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\lecture{Artificial Intelligence}{HW2: Constraint Satisfaction}{CS5300, Spring 2009}

% IF YOU'RE USING THIS .TEX FILE AS A TEMPLATE, PLEASE REPLACE
% "CS5300, Spring 2009" WITH YOUR NAME AND UID.

% Hand in at: http://www.cs.utah.edu/~hal/handin.pl?course=cs5300

\section{Crossword Puzzles}

(This is a variant of exercise 5.4 from the book.)

Consider the problem of constructing (not solving) crossword puzzles:
fitting words into a rectangular grid.  The grid, which is given as
part of the problem, specifies which squares are blank (i.e., can
contain letters) and which squares are shaded (i.e., cannot contain
letters).  Assume also that a list of words (i.e., a dictionary) is
provided.  The task is to fill in the blank squares using any subset
of the list.

% If you write your solution inline, feel free to comment out all of
% the problem definition above (especially the figure, which you
% probably won't have a copy of!).

\bee
\i Formulate this problem as a CSP where the variables are
\emph{words}.  List all the variables and constraints.

\i Formulate this problem as a CSP where the variables are
\emph{letters}.  List all the variables and constraints.

\i Suppose we wish to make sure the the crossword puzzle we generate
doesn't contain multiple words that are ``too similar.''  We'll say
that words $w_1$ and $w_2$ are too similar if \emph{either} one can be
obtained from the other by changing exactly one character \emph{or}
one is a substring of the other.  For instance ``dog'' and ``dogs''
are too similar (by the second constraint) as are ``dog'' and ``dog''
(again, by the second constraint).  Similarly, ``cats'' and ``cots''
are too similar (by the first constraint).  How would you specify
these additional constraints in both the by-word and by-letter
formulations?
\ene

\newpage
\section{Arc Consistency}

While skiing at Alta last week, Alice (our venarable computer science
student) decides that she wants to get a minor in math.  She has
already taken many of the requirements as part of her CS degree; all
that remains are the following courses.  She must take both Analysis
(A) and Analysis II (B).  She then has three electives of which she
must take two; these are: Linear Algebra (C), Number Theory (D) and
Modern Algebra (E)\footnote{This is not exactly the requirements for
  the math minor at the U, but is close.}.  Alice has three more
semester of study to go.  Fortunately, every course is offered every
semester, but she can take at most two math courses during any given
semester.

Of course, there are constraints on how she must take the courses.
Obviously she must take Analysis (A) before Analysis II (B).  She must
also take Linear Algebra (C) before Modern Algebra (D).  She can only
take Number Theory (D) once she's taken Modern Algebra (E).  Sadly,
Analysis (A) and Linear Algebra (C) are always offered at exactly the
same time, so she cannot take both in the same semester.

We will now formulate Alice's scheduling problem as a CSP in which
each variable corresponds to a class Alice must take and the values
correspond to the numbering of the remaining semesters: 1 for junior
Spring, 2 for senior Fall and 3 for senior Spring; $\infty$ for ``does
not take this course.''

% Ditto about commenting out the problem definition above.

\bee
\i Find an assignment of the variables to values (just by trial and
error) that satisfies the constraints.

\i State the domain of the variables and the constraints in
mathematical notation using the following symbols: $=, \neq, <, >,
\leq, \geq$.

\i We now solve this CSP using backtracking search with arc
consistency.  Namely, we wull run the AC-3 algorithm as a preprocessor
\emph{and} after each variable assignment.  When there is ambiguity as
to which variable to assign next, assign a value to the variable
corresponding with the course letter that comes first alphabetically.
Furthermore, when the next value to assign to a variable is ambiguous,
break ties by assigning the earliest value first.

\bee
\i What are the remaining domains after enforcing arc consistency on
the initial CSP with no assignments?

\i What are the remaining domains after assigning $A=1$ and
re-enforcing arc consistency?

\i What about after assigning $A=1$ and $B=2$?
\ene


\i {\bf (CS 6300 only)} Suppose we're running AC-3 but trying to get
Alice to finish in only two semesters instead of one.  Sketch a
complete run of AC-3 under these new constraints and say precisely
when it will fail and how many states are expanded before failing.
\ene




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