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Computability and Complexity, Spring 2016

Computability and Complexity (320352)

Jacobs University Bremen, Spring 2016, Herbert Jaeger

Class sessions: Thursdays and Fridays 14:15 – 15:30, West Hall 4

Course description. This lecture presents one half of the core material of theoretical computer science (the other half is covered in the lecture "Formal Languages and Logic''). The question: "What problems can a computer possibly solve?'', is fully answered (by characterizing those solvable problems, equivalently, through Turing machines, random access machines, recursive functions and lambda calculus). A full answer to the related question, "how much computational resources are needed for solving a given problem'' is not known today. However, the basic outlines of today's theory of computational complexity will be presented up to the most famous open problem in computer science, namely the famous "P = NP'' question: if a computer can guess the answer to a problem (and only needs to check its correctness), does that really help to speed up computation?

Course culture. Like in FLL in Fall, the basic idea is that participants pre-read the daily material, which is then rehearsed in the first part of a classroom session, after which I will reserve time to classroom interaction, by asking simple, advanced or outlandish questions, going through miniexercises, present add-on material, etc. 5% of the final grade will be based on classroom participation. A good way to reliably score high in this respect is to think of classroom questions while you are doing the pre-reading. Questions can be just mere requests for clarification of the technical contents of the reading, but also probing deep, be about the context of the taught material, challenging me, bewildering classmates. The more we get seriously challenged, the better the question. Classroom presence is mandatory.

Homeworks. Students can work in singles or as teams of two but not larger. HWs count 10 % of the course score. Homework sheets are not graded in the usual sense. Instead, a HW sheet gets a full score if all problems have been visibly and convincingly worked on. Correctness of solutions is not required for full score. It is permissible (though unwise) to copy solutions from friends or other sources.  In this case, the fact that the solutions are copied, as well as the source, must be stated on the HW sheet. TA's will annotate HW sheets that are original work (not copied) to provide useful feedback. If a student copies the solutions and does not indicate that fact and the source, it will be treated as a cheating case (nulling the sheet, notification of the Office of Academic Affairs).

Grading and exams: Grading and exams: The final course grade will be composed from homeworks (10%), presence sheets (10%), active participation in class (5%), and quizzes/exams. There will be four miniquizzes (written in class, 20 minutes), the best three of which will each account to 15% of the final grade (worst will be dropped), and one final exam, counting 30%. All quizzes and the final exam are open book. Each quiz or exam yields a maximum of 100 points. The total semester points Pts_total are computed as the weighted semester point average (each quiz points are weighted by 0.15, the final by 0.30, etc.), and from Pts_total the final grade is computed by the standard Jacobs formula. Jacobs makeup regulations apply - a makeup exam is granted if the instructor was informed before the missed exam and a medical excuse is provided. I also grant make-ups when the original exam is missed due to non-medical, good reasons on a case-by-case basis.

Lecture Notes are here (last update: May 10)

For exam preparation, previous finals with solutions from years 2004, 2006, 2008 are here. An exam in the multiple-choice format from 2010 is here (and these are the solutions)

References

  • Papadimitriou, C: Computational Complexity, (Addison-Wesley) IRC: QA267.7 .P36 1994
  • Hopcroft, John E. , Motwani, Rajeev, and Ullman, Jeffrey: Introduction to Automata Theory, 2nd (Addison-Wesley) IRC: QA267 .H56 2001
  • Garey, Michael and Johnson, David: Computers and Intractability: A guide to the Theory of NP-completeness (Macmillan) IRC: QA76.6 .G35
  • A tutorial text by  on lambda calculus by L. C. Paulson (my lecture notes owe a lot to this tutorial)

Schedule (this will be filled in synchrony with reality as we go along)

Feb 4

Introduction
Feb 5 Turing machines. Recursive and r.e. languages.
Feb 11 (no class)
Feb 12 (no class)
Feb 18 Basic properties of recursive and r.e. languages. Multi-tape TMs. Time complexity. Simulation of multitape TM by single-tape TM. Reading: Lecture notes Section 3.1 to end, skip Section 3.2, then Section 3.3 up to (including) Proposition 3.2  Exercise sheet 1
Feb 19 Polynomially related complexity functions. Tightness of quadratic loss. Space complexity. Linear speedup. Reading: Section 3.3 to end
Feb 25 Random access machines. Reading: Section 4
Feb 26 Primitive recursive functions. Reading: Section 5 up to (including) example 5.1.  Solutions to sheet 1   Exercise sheet 2
Mar 3  no class
Mar 4  no class
Mar 10 no class
Mar 11 Miniquiz 1 (CNLH!) Recursive functions and the Church-Turing hypothesis. Reading: Section 5 to end.
Mar 17 + extra make-up class 15:45 in East Hall 8  Universal TMs. Halting problem. Undecidabilty of first-order logic. Reading: LN Section 6 except 6.3  Solutions to exercises 2     Exercise sheet 3
Mar 18 Combinators: intro. Reading: Section 7.1 up to (including) Def. 7.2
Mar 31 Combinatorial algebras: evaluation order. Combinatorial completeness. Reading: Section 7.1 to end
Apr 1 Lambda abstraction. Reading: Section 7.2  Solutions to exercises 3  Exercise sheet 4
Apr 7 Miniquiz 2 (CNLH!) + extra make-up class 15:45 in East Hall 8  Booleans, lists, numerals in lambda calculus. Reading: LN section 7.3
Apr 8 The Y combinator. Lambda calculus and recursive functions. L5: A very small, very clean functional programming language. Reading: LN sections 7.4, 7.5 Solutions to exercise sheet 4  Exercise sheet 5
Apr 14 Complexity: intro. Reading: LN Section 8 Solutions to exercise sheet 5 Exercise sheet 6
Apr 15 Nondeterministic TMs. The class NP. Reading: LN Section 9
Apr 21 Miniquiz 3 (CNLH!) + extra make-up class 15:45 in East Hall 8  Relationships between complexity classes and hierarchy theorems. No reading.  Solutions to exercise sheet 6  Exercise sheet 7
Apr 22 Boolean logic re-visited under the auspices of computational complexity. Reading: LN Section 11
Apr 28 Reductions and completeness. Reading: LN Section 12.1, 12.2 up to (excluding) Theorem 12.3  Solutions to sheet 7 Exercise sheet 8
Apr 29  The world of NP-completeness.
May 6 Cook's theorem. Reading: LN Section 12.2 to end  Solutions to exercise sheet 8
May 12 Miniquiz 4 (CNLH!) + extra make-up class 15:45 in East Hall 8  Introduction to descriptive complexity part 1 (optional, classroom presence not mandatory)
May 13 Introduction to descriptive complexity part 2 (optional, classroom presence not mandatory)
   
May 21

Final exam (9:00 - 11:00 hrs, CNLH)