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CPSC 420-500 Homework
Due 10/8/07, in Class
Handwritten or printed hardcopy must be submitted
Instructor: Yoonsuck Choe
September 30, 2007
1 Uninformed Search
B
A C E D
(a) Tree 1 (b) Tree 2 (c) Tree 3
Figure 1: Search Trees.
Consider the three search trees in Figure 1. Suppose the branching factor is b and the tree is full. Nodes A, B, C, D, and E the goal nodes in the trees. Assume that nodes A, C, D, and E are at depth n; and A and C are the k-th node from the left at that depth, where k < b, i.e., they are children of the left-most node at depth n − 1. Node B is at depth m (< n). Node D is the last node to the right at depth n. Depth n is the last level of all the trees. Further assume that the exploration of each depth level proceeds from the left to the right.
Question 1 (4 pts): Which one shows a case where both depth-first and breadth-first have identical time complexity? (Tree 1, 2, or 3)
Question 2 (4 pts): Which one shows a case where depth-first can be complete but non-optimal? (Tree 1, 2, or 3) Explain why.
Question 3 (8 pts): Assume b = 4, k = 3, m = 4, and n = 20. What is the number of nodes visited in case of Tree 2 for depth-first (and breadth-first)?
Question 4 (4 pts): Assume b = 4, k = 3, m = 4, and n = 20. In which case does depth-first outperform breadth-first in terms of time complexity (= nodes visited)? (Tree 1, 2, or 3)
Question 5 (4 pts): Answer the following questions (5–6) regarding iterative-deepening search (IDS). In what sense is iterative-deepening search similar to depth-first and to breadth-first? Answer in terms of (some of) the four evaluation criteria.
Question 6 (4 pts): If space is adequate, is IDS still a good choice compared to breadth-first search (Yes/No)? Explain why.
2 Informed Search
Answer the following questions regarding informed search strategies.
Question 1 (4 pts): Why does A∗^ have an exponential space requirement (in case the heuristic function is not very good)? Compare with breadth-first search, and explain in relation to the concept of f -contour.
Question 2 (4 pts): Why is IDA∗^ more efficient in space than A∗? Explain in terms of the particular exploration strategy.
Question 3 (4 pts): What is the potential disadvantage of IDA∗^ compared to A∗^ if space is adequate?
Question 4 (4 pts): What is the difference between greedy search and A∗^ search?
I
b
c f
h i^ j
d e
g
10
20
100 80
h=
a h=100 h=
h=160 h=
h=
(^100 )
165 172 81 77
Figure 2: Search tree. An example search tree is shown with path cost on each edge and heuristic function value next to each node. The leaf nodes are goal states.
Question 5 (6 pts): Given the search tree in figure 2, calculate the f (n) value for each node (a to j).
Question 6 (12 pts): Given the search tree in figure 2, list the node visit order and the goal state reached for the two search methods: (1) greedy search, and (3) A∗.
4 Propositional Logic
4.1 Normal forms
In all of the problems in this section, show each step of the derivation and indicate which axioms (or other rules) you used: For example, distributive law, by definition, etc.
Question 1 (2 pts): Convert A ∨ (B ∧ ¬C) into conjunctive normal form.
Question 2 (4 pts): Convert A ∨ (B ∧ ¬C) ∨ (B ∧ E) into conjunctive normal form.
Question 3 (2 pts): Convert (A → B) → (C ∨ D) into disjunctive normal form.
4.2 Theorem proving
Question 1 (12 pts): Using resolution, show that G ∨ A is a logical consequence of the following:
1. C → (A ∨ B)
2. (D → E) ∧ ¬E
3. B → (G ∨ E)
4. D ∨ C
Hint: first, transform the problem into a set of clauses, and the follow the resolution steps.
