Module Contact: Dr Pierre Chardaire, CMP Copyright of the University of East Anglia Version 1

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1 UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series PG Examination ARTIFICIAL INTELLIGENCE AND ALGORITHMICS CMPSMA24 Time allowed: 3 hours Answer THREE questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. CMPSMA24 Module Contact: Dr Pierre Chardaire, CMP Copyright of the University of East Anglia Version 1

2 Page 2 1. (a) Write a Prolog definition of a predicate powers(x,n,y) which will expect X to be a list of real numbers, and N to be a positive real number, and will bind Y to the list resulting from raising every number in X to the power N. For example, powers([3,7,4],3,y) should succeed with Y=[27, 343, 64]. The solution may use the built-in arithmetic function XˆN which calculates X raised to the power N. (b) What is meant by the declarative and procedural meaning of a Prolog program? Illustrate your answer with reference to the following aspects of Prolog: (i) The ordering of clauses in a program and the ordering of subgoals within a clause. (ii) The predicates! and print. [10 marks] (c) A Prolog programmer writes the following as an attempt to implement the minimum function: min(x,y,x) :- X < Y. min(x,y,y). Does this work? Illustrate your answer by showing what will happen when each of the following subgoals is executed: (i) min(1,2,z) (ii) min(2,1,z) (iii) min(1,2,1) (iv) min(1,2,2) [12 marks] (d) How would you modify the program in 1c so that it will work whenever its first two arguments are bound to numbers? The desired program should unify its third argument with the minimum of the two numbers and fail if that is not possible. Backtracking should yield no further solutions. Explain how your program deals with the four examples. [12 marks]

3 Page 3 2. (a) (i) Give an outline of the Resolution Proof method for refutation in propositional logic. (ii) What is meant by its completeness? (iii) Show that the method is sound. [7 marks] [3 marks] [10 marks] (b) Show that = (p = q r) (r = s t) (s = t) (q = p) = (p = t) (c) Using the predicates P(x) x is a person, D(x) x is a dog, L(x,y) x likes y and constants f Fido, r Rex, b Ben translate the following statements into predicate logic: Ben is a person and Fido and Rex are dogs. Every person likes Rex. If Ben does not like Fido then he does not like any dog ; (d) Translate the FOL sentence ( ) x Q(x) = B(x,k) Q(g) A(b) A(k) ( ( ) ) B(g,b) = y A(y) B(g,y) into conjunctive normal form. (e) Show that the set of disjunctive clauses {C 1,...,C 6 }, with C 1 = {A(b}, C 3 = {B(a)}, C 5 = {D(b,g), B(y), D(b,y}, C 2 = {B(g)}, C 4 = { A(x),D(x,a)}, C 6 = { D(b,g)} is unsatisfiable. TURN OVER

4 Page 4 3. (a) Give an outline of Dijkstra s algorithm in form of a flowchart or pseudocode for the search of a tree? (b) How is the algorithm modified for searching a general graph? [10 marks] (c) What difference is there between Dijkstra s algorithm and the A algorithm? (d) What is the consistency assumption of the A algorithm? (e) The grid in figure 1 implicitly defines a graph where each blank cell represents a vertex and each cell has at most eight neighbours. Horizontal and vertical movements from one cell to a neighbour cell have a cost of 10 units, whilst diagonal movements have a cost of 14 units. The black cells represent obstacles (i.e no vertices are associated with such cells.) In order to estimate the length of the shortest path Figure 1. - Grid

5 Page 5 between cell C 1 of coordinates (i, j), and cell C 2 of coordinate (k,l) one considers using the distance d(c 1,C 2 ) defined as d(c 1,C 2 ) = 14δ + 10η where δ = min( k i, l j ) and η = max( k i, l j ) δ. (i) Justify this choice using the grids of figure 2 as illustration. Figure 2. - Grids for illustration of d (ii) Apply the A algorithm to find a shortest path from S to F by using the evaluation function f (C) = g(c) + h(c) where g(c) is the cost of the best-known path from the start cell S to cell C, and h(c) = d(c, F). Reproduce the grid of Figure 1 in your answer booklet and fill it in with triples (i, j,k) where i is the iteration number the node was evaluated, j is the value of the evaluation function f, and k is one of the strings N, S, W, E, NE, NW, SE, SW to indicate the predecessor cell. Also include inside a circle the iteration number at which the cell is selected by the algorithm for processing (examination of its neighbour cells). [10 marks] TURN OVER

6 Page 6 4. Answer the following questions in the context of Reasoning under uncertainty. (a) Explain what you understand about the uncertainties? [4 marks] (b) In general, there are two classes of reasoning approaches: Deductive and Inductive. Explain what you understand about them and how the case-based reasoning and model-based reasoning methods should be classified. (c) Assume that you are required to analyse the possible causes of the failure of an old car to start its engine. In this context, you are limited to considering the five binary variables Fu, Ba, L f, Lb and Cs defined as follows. Fu (for Fuel) is O if the amount of fuel in the tank is OK, and is L when it is low. Ba (for Battery) is G when the battery is good, and is B when it is bad. L f (for Fuel light indicator) should be T (on) when Fu = L or F (off) when Fu = O. Lb (for Battery light indicator) should be T (on) when Ba = B or F (off) when Ba = G. Cs (for Car starts) is Y if the engine starts, and N if it does not. The two light indicators L f and Lb are not reliable. The corresponding Conditional Probability Tables (CPT) are given in Table 1

7 Page 7 Lb Ba Ba T F G 0.9 G B 0.1 B Cs Ba Fu Y N Lf G O Fu Fu T F G L O 0.8 O B O L 0.2 L B L Table 1. The Conditional Probabilities Tables for five nodes: Ba (Battery), Fu (Fuel), Lb (Battery light indicator), L f (Fuel light indicator) and Cs (Car starts). (i) Construct a Bayesian network to represent the relations of the five variables (nodes): Ba, Fu, Lb, L f and Cs and explain how many types of links exist in this Bayesian network. (ii) Using the Bayesian network you have constructed, discuss under what conditions the following two pairs of variables (nodes) are conditionally independent of each other: Ba and Fu, Lb and Cs, and L f and Lb. (iii) Calculate the unconditional probabilities: P(Lb), P(L f ) and P(Cs). (iv) Suppose that your car failed to start, i.e. Cs = N. Can you reason what is the most likely cause by using the Bayesian network and why? (Show your calculations of the necessary probabilities for reasoning.) (v) If you observe that the fuel light indicator is on, i.e. L f = T, compute the probabilities that Fu is low, P(Fu L f ) and P(Cs L f ). Then you notice that the battery light indicator is also on (i.e. Lb = T ); what is the probability that the car starts? [8 marks] TURN OVER

8 Page 8 5. Answer the following questions in the context of knowledge representation and machine learning. (a) Some people claim that (i) knowledge representation (KR) is probably the most important and yet difficult task in artificial intelligence and (ii) none of the existing KR methods works satisfactorily. Do you agree or not with these two statements, and why? [8 marks] (b) Multi-Layer Perceptron (MLP) neural networks are commonly used in machine learning. An MLP can consist of any number of layers of neurons, from one layer and up to many layers in theory. (i) Discuss under what conditions a Single-Layer Perceptron (SLP) network may be used and why an SLP cannot learn an XOR problem correctly. (ii) In practice, two-layer MLP networks are commonly used for learning from data. Three-layer MLP networks can be used if necessary but there seems no need to use four-layer MLP networks. Explain why. (iii) If you found out that the test error of a trained neural network is much higher than the training error, what does that suggest and what should you do to improve it? (c) A data set shown in Table 2 contains four input attributes (Location, Education, PastCustomer, Response) and one target (Income). The aim of the analysis is to build a decision tree model to predict if a person s income is high or low.

9 Page 9 ID Location Education Past Customer Response Income 1 Village Postgraduate N Y Low 2 City University N Y High 3 City University Y N Low 4 City University N Y Low 5 Town Postgraduate N N High 6 Village University Y Y Low 7 Village Postgraduate N Y High 8 City High School Y N High 9 Town High School Y Y Low 10 Town High School N N High 11 City High School N Y Low 12 Town Postgraduate Y N High new 13 Village High School Y Y? 14 Town University N Y? 15 City Postgraduate N N? Table 2. The data of person s income. (i) Calculate the entropy value for the target Income. [3 marks] (ii) Determine which attribute the ID3 would use for the root node and why. (iii) Complete the construction of the decision tree. (iv) Use your constructed tree to classify the new cases 13, 14 and 15 in table 2. [2 marks] END OF PAPER