1) (search and logic)
Finding an explanation for observed evidence is known as abductive
reasoning. The input is a KB of definite Horn clauses (i.e. "rules",
see below), and a function that assignes each literal a
non-negative cost.
Also part of the input is a set of literals E (the observed
evidence) to be explained.
An explanation is a set of assumed literals A, that together with KB
proves E.
In this exercise, we wish to solve the weighted abduction
version: given E, find an explanation A where
the sum of the costs of the literals in A is minimal. For example, consider
the following KB:
R1: a -> b
R2: b and c -> d
R3: a -> c
R4: x -> d
The cost of assumption for literals is as follows:
cost(b) = 7
cost(c) = 6
cost(d) = 20
cost(a) = 10
cost(x) = 100
For the case where we wish to find an explanation for E = {d},
one possible explanation is A = {d} for a cost of 20.
But it is not an optimal explanation, because A = {a}
also an explanation, with a weight of 10.
a) Formalize the weighted abduction problem as a search problem,
where the state is a partial solution A', and where an operator adds
any literal not in the partial solution.
b) Same as (a), but this time a lexicographical ordering of literals
is given (say a, then b, etc.), and one can add a literal x only
if no literal y greater than x is in A'.
c) Assume that we use uniform-cost search, are we assured of finding
the minimal explanation using version a? If not, are we assured of such
using option b? You need to sketch the proof for your answer.
d) We wish to use A* to perform weighted abduction. Find an
admissible heuristic for versions a, b, (they may all be the
same heuristic, and it does NOT have to be a GOOD heuristic).
Now, show an execution of A* using your heuristic to solve the
weighted abduction problem, given as input the above KB and
the evidence to be explained E = {d}.
Use only one of the search problem versions a, b - your choice
(hint: which would result in the shortest trace?)
2) (FOL and situation calculus)
We need to represent and reason within the following scenario, called
the Wumpus shooting problem (based on the infamous Yale Shooting problem):
a) We need to represent the following facts using FOL and situation
calculus, using successor-state axioms:
A) The action of drawing the bow makes it ready, if we have an arrow. (Action name: Draw)
B) The action of waiting does nothing (Action name: Wait)
C) If the robot is facing the Wumpus and the bow is ready,
the action of shooting kills the Wumpus.
Shoot also makes the arrow disappear and the bow unready.. (Action name: Shoot)
D) In situation S0, the Wumpus is alive, the bow is not drawn, and
the robot is facing the Wumpus and has an arrow.
Use the following predicates to represent the above information:
Ready(s) : the bow is ready in situation s
Alive(s) : the Wumpus is alive in situaton s.
HasArrow(s) : the robot has an arrow in situation s.
FacingWumpus(s) : the robot is facing the Wumpus in
situation s.
Obviously, you also need to use the Result(a, s) function...
b) Now, we need to prove that the following sequence of actions (starting
from situation S0): Draw, Wait, Shoot - results in Wumpus' death.
Proceed as follows:
A) Represent the fact to be proved, and its negation.
B) Convert all the knowledge you represented in A into CNF
C) Use a refutation proof with resolution to show the Wumpus' death.
c) What would be a potential problem in the proof if we did not
have "frame axioms" in our representation (i.e. stating what
did not change)?
d) Would we be able to do the proof using only froward chaining?
3) (Propositional logic)
For each of the following sentences, determine whether they are
satisfiable, unsatisfiable, and/or valid. For each sentence that
is satisfiable, determine the number of models (over the propositional
variables in the sentence).
a) A and (A -> B) and not B
b) A -> not A
c) A or B or C or D
d) (A or B) and (C or D)
e) ((A -> B) and (B -> C)) -> (A -> C)
4) (Game trees and search)
Consider the stochastic game to be implemented in assignment 2:
grid based environment, agents can move up, down, left, or right, or shoot,
but where walking deliberately into a wall or waiting
is not possible.
You may assume a time limit as desired for the length of the game in
each scenario.
a) Show a scenario (with no bullets) where for the first agent to
move (Clarification: not "whoever is the first to move", i.e. this need be
shown only for one of the agents) under zero-sum scoring,
moving in a certain direction (say, left) loses,
but under total flags collected scoring (i.e. each player
scores its own flags, no penalty for flags collected by an opponent), the optimal action is to
move left.
b) In the game with bullets and zero-sum scoring, show a scenario
where whoever is the first agent to move, loses.
In both cases, you must PROVE the scores by generating the respective game tree, either
to a terminal position or prove that you can prune some branches.
c) Suggest heuristic evaluation functions for the total-flags scoring version of the game without bullets:
1) One that is a lower bound, but non-trivial.
2) One that is an optimistic upper bound, but non-trivial
5) (Game-tree search - alpha-beta pruning)
a) Consider the tic-tac-toe position (with O to move):
X | |
-----------------
O | |
-----------------
X | |
Give an example in ordering the complete search tree starting from this position
where alpha-beta pruning saves as little as possible, and another where the savings
is maximal. How many nodes are pruned in each case?
You may denote moves by Xxy or Oxy in order to save space.
(Clarification: in the case with no pruning, no need to specify
the tree explicitly, only estimate the number of nodes).
b) Consider the same scenario as above, except that after every move by
X, a random mark (equally probable to be X or O, uniformly distributed
over the free locations) is generated. Draw and evaluate the
search tree, and state where pruning is possible.
(Since the above is too large, you may start from the following
position for b, with O to move, instead).
X | X | O
-----------------
O | |
-----------------
X | |
6) (agents and environment simulators)
For each of the following domains, which type of agent is appropriate
(table lookup, reflex, goal based, or utility based). Also, indicate the
type of environment:
a) An agent that detects faces in images.
b) An agent that plays go.
c) An agent that can plays solitaire.
d) An autonomous robot that can win the DARPA challenge (driving in
a city).
e) An internet shopping agent (flight tickets domain).
f) An agent that plays tic-tac-toe (3X3).
Justify all answers shortly!
Deadline: Noon (12:00), Tuesday, March 11, 2008 ( strict deadline).
Submissions are to be solo.