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From: Benz Anton <anton@cis.uni-muenchen.de>
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\begin{document}
\title{Common Knowledge and its effect on the Interpretation of
 Discourse. A situation theoretic analysis of a special example.}
\author{Anton Benz\\Institut f\"ur Logik und Wissenschaftstheorie\\
  Ludwig--Max.--Universit\"at M\"unchen\\ Volkartstr.34/App.120\\ 80634
  M\"unchen\\ anton@cis.uni-muenchen.de}
\date{January 1, 1996}
\maketitle

{\scriptsize
 I look at a special example of the Conway Problems\footnote{It is  
due to
 \cite[p.201]{barwise}}. The circularity involved is represented by a
 circular structure the participants use to represent their  
knowledge. The
 reasoning is modeled as a building and revision of possible situations.
 It involves inference through constraints and controll
 through sets of infons. The aim is to give a model for the effect of an
 utterance of a sentence on the common situation. Example and  
framework are
 special but they give hints, how to handle the
 influence of common knowledge on the interpretation of discourse  
in more
 general cases.
 }

\paragraph*{The example considered}
 : (1) There are two persons, \(P_1\) and \(P_2\), to
 each of them is given a number 0 or 1. Suppose, they both get an  
1. They only
 know their own number (and know that the other knows only his own...).
 (2) So, asked whether they know the number of the other, they both  
would
 answer `no.' If asked again and again, they would still answer `no.'
 (3) Now, a third person knowing both numbers says: `The number of  
\(P_1\) is 1
 or the number of \(P_2\) is 1'.
 (4) Asked again for the number of the other one they both again would
 answer `no.' (5) But then they know.

\paragraph*{Questions}: How to represent the situations the  
participants are
 part of and the Common Knowledge involved? How do the utterances  
in (3) and
 (4) alter the situations? Why is it not altered by the answers in (2)?

 If one tries to interprete the above discourse\footnote{The intended
 discourse is not the above text but the described dialogue}
 by just adding the
 information of each new sentence to that of those uttered before, the
 answers in (4) should have no effect. They have the same meaning as
 those in (2). So, there is some context dependend discourse  
interpretation
 mechanism needed.

\paragraph*{The representation of situations}: I assume that the  
participants
 representate the situation in certain structures. The structures are
 construed from a set of {\em roles}, in the above example two roles,
 \(r_1\) and \(r_2\), for the two numbers, and two roles, \(P_1\) and
 \(P_2\), for the two participants. The {\em outer situation} seen
 by a person \(P\) is represented by a function from
 \(\sing{r_1,r_2}\) to their possible values, here \(\sing{0,1}\).  
The {\em
 total situation} is represented by a function on all roles. The  
value for a
 role \(P\) should be something I will call a {\em knowledge  
structure}, i.e.
 a function \(f\) that classifies the roles \(r\) according to  
whether P knows
 the value of \(r\) (then \(f(r)=+\)) or not (then \(f(r)=-\)); to  
the roles
 \(P_1\), \(P_2\) it again assigns a knowledge structure  
representing the way
 P represents the knowledge of \(P_1\) and \(P_2\). I write the  
values for
 the roles in a sequence, in the order \(r_1,r_2,P_1,P_2\); I put  
\(r_1,r_2\)
 in [\,] if they figure in an outer situation, and in (\,) if they  
figure in a
 knowledge structure. So, the situation after (1) and before (2) may be
 written as: \(S_1=[11]\Phi\Psi\), with \(\Phi=(+-)\Phi\Psi\),
 \(\Psi=(-+)\Phi\Psi\)\footnote{In a universe of
 Non--Well--Founded Sets, this {\em soe} has a solution, see  
\cite{Aczel}}.

\paragraph*{Perspectives, change of perspectives, assumptions}: The
 situation \(S\) seen through the perspective of a person \(P\), denoted
 \(S/\!\!\!:P\), can be derived from a knowledge state by replacing  
\(+\) by
 the appropriate value, and \(-\) by a parameter filling the gap; e.g.\
 \(S/\!\!\!:P=[1a]\Phi\Psi\) (\(a\) denotes a parameter). There are  
two basic
 operations to create new situations: {\em change of perspective}  
and {\em
 assumptions}, i.e.\ a participant can take another ones perspective and
 replace parameters by possible values. E.g.: \(P_1\) may assume  
\(a\) to be 0
 and adopt \(P_2\)'s perspective: \([1a]\Phi\Psi \to [10]\Phi\Psi \to
 [a0]\Phi\Psi\). Doing this in all directions gives an infinite
 graph\footnote{This is related to the idea of an {\em Information Flow
 Graph} in \cite{baretc}. The way how the information the nodes  
support is
 related to the other nodes and to the real situation will be made  
precise
 in a channel theoretic framework.}.

\paragraph*{Information and perspectives}: The situations in the  
above graph
 are related to the real situation \(S\). (I) All the possible  
completions of
 a perspective should support the informaton the participant has  
about \(S\),
 and (II) for every parameter there should exist two completions  
which differ
 in the relvant value. The information a participant has about  
\(S\) may be
 represented by a set of infons. In the start situation it can be  
computed
 from its perspective. So, the persons can link to every node in  
the above
 graph the information they know the participant has who's  
perspective is
 assigned to the node.

\paragraph*{Exchange of information}: By uttering a sentence the  
information
 expressed can be added by every participant to its information  
about \(S\),
 and to every set representing the information of other
 participants\footnote{So, the answers in (2) have no effect. They  
add to
 none of the representing sets new information.}.
 This rules out some of the situations in the above graph. Normaly,  
the claims
 (I) and (II) are not met. It is possible to show that there
 is a unique knowledge state which meets them. This state can be  
derived as a
 limit\footnote{This notion of a limit
 can be made precise.} of a step by step revision-process.

 {\scriptsize
 In the course of this process a
 participant can see that there can be different states according  
to different
 outer situations. This can be captured by adding a constraint to the
 knowledge structure. In the above example the new state after
 (3) can be represented as:
 \(S_3=[11]\Phi\Psi\),
 \(\Phi=\paar{(+-)\Phi a,
 \sing{\paar{[11],\Phi\Psi},\paar{[10],\Phi\Psi^*}}
 }\),
 \(\Psi=\paar{(-+)a\Psi,
 \sing{\paar{[11],\Phi\Psi},\paar{[10],\Phi^*\Psi}}}\),
 \(\Phi^*=\paar{(++)\Phi^*\Psi,\emptyset}\),
 \(\Psi^*=\paar{(++)\Phi\Psi^*,\emptyset}\), where
 \(\sing{\paar{[11],\Phi\Psi},\paar{[10],\Phi\Psi^*}}\)
 can be read as: if the outer situation is [11] then the values of  
\(P_1\),
 \(P_2\) are \(\Phi\) and \(\Psi\), if it is [10] then they are \(\Phi\)
 and \(\Psi^*\).
 }

\begin{thebibliography}{00}
 \bibitem{Aczel} Peter Aczel: {\it Non--Well--Founded Sets};  
CSLI--Lecture
 Note 14, Stanford 1988
 \bibitem{barwise} Jon Barwise: {\it On the Model Theory of Common  
Knowledge};
 in: Barwise: {\it The Situation in Logic},
 CSLI--Lecture Note 17, Stanford 1989, p.201--220.
 \bibitem{baretc} J. Barwise, J. Etchemendy:
 {\it Information, Infons and Inference};
 in: Cooper, e.a.: {\it Situation Theory and its Applications}, Vol.1,
 CSLI--Lecture Note 22, Stanford 1990, p.33--78.
\end{thebibliography}
\end{document}

