Complexity Theory

From International Center for Computational Logic

Complexity Theory

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

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.
  • 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.
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 DS2, October 14, 2019 in APB E008 Download Download
Lecture Turing Machines and Languages DS2, October 15, 2019 in APB E005 Download Download
Exercise Mathematical Foundations, Decidability, and Recognisability DS3, October 16, 2019 in APB E005 Download
Lecture Undecidability DS2, October 21, 2019 in APB E008 Download Download
Lecture Undecidability and Recursion DS2, October 22, 2019 in APB E005 Download Download
Exercise Undecidability and Rice's Theorem DS3, October 23, 2019 in APB E005 Download
Lecture Time Complexity and Polynomial Time DS2, October 28, 2019 in APB E008 Download Download
Lecture NP DS2, October 29, 2019 in APB E005 Download Download
Exercise Time Complexity, PTime, and NP DS3, October 30, 2019 in APB E005 Download
Lecture NP-Completeness DS2, November 4, 2019 in APB E008 Download Download
Lecture NP-Complete Problems DS2, November 11, 2019 in APB E008 Download Download
Lecture Space Complexity DS2, November 12, 2019 in APB E005 Download Download
Lecture Polynomial Space DS2, November 18, 2019 in APB E008 Download Download
Lecture Games/Logarithmic Space DS2, November 19, 2019 in APB E005 Download Download
Lecture The Time Hierarchy Theorem DS2, November 25, 2019 in APB E008 Download Download
Lecture Space Hierarchy and Gaps DS2, November 26, 2019 in APB E005 Download Download
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 Download Download
Exercise Space Complexity DS3, December 4, 2019 in APB E005 Download
Lecture P vs. NP and Diagonalisation DS2, December 9, 2019 in APB E008 Download Download
Lecture Alternation DS2, December 10, 2019 in APB E005 Download Download
Exercise Diagonalisation DS3, December 11, 2019 in APB E005 Download
Lecture The Polynomial Hierarchy DS2, December 16, 2019 in APB E008 Download Download
Lecture Polynomial Hierarchy / Circuit Complexity DS2, December 17, 2019 in APB E005 Download Download
Exercise Diagonalisation and Alternation DS3, December 18, 2019 in APB E005 Download
Lecture Questions and Answers DS2, January 6, 2020 in APB E008 Download Download
Lecture Circuits for Parallel Computation DS2, January 7, 2020 in APB E005 Download Download
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 Download Download
Lecture Probabilistic Complexity Classes (1) DS2, January 14, 2020 in APB E005 Download Download
Exercise Circuit Complexity DS3, January 15, 2020 in APB E005 Download
Lecture Probabilistic Complexity Classes (2) DS2, January 20, 2020 in APB E008 Download Download
Lecture Quantum Computing (1) DS2, January 21, 2020 in APB E005 Download Download
Exercise Probabilistic TMs and ComplexityClasses DS3, January 22, 2020 in APB E005 Download
Lecture Quantum Computing (2) DS2, January 27, 2020 in APB E008 Download Download
Lecture Summary, Outlook, Consultation DS2, January 28, 2020 in APB E005 Download Download
Exercise Probabilistic Complexity Classes (3) DS3, January 29, 2020 in APB E005 Download


Calendar