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{{Vorlesung
{{Vorlesung
|Title=Complexity Theory
|Title=Complexity Theory
|Research group=Automatentheorie; Knowledge Systems
|Research group=Automatentheorie; Wissensbasierte Systeme
|Lecturers=Daniel Borchmann, Markus Krötzsch
|Lecturers=Daniel Borchmann; Markus Krötzsch
|Tutors=Daniel Borchmann
|Tutors=Daniel Borchmann
|Term=WS
|Term=WS
|Year=2015
|Year=2015
|Module=INF-BAS6, INF-VERT6, INF-BAS6, INF-VERT6, MCL-TCSL, MCL-PI, MCL-KR
|Lecture series=Complexity Theory
|Module=INF-BAS6, INF-VERT6, MCL-TCSL, MCL-PI, MCL-KR
|SWSLecture=4
|SWSLecture=4
|SWSExercise=2
|SWSExercise=2
Zeile 12: Zeile 13:
|Exam type=mündliche Prüfung
|Exam type=mündliche Prüfung
|Description=This course covers the fundamental concepts as well as advanced topics of complexity theory.
|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
===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 [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.
|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=2015/10/14
|DS=DS4
|Download=Lecture-01-motivation handout.pdf,Lecture-01-motivation handout.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Turing Machines and Languages
|Room=APB E005
|Date=2015/10/16
|DS=DS4
|Download=Lecture-02-turing-machines handout.pdf,Lecture-02-turing-machines handout.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Undecidability
|Room=APB E005
|Date=2015/10/20
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Undecidability
|Room=APB E005
|Date=2015/10/21
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Mathematical Foundations
|Room=APB E005
|Date=2015/10/23
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Rice's Theorem for Recognizability
|Room=APB E005
|Date=2015/10/27
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=A Hierarchy of Undecidable Problems
|Room=APB E005
|Date=2015/10/28
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Turing Machines and Languages
|Room=APB E005
|Date=2015/10/30
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Recursion theorem
|Room=APB E005
|Date=2015/11/03
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Decidability and Logic
|Room=APB E005
|Date=2015/11/04
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Undecidability and Rice's Theorem
|Room=APB E005
|Date=2015/11/06
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Time Complexity: Polynomial Time
|Room=APB E005
|Date=2015/11/10
|DS=DS2
|Download=Lecture-06-time handout.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Time Complexity: NP
|Room=APB E005
|Date=2015/11/11
|DS=DS4
|Download=Lecture-07-np handout.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=The Recursion Theorem and Decidability in Logic
|Room=APB E005
|Date=2015/11/13
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Time Complexity: NP Completeness
|Room=APB E005
|Date=2015/11/17
|DS=DS2
|Download=CT2015-lecture-08-npcompl.pdf,CT2015-lecture-08-npcompl-handout.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Time Complexity
|Room=APB E005
|Date=2015/11/20
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=NP-Complete Problems
|Room=APB E005
|Date=2015/11/24
|DS=DS2
|Download=CT2015-lecture-09-np-problems.pdf,CT2015-lecture-09-np-problems-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Space Complexity
|Room=APB E005
|Date=2015/11/25
|DS=DS4
|Download=CT2015-lecture-10-space.pdf,CT2015-lecture-10-space-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Time Complexity
|Room=APB E005
|Date=2015/11/27
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Polynomial Space
|Room=APB E005
|Date=2015/12/01
|DS=DS2
|Download=CT2015-lecture-11-pspace.pdf,CT2015-lecture-11-pspace-nup.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Games/Logarithmic Space
|Room=APB E005
|Date=2015/12/02
|DS=DS4
|Download=CT2015-lecture-12-logspace.pdf,CT2015-lecture-12-logspace-nup.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Space Complexity
|Room=APB E005
|Date=2015/12/04
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Time Hierarchy Theorem
|Room=APB E005
|Date=2015/12/08
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=More on Hierarchy Theorems, the Gap Theorem
|Room=APB E005
|Date=2015/12/09
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Space Complexity
|Room=APB E005
|Date=2015/12/11
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Ladner's Theorem
|Room=APB E005
|Date=2015/12/15
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Baker-Gill-Solovay Theorem
|Room=APB E005
|Date=2015/12/16
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Mid-Term Consultation
|Room=APB E005
|Date=2015/12/18
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Alternating Turing Machines
|Room=APB E005
|Date=2016/01/05
|DS=DS2
|Download=CT2015-lecture-14-alternation.pdf,CT2015-lecture-14-alternation-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Alternating Complexity Classes
|Room=APB E005
|Date=2016/01/06
|DS=DS4
|Download=CT2015-lecture-15-alternation.pdf,CT2015-lecture-15-alternation-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Diagonalization
|Room=APB E005
|Date=2016/01/08
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=The Polynomial Hierarchy
|Room=APB E005
|Date=2016/01/12
|DS=DS2
|Download=CT2015-lecture-16-ph.pdf,CT2015-lecture-16-ph-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Circuit Complexity
|Room=APB E005
|Date=2016/01/13
|DS=DS4
|Download=CT2015-lecture-17-circuits.pdf,CT2015-lecture-17-circuits-2x2.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Alternation
|Room=APB E005
|Date=2016/01/15
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Circuits for Parallel Computation
|Room=APB E005
|Date=2016/01/19
|DS=DS2
|Download=CT2015-lecture-18-nc.pdf,CT2015-lecture-18-nc-nup.pdf
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Randomized Computation
|Room=APB E005
|Date=2016/01/20
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Circuit Complexity
|Room=APB E005
|Date=2016/01/22
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Randomized Computation
|Room=APB E005
|Date=2016/01/26
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Randomized Computation
|Room=APB E005
|Date=2016/01/27
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Randomized Computation
|Room=APB E005
|Date=2016/01/29
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Randomized Computation
|Room=APB E005
|Date=2016/02/02
|DS=DS2
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Übung
|Title=Consultation
|Room=APB E005
|Date=2016/02/05
|DS=DS4
}}
{{Vorlesung Zeiten
|Lehrveranstaltungstype=Vorlesung
|Title=Wrapup and Outlook
|Room=APB E005
|Date=2016/02/03
|DS=DS4
}}
}}

Aktuelle Version vom 26. Oktober 2020, 18:53 Uhr

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

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

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 Datei 1,  Datei 2
Vorlesung Turing Machines and Languages DS4, 16. Oktober 2015 in APB E005 Datei 1,  Datei 2
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 Datei
Vorlesung Time Complexity: NP DS4, 11. November 2015 in APB E005 Datei
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 Datei 1,  Datei 2
Übung Time Complexity DS4, 20. November 2015 in APB E005
Vorlesung NP-Complete Problems DS2, 24. November 2015 in APB E005 Datei 1,  Datei 2
Vorlesung Space Complexity DS4, 25. November 2015 in APB E005 Datei 1,  Datei 2
Übung Time Complexity DS4, 27. November 2015 in APB E005
Vorlesung Polynomial Space DS2, 1. Dezember 2015 in APB E005 Datei 1,  Datei 2
Vorlesung Games/Logarithmic Space DS4, 2. Dezember 2015 in APB E005 Datei 1,  Datei 2
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 Datei 1,  Datei 2
Vorlesung Alternating Complexity Classes DS4, 6. Januar 2016 in APB E005 Datei 1,  Datei 2
Übung Diagonalization DS4, 8. Januar 2016 in APB E005
Vorlesung The Polynomial Hierarchy DS2, 12. Januar 2016 in APB E005 Datei 1,  Datei 2
Vorlesung Circuit Complexity DS4, 13. Januar 2016 in APB E005 Datei 1,  Datei 2
Übung Alternation DS4, 15. Januar 2016 in APB E005
Vorlesung Circuits for Parallel Computation DS2, 19. Januar 2016 in APB E005 Datei 1,  Datei 2
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


Kalender

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