# Foundations of Complexity Theory

From International Center for Computational Logic

# Foundations of Complexity Theory

##### Course with SWS 2/2/0 (lecture/exercise/practical) in WS 2020

**Lecturer**

**Tutor**

**SWS**

- 2/2/0

**Modules**

**Examination method**

- Oral exam

**Lecture series**

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**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

### Schedule and Location

Because of the ongoing COVID-19 pandemic, we are offering this lecture as an online course. Here are all the details:

- We will host the tutorials as "live sessions" on Tuesdays from 14:50 to 16:20. See the schedule of these sessions under the "Dates and Materials" tab for more information.
- These "live sessions" will take place in a dedicated BigBlueButton room. To access this room, click on this link 10 minutes before the beginning of each session.
- Exercise sheets preparing for the tutorials and the final exams will be uploaded at least one week before the tutorial takes place.
- Every week on Tuesday, we will publish either one video (if there is a tutorial happening on that week) or two videos (if there is none) with the weekly lectures. These videos will be posted on this webpage under the "Dates and Materials" tab.

### Legacy

Simliar courses have been taught at TU Dresden by Prof. Dr. Markus Krötzsch in previous years:

Note that the lecture this year is a bit more compact than the ones offered in 2018 and 2019.

### 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.

- 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.

- 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 | DS7, October 27, 2020 in Screencast | Download 1, Download 2, Download 3 |

Lecture | Mathematical Foundations, Decidability, and Recognisability | DS8, October 27, 2020 in Screencast | Download 1, Download 2, Download 3 |

Exercise | Mathematical Foundations, Decidability, and Recognisability | DS5, November 3, 2020 in BigBlueButton | Download |

Lecture | Time Complexity and Polynomial Time | DS8, November 3, 2020 in Screencast | |

Lecture | NP | DS7, November 10, 2020 in Screencast | |

Lecture | NP-Completeness | DS8, November 10, 2020 in Screencast | |

Exercise | Time Complexity, PTime, and NP | DS5, November 17, 2020 in BigBlueButton | |

Lecture | NP-Complete Problems | DS8, November 17, 2020 in Screencast | |

Exercise | NP-Completeness and Time Complexity | DS5, November 24, 2020 in BigBlueButton | |

Lecture | Space Complexity | DS8, November 24, 2020 in Screencast | |

Lecture | Polynomial Space | DS7, December 1, 2020 in Screencast | |

Lecture | Games/Logarithmic Space | DS8, December 1, 2020 in Screencast | |

Exercise | Space Complexity | DS5, December 8, 2020 in BigBlueButton | |

Lecture | The Time Hierarchy Theorem | DS8, December 8, 2020 in Screencast | |

Exercise | Diagonalisation | DS5, December 15, 2020 in BigBlueButton | |

Lecture | Space Hierarchy and Gaps | DS7, December 15, 2020 in Screencast | |

Lecture | P vs. NP: Ladner's Theorem | DS7, January 5, 2021 in Screencast | |

Lecture | P vs. NP and Diagonalisation | DS8, January 5, 2021 in Screencast | |

Exercise | Diagonalisation and Alternation | DS5, January 12, 2021 in BigBlueButton | |

Lecture | Alternation | DS7, January 12, 2021 in Screencast | |

Exercise | Alternation and the Polynomial Hierarchy | DS5, January 19, 2021 in BigBlueButton | |

Lecture | The Polynomial Hierarchy | DS7, January 19, 2021 in Screencast | |

Exercise | Circuit Complexity | DS5, January 26, 2021 in BigBlueButton | |

Lecture | Circuit Complexity | DS7, January 26, 2021 in Screencast | |

Exercise | Probabilistic Turing Machines and Complexity Classes | DS5, February 2, 2021 in BigBlueButton | |

Lecture | Probabilistic Turing Machines and Complexity Classes | DS7, February 2, 2021 in Screencast |

### Calendar