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C00002 00002	%cs323.1[w89,jmc]		1989 Take home exam
C00004 00003	\noindent{\bf Problem 2.} Consider the circumscriptive theory with the axioms
C00005 00004	\noindent{\bf Problem 3.} Let $A$ be $∀x[P(x)⊃P(f(x))]$.
C00006 00005	\noindent{\bf Problem 4.}
C00008 00006	\noindent{\bf Problem 5.}
C00010 00007	\noindent{\bf Problem 6.} 
C00012 00008	\vfill\end
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%cs323.1[w89,jmc]		1989 Take home exam
\centerline{\bf CS323}
\bigskip
\centerline{\bf Take Home Final Examination}
\bigskip
Due 5pm Thursday March 23, in room 358, Margaret Jacks Hall.
\bigskip

\noindent{\bf Problem 1.}
The intention of the following axiom set is to describe a suite
consisting of 3 rooms:
$$∀x(x=Room1∨x=Room2∨x=Room3),$$
$$Room1≠Room2,\; Room1≠Room3,\; Room2≠Room3,$$
$$nextto(Room1,Room2),\; nextto(Room2,Room3),\; ¬nextto(Room1,Room3).$$
\medskip

(a) Describe the intended model of these axioms in the model theoretic notation
of Lifschitz's Introduction in the class notes and
in the solutions to homework 1.
\medskip

(b) Show that the axioms have unintended models (describe such a model in
model-theoretic notation).
\medskip

(c) Show how to extend this axiom set by
adding an axiom or axioms {\sl not} mentioning any of the constants $Room1$,
$Room2$, $Room3$, so that the unintended models will be eliminated, and the
new theory will be categorical.
\medskip

(d) Consider the circumscriptive theory with the extended axiom set you used
in part (c), and with the predicate $nextto$ circumscribed. Which of the axioms
are redundant (i.e., can be deleted without changing the set of models)?
\bigskip

\noindent{\bf Problem 2.} Consider the circumscriptive theory with the axioms
$$¬ ab\;x ⊃ large\;x,$$
$$red\;x ⊃ ¬ large\;x,$$
$$¬ large B↓1,$$
$$ab\;B↓2,$$
with $ab$ circumscribed and $large$ varied.
Determine the result of circumscription (find an explicit formula for $ab$).
\bigskip

\noindent{\bf Problem 3.} Let $A$ be $∀x[P(x)⊃P(f(x))]$.
\medskip

What is the result of circumscribing $P$ in $A$? Explain why your answer is
correct.
\bigskip

\noindent{\bf Problem 4.}
Consider the blocks world with only one block, that can be located either on the
table or on the floor. The current position of the block is described by the fluent
$ontable$. The action $up$ moves the block to the table, and the action $down$
moves it to the floor. Moving the block is impossible if it is dark in the room;
the current position of the lightswitch is described by the fluent $on$.
There is no information about the initial values of fluents.
Formalize the properties of actions in this world using situation calculus.
\bigskip
\noindent{\bf Problem 5.}
This concerns formalization of an extremely limited model
of people giving other people objects.
Use $has(p,x)$ to assert that
person $p$ has object $x$.  Use $canget(p,x)$ to assert that $p$
can get $x$.  Use $friend(p1,p2)$ to assert that person $p2$ is
a friend of person $p1$.  What a person can get, his friends can get.

(a) Write appropriate axioms to describe what a person can get.

(b) What should be circumscribed and what should be varied to get an
appropriate theory?  This theory is to be used in connection with
a database of axioms about who has what and who is a friend of whom.
We want to limit the necessary size of the database.
\bigskip
\noindent{\bf Problem 6.} 
This builds on problem 5.

(a) Give a language and axioms and circumscription policy for a
more comprehensive situation calculus theory in which people
sometimes give away objects to their friends.  In your theory it
should sometimes be possible to reason that an object will be
given from one person to another until a certain individual has
the object.

(b) Discuss further elaborations of the theory of (a) that includes
asking for objects.
\bigskip
\vfill\end