# Complexity Theory

# Complexity Theory

##### Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2024

**Lecturer**

**Tutor**

**SWS**

- 4/2/0

**Modules**

**Examination method**

- Oral exam

**Matrix channel**

**Lecture series**

## 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 participant limit. 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 worldwide.

### Contact

Besides the regular meetings in the lectures 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

This page will publish all dates (see *Dates & Materials* above).

- The weekly lecture sessions will take place on Mondays DS2 (9.20 - 10.50) in APB E009 and Tuesdays DS2 (9.20 - 10.50) in APB E005.
- The weekly exercise session will take place on
**Tuesday DS5 (14.50 - 16.20)**, also 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.

- 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 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.

- The

- 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

Subscribe to events of this course (icalendar)

Lecture | Introduction and Motivation | DS2, October 14, 2024 in APB E009 | |

Lecture | Turing Machines and Languages | DS2, October 15, 2024 in APB E005 | |

Lecture | Undecidability (1) | DS2, October 21, 2024 in APB E009 | |

Lecture | Undecidability (2) | DS2, October 22, 2024 in APB E005 | |

Exercise | Mathematical Foundations, Decidability, and Recognisability | DS5, October 22, 2024 in APB E005 | |

Lecture | Undecidability and Recursion | DS2, October 28, 2024 in APB E009 | |

Lecture | Time Complexity and Polynomial Time (1) | DS2, October 29, 2024 in APB E005 | |

Exercise | Undecidability | DS5, October 29, 2024 in APB E005 | |

Lecture | Time Complexity and Polynomial Time (2) | DS2, November 4, 2024 in APB E009 | |

Lecture | Nondeterministic Polynomial Time | DS2, November 5, 2024 in APB E005 | |

Exercise | Time Complexity | DS5, November 5, 2024 in APB E005 | |

Lecture | NP-Completeness | DS2, November 11, 2024 in APB E009 | |

Lecture | Space Complexity | DS2, November 12, 2024 in APB E005 | |

Exercise | NP-Completeness | DS5, November 12, 2024 in APB E005 | |

Lecture | Polynomial Space | DS2, November 18, 2024 in APB E009 | |

Lecture | Games/Logarithmic Space | DS2, November 19, 2024 in APB E005 | |

Exercise | Space Complexity | DS5, November 19, 2024 in APB E005 | |

Lecture | Hierarchy Theorems | DS2, November 25, 2024 in APB E009 | |

Lecture | Space Hierarchy and Gaps | DS2, November 26, 2024 in APB E005 | |

Exercise | Time and Space Complexity (1) | DS5, November 26, 2024 in APB E005 | |

Lecture | P vs. NP: Ladner's Theorem | DS2, December 2, 2024 in APB E009 | |

Lecture | P vs. NP and Diagonalisation | DS2, December 3, 2024 in APB E005 | |

Exercise | Time and Space Complexity (2) | DS5, December 3, 2024 in APB E005 | |

Lecture | Alternation | DS2, December 9, 2024 in APB E009 | |

Lecture | The Polynomial Hierarchy | DS2, December 10, 2024 in APB E005 | |

Exercise | Diagonalisation | DS5, December 10, 2024 in APB E005 | |

Lecture | Polynomial Hierarchy / Circuit Complexity | DS2, December 16, 2024 in APB E009 | |

Lecture | Questions and Answers | DS2, December 17, 2024 in APB E005 | |

Exercise | Alternation | DS5, December 17, 2024 in APB E005 | |

No session | Christmas Break | DS2, December 23, 2024 in APB E005 | |

No session | Christmas Break | DS2, December 24, 2024 in APB E005 | |

No session | Christmas Break | DS5, December 24, 2024 in APB E005 | |

No session | Christmas Break | DS2, December 30, 2024 in APB E005 | |

No session | Christmas Break | DS2, December 31, 2024 in APB E005 | |

No session | Christmas Break | DS5, December 31, 2024 in APB E005 | |

Lecture | Circuits and Parallel Computation | DS2, January 6, 2025 in APB E009 | |

Lecture | Probabilistic Turing Machines | DS2, January 7, 2025 in APB E005 | |

Exercise | Polynomial Hierarchy | DS5, January 7, 2025 in APB E005 | |

Lecture | Probabilistic Complexity Classes (1) | DS2, January 13, 2025 in APB E009 | |

Lecture | Probabilistic Complexity Classes (2) | DS2, January 14, 2025 in APB E005 | |

Exercise | Circuit Complexity | DS5, January 14, 2025 in APB E005 | |

Lecture | Quantum Computing (1) | DS2, January 20, 2025 in APB E009 | |

Lecture | Quantum Computing (2) | DS2, January 21, 2025 in APB E005 | |

Exercise | Randomised Computation | DS5, January 21, 2025 in APB E005 | |

Lecture | Interactive Proof Systems (1) | DS2, January 27, 2025 in APB E009 | |

Lecture | Interactive Proof Systems (2) | DS2, January 28, 2025 in APB E005 | |

Exercise | Quantum Computing (3) | DS5, January 28, 2025 in APB E005 | |

Lecture | Summary and Consultation (1) | DS2, February 3, 2025 in APB E009 | |

Lecture | Summary and Consultation (2) | DS2, February 4, 2025 in APB E005 | |

Exercise | Interactive Proof Systems (3) | DS5, February 4, 2025 in APB E005 |

### Calendar