Course on

Satisfiability of Boolean Formulas - Combinatorics and Algorithms

Spring 2015

Overview

Course ID: 252-0491-00
German title: Erfüllbarkeit logischer Formeln - Kombinatorik und Algorithmen
Credits: 7 (3V+2U+1A+1). [more information]
Lecturer: Emo Welzl
Assistant: Jerri Nummenpalo
Time and Location: Tuesday 10-12 CAB G59 (lecture), Thursday 9-10 CAB G59 (lecture), Tuesday 13-15 CAB G57 (exercise)
Exam: end-of-semester examination on Thursday, May 28, 2015, 08:00-10:00 at CAB G59
Language: English

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 sold for 10 Fr. The notes are also available online here (updated February 25, 2015).

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

In the first lecture on Tuesday, February 16, there will be an introduction to course conventions only. The first chapter of the lecture notes will be a self study task. See the table below for details on the schedule.

Three special assignments will be distributed during the semester. The final exam is on Thursday, May 28 2015, 08:00-10:00.

Week	Material covered	Exercises	Slides and Solutions
8	Tuesday: Introduction and course conventions. Self study of Chapter 1. Thursday: No lecture, but a possibility to ask questions about the probabilistic analysis sheet and Chapter 1.	No exercises. Read Basics of Probabilistic Analysis.
9	Tuesday: No lecture. Continuing the self study of Chapter 1. Thursday: Start of the lectures with Chapter 2.	1.3, 1.7, 1.9, 1.12, 1.17, 1.18, In class: 1.26
10	Tuesday: Number of clauses, Partial satisfaction, Lovász Local Lemma (proof later). Thursday: Partial satisfaction.	1.25, 1.28, 2.2, 2.3, 2.5, 2.6, (2.7 moved to next week), In class exercise about resolution
11	Tuesday: Partial satisfaction, Start of Chapter 2*. Thursday: Algorithmic Lovász Local Lemma.	2.7, 2.8, 2.11, 2.12, 2.13	Special Assignment 1 (Modified on March 11!) Special assignment 1
12	Tuesday: No lecture. Thursday: Algorithmic Lovász Local Lemma.	2.9, 2.10, 2*.1, 2*.2, 2*.3, 2*.4
13	Tuesday: Chapter 3. Thursday: Chapter 3.4	Lecture instead of exercises.
14	Tuesday: Chapters 4.1 and 4.2 Thursday: Chapter 4.2	Discussion of Special assignment 1 and exercises 3.1, 3.5, 3.6, 3.8, 3.9, 3.12, 3.15
15	--Easter Break--
16	Tuesday: Chapter 4, beginning of Chapter 5 Thursday: Chapter 5	3.17, 3.19, 3.20, 4.3, 4.4, 4.6, 4.8, 4.9	Special Assignment 2 Special assignment 2 solutions
17	Tuesday:Chapter 5 Thursday: Chapter 5	5.1, 5.4, 5.5, 5.6, 5.7, 5.9, 5.10
18	Tuesday: Chapter 6 Thursday: Chapter 6	Discussion of Special assignment 2, 5.11, 5.12, 5.13, 5.14
19	Tuesday: Chapter 7, A little about derandomising Schöning's algorithm Thursday: Exponential time hypothesis, sparsification	Lecture instead of exercises!	Special assignment 3 Special assignment 3 solutions
20	Tuesday: No lecture! Thursday: No lecture (Ascension day)	Exercises: 5.12, 5.13, 6.2, 6.4, 7.1, 7.3
21	Tuesday: Exponential time hypothesis and sparcification lemma Thursday: Sparcification lemma	Discussion of the third special assignment. Exercises: Exercise 3 from the exam in 2013, Exercise 3 from the exam in 2012. In class exercises about the exams from 2012 and 2013.
22	Tuesday: Sparcification lemma and strong exponential time hypothesis Thursday: Exam: Thursday, May 28, 08:00-10:00 at CAB G59	In class: Exercises 11.1, 11.2

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.

  Exam content is all material covered in the lecture, exercises and special assignments.

  Previous exams can be found at [1] through [11] for reference.

• 30% exercises

  At times in the course of term, we will hand out three 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

The set of exerices for the following week will be listed on this web page every Tuesday. You are encouraged to hand in solutions by the following Tuesday; 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. In addition to the exercises listed beforehand there can be at times in class exercises which will be discussed during the exercise hours. The discussed exercises will be added later to the list on this web page.

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 available here (updated February 25, 2015) along with the exercises that we suggest to be worked through. There is a chance to order the lecture slides as a printed version for 10 Fr.

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

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 | exam pdf | exam ps ]
[7] Course Fall 2009.	[ home page | exam pdf | exam ps ]
[8] Course Fall 2010.	[ home page | exam pdf ]
[9] Course Spring 2012.	[ home page | exam pdf ]
[10] Course Spring 2013.	[ home page | exam pdf ]
[11] Course Spring 2014.	[ home page | exam pdf ]