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Main points of this exam paper are: Arbitrary Decision, Information Content, Mutual Information, Arbitrary Decision, Propositional Logic, Decision Tree, Puzzle Problem
Typology: Exams
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October 22, 2008
Problem Score Max Score 1 ___________ 19 2 ___________ 18 3 ___________ 16 4 ___________ 15 5 ___________ 14 6 ___________ 18 Total ___________ 100
Question 1. Decision Trees Consider the following set of 8 training examples, each containing two attributes, A and B, and a desired binary classification, + or −. You may leave your answers in (a) – (c) as expressions containing logs of fractions. A B Class 0 10 + 0 100 + 1 100 + 1 10 + 1 100 − 1 100 − 1 10 − 1 10 − (a) [3] What is the entropy (information content) of the “Class”? (b) [3] What is the information gain (mutual information) of attribute “A” in predicting the “Class”? (c) [3] What is the information gain (mutual information) of attribute “B” in predicting the “Class”?
Question 2. Search Consider the 3-puzzle problem, which is a simpler version of the 8-puzzle where the board is 2 × 2 and there are three tiles, numbered 1, 2, and 3. There are four moves: move the blank up, right, down, and left. These moves are applied, when possible, in this order for all uninformed searches and in case of sibling ties for other searches. Break other ties by increasing time on OPEN (i.e., first attempt the one which has been on OPEN the longest). The cost of each move is 1. The start and goal states are (a) [6] Draw the entire state space for this problem, labeling nodes and arcs clearly.
(b) Assuming there is no checking for repeated states of any kind, draw the search tree produced by each of the following search methods. For each node in a tree, label it with a number indicating when it was removed from the OPEN list (and expanded or detected as a goal node). If a method does not find a solution, show the part of the search tree and then explain why no solution is found. (i) [6] Depth-First search. (ii) [6] A* search with the heuristic equal to the number of misplaced tiles (do not count the blank as a tile).
Question 4. Constraint Satisfaction Consider the problem of coloring the six areas (numbered 1…6) in the following map using three colors: R, G, and B, so that no adjacent areas have the same color. Two areas are adjacent if they share part of an edge (note: they are NOT adjacent if they only share a corner). (a) [6] Fill in the table below with the domain of each area after each of the following steps of selecting an area and assigning a color followed by forward checking (FC): 1 2 3 4 5 6 Initial domain R,G,B R,G,B R,G,B R,G,B R,G,B R,G,B After 1=R and FC After 2=G and FC After 3=G and FC (b) [3] What general conditions in a table such as the one above would indicate a “deadend” in a backtracking search with forward checking and constitute a backtrack point when searching for a solution?
(c) [6] Fill in the table below with the domain of each area after each of the following steps of applying the arc consistency algorithm to the given initial domain (note: the initial domain is not related to (a)). Write below the resulting domains for each of the six areas after each of the following arcs are processed in this order. An arc “ x y ” is consistent if for each value of x there is some value of y that is consistent with it. 1 2 3 4 5 6 Initial domain R,G,B R,G R,G,B R R,G,B R After 4 2 After 2 6 After 6 4
Question 6. Logic The propositional logic (PL) connective is defined as: P Q P Q T T F T F T F T T F F T (a) [5] Is the inference rule ¬(αβ) |- β sound for PL? Justify your answer by showing an appropriate truth table. (b) [5] You are given the following four sentences in PL defining a knowledge base: P Q (P ∨ Q) ∧ (P ∨ R) (S ∨ Q) ∧ (S ∨ ¬R) ¬R Prove S using resolution (resolution refutation).
(c) [4] Express (P ∨ Q) using only , P, and Q. (Hints: You need 3 s. What is P P?) (d) [4] Is the sentence “Any basketball player from Wisconsin is better than some basketball player from Michigan” represented accurately in First-Order Logic by ∀ x Bball-player( x ) ∧ From( x , Wisc) ∧ (∃ y (Bball-player( y ) ∧ From( y , Mich)) ⇒ Better( x , y )) If yes, explain briefly why. If no, explain why not and correct it.
