Section Marks Pre-Midterm / 32. Logic / 29. Total / 100

Transcription

1 Name: CS 331 Final Exam Spring 2011 You have 110 minutes to complete this final exam. You are only allowed to use your textbook, your notes, your assignments and solutions to those assignments during this midterm. If you find that you are spending a large amount of time on a difficult question, skip it and return to it when you ve finished some of the easier questions. Total marks for this exam is 100. Section Marks Pre-Midterm / 32 Bayesian Networks / 39 Logic / 29 Total / 100 Pg 1/11

2 Section I: Pre-Midterm questions [32 points] 1. In this question, you will answer questions about an intelligent agent that is playing Minesweeper against the computer. For each part below, circle the choice which best describes the environment. Write a one or two sentence justification. If you have never played Minesweeper before, the rules are below (taken from Wikipedia): The player is initially presented with an NxN grid of undistinguished squares. Some randomly selected squares, unknown to the player, are designated to contain mines. The game is played by revealing squares of the grid, typically by clicking them with a mouse. If a square containing a mine is revealed, the player loses the game. Otherwise, a digit is revealed in the square, indicating the number of adjacent squares (typically, out of the possible eight) that contain mines. In typical implementations, if this number is zero then the square appears blank, and the surrounding squares are automatically also revealed. a) Fully observable or Partially observable [2 points] b) Deterministic or Stochastic [2 points] c) Episodic or Sequential [2 points] d) Static or Dynamic [2 points] e) Discrete or Continuous [2 points] 2. Which of the following types of agents (simple-reflex, model-based, goal-based and utility-based) will NOT perform well in Minesweeper? Explain your answer. [4 points] Simple-reflex agents won t perform well because they only react to the last percept. You need to do some reasoning if you want to succeed in Minesweeper. Pg 2/11

3 3. Answer true or false to the following: a) Suppose I have two heuristics for the tile puzzle: h1 and h2. If h1 dominates h2, then h2 is the better heuristic to use. false b) Matrix form games in game theory can only contain one Nash Equlibrium. false c) Gradient descent cannot be applied if the objective function is not differentiable. true d) Suppose you took Iterative Deepening DFS and instead of increasing the depth limit by 1 each time, you increased it by 2. If the step cost (ie. the cost from a state to its successor) is always 1, this version of ID-DFS is optimal. false e) If you can generate the entire game tree for adversarial search and fit it in memory, then you don t need an evaluation function. true f) A strictly dominated strategy is never played in a Nash Equilibrium. true g) Expectiminimax does not consider worst case scenarios. true h) Local beam search is the same as k parallel hill-climbing searches (with random restarts). false i) In genetic algorithms, crossover and point mutations are techniques for getting out of local optima. true Pg 3/11

4 Section II: Bayesian Networks [33 points] 4. Suppose you are given the following training data set with three random variables: MidtermGrade (with values A, B, or C) HasLateAssn (with values true or false) OverallGrade (with values A, B, or C). This is the class label. MidtermGrade HasLateAssn OverallGrade A true B A true A C false C B true C A false A B false B C true C A true A B false A B true B a) Using the training data set, fill in the table below for OverallGrade). Remember to use uniform Dirichlet priors. [2 points] OverallGrade OverallGrade) A (4+1)/(10+3)=5/13=0.38 B (3+1)/(10+3)=4/13=0.31 C (3+1)/(10+3)=4/13=0.31 b) Using the training data set, fill in the table below for HasLateAssn OverallGrade). Remember to use uniform Dirichlet priors. [3 points] OverallGrade HasLateAssn HasLateAssn OverallGrade) A false ( 2+1)/(4+2)= 3/6 = 0.5 A true (2+1)/(4+2) = 3/6 = 0.5 B false (1+1)/(3+2) = 2/5 = 0.4 B true (2+1)/(3+2) = 3/5= 0.6 C False ( 1+1)/(3+2) = 2/5 = 0.4 C true (2+1)/(3+2) = 3/5 = 0.6 Pg 4/11

5 c) Suppose you use a Naïve Bayes structure to predict OverallGrade. Draw the naïve Bayes structure if OverallGrade is the class label and the features are HasLateAssn and MidtermGrade. [4 points] OverallGrade MidtermGrade HasLateAssn d) When predicting the overall grade, you need to compute OverallGrade A MidtermGrade A, HasLateAssn false) Show how to compute this expression by expressing it in terms of the conditional probabilities in the naive Bayes network (you don t have to fill in the actual probability values since you don t have all of them). I ve done the first step for you below by applying Bayes rule. OverallGrade A MidtermGrade A, HasLateAssn false) MidtermGrade A, HasLateAssn false OverallGrade A) OverallGrade MidtermGrade A, HasLateAssn false) A) Do NOT ignore the denominator you need to show how to compute it in this question. [6 points] MidtermGrade A OverallGrade A) HasLateAssn false OverallGrade A) OverallGrade A) OverallGrade o, MidtermGrade A, HasLateAssn false) o MidtermGrade A OverallGrade A) HasLateAssn false OverallGrade A) OverallGrade A) MidtermGrade A OverallGrade o) HasLateAssn false OverallGrade o) OverallGrade o) o Pg 5/11

6 5. Use the Bayesian network below to answer true or false to the following questions about conditional independence. Show the blocked paths for partial credit. A C D B G E F a) I( A, C F) [3 points] False A -> B -> E -> F <- G <- C is not blocked (because of F) b) I( D, G E) [3 points] Pg 6/11

7 False D -> E -> F -> G Blocked D -> E <- B <- C -> G Not blocked 6. The following questions deal with the Bayesian network below, along with the associated conditional probability tables. B A C A A) A B B A) A B C C A,B) false 0.25 false false 0.2 false false false 0.75 true 0.75 false true 0.8 false false true 0.25 true false 0.1 false true false 0.75 true true 0.9 false true true 0.25 true false false 0.2 true false true 0.8 true true false 0.2 true true true 0.8 a) Although the Directed Acyclic Graph structure of the Bayesian network above does not contain any visible conditional independencies, there is a conditional independence that exists in the conditional probability tables. What variables are conditionally independent of each other? Explain. [3 points] I(C,B A) You ll notice that C A,B) = C A) Pg 7/11

8 A A) A B B A) A B C C A,B) false 0.25 false false 0.2 false false false 0.75 true 0.75 false true 0.8 false false true 0.25 true false 0.1 false true false 0.75 true true 0.9 false true true 0.25 true false false 0.2 true false true 0.8 true true false 0.2 true true true 0.8 b) Compute A = true B=true, C=true). (Tables copied for your convenience) [10 points] A true, B true, C true) B true, C true) A true, B true, C true) A a, B true, C true) a A true) B true A true) C true A true, B true) A a) B true A a) C true A a, B true) a (0.75)(0.9)(0.8) (0.75)(0.9)(0.8) (0.25)(0.8)(0.25) Pg 8/11

9 7. The following question deals with the three Boolean random variables below: 1. FriendAbsent: takes the value true if your friend is not in class and false otherwise. 2. SleptIn: takes the value true if your friend slept in and false otherwise. 3. Zombie: takes the value true if your friend has been eaten by a zombie and false otherwise. You would like to model the following scenario: If you observe that your friend is absent from class, and you find out that your friend slept in, then it is less likely that your friend has been eaten by a zombie. If you don t observe that your friend is absent from class, then the event of your friend sleeping in and the event of your friend being eaten by a zombie are independent. Draw the Bayesian network for these three variables representing this scenario. [5 points] Pg 9/11

10 SleptIn Zombie FriendAbsent Section III: Logic [29 points] 8. Suppose I have a Wumpus world that consists of 2 cells: (1,1) and (2,1). In the truth table below, P i,j represents the fact that there is a Pit at (i,j) and KB represents the overall truth value of the knowledge base. P 1,1 P 2,1 KB false false false false true false true false true true true true a) Does the Knowledge Base = P 2,1? Explain your answer [2 points] No it doesn t because in every model in which the KB is true, P 2,1 is not true. b) If there are n Boolean symbols in total in the Knowledge Base, what is the time complexity (in Big-O notation) of the truth table enumeration algorithm for proving entailment? [2 points] 2 n 9. Convert the following English sentences into a propositional logic KB. Use symbols listed below. Make sure your KB is in Conjunctive Normal Form. [20 points] If a student is a CS major or an EE major, he/she is smart. If a student is an EE major, he/she owns a Tekbot. If a student is a CS major, he/she drinks coffee or Red Bull. If a student owns a Tekbot, he/she doesn t drink coffee For convenience, use the following symbols: Symbol CS EE S Meaning Student is a CS major Student is a EE major Student is smart Pg 10/11

11 T C RB Student owns a Tekbot Student drinks coffee Student drinks Red Bull CS EE S. (1) EE T...(2) CS C RB (3) T C..(4) From (1) (CS EE) S = (CS EE) S = (CS S) (EE S) From (2) EE T From (3) CS C RB From (4) T C 10. Based on the KB above, can you entail that if a student is an EE major, he/she drinks Red Bull ie. does KB = (EE => RB)? Use resolution to prove this or if it can t be proven, state that it does not resolve. [5 points] Add EE RB to the KB Does not resolve Pg 11/11