Logic Programming and Artificial Intelligence
17 May 2011
Problem Sheet 8
Due: 17:15, 24 May 2011
1. [14 marks] (from Exercise 3.1 of Nilsson’s Principles of Artificial Intelligence) Consider a sliding
block puzzle with the following initial configuration:
B B B W W W E
There are three black tiles (B), three white tiles (W), and an empty cell (E). The puzzle has the
following moves:
•A tile may move into an adjacent empty cell with unit cost.
•A tile may hop over at most two other tiles into an empty cell with a cost equal to the number
of tiles hopped over.
The goal of the puzzle is to have all of the white tiles to the left of all of the black tiles (without
regard for the position of the blank cell).
We shall use the obvious representation for this problem: the states and operators of the represen-
tation correspond directly to the configurations and moves of the puzzle.
(a) [4 marks] Specify a heuristic function, h, for this problem.
(b) [8 marks] Show the search tree that is grown by A∗search using your heuristic function. Assume
that the algorithm halts when the first goal is found. Label each node in tree by its f,g, and
hvalues, and number the nodes to indicate the order in which they are removed from the list
of nodes. Do not include any cycles in your search tree. If your search tree does not fit on a
single sheet of paper, then your heuristic function is not good enough; find a better one.
(c) [2 marks] Argue convincingly either that your heuristic function is admissible or that it is not
admissible.
2. [4 marks] Consider the following search space, in which every state is shown as a box containing the
name of the state. To the right of every state nis shown the value of h(n), the estimated distance
from state nto the nearest goal. A is the initial state, and G and H are goal states.
[A]8
/ \
/ \
3/ \3
/ \
/ \
/ \
[C]5 [B]6
/ \ |
2/ \3 |2
/ \ |
/ \ |
[D]5 [E]2 [F]2
/ \ |
7/ \2 |5
/ \ |
/ \ |
[G]0 [J]1 [H]0
(a) [3 marks] Show how each of the following search methods finds a solution in search space by
writing down, in order, the names of the nodes removed from the list of states. Assume that
the search halts when a goal state is removed from the list.
(i) Uniform-cost search
(ii) Greedy best-first search
(iii) A∗search
(b) [1 mark] Name every node in the search space for which hproduces an overestimate.
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