Complexity Theory

From International Center for Computational Logic

Complexity Theory

Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2021

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: Stay informed about current covid-19 regulations of 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.
Free lecture notes with a 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 11, 2021 in APB E008
Lecture Turing Machines and Languages DS2, October 12, 2021 in APB E005
Lecture Undecidability DS2, October 18, 2021 in APB E008
Lecture Undecidability and Recursion DS2, October 19, 2021 in APB E005
Exercise Mathematical Foundations, Decidability, and Recognisability DS3, October 20, 2021 in APB E005
Lecture Time Complexity and Polynomial Time DS2, October 25, 2021 in APB E008
Lecture NP DS2, October 26, 2021 in APB E005
Exercise Undecidability and Rice's Theorem DS3, October 27, 2021 in APB E005
Lecture NP-Completeness DS2, November 1, 2021 in APB E008
Lecture NP-Complete Problems DS2, November 2, 2021 in APB E005
Exercise Time Complexity, PTime, and NP DS3, November 3, 2021 in APB E005
Lecture Space Complexity DS2, November 8, 2021 in APB E008
Lecture Polynomial Space DS2, November 9, 2021 in APB E005
Exercise NP-Completeness and Time Complexity DS3, November 10, 2021 in APB E005
Lecture Games/Logarithmic Space DS2, November 15, 2021 in APB E008
Lecture The Time Hierarchy Theorem DS2, November 16, 2021 in APB E005
No session Day of Prayer and Repentance (Public Holiday) DS3, November 17, 2021 in APB E005
Lecture Space Hierarchy and Gaps DS2, November 22, 2021 in APB E008
Lecture P vs. NP: Ladner's Theorem DS2, November 23, 2021 in APB E005
Exercise Space Complexity DS3, November 24, 2021 in APB E005
Lecture P vs. NP and Diagonalisation DS2, November 29, 2021 in APB E008
Lecture Alternation DS2, November 30, 2021 in APB E005
Exercise Diagonalisation DS3, December 1, 2021 in APB E005
Lecture The Polynomial Hierarchy DS2, December 6, 2021 in APB E008
Lecture Circuit Complexity DS2, December 7, 2021 in APB E005
Exercise Diagonalisation and Alternation DS3, December 8, 2021 in APB E005
Lecture Circuits for Parallel Computation DS2, December 13, 2021 in APB E008
Lecture Probabilistic Turing Machines DS2, December 14, 2021 in APB E005
Exercise Alternation and the Polynomial Hierarchy DS3, December 15, 2021 in APB E005
Lecture Probabilistic Complexity Classes (1) DS2, December 20, 2021 in APB E008
Lecture Questions and Answers DS2, December 21, 2021 in APB E005
Exercise Circuit Complexity DS3, December 22, 2021 in APB E005
Lecture Probabilistic Complexity Classes (2) DS2, January 10, 2022 in APB E008
Lecture Quantum Computing (1) DS2, January 11, 2022 in APB E005
Exercise Probabilistic TMs and Complexity Classes DS3, January 12, 2022 in APB E005
Lecture Quantum Computing (2) DS2, January 17, 2022 in APB E008
Lecture Interactive Proof Systems DS2, January 18, 2022 in APB E005
Exercise Probabilistic Complexity Classes (3) DS3, January 19, 2022 in APB E005


Calendar