Foundations of Complexity Theory

Aus International Center for Computational Logic
Wechseln zu:Navigation, Suche

Foundations of Complexity Theory

Lehrveranstaltung mit SWS 2/2/0 (Vorlesung/Übung/Praktikum) in WS 2020

Dozent

Tutor

Umfang (SWS)

  • 2/2/0

Module

Leistungskontrolle

  • Mündliche Prüfung

Vorlesungsreihe

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
  • 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

Schedule and Location

Because of the ongoing COVID-19 pandemic, we are offering this lecture as an online course. Here are all the details:

  • We will host the tutorials as "live sessions" on Tuesdays from 14:50 to 16:20. See the schedule of these sessions under the "Dates and Materials" tab for more information.
  • These "live sessions" will take place in a dedicated BigBlueButton room. To access this room, click on this link 10 minutes before the beginning of each session.
  • Exercise sheets preparing for the tutorials and the final exams will be uploaded at least one week before the tutorial takes place.
  • Every week on Tuesday, we will publish either one video (if there is a tutorial happening on that week) or two videos (if there is none) with the weekly lectures. These videos will be posted on this webpage under the "Dates and Materials" tab.

Legacy

Simliar courses have been taught at TU Dresden by Prof. Dr. Markus Krötzsch in previous years:

Note that the lecture this year is a bit more compact than the ones offered in 2018 and 2019.

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.
  • 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 DS7, 27. Oktober 2020 in Screencast Download 1 Download 2 Download 3
Vorlesung Mathematical Foundations, Decidability, and Recognisability DS8, 27. Oktober 2020 in Screencast Download 1 Download 2 Download 3
Übung Mathematical Foundations, Decidability, and Recognisability DS5, 3. November 2020 in BigBlueButton Download
Vorlesung Time Complexity and Polynomial Time DS8, 3. November 2020 in Screencast
Vorlesung NP DS7, 10. November 2020 in Screencast
Vorlesung NP-Completeness DS8, 10. November 2020 in Screencast
Übung Time Complexity, PTime, and NP DS5, 17. November 2020 in BigBlueButton
Vorlesung NP-Complete Problems DS8, 17. November 2020 in Screencast
Übung NP-Completeness and Time Complexity DS5, 24. November 2020 in BigBlueButton
Vorlesung Space Complexity DS8, 24. November 2020 in Screencast
Vorlesung Polynomial Space DS7, 1. Dezember 2020 in Screencast
Vorlesung Games/Logarithmic Space DS8, 1. Dezember 2020 in Screencast
Übung Space Complexity DS5, 8. Dezember 2020 in BigBlueButton
Vorlesung The Time Hierarchy Theorem DS8, 8. Dezember 2020 in Screencast
Übung Diagonalisation DS5, 15. Dezember 2020 in BigBlueButton
Vorlesung Space Hierarchy and Gaps DS7, 15. Dezember 2020 in Screencast
Vorlesung P vs. NP: Ladner's Theorem DS7, 5. Januar 2021 in Screencast
Vorlesung P vs. NP and Diagonalisation DS8, 5. Januar 2021 in Screencast
Übung Diagonalisation and Alternation DS5, 12. Januar 2021 in BigBlueButton
Vorlesung Alternation DS7, 12. Januar 2021 in Screencast
Übung Alternation and the Polynomial Hierarchy DS5, 19. Januar 2021 in BigBlueButton
Vorlesung The Polynomial Hierarchy DS7, 19. Januar 2021 in Screencast
Übung Circuit Complexity DS5, 26. Januar 2021 in BigBlueButton
Vorlesung Circuit Complexity DS7, 26. Januar 2021 in Screencast
Übung Probabilistic Turing Machines and Complexity Classes DS5, 2. Februar 2021 in BigBlueButton
Vorlesung Probabilistic Turing Machines and Complexity Classes DS7, 2. Februar 2021 in Screencast


Kalender