Course on

Satisfiability of Boolean Formulas - Combinatorics and Algorithms

Spring 2012

Overview

Course ID: 251-0491-00
German title: Erfüllbarkeit logischer Formeln - Kombinatorik und Algorithmen
Credits: 7 (3V+2U+1A+1). [more information]
Lecturer: Emo Welzl
Assistant: Robin Moser
Time and Location: Tuesday 10-12 CAB G51 (lecture), Friday 9-10 CHN D44 (lecture), Friday 10-12 CHN D44 (exercise)
Exam: Friday, June 1, 9am-11am
Language: English. If all participants can go along with German, we will switch. The course material will be in English in any case.

Course 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.

Primary goals

• to study advanced methods in algorithms design and analysis and in discrete mathematics along a classical problem in theoretical computer science
• to introduce a number of up-to-date research topics
• to provide basis for independent research on the subject (in a semester/diploma/master/doctoral thesis).

Prerequisites

It is assumed that participants have basic knowledge in propositional logic, probability theory and discrete mathematics as 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).

Course Schedule

Date	Material covered	Exercises due on that date
Tuesday, February 21	Introduction, motivating examples (circuit verification, map labelling). Conjunctive normal form, polynomial-time conversion of general formulas into SAT-equivalent CNF formulas.
Friday, February 24	Terminology, basic observations, SAT-counting algorithms
Tuesday, February 28	SAT-counting algorithms, resolution, formulas with few clauses
Friday, March 2	Formulas with few clauses, Lovász Local Lemma	1.3, 1.5, 1.6, 1.7, 1.9, 1.12, 1.17, 1.18
Tuesday, March 6	Partial satisfaction
Friday, March 9	Partial satisfaction, algorithmic Lovász Local Lemma	1.20, 1.23, 1.28, 2.1, 2.2, 2.6, 2.7
Tuesday, March 13	Algorithmic Lovász Local Lemma, 2-SAT in polytime using resolution and unit clause reduction
Friday, March 16	2-SAT in polytime using Papadimitriou's random walk algorithm	2.9, 2.10, 2.11, 2.12, 2.13, 2.14
Tuesday, March 19	2-SAT in linear time; applications of 2-SAT to 3-SAT and 3-coloring; introduction to falsifiers, reduction from coloring to SAT	NOTE: We have distributed Special Assignment 1 today, due on April 3.
Friday, March 23	Verifiers and falsifiers	2*.1, 2*.2, 2*.5, 3.3, 3.6, 3.9, 3.12, 3.13
Tuesday, March 26	Verifiers and falsifiers, complexity classes, NP-completeness of SAT, introduction to the Boolean n-cube, inequalities
Friday, March 30	Inequalities in the Boolean n-cube, Hamming Balls	3.15, 3.17, 3.19, 3.20, 4.4
Tuesday, April 3	Satisfiability coding and the PPZ algorithm	Special Assignment 1 is due
--Easter Break--
Tuesday, April 17	Satisfiability coding and the PPZ algorithm
Friday, April 20	Hamming Balls and k-SAT algorithms	No exercises; Special Assignment 1 will be returned and discussed NOTE: Moreover, we have distributed Special Assignment 2, due on Friday, May 4.
Tuesday, April 24	Hamming Balls and k-SAT algorithms, Schöning's Algorithm
Friday, April 27	Derandomization of Schöning's Algorithm, Preliminary part: typical executions	5.7, 5.8, 5.9, 5.10, 5.14, 6.2, 6.3, 6.4
-- Tag der Arbeit --	--
Friday, May 4	Derandomization of Schöning's Algorithm, algorithm and proof	7.1, 7.2, 7.3 + inclass exercises Special Assignment 2 is due.
Tuesday, May 8	The PPSZ Algorithm, Part 1: Algorithm, Basics of Analysis, Critical Clause Trees
Friday, May 11	The PPSZ Algorithm, Part 2: Preliminaries for a Probabilistic Analysis	No regular exercises. NOTE: We have distributed Special Assignment 3, due on Friday, May 25.
Tuesday, May 15	The PPSZ Algorithm, Part 3: Proof of Theorem 8.6.
Friday, May 18	The PPSZ Algorithm, Part 4: Multiple Satisfying Assignments - What's the problem?
Tuesday, May 22	The PPSZ Algorithm, Part 5: Multiple Satisfying Assignments - Analysis via a Cost Function
Friday, May 25	The PPSZ Algorithm, Part 6: Multiple Satisfying Assignments - Bounding Correlations in Weighted Sums	Special Assignment 3 is due.
Tuesday, May 29	The PPSZ Algorithm, Part 7: Multiple Satisfying Assignments - Putting Things Together
Friday, June 1	EXAM in NO C 60

Exam and grades

Your final grade will be calculated as the the weighted average of:
• 70% final written exam

  The exam is open book, i.e. you are allowed to consult any books, handouts and personal notes of your choice. The use of electronic devices is not allowed.

  Anybody other than D-INFK students should indicate participation in the exam by email both to Emo Welzl and Robin Moser before (deadline to be announced). D-INFK students are expected to subscribe following official procedures.

  Previous exams can be found at [1] through [4] for reference.

• 30% exercises

  At times in the course of term, we will hand out specially marked exercises the solution of which (typeset in LaTeX or similar) is due two weeks later.

  Your solutions will be graded, and each grade will account for 10% of your final grade.

  You are welcome to discuss these exercises with your colleagues, but we expect you to hand in your own writeup.

Regulations for PhD students

For PhD students, the same rules apply for obtaining credit points as for all other participants. Taking the exam and achieving an overall grade of at least 4.0 (computed as a weighted average of grades for special exercises and the written final exam as detailed above) qualifies for receiving credits. In order to comply with new regulations recently issued by the department, merely attending the course and/or handing in exercises is no longer sufficient.

Exercises

We will hand out weekly exercises each Friday. You are encouraged to hand in solutions by the following Friday; submitted solutions will be looked at and returned with comments. Note, however, that only the specially marked exercises (see above) will count towards your grade. Nevertheless, we strongly recommend that you solve all weekly problems and attend the exercise classes.

What are the "sixth hour" in the Course Catalogue and the seventh credit point supposed to mean?

Apart from three regular lecture (V) hours and two hour of exercises (U), this course comes with one extra sixth hour of independent work (A); this makes 7 credit points. You will deserve the credits for the A-unit by independent work, i.e. studying and learning material (also prerequisistes) on your own. Details will be announced at the beginning of the course the latest.

Lecture Notes

Lecture notes are distributed during the course, including the exercises that we suggest to be worked through. There is no online version of the lecture notes.

Errata

Please check this section for updates on a regular basis. If you discover any mistakes, typos or similar in any of the material we hand out, please advise us as quickly as possible.

where	correction
Chapter 8, page 5	construction of T_x, step 2.(b), set beta(z) to 1-alpha(z) if z is in W [instead of: U]
Chapter 8, page 5	last sentence: ... there is no alpha-violated literal over y ... [rather than: beta]
Chapter 8, page 8	definition of S : integrand should be p^2/(1-p)^2 [rather than: 1-p^2]
Chapter 8, page 8	Theorem 8.6: although the theorem itself is fine this way, we would like to define SD differently, because it should not depend on T. The theorem should simply read "Then the probability that the resultant tree T' contains more than Dε nodes satisfies Pr[|V(T')| > Dε] <= S + ε. After the theorem and independently, define SD to be the supremum over all T of Pr[|V(T')| > D].

Links and Downloads

[1] Course Winter 2003/2004.	[ home page | exam pdf | exam ps ]
[2] Course Winter 2004/2005.	[ home page | exam pdf | exam ps ]
[3] Course Winter 2005/2006.	[ home page | exam pdf | exam ps ]
[4] Course Winter 2006/2007.	[ home page | exam pdf | exam ps ]
[5] Course Winter 2007/2008.	[ home page | exam pdf | exam ps ]
[6] Course Fall 2008.	[ home page | | ]
[6] Course Fall 2009.	[ home page | exam pdf | exam ps ]

last change: February 19, 2012 by Robin Moser