# Complexity Theory

##### Lehrveranstaltung mit SWS 4/2/0 (Vorlesung/Übung/Praktikum) in WS 2023

Dozent

Tutor

Umfang (SWS)

• 4/2/0

Module

Leistungskontrolle

• Mündliche Prüfung

Matrix-Kanal

Vorlesungsreihe

## Announcements

• Regarding the oral examinations, please register with your respective examination offices and, afterward, make an appointment with Kati Domann.

## Contents

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

### Mode of Teaching and Registration

The course generally does not require a special registration and there is no limit for participants. However, students in programmes that use the Selma system (esp. students in CMS Master) will need to register there to obtain credits. Most of the materials will be freely available world-wide.

### Contact

Besides the regular meetings in the lectur(Habs heute auch nochmal in der Übung erwähnt.) ￼ es and exercise classes, you can also contact the teachers and other students in the public discussion channel on Matrix shown on the side.

### 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) in REC C118 (Recknagelbau C118, Haeckelstraße 3) and Tuesdays DS2 (9.20 - 10.50) in APB E005.
• The weekly exercise session will take place on Wednesdays DS3 (11.10 - 12.40) in APB E005.
• Important: Stay informed about current covid-19 regulations of 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 a 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 and Motivation DS2, 9. Oktober 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Turing Machines and Languages DS2, 10. Oktober 2023 in APB E005 Datei 1,  Datei 2 Vorlesung Undecidability DS2, 16. Oktober 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Undecidability and Recursion DS2, 17. Oktober 2023 in APB E005 Datei 1,  Datei 2 Übung Mathematical Foundations, Decidability, and Recognisability DS3, 18. Oktober 2023 in APB E005 Datei Vorlesung Time Complexity and Polynomial Time (1) DS2, 23. Oktober 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Time Complexity and Polynomial Time (2) DS2, 24. Oktober 2023 in APB E005 Übung Undecidability DS3, 25. Oktober 2023 in APB E005 Datei Vorlesung Nondeterministic Polynomial Time DS2, 30. Oktober 2023 in REC C118 Datei 1,  Datei 2 Entfällt Reformation Day DS2, 31. Oktober 2023 in APB E005 Übung Time Complexity DS3, 1. November 2023 in APB E005 Datei Vorlesung NP-Completeness DS2, 6. November 2023 in REC C118 Datei 1,  Datei 2 Vorlesung NP-Complete Problems DS2, 7. November 2023 in APB E005 Datei 1,  Datei 2 Übung NP-Completeness DS3, 8. November 2023 in APB E005 Datei Vorlesung Space Complexity DS2, 13. November 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Polynomial Space DS2, 14. November 2023 in APB E005 Datei 1,  Datei 2 Übung Space Complexity DS3, 15. November 2023 in APB E005 Datei Vorlesung Games/Logarithmic Space DS2, 20. November 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Hierarchy Theorems DS2, 21. November 2023 in APB E005 Datei 1,  Datei 2 Entfällt Day of Prayer and Repentance DS3, 22. November 2023 in APB E005 Vorlesung Space Hierarchy and Gaps DS2, 27. November 2023 in REC C118 Datei 1,  Datei 2 Vorlesung P vs. NP: Ladner's Theorem DS2, 28. November 2023 in APB E005 Datei 1,  Datei 2 Übung Space Complexity (cont'd) DS3, 29. November 2023 in APB E005 Vorlesung P vs. NP and Diagonalisation DS2, 4. Dezember 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Alternation DS2, 5. Dezember 2023 in APB E005 Datei 1,  Datei 2 Übung Diagonalisation DS3, 6. Dezember 2023 in APB E005 Datei Vorlesung The Polynomial Hierarchy DS2, 11. Dezember 2023 in REC C118 Datei 1,  Datei 2 Vorlesung Polynomial Hierarchy / Circuit Complexity DS2, 12. Dezember 2023 in APB E005 Datei 1,  Datei 2 Übung Alternation DS3, 13. Dezember 2023 in APB E005 Datei Vorlesung Circuits and Parallel Computation DS2, 18. Dezember 2023 in REC C118 Datei 1,  Datei 2 Übung Polynomial Hierarchy DS2, 19. Dezember 2023 in APB E005 Datei Entfällt moved to December 19 DS3, 20. Dezember 2023 in APB E005 Entfällt Christmas Break DS2, 25. Dezember 2023 in REC C118 Entfällt Christmas Break DS2, 26. Dezember 2023 in APB E005 Entfällt Christmas Break DS3, 27. Dezember 2023 in APB E005 Entfällt Christmas Break DS2, 1. Januar 2024 in REC C118 Entfällt Christmas Break DS2, 2. Januar 2024 in APB E005 Entfällt Christmas Break DS3, 3. Januar 2024 in APB E005 Vorlesung Probabilistic Turing Machines DS2, 8. Januar 2024 in REC C118 Datei 1,  Datei 2 Vorlesung Probabilistic Complexity Classes (1) DS2, 9. Januar 2024 in APB E005 Datei 1,  Datei 2 Übung Circuit Complexity DS3, 10. Januar 2024 in APB E005 Datei Vorlesung Probabilistic Complexity Classes (2) DS2, 15. Januar 2024 in REC C118 Datei 1,  Datei 2 Vorlesung Quantum Computing (1) DS2, 16. Januar 2024 in APB E005 Datei 1,  Datei 2 Übung Randomised Computation DS3, 17. Januar 2024 in APB E005 Datei Vorlesung Quantum Computing (2) DS2, 22. Januar 2024 in REC C118 Datei 1,  Datei 2 Vorlesung Interactive Proof Systems (1) DS2, 23. Januar 2024 in APB E005 Datei 1,  Datei 2 Übung Quantum Computing DS3, 24. Januar 2024 in APB E005 Datei Vorlesung Interactive Proof Systems (2) DS2, 29. Januar 2024 in REC C118 Vorlesung Summary and Consultation DS2, 30. Januar 2024 in APB E005 Datei 1,  Datei 2 Übung Quantum Computing (cont.) / Exercises from IPS Lecture DS3, 31. Januar 2024 in APB E005