On representation and approximation of operations in Boolean algebras

Several universal approximation and universal representation results are known for non-Boolean multivalued logics such as fuzzy logics. In this paper, we show that similar results can be proven for multivalued Boolean logics as well. © 2001 John Wiley & Sons, Inc.     

Interval Methods and Their Applications: Invited Sessions at the World Automation Congress (WAC'2000) Maui,Hawaii, June 10–16, 2000

Reliable Computing -     

Applications of interval computations to earthquake-resistant engineering: How to compute derivatives of interval functions fast

One of the main sources of destruction during earthquake is resonance. Therefore, the following idea has been proposed. We design special control linkages between floors that are normally unattached to the building but can be attached if necessary. They are so designed that adding them changes the building's characteristic frequency. We continuously monitor displacements within the structure, and when they exceed specified limits, the linkages are engaged in a way to control structural motion. This idea can also be applied to avoid vibrational destruction of large aerospace structures.     

A new characterization of the set of all intervals,based on the necessity to check consistency easily

The purpose of this paper is to present a new characterization of the set of all intervals. This characterization is based on several natural properies useful in mathematical modeling; the main of these properties is the necessity to easily check consistency of incompletes knowledge. This characterization is obtained both for one-dimensional and for multidimensional cases.     

Why intervals? A simple limit theorem that is similar to limit theorems from statistics

What is the set of possible values of a measurement error? In the majority of practical applications, an error is caused not by a single cause; it is caused by a large number of independent causes, each of which adds a small component to the total error. This fact is widely used in statistics: Namely, since it is known that the distibution of the sum of many independent small random variables is close to one of the so-called infinitely distribution of the total error is infinitely divisible. This assumption is used in the majority of the statistical applications. In this paper, we prove a similar result for the set of possible values of an error. Namely, if an error equals the sum of many small independent components, then its set of possible values is close to an interval; the smaller the components, the closer this set to an interval. This result provides one more justification for using intervals in data processing.     

How to improve mamdani's approach to fuzzy control

Fuzzy control is a methodology that translates “if”-“then” rules, A_ji (x₁) &…& A_jn(x_n) → B_j(u), formulated in terms of a natural language, into an actual control strategy u(x). Implication of uncertain statements is much more difficult to understand than “and,” “or,” and “not.” So, the fuzzy control methodologies usually start with translating “if”-“then” rules into statements that contain only “and,” “not,” and “or.” the first such translation was proposed by Mamdani in his pioneer article on fuzzy control. According to this article, a fuzzy control is reasonable iff one of the rules is applicable, i.e., either the first rule is applicable (A₁₁(x₁) &…& A_1n(x_n) & B₁(u)), or the second one is applicable, etc. This approach turned out to be very successful, and it is still used in the majority of fuzzy control applications. However, as R. Yager noticed, in some cases, this approach is not ideal: Namely, if for some x, we know what u(x) should be, and add this crisp rule to our rules, then the resulting fuzzy control for this x may be different from the desired value u(x). to overcome this drawback, Yager proposed to assign priorities to the rules, so that crisp rules get the highest priority, and use these priorities while translating the rules into a control strategy u(x). In this article, we show that a natural modification of Mamdani's approach can solve this problem without adding any ad hoc priorities. © 1995 John Wiley & Sons, Inc.