Logic, Games & Computation:
'the story behind the
scenery'
Lecturer: Johan van Benthem
& guest speakers
Autumn 2006, Master of Logic Seminar,
Amsterdam
Sunday26 November: FINAL SCHEDULE
Homework
The deadline for all homeworks is Monday December 4
 'this is it', and on that date, you can give me all assignments that you
have not handed in yet. The last homework is that on epistemic temporal logic
by Eric Pacuit (Week 11, see below).
Workshops
These are working sessions where new topics and open problems are presented
which you can use for a final paper. You can work together with other students
on a topic if you want to. This replaces the original paper assignment
for your final requirement. However, if you have an individual paper topic
already, then it would also be fine to do that, and skip the workshop. The
deadline for both variants is Friday December 15.
23 November, Workshop 1:
We discussed two topics in some technical
detail:
(a) Frame correspondence arguments
for DEL axioms. The first task would be to define a suitable general
setting that makes the best sense of the frame correspondence arguments in
my paper 'Dynamic Logic of Belief Revision'. This requires some formal setting
up of families of epistemic (plausibility) frames and transitions between
them, plus maybe abstract functions or relations between world across frames.
Then you could sharpen up some correspondence arguments. E.g., we found during
our session that, to fully capture
eliminative update, we needed
the reduction axiom for knowledge, but also one for an existential modality,
plus a basic equivalence <!P>T iff P. A further desideratum:
analyze the reduction axioms for knowledge in full DEL, and find a correspondence
proof showing to which extent this forces us to adopt product update. Finally,
all this led to a more general mathematical question: what happens to existing
modal correspondence theory when we consider modal languages with dynamic
modalities referring to changing models? For instance, the axioms that we
analyzed all have a simple 'Sahlqvist form', but what is the proper generalization?
(b) Topological models
for DEL. We discussed the basics of topological semantics for
modal and epistemic languages, whose details (including bisimulation, and
all that) you can find explained in several papers on my homepage under Research.
Topological models add new, nongraphbased structures that can represent
information, and hence they can also model information flow in new ways.
Concrete question: Generalize product update MxE over topological
static models M and topological event models E.
You need to generalize the formulation of the basic reduction axiom for knowledge.
The generalization should reduce to standard product update on relational
models (which correspond to special 'Alexandroff topologies'), and it should
validate the generalized reduction axiom. My prediction is that you will
need something close to standard 'topological product spaces', the ones you
find in basic results in topology like Tychonoff's Theorem.
There are further topics along these
lines.
E.g., the representation theorem for
DEL models inside branching temporal universes presented in
class by Eric Pacuit also has a 'correspondence' flavour, describing what
conditions on temporal universes are enforced by what DEL axioms. One
obvious question is how this result and its proof can be extended to a temporal
logic representation theorem for epistemic plausibility models allowing
for both epistemic update and belief revision. You would need to think
about translations from DEL and its belief versions into ETL and its belief
versions. Moreover, conceptually, you will encounter the fact that 'belief'
in a temporal setting has two distinct aspects: beliefs about the world based
on observation of events so far, and belief based on prior expectations about
what still has to happen in the future. These may require different formal modeling.
29
November, Workshop 2
Wednesday 11:15  13 hours, usual place Diamantslijperij)
The second workshop will be led by
our Ph.D. students who presented in class in October and November, and they
will provide materials on topics and open problems in research right now.
Theme: "From Constraints to Preferences
and Back: the General Picture"
We are going to study the connections between constraint based preferences,
as presented earlier by Fenrong Liu, and "possible world"based preferences,
as presented by Olivier Roy and Patrick Girard.
A very suggestive parallel can be drawn between the two approaches. The essence
of what Fenrong presented is to propose various ways (there's 3 in the paper)
to construct a preference relation over objects (i.e. c > c') from the
a defined binary relation between predicates, that she calls constraints.
It's a "topdown" procedure. For example, given two constraints C and C'
such that C >> C' : c > c' iff = C(c) or ((C (c) iff C(c')) and
C'(c). The "possibleworld"based preferences approach goes the other way
around. There, various procedures (the number depends on the properties of
>) are proposed
to construct a preference relation between predicate (or formulas) from the
defined binary relation between objects/worlds. It's a "bottomup" procedure.
One example: P Pref P' iff forall x,y if P(x) and P'(y) then x > y. Mathematically,
the two approaches appear to be highly comparable. Just look at formulas
P, P' and constraint C, C' abstractly, as predicates, and at world as objects.
In this workshop, we are going to investigate aspects of this connection,
through the following questions:
Is one of the topdown procedures "transformationequivalent" to one of the
bottomups? By "transformation equivalence" we will mean: take an ordering
on predicate >>, transfer it to an ordering between worlds > using
a given topdown procedure, then can we find a bottomup procedure in preference
logic that would lift back > to a new >>' such that C >> C'
iff C >>' C'? The same question aries mutatis mutandis from a defined
> back to >'.
Two of the topdown procedures presented in Fenrong's lecture involved beliefs
of the agents. How can we translate these requirements in preference logic?
If time permits, we will also look at the connection between the completeness
results obtained in Fenrong's work, and the ones for preference logic.
========================================
Aim of the Seminar
This seminar
is about dynamic logics that describe events where information
flows. Our special focus will be games, which form a natural
interface with modern logic, which is broadening year by year.
You will learn about some current topics, basic techniques, and
ongoing research. About half of our presentations will be
by guest speakers active in the area.
Time and Place
Please note: we have kept
the Wednesday slot 11  13 hours, in the 'Diamantslijperij',
and for a fallback, we also have Thursdays 18  20 hours, 'Euclides'
building.
Schedule
September
6 Games in logic, and
logic in games
Logical notions
can be cast in the form of 'logic games', and this leads to significant
links with game theory. But also more generally, the structure of arbitrary
games can be analyzed by logical means. For this purpose, we need to
develop 'game logics' that combine ideas from computation (dynamic
logics) and philosophy (epistemic logics). Here is a general Introduction.
Two useful chapters with background: Evaluation
Games, Comparison
Games.
13 + 14 Philosophy meets computer
science: epistemic and dynamic logic
We have surveyed
the main ideas from epistemic logic and dynamic logic (ßsee
handout with points): language, models, truth definition, validities,
correspondence, bisimulation and invariance, and complexity. A
text containing the whole story in more detail is the chapter on
modal logic in a semantic perspective by Blackburn
& van Benthem in the forthcoming Handbook of Modal Logic.
20 From public annoucement
to dynamic epistemic logic
Read the introduction
to the logic PAL of public announcement in the
paper 'One is a Lonely Number'. We covered: basic epistemicdynamic
language, models, reduction axioms, completeness, decidability,
adding common knowledge, bisimulation invariance. Two still open
questions were mentioned: describing the substitutionclosed validities,
characterizing the persistent formulas whose public announcement
turns them into (common) knowledge. A brief introduction was given
to product update with event models. You can read the definitions later
on in the same paper.
27 A logic of communication and
change Barteld Kooi
Read the definition of DEL
in the paper 'A Logic of Communication and Change', and get a sense of completeness
theorems and other results. Some recent further literature was
mentioned in class.
October
4 Dynamics of belief
change and revision
We took two hours on
Wednesday, going through the following handout.
I hope that the apple symbols, which of course stand for the Tree
of Knowledge, come through this time.
Read the paper 'Dynamic Logic of Belief Revision', as far as
you can get. Background on belief revision is not needed, but you
may want to check any standard source on 'AGM theory' and 'Grove
models' (Google should help). By now, you should see the same dynamic
logic ideas returning!
Our next block of three classes works with essentially the same
epistemic accessibility + plausibility models which were used this week. But
now they serve as a basis for new developments at ILLC in static and dynamic
preference logics, which are related to plausibility reasoning in several
ways.
11 Preference logic and games Olivier Roy
18 Preference change: qualitative
and quantitative Fenrong Liu
Here is special webpage
for this presentation, including a homework question.
25 Belief Revision and (Ceteris
Paribus) Preference Logic Patrick
Girard
Here is a handout
of this presentation.
November
1 Dynamic logics of game structure
Johan van Benthem
This is based on two papers that you
can download under Research on Logic & Games at my website.
They are called 'Extensive Games as Process Models'
and 'Games in Dynamic Epistemic Logic'. We mainly discussed
two basic levels for studying games: bisimulation and modal logic, and
power equivalence and modal 'forcing languages'. Both introduce finestructure
into game theory, and allow us to balance expressive power versus computational
complexity in describing players, and game solutions.
8 Complexity and games Merlijn Sevenster
This lecture will mainly evolve around the following two questions:
1. How can one measure the complexity of a game? 2. What properties
make games harder? We will discuss winloss games, and their
computational complexity, in terms of some complexity theory: P, NP,
PSPACE, EXP, NEXP. Papadimitriou's Theorem: Games that meet the following
conditions are in PSPACE: 1. any legal sequence of moves is polynomially
bounded, 2. all successors of a position can be enumerated in PSPACE;
if there are none, it can be decided in PSPACE to whom the position belongs.
Logical games of increasing complexity SAT  NP  1 player game, QBF
 PSPACE  2 player game, IFQBF  NEXPTIME  2 player, impf. inf. game,
IFQBF + Perfect Recall  PSPACE. "PlayGames" Minesweeper, Sudoku, Battleships
 NP, Geography  PSPACE, Checkers, Chess  EXPTIME. The effect of impf.
inf. is not well understood, but: SY  PSPACE. Reflections on the complexity
theory measure for games: Advantages, Disadvantages
References: (a) C.H. Papadimitriou, 1994, Computational
complexity  especially chapter 19, (b) L. J. Stockmeyer and A. R. Meyer,
1973, Word problems requiring exponential time, Proceedings of the 5th ACM
Symposium on the Theory of Computing (STOC 73), 19  here PSPACEcompleteness
of QBF is established. (c) R.J. Nowakowski, 2002, More games of no chance,
Cambridge University Press  gives a nice overview and uptodate
papers on combinatorial game theory. Don't forget to check out the
list of open problems! (d) M. Sevenster, 2006, Branches
of imperfect information: logic, games, and computation  especially
section 3.6 and chapter 6.
15 Epistemictemporal logic Eric Pacuit
See the paper The
Tree of Knowledge in Action, presented at AiML Melbourne 2006..
23 (Thursday 18;00 
20 hours!) Workshop 1: Belief revision, correspondence,
and temporal representation
29 (Wednesday 11:15
 13 hours!) Workshop 2: Preference logic and games
December
Date to be arranged: paper/workshop
presentations.
Prerequisites
A working knowledge of modal logic,
plus some general logical sophistication.
Homework Assignments
Weeks 1: A question on firstorder evaluation games
Look up the paper 'An Essay on Sabotage and Obstruction', ILLC
Research Report 2005. Explain the sabotage game with some new
example, verify the given translation for Traveler's being able
to win into the truth of some matching firstorder formula, and show
that evaluation games for firstorder formulas are determined, with
an upper bound of polynomial space for their solution complexity.
Week 2A: A question on modalepistemic frame correspondence
Consider the epistemicmodal principle [a]K phi <>
K[a] phi. Determine which relational conditions on epistemicmodal frames
correspond to validity of the two separate implications, and prove this.
Discuss the connection with one particular reduction axiom (which one?) of
the logic of public announcement. What does this express about agents' powers?
The [a]K/K[a] equivalence is almost valid in DEL product update, but
with an important modification: explain.
Week 2B: A question on bisimulation and invariance
State and prove the Adequacy Lemma for modal bisimulation games,
which says that Duplicator has a winning strategy in the game
over k rounds if the initial pointed models have the same modal
theory up to operator depth k. Also state the equivalent version for
Spoiler, and compare the two versions. Explain the difference between
the usual 'Esick' statement of the result, and the more constructive
information that we discussed in class.
Week 3: A question about completeness for public announcement
logic
Look up the completeness proof for epistemic logic with conditional
common knowledge in the paper 'A Logic of Communication and Change'.
Explain the overall structure, and supply the missing steps.
Week 4: Question on DEL supplied by our guest speaker
Barteld Kooi: assignment.
You can hand this in either by email by next Monday morning (get it out of
your system!), or as part of the second batch of homeworks end
of October.
Week 5: Homework question for dynamic logic of belief revision:
Give a complete soundness argument for the reduction axioms
given in the paper for the 'conservative' variant of lexicographic
upgrade. Also, give the complete modal correspondence argument analyzing
the axiom for knowledge of agents after public announcements (you will
encounter some technical issues to be resolved, but they are not major).
Bonus: Explain how the uniform update rule of Baltag &
Smets covers all cases dealt with in class, as well as those in van Benthem
& Liu.
Week 6: Here is the assignment
by Olivier Roy
Week 7: Here is the assignment
by Fenrong Liu.
Week 8: Here is the assignment
by Patrick Girard.
Week 9: (a) Complete the details of the representation argument
for players' powers in terms of games given in class. (b) Look up
the axioms for Rohit Parikh's 'Dynamic Logic of Games', either
in the old lecture notes "Logic and Games", or in the Pauly/Parikh survey
paper (check via Google). Explain the intended formal semantics, and
show how the modality differs from the standard ones in relational semantics.
Then show that the axioms are sound. Explain also how the
Parikh modality could be formulated inside standard propositional dynamic
logic, at least within a given game. (c) Try to define
the Backward Induction solution to a finite game with values for outcomes
using a combination of modal logic of actions and the preference logics
you have learnt about in October. You can either look up a solution in
the literature, or think up one for yourself.
If you have questions about the formulation or content
of these Exercises, do send email.
Week 10 (Merlijn Sevenster): Please check
with sevenstr@science.uva.nl for any questions about the following
assignment.
Select two nonstarred questions or one starred question
and submit your answer to the selected problems electronically to Merlijn
Sevenster. Note that the starred questions are challenging  it would most
certainly be appreciated if you gave one of them a try.
A. In class we discussed the complexity measure provided by computational
complexity theory. But many more are available on the market. Give a short
overview of two other measures of complexity for games (with examples!).
B. Give a detailed account (using pseudo code) of Papadimitriou's algorithm
solving any game meeting the conditions 1. and 2 mentioned in class.
C. Prove that DQBF is computable in EXPTIME.
(*) D. Prove NEXPhardness for DQBF by reducing from TILING, see Problem
20.2.10 on pg. 501 of Papadimitriou (1994).
E. Prove the equivalence: QBF = Gamified QBF. That is, show that a QBF instance
F holds iff Eloise has a winning strategy in G(F).
(*) F. Let DQBF+PR be the fragment of DQBF, such that every F gives rise
to a game G(F) that has perfect recall. Show that for every DQBF+PR instance
F, there is a QBF instance G such that F holds iff G holds.
G. Extend the ladder SAT  QBF  DQBF+PR  DQBF by another SATlike problem,
that can be associated with a gametheoretic property. Here you can start
from either game theory or logic. What, you guess, would be the problem's
computational complexity?
H. It is sometimes informally argued that NPcompleteness is a necessary
property for a game to be interesting (for instance pg. 476 of A.S. Fraenkel
(2002) in R.J. Nowakowski (2002) and http://www.ics.uci.edu/~eppstein/cgt/hard.html).
Please give your opinion on this proposition supported by three arguments.
References
C.H. Papadimitriou, 1994, ComputationalComplexity  especially chapter
19.
L. J. Stockmeyer and A. R. Meyer, 1973, Word problems requiring exponential
time, Proceedings 5th ACM Symposium on the Theory of Computing (STOC 73),
19  here PSPACEcompleteness of QBF is established.
R.J. Nowakowski, 2002, More games of no chance, Cambridge
University Press  gives a nice overview and uptodate papers on combinatorial
game theory. Don't forget to check out the list of open problems!
M. Sevenster, 2006, Branches of imperfect information: logic,
games, and computation  especially section 3.6 and chapter 6.
Week 11 (Eric Pacuit):Check with epacuit@science.uva.nl
if clarification is needed.
For the precise definition of the no miracles and perfect recall
properties, refer to the paper
`The Tree of Knowledge in Action: Towards a Common Perspective'.
(1) The no learning and no forgetting property from the Halpern
and Vardi paper are as follows:
(nf) for all histories H, H', n,n',k\ge 0, if H_n\sim_i H'_n' and
k\le n, then there is a k'\le n'
such that $H_k\sim_i H'_k'
(nl) for all histories H, H', if H_n \sim_i H'_m, then for all k\ge
n, there is a l\ge m such that
H_k\sim_i H_l
Answer the following questions:
a. What is the precise connection between the no miracles property
and the no learning property? Are they equivalent? Does one imply the other?
b. What is the precise connection between the perfect recall property and
the no forgetting property? Are they equivalent? Does one imply the other?
c. Find ETL formulas that are validated in frames with the (nf) and the
(nl) properties respectively.
2. Let M be a Kripke structure and E an event model. Prove that Tree(M,
E)= ((M x E) x E) x E x ...
has the perfect recall and uniform no miracles property.
3. Find an event model E and an initial model M such that in the generated
tree Tree(M, E)= ((M x E) x E) x E x ..., the number of nodes at each moment
continues to grow (that is for each t\ge 0, the number of finite histories
of length t+1 is greater than the number of finite histories of length t).
Are the structures at each stage ``really different'' or is there a momen
after which all the models are epistemically bisimilar?
Course Materials
General material on
update logics and dynamic logics of games can be found
on the web page Logic,
Games and Computation at Amsterdam . You can also check
my own website under Research, and under Teaching
for some earlier seminars. We will post relevant documents later
on a weekly basis.
Some background literature
Earlier course webpage:
Logic and
Games. Check the electronic lecture notes Logic in Games
, which will be updated periodically. Here is the Introduction
to the 'Logic in Games' seminar from 2002. Update for everybody: Farewell
to Loneliness. Recent challenge papers: Open
Problems in Logical Dynamics, and Open Problems in Logic
and Games. Publication details and further references:
check again my webpage under Research. My own philosophical
views: Epistemic
Logic and Epistemology.
Here are some links
for specific topics, but these will be updated as we proceed:

Paper on epistemic logic
and dynamic logic of public announcement: One is a Lonely
Number (appeared in Logic
Colloquium '02, ASL Publications, 2006).

Paper on dynamicepistemic logic
of general events, with Jan van Eijck & Barteld Kooi: Logics
for Communication and Change, TARK Singapore 2005 & to appear
in Information & Computation.

Paper on topological models for
epistemic logic, with Darko Sarenac: The
Geometry of Knowledge, appeared in Universal Logic 2004.






Typical paper on dynamic logic for
game solution: Rational
Dynamics, Liverpool & Siena 2002; to appear in International Journal of Game Theory.

Contact
Write to johan@science.uva.nl
with comments or suggestions.
