# Complexity Theory

From International Center for Computational Logic

# Complexity Theory

##### Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2019

**Lecturer**

**Tutor**

**SWS**

- 4/2/0

**Modules**

**Examination method**

- Oral exam

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:**as indicated in the class calendar below, the lecture sessions start on the 15th of October. Furthermore, there will be no exercise sessions on the 6th, 13th, and 20th of November.

**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, 2019 in APB E005 | Download, Download |

Exercise | Mathematical Foundations, Decidability, and Recognisability | DS3, October 16, 2019 in APB E005 | Download, Download |

Lecture | Turing Machines and Languages | DS2, October 21, 2019 in APB E008 | Download, Download |

Lecture | Undecidability | DS2, October 22, 2019 in APB E005 | Download, Download |

Exercise | Undecidability and Rice's Theorem | DS3, October 23, 2019 in APB E005 | Download, Download |

Lecture | Undecidability and Recursion | DS2, October 28, 2019 in APB E008 | Download, Download |

Lecture | Time Complexity and Polynomial Time | DS2, October 29, 2019 in APB E005 | Download, Download |

Lecture | NP | DS3, October 30, 2019 in APB E008 | Download, Download |

Lecture | NP-Completeness | DS2, November 4, 2019 in APB E008 | Download, Download |

Lecture | NP-Complete Problems | DS2, November 5, 2019 in APB E005 | Download, Download |

Exercise | Time Complexity, PTime, and NP | DS2, November 11, 2019 in APB E008 | Download |

Lecture | Space Complexity | DS2, November 12, 2019 in APB E005 | Download, Download |

Lecture | Polynomial Space | DS2, November 18, 2019 in APB E008 | |

Lecture | Games/Logarithmic Space | DS2, November 19, 2019 in APB E005 | |

Lecture | The Time Hierarchy Theorem | DS2, November 25, 2019 in APB E008 | |

Lecture | Space Hierarchy and Gaps | DS2, November 26, 2019 in APB E005 | |

Exercise | NP-Completeness and Time Complexity | DS3, November 27, 2019 in APB E005 | Download |

Lecture | Space Hierarchy and Gaps (continued) | DS2, December 2, 2019 in APB E008 | |

Lecture | P vs. NP: Ladner's Theorem | DS2, December 3, 2019 in APB E005 | |

Exercise | Space Complexity | DS3, December 4, 2019 in APB E005 | Download |

Lecture | P vs. NP and Diagonalisation | DS2, December 9, 2019 in APB E008 | |

Lecture | Alternation | DS2, December 10, 2019 in APB E005 | |

Exercise | Diagonalisation | DS3, December 11, 2019 in APB E005 | Download |

Lecture | The Polynomial Hierarchy | DS2, December 16, 2019 in APB E008 | |

Lecture | Polynomial Hierarchy / Circuit Complexity | DS2, December 17, 2019 in APB E005 | |

Exercise | Diagonalisation and Alternation | DS3, December 18, 2019 in APB E005 | Download |

Lecture | Questions and Answers | DS2, January 6, 2020 in APB E008 | |

Lecture | Circuits for Parallel Computation | DS2, January 7, 2020 in APB E005 | |

Exercise | Alternation and the Polynomial Hierarchy | DS3, January 8, 2020 in APB E005 | Download |

Lecture | Probabilistic Turing Machines | DS2, January 13, 2020 in APB E008 | |

Lecture | Probabilistic Complexity Classes (1) | DS2, January 14, 2020 in APB E005 | |

Exercise | Circuit Complexity | DS3, January 15, 2020 in APB E005 | Download |

Lecture | Probabilistic Complexity Classes (2) | DS2, January 20, 2020 in APB E008 | |

Lecture | Quantum Computing (1) | DS2, January 21, 2020 in APB E005 | |

Exercise | Probabilistic TMs and ComplexityClasses | DS3, January 22, 2020 in APB E005 | Download |

Lecture | Quantum Computing (2) | DS2, January 27, 2020 in APB E008 | |

Lecture | Summary, Outlook, Consultation | DS2, January 28, 2020 in APB E005 | |

Exercise | Probabilistic Complexity Classes (3) | DS3, January 29, 2020 in APB E005 | Download |

### Calendar