DS4, 29. Januar 2016 in APB E005
Vorlesung Randomized Computation DS2, 2. Februar 2016 in APB E005
Vorlesung Wrapup and Outlook DS4, 3. Februar 2016 in APB E005
Übung Consultation DS4, 5. Februar 2016 in APB E005


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