CS 188 Introduction to AI
Fall 1999 Stuart Russell
Final solutions
1. (10 pts.) True/False
(a) (2) False; a feedforward network has no internal state and hence no memory.
(b) (2) True; both return the \leftmost" among the shallowest solutions.
(c) (2) True; although the solution to an MDP is a policy rather than just the solution path returned by A*, the
rest of the policy besides the solution path is irrelevant b ecause those states are never reached.
(d) (2) True; a neural net with enough hidden nodes can represent any Boolean function.
(e) (2) False; the entailment is the other way.
2. (18 pts.) Logic
True/false
:
(a) (3) True; otherwise we can assign each distinct literal to be false and falsify the clause.
(b) (3) False;
Q
(
w; A
) could only resolve against the negative literal
:
Q
(
x; F
(
x
)), leaving a positive literal.
(c) (4) True;
C
1
j
=
C
1
for any
; if
C
1
C
2
then
C
1
j
=
C
2
, by the semantics of disjunction.
(d) (4) False; the dreaded
9
:::
)
:::
.
(e) (4) True; the search tree is linear and nite, and resolution is complete.
3. (14 pts.) Planning and MDPs
(a) (2)
Op
(
Action:
T urnOn
(
b
)
;
Precond:
Of f
(
b
)
;
Effect:
On
(
b
)
Op
(
Action:
T urnOff
(
b
)
;
Precond:
On
(
b
)
;
Effect:
Of f
(
b
)
(b) (2) Start with postconditions
Of f
(1),
Of f
(2),
Of f
(3) and End with preconditions
On
(2) and
On
(3).
(c) (5) The open conditions are
On
(2) and
On
(3). These are not achieved by Start so a new step must be added.
T urnOn
(2) and
T urnOn
(3) are added with preconditions
Of f
(2) and
Of f
(3). These are achieved by causal
links from Start.
(d) (4) An MDP requires the following:
states
are all 8 settings of the three bits;
actions
are all applicable
T urnOn
and
T urnOff
actions in each state (3 actions per state but the 2 goal states are absorbing);
rewards
are +ve (say +1) for goal states, -ve for others to ensure shortest solution;
transition model
is deterministic:
T urnon
(
b
) turns the bit
b
on with probability 1 where applicable.
(e) (1) Just need to remember a p olicy for
all
states: if bit 2 is o, turn it on; if bit 3 is o, turn it on.
4. (16 pts.) Probabilistic inference
(a) (3) (ii) is asserted, by the local semantics of BNs: a no de is conditionally independent of its nondescendants
given its parents. (i) is not asserted since H and S are linked by an arc. (iii) is not asserted by the structure
alone, because arcs do not
deny
independence. (The CPTs can deny it, however.)
(b) (3)
P
(
h; s;
:
p;
:
e
) =
P
(
h
)
P
(
s
j
h
)
P
(
:
p
j
h; s
)
P
(
:
e
j:
p
) = 0
:
1
0
:
3
0
:
1
0
:
9 = 0
:
00027
(c) (4) Probably the simplest way to do this is to construct the part of the full joint for
H
true (8 rows) and
then add up. The following is the enumeration algorithm:
P
(
E
j
h
) =
P
(
h
)
X
sP
(
s
j
h
)
X
pP
(
p
j
h; s
)
P
(
E
j
p
)
=
0
:
1[0
:
3
(0
:
9
h
0
:
6
;
0
:
4
i
+ 0
:
1
h
0
:
1
;
0
:
9
i
) + 0
:
7
(0
:
5
h
0
:
6
;
0
:
4
i
+ 0
:
5
h
0
:
1
;
0
:
9
i
)] =
h
0
:
41
;
0
:
59
i
(d) (6) Let's assume honesty doesn't inuence fundraising ability, but slickness does. Funds support advertising
which increases popularity, but do not directly aect electability otherwise. So
L
should be a child of
S
and
parent of
P
. We would need a CPT for
P
(
L
j
S
) and an augmented CPT
P
(
P
j
H; S; L
). Any CPTs refelcting
the abovementioned inuences will do.
5. (10 pts.) Vision
(a) (4) (i) A appears bigger and cars are usually roughly similar in size; (ii) since A and B are both on the same
horizontal plane and B appears
above
A, it
must
be further away.
1
