# Complexity Theory

# Complexity Theory

##### Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2022

**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 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 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

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 APB E008 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.

- 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 | DS2, October 10, 2022 in APB E008 | File |

Lecture | Turing Machines and Languages | DS2, October 11, 2022 in APB E005 | File |

Lecture | Undecidability | DS2, October 17, 2022 in APB E008 | File |

Lecture | Undecidability (continued) | DS2, October 18, 2022 in APB E005 | |

Exercise | Mathematical Foundations, Decidability, and Recognisability | DS3, October 19, 2022 in APB E005 | File |

Lecture | Undecidability and Recursion | DS2, October 24, 2022 in APB E008 | File |

Lecture | Time Complexity and Polynomial Time | DS2, October 25, 2022 in APB E005 | File |

Exercise | Undecidability | DS3, October 26, 2022 in APB E005 | File |

No session | Reformation Day (Public Holiday) | DS2, October 31, 2022 in APB E005 | |

Lecture | Time Complexity and Polynomial Time (continued) | DS2, November 1, 2022 in APB E005 | |

Exercise | Time Complexity | DS3, November 2, 2022 in APB E005 | File |

Lecture | NP | DS2, November 7, 2022 in APB E008 | File |

Lecture | NP-Completeness | DS2, November 8, 2022 in APB E005 | File |

Exercise | NP and NP-Completeness (continued) | DS3, November 9, 2022 in APB E005 | |

Lecture | NP-Complete Problems | DS2, November 14, 2022 in APB E008 | |

Lecture | Space Complexity | DS2, November 15, 2022 in APB E005 | File |

No session | Day of Prayer and Repentance (Public Holiday) | DS3, November 16, 2022 in APB E005 | |

Lecture | Polynomial Space | DS2, November 21, 2022 in APB E008 | File |

Lecture | Polynomial Space (continued) | DS2, November 22, 2022 in APB E005 | |

Exercise | NP-Completeness and Time Complexity | DS3, November 23, 2022 in APB E005 | File |

Lecture | Games/Logarithmic Space | DS2, November 28, 2022 in APB E008 | File |

Exercise | Space Complexity | DS3, November 29, 2022 in APB E005 | File |

Lecture | The Time Hierarchy Theorem | DS2, November 30, 2022 in APB E005 | File |

Lecture | Space Hierarchy and Gaps | DS2, December 5, 2022 in APB E008 | File |

Lecture | P vs. NP: Ladner's Theorem | DS3, December 6, 2022 in APB E005 | File |

Exercise | Space Complexity (continued) | DS3, December 7, 2022 in APB E005 | |

Lecture | P vs. NP and Diagonalisation | DS2, December 12, 2022 in APB E008 | File |

Lecture | P vs. NP and Diagonalisation (continued) | DS3, December 13, 2022 in APB E005 | |

Exercise | Diagonalisation | DS3, December 14, 2022 in APB E005 | File |

Lecture | Alternation | DS2, December 19, 2022 in APB E008 | |

Lecture | The Polynomial Hierarchy | DS2, December 20, 2022 in APB E005 | |

Exercise | Diagonalisation and Alternation | DS3, January 4, 2023 in APB E005 | File |

Lecture | The Polynomial Hierarchy / Circuit Complexity | DS2, January 9, 2023 in APB E008 | |

Lecture | Circuits for Parallel Computation | DS2, January 10, 2023 in APB E005 | |

Exercise | Polynomial Hierarchy | DS3, January 11, 2023 in APB E005 | File |

Lecture | Probabilistic Turing Machines | DS2, January 16, 2023 in APB E008 | |

Lecture | Probabilistic Complexity Classes (1) | DS2, January 17, 2023 in APB E005 | |

Exercise | Alternation (continued) | DS3, January 18, 2023 in APB E005 | |

Lecture | Probabilistic Complexity Classes (2) | DS2, January 23, 2023 in APB E008 | |

Lecture | Quantum Computing (1) | DS2, January 24, 2023 in APB E005 | |

Exercise | Circuit Complexity | DS3, January 25, 2023 in APB E005 | File |

Lecture | Quantum Computing (2) | DS2, January 30, 2023 in APB E008 | |

Lecture | Interactive Proof Systems | DS2, January 31, 2023 in APB E005 | |

Exercise | Probabilistic TMs and Complexity Classes | DS3, February 1, 2023 in APB E005 | File |

### Calendar