Complexity Theory (WS2020)

Foundations of Complexity Theory

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

Dozent

• David Carral

Tutor

• Stephan Mennicke

Umfang (SWS)

• 2/2/0

Module

• CMS-LM-MOC
• CMS-LM-ADV
• INF-B-510
• INF-B-520
• INF-BAS6
• INF-VERT6
• MCL-KR
• MCL-PI
• MCL-TCSL

Leistungskontrolle

• Mündliche Prüfung

Vorlesungsreihe

• Complexity Theory

• Information
• Literatur
• Termine und Unterlagen

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 the Videokonferenz link associated with the tutorial session under "Dates and Materials". The room will be open 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:

• Complexity Theory 2018
• Complexity Theory 2019

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.

• Erich Grädel: Complexity Theory; Lecture Notes, Winter Term 2009/10. Available online at https://logic.rwth-aachen.de/Teaching/KTQC-WS09/index.html.en

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

Kalender