dr. Peter van Emde Boas
hours:
Monday 09.-- -- 10.45
Room P 019
Wednesday 17.-- -- 18.45 Room P 015A
Beware ! The Course
is Finished ! Everything went much faster comparted to last year. Time
to pack my stuff.
THE EXCERCISES PAGE HAS BEEN OPENED.
Scheme of classes (with tentative subjects):
Week Tuesday
Wednesday
06
Feb 02 Games and Computation models
Feb
04 Game Models
07 Feb 09 Game
models cont.
Feb 11 Games as acceptors
& Know thy Numbers
08 Feb 16 Complexity
of Backward Induction
Feb 18 PSPACE and Games , QBF and the
Savitch Game
09 Feb 23 parallelism
& recursion, EditRAM, Alternation Feb 25 Alternation
Theorems, Combinatorial games hard for PSPACE
10 Mar 02 The
pebbling Game
Mar 04 Speed-up
and Diagonalization
11 Mar 09 Hopcroft
Paul Valiant, GTL theorem
Mar 11 The Evasiveness
Game
12 Mar 16 Algebra
101 ; orbits in the evasiveness Game Mar 18
The Rivest Vuillemin proof
13
Mar 23 No
Class (Exam week)
Mar 25 No Class (exam week)
14
Mar 30 Interactive protocols; the models
Apr 01
Protocols for the permanent and QBF (Shamir)
15 Apr 06 Games against
nature
Apr 08 No Class (Course
finished)
16 Apr 13 No
class (Easter monday)
Apr 15 No Class
(Course finished)
17 Apr 20 No Class
(Course finished)
Apr 22 No Class
(Course finished)
18 Apr 27 No Class
(Course finished)
Apr 29 No Class
(Course finished)
19 May 04 No Class
(Course finished)
May 06
No Class (by agreement)
20 May 11 No Class (Move
institute)
May 13 No Class (Move Institute)
21 May 18 No Class (TAMC
in China)
May 20 No Class (TAMC in China)
22 May 25
No Class (exam week)
May 27
No Class (exam week)
This course is offered in the
master programs Logic, Grid Computing, Artificial Intelligence
in all cases as an elective course. The course is open as a choice
subject for
students of other programs.
Reading Material for the classes on Evasiveness
and Interactive Protocols now has been placed in the Library
(folders with copies in the shelves
next to the front desk in the library)
The course is a resurrection
from a course from the old program: Theoretische
Modellen ,
both offered for the last time in the year 2002 - 2003. During the
years 2003-06 this course was
incorporated into a course Game Theory for Information
Sciences, but this merger has been undone.
During the year 2007 - 2008
a number of other game related courses will be offered in the Master of Logic
Program:
Semester 2:
U. Endriss: Computational
Social Choice
K.R. Apt: Cooperative games
Course Contents:
Game theory originates from Economical
Sciences; it
treates theories about strategic
interaction and
related rationality concepts.
The subject is currently attracting researchers from
Computer Science and Technology,
particularly involving
themes about controling behavior
of unrelyable agents on the
Internet.
Fact is that concepts from game
theory have been used in
Computer Science already much
longer. Several of the models
used in theoretical Computer
Science for analysing problems with regards to
effectivity and efficiency
can be quite reasonably be described and analysed
in terms of games.
This link will be illustrated
in this course on the basis of a number
of models which originally
date from the 70-ies and 80-ies.
The main target is to illustrate
that games represent a computational
model in the same way nondeterministic
computations do.
Examples of models connecting games and computation theory are:
The theory of Traub and Wozniakowski
on the solution of numerical
problems.
Deciding Graph properties from
Adjacency Matrices ("Twenty Questions");
theory of Evasiveness.
The pebling game, used as a model
both for register allocation
and in the context of machine
model theory.
The Tiling game and its use in Reduction Theory
The Alternating Machine model
and its connection to
logic description languages
and games.
Interactive protocols for convincing
opponents about the presence of
information, without leaking
more information than
necessary: (Interactive proofs,
Arthur Merlin games, Zero Knowledge
proofs, ...)
The design of mechanisms for tweaking
agents on the internet to behave
in a truthful manner.
Except for the latter problem
the game theory involved deals combinatorial
games rather than the more
stochasitic real valued games studied in
Econmical Sciences. The game
theory involved therefore is rather
elementary.
Sheets available: (from the current
course Games and Complexity)
Part 1:
Topics: Introduction, Games and Tilings:
Updated version Available in Powerpoint00 format and pdf format
Part 2: Know thy numbers;
the impact of complexity.
Available in Powerpoint98
format
Part 3: Topics:
Endgame Analysis, PSPACE and Parallellism:
Updated version Available in Powerpoint00 format and
pdf format
Part 4:
Topics: PSPACE, Alternation, Games:
Updated version Available in Powerpoint00 format and
pdf format
Part 5 : Topics:
the Pebble Game:
Last version Available in Powerpoint00 Format and
pdf format
Part 6:
Topics: Diagonalization and Compression:
Updated and Extended version Available in Powerpoint00 format and
pdf format
Part 7: The evasiveness problem.
Available in
Powerpoint98 format
Part 8:
Topics: Interactive Proof Systems:
Available in Powerpoint00 format and pdf format
Sheets available: (from the current
course Introduction Game Theory)
Part 1:
Topics: Introduction, Games and Tilings:
Updated version Available in Powerpoint00
format and pdf format
Part 2:
Topics: Introductionary Examples
Updated version Available in Powerpoint00 format
and pdf format
Part 3:
Topics: Games with chance moves
Updated version Available in Powerpoint00 format and
pdf format
Part 4:
Topics: Utility Theory
with an extension into the axiomatic theory of Utility
based on material by Wakker and Meyerson listed below.
Updated version Available in Powerpoint00 format and
pdf format
Part 5:
Topics: Domination of strategies, with an application to
Kuhhandel
Updated version
Available in Powerpoint00
format and pdf format
Part 6:
Topics: Cooperative games: Nash Bargaining Theory
Most recent version
Available in Powerpoint00 format
and pdf format
Extra:
Topics: Chaotic Elections (after Donald G. Saari):
Available in Powerpoint00 format and pdf format
Part 7:
Topics: Mixed Strategies and von Neuman Min-Max Theorem
Most recent version Available in Powerpoint00 format and
pdf format
Part 8: Topics:
More on equilibria and correlated equilibria
Most recent version Available in Powerpoint00 format and
pdf format
Part 9: Topics:
Iterated games
New slides Available in Powerpoint00 format and
pdf format
Part 11: Topics:
Mechanism Design after Nissan & Ronen
Latest version Available in Powerpoint00 format and
pdf form
Part 12: Topics:
Halpern, Moses & Dolev on the Cheating Husbands problem
Version Available in Powerpoint00
format and pdf form
Sheets available: (from the previous
course Game Theory for Information Sciences;
Material from the reconstructed course will become available during the course)
Part 1:
Topics: Introduction, Games and Tilings:
Updated version Available in Powerpoint00 format and pdf format
Part 2:
Topics: Endgame Analysis, PSPACE and Parallellism:
Updated version Available in Powerpoint00 format
and pdf format
Part 3:
Topics: PSPACE, Alternation, Games:
Updated version Available in Powerpoint00 format and
pdf format
Part 4:
Topics: Introductionary Examples
Updated version Available in Powerpoint00 format and pdf format
Part 5:
Topics: Games with chance moves
Updated version Available in Powerpoint00 format and pdf format
Part 6:
Topics: Utility Theory
with an extension into the axiomatic theory of Utility
based on material by Wakker and Meyerson listed below.
Updated version Available in Powerpoint00
format and pdf format
Extra:
Topics: Chaotic Elections (after Donald G. Saari):
Available in Powerpoint00 format and pdf format
Part 7:
Topics: Domination of strategies
Updated version
Available in Powerpoint00 format and pdf format
Part 8:
Topics: Cooperative games:
Most recent version
Available in Powerpoint00 format and
pdf format
Part 9:
Topics: Mixed Strategies and von Neuman Min-Max Theorem
Most recent version Available in Powerpoint00 format and pdf
format
Part 10: Topics:
More on equilibria and correlated equilibria
Most recent version Available in Powerpoint00 format and pdf
format
Part 11: Topics:
Mechanism Design after Nissan & Ronen
Old version Available in Powerpoint00 format and
pdf form
Part 12: Topics:
the Pebble Game:
Last version Available in Powerpoint00 Format
and pdf format
Part 13:
Topics: Diagonalization and Compression:
Updated and Extended version Available in Powerpoint00 format and
pdf format
Sheets available: (from the previous course Theoretical Models)
Part 1:
Topics: Introduction, Games and Tilings:
Available in Powerpoint00 format and
pdf format
Part 2:
Topics: Endgame Analysis, PSPACE and Parallellism:
Available in Powerpoint00 format and
pdf format
Part 3:
Topics: PSPACE, Alternation, Games:
Available in Powerpoint00 format and
pdf format
Part 4:
Topics: the Pebble Game:
Available in Powerpoint00 Format and pdf format
Part 5:
Topics: Diagonalization and Compression: (version 2000)
Available in Powerpoint00 format and
pdf format
Part 6:
Topics: Interactive Proof Systems:
Available in Powerpoint00 format and pdf format
Part 7: Controling
Selfish Agents on the Internet:
Available in Powerpoint00 format and pdf format
Sheets available: (from the last course Intelligent
Databases)
Chapter 0:
Topics: Introduction.
Available in Powerpoint98 format and pdf format
Chapter 1:
Topics: Basic Concepts
Available in Powerpoint98 format and
pdf format:
Chapter 2:
Topics: Games with chance moves
Available in Powerpoint98 format and pdf format
Chapter 3:
Topics: Utility Theory
with an extension into the axiomatic theory of Utility
based on material by Wakker and Meyerson listed below.
Available in Powerpoint98 format and pdf format
Chapter 4:
Topics: Domination of strategies
Available in Powerpoint98 format and pdf format
Chapter 5:
Topics: Cooperative games:
Available in Powerpoint98 format and pdf format
Chapter 6:
Topics: Mixed Strategies
Available in Powerpoint98 format and pdf format
Chapter 7:
Topics: On Equilibria: cheap talk and correlated equilibria
Available in Powerpoint98 format and pdf format
Contents
of previous courses 2002 and 2003 are almost identical, but beware for
obsolete URL references !
Aditional Material related to
games available as slides.
Extra: Topics: The Game of Chaos:
Presentation at the 34 Dutch Mathematics Conference April 1999
Nov 19-20 1999. Available in Powerpoint98 Format
and pdf format
See also: P. van Emde Boas &
E.H. van Emde Boas, The Game of Chaos, as included in the
bibliography below.
Extra:
Topics: Games in the Classroom:
Available in Powerpoint98 format and pdf format
Extra:
Topics: The connection between Games and Computer Science,
Talks prepared for a trip to China, April 2000:
Available in Powerpoint98 format and pdf format
Extra:
Topics: The Games of Computer Science,
Talk at TU Delft, Feb 23 2001:
Available in Powerpoint98 format and pdf format
Extra:
Topics: Playing Savage:
Available in Powerpoint98 format and pdf
format
Manuscript available in Postscript
Extra: Topics: Imperfect
Information Games, looking for the right model.
Talk at Algemeen Wiskunde Colloquium, Feb 27 2002:
Available in Powerpoint2000 format
Extra: Topics: Imperfect
Information Games, what makes them hard to decide.
Talk at Amsterdam Aachen Exchange Feb 15, 2002:
Available in Powerpoint2000 format
Extra: Topics: IF Logic;
slides of the presentation of Merlijn Sevenster on Nov 29 2004
Available in pdf format
Extra:
Topics: Games in Computation and Complexity Theory,
Tutorials at the workshop Games and Logic Kazimierz Dolny (PL), Sep 25-26
2006
Available in Powerpoint2000 format and pdf format
Literature: (This is a joint list for both my game related courses).
Game Theory.
Ken Binmore, Fun and Games,
Houghton Mifflin Company, 1992;
this is a textbook which was
used for earlier editions of my other course
Ken Binmore, Playing for Real,
a text on game theory, Oxford University Press, 2007;
this is a textbook which
is used for my other course
Elwyn R. Berlekamp, John H. Conway
& Richard K. Guy,
Winning Ways (Vol 1
and Vol. 2), Academic Press, 1982.
John H. Conway, On Numbers and Games, Academic Press 1976.
the above two references develop the mathematics of combinatorial games in great depth.
V.W. Gijlswijk, G.A.P. Kindervater,
G.J. van Tubergen & J.J.O.O Wiegerinck,
Computer Analysis of E. de
Bono's L-Game, Rep. Math, Inst. UvA 76-18
(an early computerized backward
analysis of a non-trivial game; also an early
student project in our department...)
M.J. Osborne & A. Rubinstein, A Course in Game Theory, MIT Press 1994.
Roger B. Myerson, Game Theory;
Analysis of Conflict, Harvard University Press 1991.
see chapter
1 for the axiomatic treatement of Utility Theory.
Alexander Mehlman, The
Game's Afoot!, Game Theory in Mythand Paradox,
Amer. Math. Soc. Student Math.
Library 5, 2000
Introduction, with many examples
drawn from the literature and
mythology; however, in final
sections rather difficult.
Steven J. Brams, Superior
Beings, Springer Verlag 1983;
Game theory applied to Theology.
Boudewijn de Bruin, Explaining Games, On the
Logic of Game Theoretic Explanations,
Ph.D. Thesis ILLC UvA 20041207,
ILLC Dissertation Series DS-2004-03
An in depth study of the connections between Backward Induction, Elimination
of dominated strategies,
and Common Knowledge of Rationality.
K.R.
Apt, "Order Independence
and Rationalizability'' ,
Proc. of the 10th Conference on Theoretical Aspects of Rationality
and Knowledge (TARK X),
pp. 22-38, 2005.
K.R. Apt, "The Many Faces of Rationalizability''.
Manuscript, August 2006.
Yoram Moses, Danny Dolev
and Joseph Y. Halpern,
Cheating Husbands and other stories;
a case study on knowledge, action, and communication,
Distributed Computing (1986) vol. 1 167-176.
Preprint in procs PODC 4, 1985.
An in depth study explaining
the importance of synchronization for solving the muddy children problem.
Computation Theory.
John E. Hopcroft & Jeffrey
D. Ullman, Intorduction to Automata Theory,
Languages and Computation,
Addison Wesley, 1979.
David Harel, Algorithmics;
the Spirit of Computing, (second Edition),
Addison Wesley 1992.
Thomas A. Sudkamp, Languages
and Machines; An Introduction to the
Theory of Computer Science,
(second Edition), Addison Wesley 1997.
a formerly used textbook for
the course Automata and Complexity Theory
Harry R. Lewis & Christos
H. Papadimitriou,
Elements of the Theory
of Computation, Prentice Hall 1981.
Christos H. Papadimitriou, Computational
Complexity,
Addison Wesley 1995.
Chapter 19 covers many
of the computation theory subjects dealt with in this course!
Cees Slot & Peter van Emde
Boas, The Problem of Space Invariance for
Sequential Machines,
Inf. and Comp. 77 (1988) 93--122.
Peter van Emde Boas, Space
Measures for Storage Modification Machines,
Inf. Proc. Letters 30
(1989) 103--110.
Peter van Emde Boas, Machine
Models and Simulations, in
J. van Leeuwen, Handbook of
Theoretical Computer Science vol A,
Algorithms and Complexity,
Elsevier, 1990, pp 3--66;
preprint: ITLC-CT-89-02.
The last three papers concern the Invariance Thesis.
The pebling game, used as a model
both for register allocation
and in the context of machine
model theory.
J.E. Hopcroft, W. Paul & L.
Valiant, On Time versus Space,
J. Assoc. Comput. Mach., 24
(1977) 332--337.
A. Lingas, A PSPACE Complete
Problem related to a Pebble Game,
G. Ausiello & C Böhm
eds., Proc. ICALP'78,
Springer LNCS 62, 1978, pp.
300--321.
P. van Emde Boas & Jan Van
Leeuwen, Move Rules and
Trade-Offs in the Pebble
Game, in K. Weihrauch ed.,
Proc. 4th GI Theoretical Computer
Science Conference,
Springer LNCS 67 1979, pp.
101--112.
John R. Gilbert, Thomas Lengauer
& Robert E. Tarjan,
The Pebbling Problem is
Complete in Polynomial Space,
SIAM J. Comput. 9 (1980) 513--524.
Hiroaki Tohyama & Akeo Adachi,
Complexity of path discovery
game problems,
Theor. Comp. Sci.. 237, 2000,
381--406.
another PSPACE-hard solitaire
game.
Tiling Game and its use in Reduction Theory
Bogdan S. Chlebus, Domino-Tiling
Games, J. Comput. Syst. Sci. 32
(1986), 374--392.
Martin. P.W. Savelsberg &
Peter van Emde Boas, BOUNDED TILING,
an alternative to SATISFIABILITY?, in G.Wechsung ed., proc. 2nd
Frege Memorial Conference,
Schwerin, Sep 1984, Akademie Verlag,
Mathematische Forschung vol.
20, 1984, pp. 401--407.
preprint: rep. CWI-OS-R8405.
Peter van Emde Boas, The Convenience
of Tilings, in: Andrea Sorbi, ed.,
Complexity, Logic and Recursion
Theory, lecture notes in pure and
applied mathetaics vol 187,
1997, pp. 331--363. (for ps. version of preprint
).
The sheets of this lecture
are available at sheets in postscript . However
beware: on behalf of its origin as a (by now obsolete MacWrite document)
the Postscript pages are sorted
in reverse order.
Peter van Emde Boas, Is elf plus één
twaalf?; over rekenen en puzzelen.
Explanation of the construction of the tiling puzzle
demonstration
model for precollege students (in Dutch). Text available
in Postscript
.
The corresponding figures are available in Postscript
also: pict1pict2pict3pict4
.
David Harel, Recurring Dominos:
making the Highly Undecidable
Highly Understandable,
Ann. Discrete Math. 24 (1985) 51--72.
David Harel, Dynamic Logic,
in D. Gabbay & F. Guenthner (eds.)
Handbook of PhilosophicalLogic,
Vol II, D. Reidel 1984, pp. 497--604.
Background information on
Dynamic Logic for the reduction to
PDL-Satisfiability from the
two person tiling game as invented by
Chlebus.
The Alternating Machine model
and its connection to
logic description languages
and games.
A.K. Chandra, D. Kozen & L.J.
Stockmeyer, Alternation,
J.Assoc. Comput. Mach. 28
(1981) 114--133.
Christos H. Papadimitriou, Computational
Complexity,
Addison Wesley 1995. See chapter 19 in particular.
L.J. Stockmeyer & A.R. Meyer,
Word problems requiring exponential time,
Proc STOC 5 (1973), pp 1--9.
An important early paper and
the source of the QBF problem.
T.J. Schäfer, Complexity
of some two-person perfect-information games,
J.Comput. Syst. Science, 16
(1978) 185--225.
The first PSPACE complete
game: Geography.
D. Lichtenstein & M. Sipser,
GO is polynomial-space hard,
J. Assoc. Comput. Mach, 27
(1980) 393--401.
S. Even & R.E. Tarjan, A
combinatorial game which is complete for polynomial
space, J. Assoc. Comput.
Mach, 23 (1976) 710--719.
S. Reisch, HEX ist PSPACE-volständig, Acta Informatica 15 (1981) 167--191.
A.S. Fraenkel, M.R. Garey, D.S.
Johnson, T. Schäfer & Y. Yesha,
The complexity of checkers
on an N X N board - preliminary report,
Proc IEEE FOCS 19 (1978) pp.
55--64.
A.S. Fraenkel & D. Lichtenstein,
Computing a perfect strategy for n x n chess
requires time exponential
in n, J. Combin. Theory series A 31 (1981) 119--213.
G.W. Flake & E.B. Baum, Rush
Hour is PSPACE-complete, or "Why you should generously tip
parking lot attendants",
Theoretical Computer Science, 270 (2002) 895--911.
A very simple combinatorial
solitaire game which is PSPACE-complete. For a yet simpler version
see this note by John Tromp.
Erik D. Demaine, Playing
Games with Algorithms: Algorithmic Combinatorial Game Theory ,
MFCS 2001, Springel LNCS 2136,
pp. 18-32 .
Much more information is available
from his impressive website .
The above five papers involve games played by real people.
P. van Emde Boas, The Second
Machine Class 2, an Encyclopedic View on the
Parallel Computation Thesis.
in: H. Rasiowa ed., Mathematical Problems in
Computation Theory, Banach
Center Publications, vol 20, PWN Warsaw 1988,
pp. 235--256.
An older version of sections
3 and 4 in my Handbook Chapter.
Slides about this subject
are available in Postscript
(but again in reverse order...).
Robert A. Stegwee, Leen Torenvliet & Peter van Emde
Boas,
The Power of your Editor, Rep. IBM RJ 4711
(50179) 5/21/85;
also: Report FVI-UvA-85-03.
Sheets have been placed on
the Web; once more in reverse order Postscript
Interactive Proof systems and other models of interaction and/or randomized computation
Carsten Lund, Lance Fortnow &
Howard Karloff, Algebraic Methods for Interactive Proof Systems,
J. ACM. 39 (1992) 859-868.
Adi Shamir, IP = PSPACE, J.ACM. 93 (1992) 869-877.
A. Shen, IP= PSPACE: Simplified Proof, J.ACM. 93 (1992) 878-880.
L. Babai, E-mail and
the unexpected power of interaction, Proc. IEEE Symp. Structure in
Complexity Theory 5, Barcelona,
July 08-11 1990, pp. 30--44.
Shafi Goldwasser, Silvio Micali
& Charles Rackoff, The Knowledge Complexity of
Interactive Proof Systems,
SIAM, J. Comput. 18 (1989), 186-208.
Christos H. Papadimitriou, Games against Nature, Proc. IEEE FOCS 1983, pp. 446-450
L. Babai &S. Moran, Arthur-Merlin
Games: a Randomized Proof System and a Hierarchy
of Complexity Classes,
J. Comput. Syst. Sci. 36 (1988) 254-276.
G. Peterson, J. Reif and S.
Azhar, Lower Bounds for Multiplayer Nocooperative Games of Incomplete
Information,
Computers and Mathematics with Applications 41 (2001) 957-992.
Preprint vailable here
The design of mechanisms for tweaking
agents on the internet to behave
in a truthful manner.
Noam Nissan & Amir Ronen,
Algorithmic Mechanism Design,
proc. ACM STOC 31 (1999).
Noam Nisan, Algorithms for
Selfish Agents,
in C. Meinel & S. Tison,
eds., Proc. STACS'99,
Springer LNCS 1563, 1999,
pp. 1--15.
Amir Ronen, Algorithms for
Rational Agents,
proc. SOFSEM 2000, Springer
LNCS 1963, 56--70.
Evasiveness problem ("twenty questions")
M.R. Best, P. van Emde Boas and
H.W. Lenstra, jr.,
A Sharpened version of
the Aandera-Rosenberg Conjecture,
rep. MC-ZW-30-74, October
1974.
Ronald L. Rivest & Jean Vuillemin,
On recognizing Graph properties
from Adjacency Matrices,
Theor. Comp. Sci. 3 (1976)
371-384.
J.Kahn, M. Saks, D. Sturtevant, A topological Approach
to Evasiveness,
Combinatorica 4 (1984) 297-306.
N. Illies, A counterexample to the Generalized Aanderaa-Rosenberg
Conjecture,
Inf. Proc. Letters, 7 (1978) 154-155.
L. Lovasz, Lecture
notes on Evasive Graph properties, notes taken by Neal Young around fall 1990
at Princeton University.
A. Chakrabarti, S. Khot & Y. Shi, Evasiveness of subgraph containment and related
properties,
SIAM J. of Computing 31(3) (2002) 866-875.
V. Welker, Constructions
preserving evaisiveness and collapsability,
Discrete Math. 207 (1999) 243-255.
A. Aggarwal, D. Coppersmith & D. Kleitman, A generalized model for understanding Evasiveness,
Inf. proc. letters 30 (1989) 205-208.
F.H. Lutz, Some results
Related to the Evasiveness Conjecture,
J. Comb. theory B 81 (2001) 110-124.
Unrelated but still about Games and/or Computer Science.
Peter van Emde Boas, Games
in the Classroom,
position paper at OOPSLA'99
Workshop #2, Quest for Effective Classroom Examples.
Postscript version Available .
H. Jaap van den Herik, Jos W.H.M.
Uiterwijk & Jack van Rijswijck,
Games solved: Now and in
the future,
Artificial Intelligence 134
(2002), 277-311.
A survey on the state of the
art in real life game analysis by computers;
the entire issue of Artificial
Intelligence is dedicated to this problem area in AI.
Donald G. Saari, Chaotic
Elections! A mathematical Looks at Voting,
AMS 2001.
A thorough analysis on how
voting systems fail to implement democracy.
Mandatory reading material
for voters.
Course Evaluation:
Except for specific student contributions
(with previous arrangements
with the teacher) the students
will be graded on their solutions of
homework exercises which will be made public on the
website gradually
during the course. Answers
should be returned within the deadline as listed with
the exercise, only on paper.
Even when the answers are prepared electronically
the solution must be submitted
on paper.