Concrete Epistemic Modal Logic: Flatland
Olivier Gasquet and François Schwarzentruber
IRIT (Institut de Recherche en Informatique de Toulouse)
Université de Toulouse
Abstract. In this paper, we give a logic for perception and knowledge:
Flatland. This semantics of this framework is a concrete Kripke model
so that it is an easy-to-understand toy example for students. We present
a piece of software called Plaza’s world enabling to check formulas in
such a concrete Kripke model and to announce formulas.
Keywords: Epistemic modal logic, Public announcements, Flatland.
This work is directly inspired by some aspects of the thesis of one of the authors
of this paper . The initial idea was inspired by the famous Tarski’s world
, a software that allow undergraduate students to practice predicate logic in
a concrete situation1.
Here, we aimed at developing a concrete example of application of epistemic
logic  to help graduate students to understand and to practice it with the help
of announcements as in the logic of announcement of  who gave his name to
our project. This idea lead to several theoretical developments (axiomatization,
completeness, complexity) and it is now time to go back to the initial aim: to
offer a tool for teaching epistemic logic. In this paper, we will deliberatelay omit
theoretical considerations and focus on the tool itself, all details can be found in
the thesis. Also, we consider the reader more or less familiar with modal logic,
this paper is written for teachers and not really for students.
We consider a framework where some artificial agents have some knowledge2
and, can see or cannot see both other agents and where they are looking at. We
knowledge that each of them sees c?", ...To this aim, we will consider a concrete
situation in the plane3where we suppose that any agent sees the entire half plane
in front of her. At the moment, Plaza’sworldopens on a window where all differ-
ent dispositions of three agents are visible, each of these dispositions representing
a possible state of affair. In some state, let us say that of Figure 1:
1A universe of coloured geometrical figures in which students interpret or write for-
mulas like ∀x.(square(x) → ∃y.(triangle(y) ∧ grey(y)∧ on_left(x,y))).
2As often in the literature, our logic for knowledge is S5.
3We do not present here the version where agents are situated on a line which has been
the first framework investigated in  and .
P. Blackburn et al. (Eds.): TICTTL 2011, LNAI 6680, pp. 70–76, 2011.
c ? Springer-Verlag Berlin Heidelberg 2011
Concrete Epistemic Modal Logic: Flatland71
Fig.1. A possible disposition of agents
Ann sees both Bob and Chris, Chris also sees everybody but Bob
sees no one. But more, “Ann knows that Chris sees Bob”, and also
“Ann knows that Chris knows that Bob does see Ann’, ...What does
Bob know? In fact nothing beside that Ann and Chris either sees
him or does not see him. Bob can imagine any possible situations of
Ann and Chris, but “Ann knows that Bob doesn’t know she sees him” (*).
Any agent can imagine that the actual situation is any situation that is compat-
ible with her partial view, with what she sees, imagination concerns what she
does not see. Plaza’s world allows two kind of actions:
1. the user may test whether such sentence may be true in some situations: by
entering a logical formula representing the sentence, then Plaza’s world
will indicate those situations where the formula is true: in situation of Fig-
ure 1, the sentence (*) is true;
2. the user may publically announce a sentence like “Ann knows that Bob does
not see Chris”(**), this would provoke the deletion from the model of situa-
tions which are not compatible with this sentence.Of course this may change
the result of testing the truth of some sentences, e.g. after such an announce,
it is true that “Bob knows that Ann sees him” in any situation since it is
a logical consequence of the announcement which, of course has become
Note that at the beginning, knowledge of agents is only based on what they see,
but evolve as announcements happen.
With these two possibilities, students may play with epistemic logic: it helps in
understanding formal truth-conditions in Kripke models by interpreting formulas
in a concrete and intuitive situation, and modal subtleties by experimenting the
effects of various announces, in particular the classic Moore sentence : “Ann
sees Bob and Bob does not know Ann sees him” which becomes false after being
The piece of software Plaza’s world can been launched from the web site
72O. Gasquet and F. Schwarzentruber
2The Logic Framework of Flatland
The syntax is the following: formulas describing questions or announces are built
with the usual boolean connectives (∧, ∨, ¬) together with Ka(agent a knows
that...) and the only propositions of the form (a ? b) (agent a sees agent b).
Thus sentence (*) is represented by the formula: KAnn¬KBob(Ann ? Bob).
Concerning the semantics, formulas are interpreted in Kripke models corre-
sponding to all situations and their links (W,RAnn,RBob,RChris,m) where:
– W is the set of all possible situations and is depicted in Figure 2;
– m says in each situations who sees who;
– each Ra links each situation to those that agent a cannot distinguish on
the basis of what she sees or knows because of an announce, they are called
accessible situations for a.
The interpretation of boolean connectives is classical and that of Kaϕ is as usual
in epistemic modal logic: Kaϕ is true in a situation iff ϕ is true in any accessible
situation for a. For more details, the reader is invited to look at .
This section deals with a running example with the model-checker Plaza’s
world. Let us start with the Kripke model of Flatland with 3 agents depicted in
Figure 2. As you can see, the Kripke graph is not planar4.
Fig.2. 3-agents Flatland model
4For graph theorists: it contains the graph K3,3.
Concrete Epistemic Modal Logic: Flatland 73
The graphical user interface enables us to see a Kripke model on the top of
the screen. The bottom of the screen is devoted to write an epistemic formula.
Then one button “model-checking” enables to check where this formula is true
whereas the other button enables to announce the formula.
Now let us check the formula Ann?Bob on this model. In order to this formula,
the graphical user interface provides some buttons for each constructions. We
simply click on those buttons to enter the formula:
By clicking on the button “model-checking”, the software highlights the worlds
in which the formula Ann ? Bob is true.
74 O. Gasquet and F. Schwarzentruber
The button “announce” updates the model by deleting all worlds in which the
formula Ann ? Bob is false. We obtain:
Now we rearrange the disposition of the worlds and we check the Moore’s
sentence Bob ? Chris ∧ ¬KChrisBob ? Chris on the model.
We can announce this Moore sentence and check that this formula is now false
in all worlds of the updated model:
Concrete Epistemic Modal Logic: Flatland 75
Of course now the student can check that KChris(Ann ? Bob ∧ Bob ? Chris)
in all worlds.
4 Beyond the Scene
The piece of software Plaza’s world has been developped in JAVA for one
major reason: JAVA enables the program to run on several multi-platforms.
Even more: JAVA enables the software to run without any installation. Indeed
the program can been launched directly from the Web via Java Web Start. This
is a good requirement for students.
The other advantage of JAVA is that the language is object-oriented and many
libraries already exists. For instance:
– itwaspossible to use
(http://forge.scilab.org/index.php/p/jlatexmath/) in order to dis-
play epistemic modal logic formulas correctly;
– it was easy to extend the class JTextField for colouring parenthesis and key
words as “sees” and “knows”;
– the software is easy to extend: for instance the class KripkeGraph represents
an abstract Kripke model that can be displayed. Our 3-agents Kripke model
of Flatland is only a specific implementation of this class.
the library jLatexMath
We found it pleased master students who could “see” epistemic logic in action in
a concrete and intuitive application. It also ease the understanding of what an
76O. Gasquet and F. Schwarzentruber Download full-text
epistemic relation stands for because here the epistemic relation is naturally de-
fined. For sure Plaza’s world is still to be improved and is under development.
We already plan to do the following improvement:
– to enhance the layout of the graph especially when the Kripke graph is
complicated, non-planar. Solutions are multiple: using 3D, algorithms with
spring and gravity forces etc. Maybe, we are going to implement it using a
graph visualization library like Jung (http://jung.sourceforge.net/);
– to allow the editing of the model: add nodes, remove edges by mouse clicking
– extending the language with common knowledge and allow to test or an-
nounce sentences like “All members of a group commonly knows that ϕ is
true” (see  for details on common knowledge);
– add announcements in the language and allow formulas of the form “Ann
knows that after she will have said that she knows that Bob does not see
him, Bob will know she sees him” both for testing and announcement.
Of course there are also still open problems concerning the logical framework
itself. For instance some researchers have introduced arbitrary announcement
logic  in order to model the “knowability” of agents. Unfortunately it has been
proved to be undecidable . But what about arbitrary announcement logic in
Flatland? Is it still undecidable? Can we implement it in Plaza’s world?
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