General comments:

• Unless explicitly specified, each homework should be turned in by email.
• For a programming assignment, send:
  • the softcopy of your program,
  • if needed, the softcopy of the code that you used to test this method, and
  • the softcopy of the results.
  Feel free to use Java, C, C++, Fortran, or any programming language in which the code is understandable.
• Homeworks are due at the beginning of the class on the day when the homework is due.
• Solutions to the homeworks will be posted two classes later, so:
  • if there are objective reasons why you cannot turn in your homework on time,
  • you can submit it before the solution is posted.
  Together with your submission, please provide an explanation of why you are submitting this homework late. If no explanation is provided, some points will taken off for late submission.
• If, for objective reasons, you are very late -- did not submit yet, and the solution is already posted, let the instructor know. We will prepare an alternative assignment for you, so you can still earn credit for this homework.

1. (Due July 8)

• Why do we need explainable AI in the first place?
• Why is it a reasonable idea to try to use fuzzy techniques when designing explainable AI?
• Why were fuzzy technique invented in the first place -- and who was their inventor?

Solution

2. (Due July 8) We know that:

• for x₁ = 1, we have y₁ = 2, and
• for x₂ = 2, we have y₂ = 3.

• Use the general linear interpolation formula that we had in class to come up with the expression y = f(x) for the dependence of y on x.
• For your expression f(x), what is the value of f(1.5)?
• How is linear interpolation used in fuzzy techniques?
• Explain, step by step, how we can derive the general formula for the linear interpolation.

Solution

3. (Due July 8) If the degree of confidence in a statement A is 0.7 and the degree of confidence in a statement B is 0.8, then what are the estimated degrees of confidence in statements A & B and A \/ B? Consider two cases:

• "and"-operation is min(a,b) and "or"-operation is max(a,b);
• "and"-operation is a*b and "or"-operation is a + b − a*b.

Solution

4. (Due July 12) Let us consider the following expert rules:

• if a cat is somewhat bored and you have a little bit of time, play with it a little bit;
• if a cat is very bored and you have a lot of time, play with it for a long time.

Let us now assume that:

• the cat is somewhat bored with degree 0.3 and very bored with degree 0.7,
• t = p = 1 hour;
• the degree to which this time is a little bit is 0.4, the degree to which this is a long time is 0.6;
• we use a*b as the "and"-operation and a + b − a*b as the "or"-operation.

Solution

5. (Due July 9) Prepare a project that you would like to do for the class.

• If you selected a project recommended by your advisor, let me know what exactly you plan to do (and who is your advisor).
• One of the options is to review a paper which is related either to explainable AI or to fuzzy. The relation may be weak, it is OK, the main idea is that you yourself are enthusiastic about this paper.
  • If you selected a paper for review, let me know whether this paper was proposed by your advisor or you selected it on your own.
  • In both cases, send me a softcopy of your paper.
  • If you selected the paper yourself, I will check:
    • that this paper is sufficiently technical -- i.e., has at least some formulas or algorithms, and
    • that this paper is not too complex -- can be understood and reviewed in a month.
• If you decided to work on a yet-to-come challenging theoretical project, let me know, and, if possible, indicate what interests you, so that we can jointly select a project that will be of interest to you.

6. (Due July 12) What is the distance D(a,b) between the points a = (1, 2) and b = (6, −10)?

Solution

7. (Due July 12) What is the squared distance D²(a,b) between the points a = (1, 2, 3) and b = (−1, −2, −3)?

Solution

8. (Due July 12) Use differentiation to find the minimum (= smallest value) of the expression (2x − 5)² + 3x − 6.

Solution

9. (Due July 12) Suppose that we have the following reasonableness degrees:

• for u₁ = 0, we have μ(u₁) = 0.5;
• for u₂ = 1, we have μ(u₂) = 1.0;
• for u₃ = 2, we have μ(u₃) = 0.5.

Solution

10. (Due July 14) Write a program that simulates fuzzy control with the three rules that we had in class:

• if ΔT is negligible, then u is negligible;
• if ΔT is small positive, then u is small negative;
• if ΔT is small negative, then u is small positive.

• input ΔT, and
• return the corresponding control u.

Use separate methods for computing the corresponding membership functions, for computing the "and"-operation, and for computing the "or"-operation, so that if you will need to change one of these operations, all you would have to do it replace the corresponding method without having to change the main method.

Test your program on the example of membership functions that we had in class and values ΔT = +3 and ΔT = −3. For each of these two values of ΔT, use your program to compute the resulting control value corresponding to the following two pairs of "and"- and "or"-operations: cases:

• "and"-operation is min(a,b) and "or"-operation is max(a,b);
• "and"-operation is a*b and "or"-operation is a + b − a*b.

Solution

11. (Due July 13) Write a general program for computing an integral of a given function over a given range. Test it by showing how to compute the integral of x² on the interval [0,1].

Solution

12. (Due July 14) Reproduce, in all detail, the proof that linear interpolation is the most robust.

Solution: see Section 3 of Lecture 3.

13. (Due July 14) Different marks on a 0-to-5 scale correspond to different degrees of confidence. For each possible degree corresponding to marks on a scale from 0 to 5, find the mark on a 0-to-4 scale which leads to the closest value of the degree.

Solution

14. (Due July 14) Reproduce, in all detail, the proof that algebraic product is the most robust "and"-operation.

Solution: se Section 5 of Lecture 3.

15. (Due July 14) As you know, in the usual 2-valued logic, negation is defined by the formulas f(0) = 1 and f(1) = 0. We would like to extend this function f(x) to all possible values x from the interval [0,1]. Such an extension is known as the negation operation. What is the most robust negation operation?

Solution

16. (Due July 16) Prove that for the case when f(a) > f(b), linear interpolation is also the only maximally individually robust interpolation.

Solution

17. (Due July 20) Use the least squares method to find the dependence y = c₁ * x + c₂ for the case when we have the following three measurements:

• x⁽¹⁾ = −2, y⁽¹⁾ = 1;
• x⁽²⁾ = 0, y⁽²⁾ = −1;
• x⁽³⁾ = 2, y⁽³⁾ = −1.

Solution

18. (Due July 20) Suppose that we know that f(0) = 2 and f(2) = 1. Use variational derivative to find the value f(1) for which the sum (f(1) − f(0))² + (f(2) − f(1))² attains its smallest possible value.

Solution

19. (Due July 20) Let us assume that we know the values f(a) and f(b) for some a and b, and we want to interpolate, i.e., to find the values f(x) for all x between a and b. By definition, the maximally individually robust interpolation f(x) must satisfy the inequality |f(x) − f(y)| ≤ r * |x − y| for all x and y, where r = |f(b) − f(a)|/|b − a|. Provide an example of the values x and y showing that when a = 0, b = 1, f(a) = 0, and f(b) = 1, the function f(x) = x² is not a maximally individually robust interpolation. Hint: it is sufficient to consider values 0, 0.5, and 1.

Solution

20. (Due July 20) We have shown that the only maximally individually robust "or"-operation is max(a,b). Maximally individually robust means, in this case, that for all possible values a, b, a′, and b′, we must have |f_\/(a,b) − f_\/(a′,b′)| ≤ max(|a − a′|, |b − b′|). Provide an example of the values a, b, a′, and b′ showing that the "or"-operation a + b − a * b is not maximally individually robust. Hint: it is sufficient to consider values 0, 0.5, and 1.

Solution

21. (Due July 20). Which "and" and "or"-operations should we use in the following two situations:

• if we are controlling a group of objects, and malfunctioning of one of them is OK as long as, on average, they all fulfil their mission;
• if we are controlling a single object.

Solution

22. (Due July 20). So how can we use fuzzy techniques to come up with explainable AI?

Solution

23. (Due July 20) What is tuning? What is the difference between tuning and machine learning?

Solution

24. (Due July 21) Write a program that, given two arrays of values x⁽¹⁾, ..., x^(K) and y⁽¹⁾, ..., y^(K), uses the least squares method to find the values of the parameters c₁ and c₂ of the linear dependence y = c₁ * x + c₂. Test your method on two examples:

• an example when you have x⁽¹⁾ = 0, y⁽¹⁾ = 0, x⁽²⁾ = 1, and y⁽²⁾ = 1; in this example, your program must return c₁ = 1 and c₂ = 0; and
• an example from Lecture 4.

25. (Due July 26) Suppose that a 2-layer neural network has two inputs x₁ = 0 and x₂ = 1.

• In the first layer, we perform a linear transformation and compute the value y = w₀ + w₁ * x₁ + w₂ * x₂.
• In the second layer, we apply, to the result of the first layer, the rectified linear activation function and get z = F(y).

• when w₀ = w₁ = w₂ = 1, and
• when w₀ = w₁ = w₂ = −1.

Solution

26. (Due July 26) What will be the result of max-pooling three values x₁ = 0, x₂ = 1, and x₃ = −1? of sum-pooling these three values?

Solution