Unformatted text preview: CS 188 Spring 1996 Introduction to AI Stuart Russell Final examination You have 2 hours 50 min. The exam is openbook, opennotes. There a total of 100 points available. Write your answers in blue books. Hand them all in. Several of the questions on this exam are true/false or multiple choice. In all the multiple choice questions more than one of the choices may be correct. Give all correct answers. Each multiple choice question will be graded as if it consisted of a set of true/false questions, one for each possible answer. 1. (12 pts.) True/False 2. (12 pts.) Gameplaying Decide if each of the following is true or false. If you are not sure you may wish to provide some explanation to follow your answer. (a) (2) A feedforward neural network connecting sensors to e ectors can implement a re ex agent with state. (b) (2) No internal state is required for an agent to be successful in a perfectly accessible environment. (c) (2) In a nite state space containing no goal state, A* will always explore all states. (d) (2) 8x (x = x) _ (x > x) _ (x < x) is a valid sentence. (e) (2) Any MDP model can be translated directly into situation calculus. (f) (2) Since the value of information is nonnegative, the outcome of acting on more information will always be at least as good as the outcome of acting on less information. (a) (2) For a game with branching factor b and search depth d, what are the space requirements of alphabeta search? Explain (brie y). (b) (4) Roughly speaking, Go is played by placing stones anywhere on a 19 19 board until the board is full, and takes up to six hours. Make a very rough estimate of how deep alphabeta can search in Go, stating all your assumptions. (c) (2) Given a choice between moves A and B in the following tree, which would minimax choose (assuming MAX is to move at the root)?
A B 100 100 100 100 100 100 100 100 100 100 100 100 100 99 500 3. (15 pts.) Logic True/false:
(a) (b) (c) (d) (d) (4) It has often been argued that minimax backup is designed on the assumption that leaf node evaluations are correct. Clearly, leaf node evaluations (except for terminal states) are subject to signi cant errors that is, states with high values may be bad, and states with low values may be good, etc. Given the above choice, where all the leaves are nonterminals, which would you choose? Why? (2) Resolution can always be used to decide if two rstorder sentences are consistent. (2) The results of exhaustive backwardchaining (to nd all solutions) are independent of the search order. (2) A universally quanti ed sentence can be proved using a su cient number of examples. (2) > (sqrt(x) sqrt(z )) uni es with > (y sqrt(y)). 1 4. (11 pts.) Probabilistic inference (e) (2) \Newt makes more royalties than anyone" is a good translation of 9r Royalty(r) ^ Makes(Newt r) ^ 8x s :(x = Newt) ) Royalty(s) ^ Makes(x s) ^ r > s (f) (2) 8a x s Sane(x s) ^ :(a = TakeCS 162) ) Sane(x Result(a s)) is a correctly formed e ect axiom in situation calculus. (g) (3) The STRIPS operator Op(Action:Go(there) Precond:At(here) ^ Path(here there) Effect:At(there) ^ :At(here)) is entirely equivalent to the successorstate axiom 8a p q s At(q Result(a s)) , (a = Go(q) ^ At(p s) ^ Path(p q)) _ (At(q s) ^ :9r a = Go(r)) ] Consider the belief network shown in the following gure: C A B D
(a) (3) Which of the following statements are implied by the network structure? i. P (B jA C D) = P (B jA) ii. P (DjB ) = P (DjB C ) iii. P (B jA) 6= P (B ) (b) (3) Suppose we have evidence at B . Use Bayes' rule to solve the query P (AjB ) in terms of probabilities directly available in the network. (You may assume the usual normalization constant .) (c) (5) Suppose instead that we have evidence at C and D. Use Bayes' rule followed by conditioning on B to solve the query P (AjC D) in terms of probabilities directly available in the network. Given an expression P (X jZ ), conditioning on Y gives P (X jZ ) = Py P (X jY = y Z )P (Y = yjZ ).] 5. (10 pts.) Markov decision problems In this question we will look at how to describe MDPs in the language of probability theory. Consider an agent moving in a 4 4 rectangular grid with locations de ned as 1,1] through 4,4]. Let St be the actual location at time step t. Let Et be the location that the agent perceives it is in at time t. Let At be the action taken at time t. (a) (1) State precisely the condition required for this to be an MDP (rather than a POMDP). (b) (3) Let us assume that the agent has four actions (up, down, left, right), and that they are deterministic. They can be described by a collection of probability statements describing the probability of speci c values of St+1 given speci c values of St and At . Write down one such statement for a speci c location pair and action. How many such statements are needed in all? (c) (3) Suppose the location is de ned by the variables Xt and Yt , each of which takes on values 1 through 4. Show by writing down a probability statement for the probability of a speci c value Xt+1 given Xt and At that this decomposes the problem of describing the MDP. How many such statements do we need now? (d) (3) Draw a belief net structure relating the variables Xt , Yt, At, Et, Xt+1 , Yt+1 , Et+1 2 6. (12 pts.) Inductive learning
(a) (3) Explain why no representation scheme for Boolean functions can provide a compact representation for all possible functions. (b) (3) Draw a decision tree to represent the \two or more" function for three inputs. (c) (3) Let us consider neural nets with inputs in the range 0 1] and with g a step function. A network is de ned by the weights on the links and the threshold value of g at each node. Draw a network to represent the \two or more" function for three inputs. (d) (3) The standard learning algorithm for NNs uses hillclimbing in weight space. Explain what practical problem would make it di cult to use best rst search instead. 7. (13 pts.) Natural language Consider the following contextfree grammar: S !Question ? Pronoun !I jyoujhej him Question !Aux NP VP QPronoun !who j what Question !QNP VP Determiner !a j the j some Question !QNP Aux NP Verb QDeterminer!which j what NP !Determiner NC j Pronoun Adjective !big NC !Adjective* Noun Noun !dog j cat j table NC !Adjective* Noun RelClause Verb !see j eat j is QNP !QDeterminer NCj QPronoun Aux !did j will j can RelClause !that VP RelClause !that NP Verb VP !Verb NP (a) (4) Multiple choice: Which of the following sentences are generated by the grammar? i. Did you see the dog eat the cat ? ii. Can the table is me ? iii. Which big big table that the cat see will you eat ? iv. What cat eat the table that I see ? (b) (2) Write down at least one other English sentence generated by the grammar above. It should be signi cantly di erent from the above sentences, and should be at least ve words long. Do not use any of the openclass words from the above sentences instead, add grammatical rules of your own, of the form (grammatical category) !(speci c word)for instance, Noun ! bottle. (c) (2) Show the parse tree for your sentence. (d) (3) The grammar as given does not accept sentences such as \Is the cat a dog?"? Suggest a (sensible) modi cation so that it will. (e) (2) Suppose that the rule for NP is replaced by NP !Determiner Adjective* Noun RelClause Explain why the resulting grammar contains no sentences. 8. (5 pts.) Perception Brie y list the reasons why it is hard to wreck a nice beach (recognize speech). Exam continues overleaf : : : 3 9. (10 pts.) Robotics Consider the very simple robot arm shown in the following gure, where the shaded regions represent obstacles and the arm rotates around the xed pivot:
A B (a) (1) How many degrees of freedom does the robot have? (b) (5) Draw the con guration space of the robot, labelling each axis with the corresponding degree of freedom and showing the con guration space obstacles. (c) (4) Suppose that the robot arm can telescope, i.e., shorten or elongate itself to an arbitrary degree. Draw the new con guration space, and show how the robot gets its ende ector from A to B. 4 ...
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 Spring '08
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 Computer Science, pts, leaf node evaluations, Aux NP VP, Adjective* Noun RelClause, NP Verb VP

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