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|Title=Complexity Theory
|Title=Complexity Theory
|Research group=Wissensbasierte Systeme
|Research group=Wissensbasierte Systeme
|Lecturers=Markus Krötzsch; David Carral
|Lecturers=Markus Krötzsch
|Tutors=David Carral
|Term=WS
|Term=WS
|Year=2019
|Year=2019
|SWSLecture=0
|Lecture series=Complexity Theory
|SWSExercise=0
|Module=INF-B-510, INF-B-520, INF-BAS6, INF-VERT6, MCL-TCSL, MCL-PI, MCL-KR,INF-E-3,INF-PM-FOR
|SWSLecture=4
|SWSExercise=2
|SWSPractical=0
|SWSPractical=0
|Exam type=mündliche Prüfung
|Exam type=mündliche Prüfung
|Description=Uh
|Description=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 <!--; NTIME(n) ⊄ TISP(n¹·², n⁰·²)-->
* '''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 [http://logic.las.tu-berlin.de/Members/Kreutzer/ Stefan Kreutzer] and [http://www.cs.ox.ac.uk/people/ian.horrocks/ Ian Horrocks] for that course.
 
Further material has been prepared first by [[Daniel Borchmann/en|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.
* <b>Important:</b> as indicated in the class calendar below, the lecture sessions start on the 15th of October. Furthermore, there will be no exercise sessions on the 6th, 13th, and 20th of November.
|Literature=* '''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
 
<!--* Neil Immerman: ''Descriptive Complexity''; Springer Verlag 1999-->
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Introduction
|Room=APB E005
|Date=2019/10/15
|DS=DS2
|Download=CT2019-Lecture-01-overlay.pdf,CT2019-Lecture-01-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Mathematical Foundations, Decidability, and Recognisability
|Room=APB E005
|Date=2019/10/16
|DS=DS3
|Download=CT-WS2019-exercise-01.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Turing Machines and Languages
|Room=APB E008
|Date=2019/10/21
|DS=DS2
|Download=CT2019-Lecture-02-overlay.pdf,CT2019-Lecture-02-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Undecidability
|Room=APB E005
|Date=2019/10/22
|DS=DS2
|Download=CT2019-Lecture-03-overlay.pdf,CT2019-Lecture-03-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Undecidability and Rice's Theorem
|Room=APB E005
|Date=2019/10/23
|DS=DS3
|Download=CT-WS2019-exercise-02.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Undecidability and Recursion
|Room=APB E008
|Date=2019/10/28
|DS=DS2
|Download=CT2019-Lecture-04-overlay.pdf,CT2019-Lecture-04-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Time Complexity and Polynomial Time
|Room=APB E005
|Date=2019/10/29
|DS=DS2
|Download=CT2019-Lecture-05-overlay.pdf,CT2019-Lecture-05-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=NP
|Room=APB E008
|Date=2019/10/30
|DS=DS3
|Download=CT2019-Lecture-06-overlay.pdf,CT2019-Lecture-06-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=NP-Completeness
|Room=APB E008
|Date=2019/11/04
|DS=DS2
|Download=CT2019-Lecture-07-overlay.pdf,CT2019-Lecture-07-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=NP-Complete Problems
|Room=APB E005
|Date=2019/11/05
|DS=DS2
|Download=CT2019-Lecture-08-overlay.pdf,CT2019-Lecture-08-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Time Complexity, PTime, and NP
|Room=APB E008
|Date=2019/11/11
|DS=DS2
|Download=CT-WS2019-exercise-03.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Space Complexity
|Room=APB E005
|Date=2019/11/12
|DS=DS2
|Download=CT2019-Lecture-09-overlay.pdf,CT2019-Lecture-09-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Polynomial Space
|Room=APB E008
|Date=2019/11/18
|DS=DS2
|Download=CT2019-Lecture-10-overlay.pdf,CT2019-Lecture-10-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Games/Logarithmic Space
|Room=APB E005
|Date=2019/11/19
|DS=DS2
|Download=CT2019-Lecture-11-overlay.pdf,CT2019-Lecture-11-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Time Hierarchy Theorem
|Room=APB E008
|Date=2019/11/25
|DS=DS2
|Download=CT2019-Lecture-12-overlay.pdf,CT2019-Lecture-12-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Space Hierarchy and Gaps
|Room=APB E005
|Date=2019/11/26
|DS=DS2
|Download=CT2019-Lecture-13-overlay.pdf,CT2019-Lecture-13-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=NP-Completeness and Time Complexity
|Room=APB E005
|Date=2019/11/27
|DS=DS3
|Download=CT-WS2019-exercise-04-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=P vs. NP: Ladner's Theorem
|Room=APB E008
|Date=2019/12/02
|DS=DS2
|Download=CT2019-Lecture-14-overlay.pdf,CT2019-Lecture-14-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=P vs. NP and Diagonalisation
|Room=APB E005
|Date=2019/12/03
|DS=DS2
|Download=CT2019-Lecture-15-overlay.pdf,CT2019-Lecture-15-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Space Complexity
|Room=APB E005
|Date=2019/12/04
|DS=DS3
|Download=CT-WS2019-exercise-05.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Alternation
|Room=APB E008
|Date=2019/12/09
|DS=DS2
|Download=CT2019-Lecture-16-overlay.pdf,CT2019-Lecture-16-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Polynomial Hierarchy
|Room=APB E005
|Date=2019/12/10
|DS=DS2
|Download=CT2019-Lecture-17-overlay.pdf,CT2019-Lecture-17-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Diagonalisation
|Room=APB E005
|Date=2019/12/11
|DS=DS3
|Download=CT-WS2019-exercise-06-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Questions and Answers
|Room=APB E008
|Date=2019/12/16
|DS=DS2
|Download=CT2019-Lecture-18-overlay.pdf,CT2019-Lecture-18-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Diagonalisation and Alternation
|Room=APB E005
|Date=2019/12/17
|DS=DS2
|Download=CT-WS2019-exercise-07-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=---
|Room=APB E005
|Date=2019/12/18
|DS=DS3
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Circuit Complexity
|Room=APB E008
|Date=2020/01/06
|DS=DS2
|Download=CT2019-Lecture-19-overlay.pdf,CT2019-Lecture-19-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Circuits for Parallel Computation
|Room=APB E005
|Date=2020/01/07
|DS=DS2
|Download=CT2019-Lecture-20-overlay.pdf,CT2019-Lecture-20-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Alternation and the Polynomial Hierarchy
|Room=APB E005
|Date=2020/01/08
|DS=DS3
|Download=CT-WS2019-exercise-08-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Probabilistic Turing Machines
|Room=APB E008
|Date=2020/01/13
|DS=DS2
|Download=CT2019-Lecture-21-overlay.pdf,CT2019-Lecture-21-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Probabilistic Complexity Classes (1)
|Room=APB E005
|Date=2020/01/14
|DS=DS2
|Download=CT2019-Lecture-22-overlay.pdf,CT2019-Lecture-22-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Circuit Complexity
|Room=APB E005
|Date=2020/01/15
|DS=DS3
|Download=CT-WS2019-exercise-09-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Probabilistic Complexity Classes (2)
|Room=APB E008
|Date=2020/01/20
|DS=DS2
|Download=CT2019-Lecture-23-overlay.pdf,CT2019-Lecture-23-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Quantum Computing (1)
|Room=APB E005
|Date=2020/01/21
|DS=DS2
|Download=CT2019-Lecture-24-overlay.pdf,CT2019-Lecture-24-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Probabilistic TMs and ComplexityClasses
|Room=APB E005
|Date=2020/01/22
|DS=DS3
|Download=CT-WS2019-exercise-10-2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Quantum Computing (2)
|Room=APB E008
|Date=2020/01/27
|DS=DS2
|Download=CT2019-Lecture-25-overlay.pdf,CT2019-Lecture-25-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Entfällt
|Title=(no lecture)
|Room=APB E005
|Date=2020/01/28
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Interactive Proof Systems
|Room=APB E005
|Date=2020/02/03
|DS=DS2
|Download=CT2019-Lecture-26-overlay.pdf,CT2019-Lecture-26-print.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Summary, Outlook, Consultation
|Room=APB E005
|Date=2020/02/04
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Probabilistic Complexity Classes (3)
|Room=APB E005
|Date=2020/02/05
|DS=DS3
|Download=CT-WS2019-exercise-11-2.pdf
}}
}}

Aktuelle Version vom 11. Oktober 2021, 18:56 Uhr

Complexity Theory

Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2019

Dozent

Tutor

Umfang (SWS)

  • 4/2/0

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
  • 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: as indicated in the class calendar below, the lecture sessions start on the 15th of October. Furthermore, there will be no exercise sessions on the 6th, 13th, and 20th of November.
  • 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 2019 in APB E005 Datei 1,  Datei 2
Übung Mathematical Foundations, Decidability, and Recognisability DS3, 16. Oktober 2019 in APB E005 Datei
Vorlesung Turing Machines and Languages DS2, 21. Oktober 2019 in APB E008 Datei 1,  Datei 2
Vorlesung Undecidability DS2, 22. Oktober 2019 in APB E005 Datei 1,  Datei 2
Übung Undecidability and Rice's Theorem DS3, 23. Oktober 2019 in APB E005 Datei
Vorlesung Undecidability and Recursion DS2, 28. Oktober 2019 in APB E008 Datei 1,  Datei 2
Vorlesung Time Complexity and Polynomial Time DS2, 29. Oktober 2019 in APB E005 Datei 1,  Datei 2
Vorlesung NP DS3, 30. Oktober 2019 in APB E008 Datei 1,  Datei 2
Vorlesung NP-Completeness DS2, 4. November 2019 in APB E008 Datei 1,  Datei 2
Vorlesung NP-Complete Problems DS2, 5. November 2019 in APB E005 Datei 1,  Datei 2
Übung Time Complexity, PTime, and NP DS2, 11. November 2019 in APB E008 Datei
Vorlesung Space Complexity DS2, 12. November 2019 in APB E005 Datei 1,  Datei 2
Vorlesung Polynomial Space DS2, 18. November 2019 in APB E008 Datei 1,  Datei 2
Vorlesung Games/Logarithmic Space DS2, 19. November 2019 in APB E005 Datei 1,  Datei 2
Vorlesung The Time Hierarchy Theorem DS2, 25. November 2019 in APB E008 Datei 1,  Datei 2
Vorlesung Space Hierarchy and Gaps DS2, 26. November 2019 in APB E005 Datei 1,  Datei 2
Übung NP-Completeness and Time Complexity DS3, 27. November 2019 in APB E005 Datei
Vorlesung P vs. NP: Ladner's Theorem DS2, 2. Dezember 2019 in APB E008 Datei 1,  Datei 2
Vorlesung P vs. NP and Diagonalisation DS2, 3. Dezember 2019 in APB E005 Datei 1,  Datei 2
Übung Space Complexity DS3, 4. Dezember 2019 in APB E005 Datei
Vorlesung Alternation DS2, 9. Dezember 2019 in APB E008 Datei 1,  Datei 2
Vorlesung The Polynomial Hierarchy DS2, 10. Dezember 2019 in APB E005 Datei 1,  Datei 2
Übung Diagonalisation DS3, 11. Dezember 2019 in APB E005 Datei
Vorlesung Questions and Answers DS2, 16. Dezember 2019 in APB E008 Datei 1,  Datei 2
Übung Diagonalisation and Alternation DS2, 17. Dezember 2019 in APB E005 Datei
Übung --- DS3, 18. Dezember 2019 in APB E005
Vorlesung Circuit Complexity DS2, 6. Januar 2020 in APB E008 Datei 1,  Datei 2
Vorlesung Circuits for Parallel Computation DS2, 7. Januar 2020 in APB E005 Datei 1,  Datei 2
Übung Alternation and the Polynomial Hierarchy DS3, 8. Januar 2020 in APB E005 Datei
Vorlesung Probabilistic Turing Machines DS2, 13. Januar 2020 in APB E008 Datei 1,  Datei 2
Vorlesung Probabilistic Complexity Classes (1) DS2, 14. Januar 2020 in APB E005 Datei 1,  Datei 2
Übung Circuit Complexity DS3, 15. Januar 2020 in APB E005 Datei
Vorlesung Probabilistic Complexity Classes (2) DS2, 20. Januar 2020 in APB E008 Datei 1,  Datei 2
Vorlesung Quantum Computing (1) DS2, 21. Januar 2020 in APB E005 Datei 1,  Datei 2
Übung Probabilistic TMs and ComplexityClasses DS3, 22. Januar 2020 in APB E005 Datei
Vorlesung Quantum Computing (2) DS2, 27. Januar 2020 in APB E008 Datei 1,  Datei 2
Entfällt (no lecture) DS2, 28. Januar 2020 in APB E005
Vorlesung Interactive Proof Systems DS2, 3. Februar 2020 in APB E005 Datei 1,  Datei 2
Vorlesung Summary, Outlook, Consultation DS2, 4. Februar 2020 in APB E005
Übung Probabilistic Complexity Classes (3) DS3, 5. Februar 2020 in APB E005 Datei


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