How Our Beliefs Contribute to Interpret Actions' CEEMAS05

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How Our Beliefs Contribute to Interpret
Actions.(CEEMAS05)

• Guillaume Aucher
• PhD student under the supervision of Hans van
  Ditmarsch and Andreas Herzig.
• University of Otago (NZ) University Paul
  Sabatier (F)

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Introduction.

• Often in everyday life we interpret an action on
  the basis of our beliefs about the situation.
• For example, assume that you see agent A drawing
  a ball from an urn containing black and white
  balls and you do not know whether he withdraws a
  black ball or a white ball. If you do not believe
  that there is a particular distribution in the
  urn, then you will believe equally that agent A
  draws a black or a white ball. But if you believe
  that there are more black balls than white balls
  in the urn, then you will believe that agent A
  draws a black ball with more probability than a
  white ball.
• Also, actions often change facts of the
  situation.
• For example, if a ball is withdrawn from the urn
  then there is one ball less in the urn.
• Finally, the notion of surprise is often relevant
  to describe the static epistemic states of mind
  of agents what contradicts their beliefs often
  surprises them.

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Mathematical Preliminaries

• We will use hyperreal numbers and more
  specifically infinitesimal numbers to express
  what would surprise the agents.
• Roughly speaking, hyperreal numbers are an
  extension of the real numbers to include certain
  classes of infinite and infinitesimal numbers. A
  hyperreal number, typically denoted e is said to
  be infinitesimal if e lt 1/n for all integers
  n
• 0ltlte2lteltlt0.0001lt250lt
• We would want to approximate our expressions.
  That is to say, when an expression a is
  infinitely smaller than an expression b, then we
  would want abb. For example, 1 e 1, e e2
  e,
• We then define a particular structure V as the
  quotient structure of the positive hyperreals
  numbers by a particular equivalence relation.

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The core of Update Logic

• Classically, we divide our task in three parts
• Representation of the situation.
• Representation of the action.
• Update.

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Representation of the situation
Within an equivalence class wjvvjw, some
worlds are assigned by Pj an infinitesimal value
e and are called surprising worlds. Some other
worlds are assigned by Pj a real value and are
called conceived worlds. The surprising worlds
are the ones that the agent j would be surprised
to hear somebody claiming that they correspond to
the world in which she dwells. The conceived
worlds are the ones that the agent conceives
consciously as possible candidates for the world
in which she dwells.
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Static Language
With this simple language we can express what
would surprise the agents and by how much. For
example, means that
in world w, agent j would be surprised with
intensity e to hear somebody claiming that ? is
true. Note that the smaller e is the more agent j
is surprised. Note that if we define Bj? by
Pj(?)gt0.5, then Bj and Cj correspond exactly to
the notions of weak belief and conviction of
Lenzen.
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Examples of pd-models
There are two agents A and B and an urn
containing 0 black balls and n2.k white balls.
A, unlike B, knows how many black balls there are
in the urn. Example 1 In this example, B does
not believe that there is a particular
distribution in the urn. This situation is
depicted in Fig.1. pk stands for there are k
black balls in the urn and the double bordered
world is the actual world.
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Example 2 In this example, B is convinced (sure)
that there are more black balls than white balls.
So she would be surprised to hear somebody
claiming that there are actually less black balls
than white balls.
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Representation of the Action
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Example of Generic Action Model
stands for A draws a black ball and
stands for A draws a white ball. From now on,
we note for the
unique such that
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Update
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Remark

• - can be interpreted as the
  probability for the agent j that action is
  happening when j considers what she believes in
  world w.
• - can be interpreted as the
  probability for the agent j that action is
  happening when j considers what is true in world
  w.

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Example 1
In example 1, B believed that there was not any
particular distribution in the urn before A drew
a ball. So, B should believe equally that A is
drawing a white ball and that A is drawing a
black ball. Indeed
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Example 2
In example 2, B believed that there were more
black balls than white balls before A drew a
ball. So, B should believe that A is drawing a
black ball with a higher probability than a white
ball. Indeed,
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Conclusion

• We introduced the notion of surprise. This
  refines and sharpens the static description of
  the agents epistemic state of mind.
• Our probabilistic update satisfies the AGM
  postulates if we pay attention as in AGM only to
  propositional formulas. This provides a new
  probabilistic approach to belief revision.
• We dealt formally with actions which change facts
  of the situation.
• We showed formally how our beliefs contribute to
  interpret actions. This complements and reverses
  the traditional approach of belief revision.
• Our work is based on the update logic of Baltag,
  Moss, Solecki.