Satisfiablity (SAT) Course, 2006
Course on
Satisfiability of Boolean Formulas - Combinatorics and Algorithms
Winter 06/07 (251-491-00 Erfüllbarkeit logischer Formeln -
Kombinatorik und Algorithmen,
2V1U, 5 credits)
Lecturer: Emo Welzl
Assistant: Dominik Scheder
Time: Friday, 10-12 (lecture), 13-14 (exercise)
Location: CAB G52
Schedule: here
Exam: Wednesday, February 21, 2007, 09:00-11:00, IFW A 36
Language: German, in case nobody expresses preference for English.
All accompanying material will be supplied in English.
Contents:
Satisfiability (SAT) is the problem of deciding whether a boolean
formula in propositional logic has an assignment that evaluates to
true. SAT occurs as a problem and is a tool in applications (e.g.
Artificial Intelligence and circuit design) and it is considered a
fundamental problem in theory, since many problems can be naturally
reduced to it and it is the 'mother' of NP-complete problems.
Therefore, it is widely investigated and has brought forward a
rich body of methods and tools, both in theory and practice
(including software packages tackling the problem).
This course concentrates on the theoretical aspects of the problem.
We will treat basic combinatorial properties (employing the
probabilistic method including a variant of the Lovász Local Lemma),
recall a proof of the Cook-Levin Theorem of the NP-completeness
of SAT, discuss and analyze several deterministic and randomized
algorithms and treat the threshold behavior of random formulas.
In order to set the methods encountered into a broader context,
we will deviate to the more general set-up of constraint
satisfaction and to the problem of proper k-coloring of graphs.
Goal: Studying of advanced methods in algorithms design
and analysis, and in discrete mathematics along a classical
problem in theoretical computer science.
Introduction to(wards) a number of up-to-date research
topics. Provide basis for independent research on the
subject (in a semester/Diplom/master/doctoral thesis).
Prerequisites: The course assumes basic knowledge
in propositional logic, probability theory and discrete mathematics,
as it is supplied in the Vordiplomstudium.
Literature: There exists no book that covers the
many facets of the topic. Lecture notes covering the
material of the course will be distributed (Errata, updates, see below).
Here is a list of books with material related to the course.
They can be found in the textbook collection
(Lehrbuchsammlung)
of the Computer Science Library:
-
George Boole,
An Investigation of the Laws of Thought on which are Founded
the Mathematical Theories of Logic and Probabilities,
Dover Publications
(1854, reprinted 1973).
-
Peter Clote, Evangelos Kranakis,
Boolean Functions and Computation Models,
Texts in Theoretical Computer Science,
An EATCS Series, Springer Verlag, Berlin (2002).
-
Nadia Creignou, Sanjeev Khanna, Madhu Sudhan,
Complexity Classifications of Boolean Constrained
Satisfaction Problems,
SIAM Monographs on Discrete Mathematics and Applications, SIAM (2001).
-
Harry R. Lewis, Christos H. Papadimitriou,
Elements of the Theory of Computation,
Prentice Hall (1998).
-
Rajeev Motwani, Prabhakar Raghavan,
Randomized Algorithms,
Cambridge University Press, Cambridge, (1995).
-
Uwe Schöning,
Logik für Informatiker,
BI-Wissenschaftsverlag (1992).
-
Uwe Schöning,
Algorithmik,
Spektrum Akademischer Verlag, Heidelberg, Berlin (2001).
-
Michael Sipser,
Introduction to the Theory of Computation,
PWS Publishing Company, Boston (1997).
-
Klaus Truemper,
Design of Logic-based Intelligent Systems,
Wiley-Interscience, John Wiley & Sons, Inc., Hoboken (2004).
Previous Years.
web-page course 2003,
web-page course 2004,
web-page course 2005.
Course Schedule
[27 Oct] (For an exception, lectures extend to the exercises
in the afternoon.)
Material covered: Cunjunctive normal forms; transformation into CNF; deciding
satisfiability; counting satisfying assignments
Exercises: 1.3, 1.6, 1.12, 1.17, 1.18
[3 Nov] Material covered: Counting satisfying assignments (continued); resolution; Dominik gives the lecture
Exercises: 1.20, 1.23, 1.25, 1.29
[10 Nov] Material covered: Extremal properties
(number of clauses/of dependent clauses);
partial satisfaction
Exercises: 2.1, 2.2, 2.4
[17 Nov] Material covered: partial satisfaction, unweighted formulas
Exercises: 2.5, 2.6 (Hint: use Hall's theorem), 2.18, 2.19
[24 Nov] Material covered: polynomial algorithms for 2-SAT, symmatric random
walk
Exercises: 3.6, 3.11, 3.12, 3.13
[1 Dec] Material covered: linear time algorithm for 2-SAT, algorithms for
graph coloring
Exercises: 3.17, 3.18, 3.20, 4.2
[8 Dec] Material covered: NP, polynomial constant falsifiers
Exercises: 4.2 (hint: first try for directed acyclic graphs, then generalize),
4.4, 4.9
[15 Dec] Material covered: NP = FP3, Turing machines;
Dominik gives the lecture
Exercises: 4.5, 4.10
[22 Dec] Material covered: the
n-dimensional cube;
faces, packings, covering
Exercises: 2.3, 5.1, 5.3, 5.4
---(X-mas break)---
[5 Jan] Material covered: Hamming balls, bounds on the volume,
covering codes
Exercises: 5.9, 5.10, 5.11, 5.12
[12 Jan] Material covered: Satisfiability Coding Lemma, PPZ algorithm;
Dominik gives the lecture
Exercises: 5.14, 5.17
[19 Jan] Material covered: searchball algorithm, deterministic construction
of approximately optimal covering codes
Exercises: 6.3, 6.4, 7.3
[26 Jan] Material covered: Schöning's Algorithm for 3-SAT,
Random
k-SAT
Exercises:
[2 Feb] Material covered: Random 2-SAT (bicycles), Improved Bound for
Unsatisfiability in Random 3-SAT;
Dominik gives the lecture
Exercises:
---(end of semester)---
Exam: Wednesday, February 21, 2007
Check out previous years exams,
(February 2004
.pdf or
.ps,
February 2005
.pdf or
.ps,
March 2006
.pdf or
.ps).
See below for further information.
Lecture Notes.
Lecture notes are distributed during the course, including the
exercises that we suggest to work through. There is no on-line
accessible version of the lecture notes.
Errata:
Exam:
There is a written 2 hours exam (exact time and place to be announced),
also for the students from D-MATH. You are allowed to take any kind of written material
with you, but no electronic devices of whatever kind.
Time and Location: February 21, 09:00-11:00, IFW A 36
Anybody other than D-INFK students should indicate participation in the exam
by email both to Emo Welzl and Dominik Scheder before to be announced.
(D-INFK students are assumed to have subscribed for the exam through the usual procedures.)
Material relevant for the exam is as given in the lecture notes handed
out...