Complexity Theory

From International Center for Computational Logic

Complexity Theory

Course with SWS 4/2/0 (lecture/exercise/practical) in WS 2015

Lecturer

  • Daniel Borchmann, Markus Krötzsch

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

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.
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 Download
Lecture Turing Machines and Languages DS4, October 16, 2015 in APB E005 Download
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 Download
Lecture Time Complexity: NP DS4, November 11, 2015 in APB E005 Download
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 Download Download
Exercise Time Complexity DS4, November 20, 2015 in APB E005
Lecture NP-Complete Problems DS2, November 24, 2015 in APB E005 Download Download
Lecture Space Complexity DS4, November 25, 2015 in APB E005 Download Download
Exercise Time Complexity DS4, November 27, 2015 in APB E005
Lecture Polynomial Space DS2, December 1, 2015 in APB E005 Download Download
Lecture Games/Logarithmic Space DS4, December 2, 2015 in APB E005 Download Download
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 Download Download
Lecture Alternating Complexity Classes DS4, January 6, 2016 in APB E005 Download Download
Exercise Diagonalization DS4, January 8, 2016 in APB E005
Lecture The Polynomial Hierarchy DS2, January 12, 2016 in APB E005 Download Download
Lecture Circuit Complexity DS4, January 13, 2016 in APB E005 Download Download
Exercise Alternation DS4, January 15, 2016 in APB E005
Lecture Circuits for Parallel Computation DS2, January 19, 2016 in APB E005 Download Download
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