# AI代写 | COMP0024 Coursework 3, 2020/21

Artificial Intelligence and Neural Computing, COMP0024 (A6U, A7P) Coursework 3, 2020/21

Suitable for Cohorts: 2020/21, 2019/20

This assessment consists of TWO questions, each worth a total of 50 marks. Answer BOTH questions.

Marks for each part of each question are indicated in square brackets. Standard calculators are permitted.

1.
a. A boolean function !(#\$, #&, #’) is defined by the truth table below:

x1 x2 x3 !(#\$, #&, #’) 0000 0011 0101 0110 1001 1010 1100 1111

1. In a two-layer binary decision neuron (BDN) net realisation of this function, based on the expression of the above definition in first canonical (‘sum of products’) form, how many hidden units would be needed?

[1 mark]

2. Write down an algebraic expression in product form for each of the functions that would need to be performed by these hidden units.

[4 marks]

3. Using the constructive rule derived from digital circuit theory, which is based on the representation of a Boolean function in first canonical form, obtain for each hidden unit a weight vector * = (,\$, ,&, ,’) and a threshold s that would allow the required function to be performed.

[8 marks] [Question 1 cont. over page]

COMP0024 1 TURN OVER

[Question 1 cont.]

iv. Couldthebooleanfunctioncomputedbythistwo-layernetbecomputedbya single 3-input BDN neuron? Give a clear justification for your answer.

[3 marks]

1. Explain why multiple minima of the Hopfield energy function are considered valuable, while in contrast multiple minima of a mean-squared error function are considered undesirable.

[4 marks]

2. It is desired to store the two binary patterns (0,1,0) and (1,1,0) in a 3-node Hopfield net. Using the version of the Hopfield parameter setting rule that calculates values for the neuron thresholds as well for their weights
1. calculate an algebraic form for the Hopfield energy function -(#\$, #&, #’);
[4 marks]
2. draw a state transition diagram for the system, labelling all transitions with their probabilities and showing the energy levels of the system;

[8 marks]

3. Comment on the content addressable memory (CAM) structure associated with your solution—have the two patterns been ideally stored?

[4 marks]

[Question 1 cont. over page]

COMP0024 2 CONTINUED

[Question 1 cont.]

1. This question considers the conditions under which one might choose to use reinforcement learning, as opposed to choosing supervised learning. Under what general circumstances would one be forced to use the reinforcement training option? Under what kind of circumstances might one prefer to use reinforcement training, even though a supervised algorithm could also in this case be used?

[4 marks]

2. The Kohonen net below

having been trained on an appropriate set of 3-dimensional pattern vectors, has acquired the weight vectors

COMP0024

3 TURN OVER

i.

ii.

*. = (0,0,0),*0 = 1\$,\$,\$2,*3 = 1&,&,&2,*4 = (1,1,1) ”’ ”’

For each of the eight possible 3-bit binary vectors (0,0,0)…(1,1,1), determine which of the four neurons would be the ‘winner’ for that pattern. (Hint: use the symmetry of the weight vectors to reduce the number of independent calculations needed.)

[8 marks] Consider the separation of the patterns that you have discovered. On what basis

have the patterns been topologically mapped?

[2 marks]

[Total for Question 1: 50 marks]

2. a. Givetheexpectedutilitywithcalculationsforthefollowinglotteriesusingtheutility function U such that U(goodmeal) = 10, U(okmeal) = 2, and U(badmeal) = 5.

stayhome [1.0, okmeal]
meetfriend [0.8, restaurantlunch; 0.2, cafelunch] restaurantlunch [0.4,goodmeal;0.4,okmeal;0.2,badmeal] cafelunch [0.1, goodmeal; 0.6, okmeal; 0.3, badmeal]

Assuming you agree with the above utility function, should you choose stayhome or meetfriend? Give your calculations with your answer.

COMP0024

4 CONTINUED

b. i.

[4 marks]

Explain the following axiom of utility theory. What does it mean and what is a justification for it?

↵ ! (p q \$ [p, ↵; 1 p, ] ⌫ [q, ↵; 1 q, ])
[4 marks]

1. Explain the following axiom of utility theory. What does it mean and what is a justification for it?

((↵ ) and ( )) ! 9p[p, ↵; 1 p, ] ⇠

[4 marks]

2. Consider following example: ↵ is the outcome where you get \$10, is the outcome where you nothing, and is the outcome where you get killed. Give an axiom of utility theory for which this example could be regarded as problematic. Fully explain the nature of the problem with the axiom.

[5 marks]

c. For each of the following argument graphs, give each of the preferred extensions, grounded extensions, and stable extensions (if they exist).

[9 marks]

AAB BCDCDCBA

G1 G2 G3

d. i. The following two bpas are defined for the frame of discernment = {↵, , }. Give the combined bpa (after normalization), the resulting belief function, and

the resulting plausibility function.

[6 marks]

{↵} {} {} {↵,} {↵,} {,} {↵,,} m1 0.5 0.0 0.5 0.0 0.0 0.0 0.0 m2 0.0 0.5 0.0 0.0 0.5 0.0 0.0

ii. Ifm1 =m2 (i.e. forallA✓⌦,m1(A)=m2(A),ism1m2 =m1 =m2 (i.e. for all A ✓ ⌦, m1 m2(A) = m1(A) = m2(A)? If yes, give a proof,

otherwise give a counterexample.

[6 marks]

e. Let L be a logical language and let Cn() = {↵ | ` ↵} be the consequence closure of ✓ L for classical propositional logic. Assume has no models for each of the following questions. For each question, if the answer is yes, then give a proof, otherwise give a counterexample.

i. Does Cn() = ;? ii. Does Cn() = L?

iii. Does Cn(Cn()) = Cn()?

[4 marks] [4 marks] [4 marks]

[Total for Question 2: 50 marks]

COMP0024

5

END OF QUESTIONS

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