CMSI 485
Final Exam
Spring 2006
Partial Answers
========================================================

PROBLEM 1
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The variables are Xi for i in 0..n-1.
The domain of each Xi is 0..n-1
The constraints are everything we can make by instantiating the template
    forall i j. (i != j) ==>
        (Xi != Xj /\ (abs(i-Xi) != abs(j-Xj)))
for all is and js.

More formally the set of constraints is

    {Xi != Xj /\ (abs(i-Xi) != abs(j-Xj)) | 0<=i<n /\ 0<=j<n /\ i!=j}

Complete state formulations (that allow local search) do better
since value and variable ordering are actually useful.  In the incremental
state formulation you lay out the queens one column at a time.  There
aren't that many choices as to what to do next.

PROBLEM 2
---------
First convert the sentences in the KB, and the negation of term you
are asking about into CNF.  This gives
    p \/ r
    q \/ s
    ~p \/ ~q
    ~r \/ ~s
    p \/ s
    q \/ r
Then start resolving them.  It works out.

PROBLEM 3
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You should end up with one clause

    ~P(A, y) \/ P(f(y, z), z)

But showing your work is important.

PROBLEM 4
---------
You only need one. Here's my plain text formulation.  You'll want to use the
special symbol for Subset.  The idea here is that b is a particular set of
things from the category s.  Then remember BunchOf(b) is then the composite
object made from those items; this represents the items in the bag.

    forall i s n c.
        Subset(b, s) /\
        PricePer(s, Kilograms(1)) = Dollars(c) /\
        Weight(BunchOf(b)) = Kilograms(n) ==>
        Price(BunchOf(b)) = Dollars(Times(n, c))

PROBLEM 5
---------
Here are the answers, but you have to show your work to get credit.

    P(a) = .69

    P'(C) = <.72,.28>

    P'(A|c) = <.806, .194>

    P'(C | a\/b) = <.785, .215>

PROBLEM 6
---------
Answers are online.

PROBLEM 7
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You can generate:
    I is well.
    They plays badly.
    Five chair runs quickly slowly.

You cannot generate:
    Awesome!
    They swim.
    At the gym we played basketball.

PROBLEM 8
---------
public static double G(double s, double x, double y) {
    return Math.exp(-(x*x + y*y) / 2.0*s*s)
            / (s*s * Math.sqrt(Math.PI + Math.PI));
}

sub convolute {
    my ($f, $g) = @_;
    integral(
        sub {
            my $u = shift;
            integral(
                sub {
                    my $v, $x, $y = @_;
                    $f->($u, $v) * $g($x - $u, $y - $v);
                }
            )
        }
    );
}

PROBLEM 9
---------
a) 49
b) 4
c) 3
d) 40, under the restriction that column 4 is a one-way street
   moving up.
e) One order that they can be generated is
      10,00,20,21,30,31,01,02,03,04,05,06,15,25,35,32,42,43,52,34,45
   but you have to show the tree


PROBLEM 10
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a) A* is fine, you won't run out of memory here.
b) Do a backtracking search, terminating paths on bumps.  Always order
   the action choices according to which direction points closest
   to the goal.  Break ties randomly.
c) 12 actions (1,0)-(2,1)-(3,2)-(4,3)-(4,4)-(4,5)-(5,6)-(6,7)
               -(6,8)-(5,9)-(4,9)-(3,9)-(2,9)