# Complexity Theory

From International Center for Computational Logic

# Complexity Theory

##### Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2018

**Lecturer**

**Tutor**

**SWS**

- 4/2/0

**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**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:**There will be neither lectures nor exercises on the first week of the semester. Therefore, the first lecture of this course will be on Monday, 15th of October.

**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 | DS2, October 15, 2018 in APB E008 | File 1, File 2 |

Lecture | Turing Machines and Languages | DS2, October 16, 2018 in APB E005 | File 1, File 2 |

Exercise | Mathematical Foundations, Decidability, and Recognisability | DS3, October 17, 2018 in APB E005 | File |

Lecture | Undecidability | DS2, October 22, 2018 in APB E008 | File 1, File 2 |

Lecture | Undecidability and Recursion | DS2, October 23, 2018 in APB E005 | File 1, File 2 |

Exercise | Undecidability and Rice's Theorem | DS3, October 24, 2018 in APB E005 | File |

Lecture | Time Complexity and Polynomial Time | DS2, October 29, 2018 in APB E008 | File 1, File 2 |

Lecture | NP | DS2, October 30, 2018 in APB E005 | File 1, File 2 |

Lecture | NP-Completeness | DS2, November 5, 2018 in APB E008 | File 1, File 2 |

Lecture | NP-Complete Problems | DS2, November 6, 2018 in APB E005 | File 1, File 2 |

Exercise | Time Complexity, PTime, and NP | DS3, November 7, 2018 in APB E005 | File |

Lecture | Space Complexity | DS2, November 12, 2018 in APB E008 | File 1, File 2 |

Lecture | Polynomial Space | DS2, November 13, 2018 in APB E005 | File 1, File 2 |

Exercise | NP-Completeness and Time Complexity | DS3, November 14, 2018 in APB E005 | File |

Lecture | Games/Logarithmic Space | DS2, November 19, 2018 in APB E008 | File 1, File 2 |

Lecture | The Time Hierarchy Theorem | DS2, November 20, 2018 in APB E005 | File 1, File 2 |

Lecture | Space Hierarchy and Gaps | DS2, November 26, 2018 in APB E008 | File 1, File 2 |

Lecture | Space Hierarchy and Gaps (continued) | DS2, November 27, 2018 in APB E005 | |

Exercise | Space Complexity | DS3, November 28, 2018 in APB E005 | File |

Lecture | P vs. NP: Ladner's Theorem | DS2, December 3, 2018 in APB E008 | File 1, File 2 |

Lecture | P vs. NP and Diagonalisation | DS2, December 4, 2018 in APB E005 | File 1, File 2 |

Exercise | Diagonalisation | DS3, December 5, 2018 in APB E005 | File |

Lecture | Alternation | DS2, December 10, 2018 in APB E008 | File 1, File 2 |

Lecture | The Polynomial Hierarchy | DS2, December 11, 2018 in APB E005 | File 1, File 2 |

Exercise | Diagonalisation and Alternation | DS3, December 12, 2018 in APB E005 | File |

Lecture | Polynomial Hierarchy / Circuit Complexity | DS2, December 17, 2018 in APB E008 | File 1, File 2 |

Lecture | Questions and Answers | DS2, December 18, 2018 in APB E005 | File 1, File 2 |

Exercise | Alternation and the Polynomial Hierarchy | DS3, December 19, 2018 in APB E005 | File |

Lecture | Circuits for Parallel Computation | DS2, January 7, 2019 in APB E008 | File 1, File 2 |

Lecture | Probabilistic Turing Machines | DS2, January 8, 2019 in APB E005 | File 1, File 2 |

Exercise | Circuit Complexity | DS3, January 9, 2019 in APB E005 | File |

Lecture | Probabilistic Complexity Classes (1) | DS2, January 14, 2019 in APB E008 | File 1, File 2 |

Lecture | Probabilistic Complexity Classes (2) | DS2, January 15, 2019 in APB E005 | File 1, File 2 |

Exercise | Probabilistic TMs and ComplexityClasses | DS3, January 16, 2019 in APB E005 | File |

Lecture | Quantum Computing (1) | DS2, January 21, 2019 in APB E008 | File 1, File 2 |

Lecture | Quantum Computing (2) | DS2, January 22, 2019 in APB E005 | File 1, File 2 |

Exercise | Probabilistic Complexity Classes (3) | DS3, January 23, 2019 in APB E005 | File |

Lecture | Summary, Outlook, Consultation | DS2, January 29, 2019 in APB E005 | File 1, File 2 |

### Calendar