"And"- and "Or"-operations for "double", "triple", etc. fuzzy sets

In the traditional fuzzy logic, the expert’s degree of confidence d(A&B) in a complex statement A&B (or A∨B) is uniquely determined by his/her degrees of confidence d(A) and d(B) in the statements A and B, as f&(d(A), d(B)) for an appropriate “and”-operation (t-norm). In practice, for the same degrees d(A) and d(B), we may have different degrees d(A&B) depending on the relation between A and B. The best way to take this relation into account is to explicitly elicit the corresponding degrees d(A&B) and d(A∨B), i.e., to come up with a “double” fuzzy set. If we only elicit information about pairs of statements, then we still need to estimate, e.g., the degree d(A&B&C) based on the known values d(A), d(B), d(C), d(A&B), d(A&C), and d(B&C). In this paper, we explain how to produce such “and”-operations for “double” fuzzy sets – and how to produce similar “or”-operations. I. TRADITIONAL FUZZY TECHNIQUES: A BRIEF REMINDER Need for fuzzy techniques: reminder. Experts often describe their knowledge by using imprecise (“fuzzy”) words from a natural language like “small” or “fast”. One of the most widely used describe this knowledge in computerunderstandable terms is to use fuzzy techniques, in which, for each imprecise property P and for each possible value x of the corresponding property, we store the degree μP (x) to which the expert believes that x satisfies the property P ; see, e.g., [7], [8], [10]. Each of these values can be obtained, e.g., by asking the expert to mark his or her degree of certainty that x satisfies P by a mark on a scale from 0 to to some integer n. If the expert marks m on a scale from 0 to n, we take μP (x) = m/n. Another possibility is to use polling: we ask n experts and if m of them think that x satisfies the property P , we take μP (x) = m/n. Thus obtained degree can be interpreted as a probability: namely, as a probability that a randomly selected expert thinks that x satisfies the property P . Need for “and”and “or”-operations. One of the main objectives of storing the expert knowledge is to enable the computer to use expert rules – rules formulated in terms of imprecise natural-language words. The conditions of such rules often include several properties: e.g., if the car in front Hung T. Nguyen is with Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA, and with Department of Economics, Chiang Mai University, Thailand (email: [email protected]); Vladik Kreinovich and Olga Kosheleva are with the University of Texas at El Paso, 500 W. University, El Paso, TX 79968, USA (emails: [email protected], [email protected]). This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant N62909-12-1-7039 from the Office of Naval Research. is close and it is going fast, then . . . To figure out to what extend such rules are applicable in given situations, we need not only to describe the degree to which a given distance is close and the degree to which a given velocity is fast, we also need to find the degree to which the expert believes in the corresponding composite “and”-statement. Ideally, we should ask the expert’s opinion about all such combinations. However, in principle, many such combinations are possible, and it is not possible to ask the expert’s opinion about all such combinations. It is therefore necessary to estimate our degree of belief in a propositional combination like A&B or A ∨B in the situation when the only information that we have is the expert’s degrees of belief d(A) and d(B) in statements A and B. For each of these two propositional connectives & and ∨, we thus need to come up with an algorithm that transform the degrees d(A) and d(B) into a reasonable estimate for d(A&B) or d(A ∨B). Let us denote the algorithm corresponding to & by f&(a, b), and the algorithm corresponding to ∨ by f∨(a, b). Once we use these algorithms, we estimate d(A&B) as f&(d(A), d(B)) and d(A∨B) as f∨(d(A), d(B)). We want these algorithms to be reasonable. For example, since A&B is equivalent to B&A, it is reasonable to require that these two formulas lead to the same estimate for d(A& b), i.e., that the equality f&(d(A), d(B)) = f&(d(B), d(A)) be true for all possible values of d(A) and d(B). In mathematical terms, it is reasonable to require that the operation f&(a, b) is commutative. Similarly, since A ∨B also means the same as B ∨ A, it is also reasonable to require that the operation f∨(a, b) is commutative. Similarly, since A&(B&C) is equivalent to (A&B)&C, it makes sense to require that the corresponding estimates coincide, i.e., that f&(d(A), f&(d(B), d(C))) = f&(f&(d(A), d(B)), d(C)). In mathematical term, this means that the operation f&(a, b) is associative. Similarly, it is reasonable to require that the operation f∨(a, b) is associative. Together with additional reasonable requirements like monotonicity, continuity, etc., these properties form the definitions of “and”-operations (also known as t-norms) and “or”-operations (also known as t-conorms. Similarly, we use a negation operation f¬(a) to estimate the degree to which the negation ¬A is true as