Complexity Theory
From International Center for Computational Logic
Complexity Theory
Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2015
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
Legacy
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.- 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.
- 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 | DS4, October 14, 2015 in APB E005 | File 1, File 2 |
| Lecture | Turing Machines and Languages | DS4, October 16, 2015 in APB E005 | File 1, File 2 |
| Lecture | Undecidability | DS2, October 20, 2015 in APB E005 | |
| Lecture | Undecidability | DS4, October 21, 2015 in APB E005 | |
| Exercise | Mathematical Foundations | DS4, October 23, 2015 in APB E005 | |
| Lecture | Rice's Theorem for Recognizability | DS2, October 27, 2015 in APB E005 | |
| Lecture | A Hierarchy of Undecidable Problems | DS4, October 28, 2015 in APB E005 | |
| Exercise | Turing Machines and Languages | DS4, October 30, 2015 in APB E005 | |
| Lecture | The Recursion theorem | DS2, November 3, 2015 in APB E005 | |
| Lecture | Decidability and Logic | DS4, November 4, 2015 in APB E005 | |
| Exercise | Undecidability and Rice's Theorem | DS4, November 6, 2015 in APB E005 | |
| Lecture | Time Complexity: Polynomial Time | DS2, November 10, 2015 in APB E005 | File |
| Lecture | Time Complexity: NP | DS4, November 11, 2015 in APB E005 | File |
| Exercise | The Recursion Theorem and Decidability in Logic | DS4, November 13, 2015 in APB E005 | |
| Lecture | Time Complexity: NP Completeness | DS2, November 17, 2015 in APB E005 | File 1, File 2 |
| Exercise | Time Complexity | DS4, November 20, 2015 in APB E005 | |
| Lecture | NP-Complete Problems | DS2, November 24, 2015 in APB E005 | File 1, File 2 |
| Lecture | Space Complexity | DS4, November 25, 2015 in APB E005 | File 1, File 2 |
| Exercise | Time Complexity | DS4, November 27, 2015 in APB E005 | |
| Lecture | Polynomial Space | DS2, December 1, 2015 in APB E005 | File 1, File 2 |
| Lecture | Games/Logarithmic Space | DS4, December 2, 2015 in APB E005 | File 1, File 2 |
| Exercise | Space Complexity | DS4, December 4, 2015 in APB E005 | |
| Lecture | The Time Hierarchy Theorem | DS2, December 8, 2015 in APB E005 | |
| Lecture | More on Hierarchy Theorems, the Gap Theorem | DS4, December 9, 2015 in APB E005 | |
| Exercise | Space Complexity | DS4, December 11, 2015 in APB E005 | |
| Lecture | Ladner's Theorem | DS2, December 15, 2015 in APB E005 | |
| Lecture | The Baker-Gill-Solovay Theorem | DS4, December 16, 2015 in APB E005 | |
| Exercise | Mid-Term Consultation | DS4, December 18, 2015 in APB E005 | |
| Lecture | Alternating Turing Machines | DS2, January 5, 2016 in APB E005 | File 1, File 2 |
| Lecture | Alternating Complexity Classes | DS4, January 6, 2016 in APB E005 | File 1, File 2 |
| Exercise | Diagonalization | DS4, January 8, 2016 in APB E005 | |
| Lecture | The Polynomial Hierarchy | DS2, January 12, 2016 in APB E005 | File 1, File 2 |
| Lecture | Circuit Complexity | DS4, January 13, 2016 in APB E005 | File 1, File 2 |
| Exercise | Alternation | DS4, January 15, 2016 in APB E005 | |
| Lecture | Circuits for Parallel Computation | DS2, January 19, 2016 in APB E005 | File 1, File 2 |
| Lecture | Randomized Computation | DS4, January 20, 2016 in APB E005 | |
| Exercise | Circuit Complexity | DS4, January 22, 2016 in APB E005 | |
| Lecture | Randomized Computation | DS2, January 26, 2016 in APB E005 | |
| Lecture | Randomized Computation | DS4, January 27, 2016 in APB E005 | |
| Exercise | Randomized Computation | DS4, January 29, 2016 in APB E005 | |
| Lecture | Randomized Computation | DS2, February 2, 2016 in APB E005 | |
| Lecture | Wrapup and Outlook | DS4, February 3, 2016 in APB E005 | |
| Exercise | Consultation | DS4, February 5, 2016 in APB E005 |
Calendar
