# 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
 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