## Interval Computations, Test 2 for the course CS 5351/CS 4365, Fall 2017

Name: ___________________________________________________________

1a. What is indirect measurement?

1b. Why do we need indirect measurements? Provide an example.

1c. What is data processing? Give an example.

1d. What is knowledge processing? Give an example.

1e. Why do we need data processing and knowledge processing?

1f. What is the difference between data processing and knowledge processing?

2a. What is constraint satisfaction? Why do we need it? Provide an example.

2b. What is optimization? Why do we need it? Provide an example.

2c. What is the difference between constraint satisfaction and optimization?

3a. What is fuzzy processing? Why is it called "fuzzy"?

3b. Why do we need fuzzy processing? Give an example.

3c. Who was the first to develop fuzzy techniques?

4a. Why degree of belief 1 usually corresponds to absolute confidence, and 0 to absolute lack of confidence?

4b. Describe two ways to elicit degrees from experts.

4c. Suppose that out of 20 experts, 14 think that 1.8 m is tall. What will be the corresponding degree of belief that 1.8 m is tall?

4d. Suppose that an expert marks her degree of confidence that 1.8 m is tall as 4 on a scale from 0 to 5. What is the resulting degree of belief?

4e. What is a membership function? How can we determine the membership function?

4f. We know that m(1) = 0.5 and m(3) = 0.8. Use linear interpolation to find the value m(2).

5a. What is an "and"-operation? Why do we need "and"-operations? Why cannot we explicitly elicit all the degrees from the experts?

5b. What is an "or"-operation? Why do we need "or'-operations? Why cannot we explicitly elicit all the degrees from the experts?

5c. Why should "and"- an "or"-operations be commutative and associative? Explain.

5d. Why should "and"- and "or"-operations be idempotent? Explain.

5e. What is the only idempotent "and"-operation? the only idempotent "or"-operation? Who invented them? Are they used in practice?

6a. Describe the main problem of fuzzy data processing in precise terms.

6c. For extra credit: explain how Zadeh's extension principle is derived.

7a. What is an α-cut? What is the relation between α-cuts corresponding to different α?

7b. For μ(x) = max(0, 1 − |x − 1|) and α = 0.4, what is the α-cut?

7c. What is a fuzzy number? How to describe a fuzzy number in terms of α-cuts?

7d. How to describe "and" and "or" in terms of α-cuts?

7e. If x1(0.6) = [−1, 1] and x2(0.6) = [−2, 3], what is the α-cut y(0.6) for y = x1 − x2?

8a. Why do we need interval-valued degrees of belief?

8b. If an expert marked his/her degree of confidence as 7 on a 0 to 8 scale, what is the corresponding interval?

8c. If out of 20 experts, 6 say "yes", 8 say "no", and the rest are undecided, how will this be represented in intuionistic fuzzy logic? in Dempster-Shafer approach?

8d. For two statements S1 and S2, we have d1 = [0.5, 0.7] and d2 = [0.4, 0.8]. What is the degree of confidence in S1 & S2? in S1 \/ S2?

9. Suppose that we have three alternatives, for which the gains are in the intervals [0, 110], [40, 70], and [80, 90].
• is there an alternative which is guaranteed to be optimal (i.e., for which the gain is the largest)?
• list all alternatives which can be optimal;
• which alternative(s) should we select if we use Hurwicz optimism-pessimism criterion with α = 0, 0.5, and 1.

10. If we only use 2-digit decimal numbers and we use rounding that preserves guaranteed bounds, what will be the result of multiplying the intervals [0.60, 0.70] and [0.13, 1.21]