Abstract. In many engineering applications, we have to combine probabilistic and interval errors. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. In environmental measurements, we often only know the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data. Such modification are described in this paper.

Abstract—In traditional interval computations, we assume that the interval data corresponds to guaranteed interval bounds, and that fuzzy estimates provided by experts are correct. In practice, measuring instruments are not 100% reliable, and experts are not 100 % reliable, we may have estimates which are “way off”, intervals which do not contain the actual values at all. Usually, we know the percentage of such outlier un-reliable measurements. However, it is desirable to check that the reliability of the actual data is indeed within the given percentage. The problem of checking (gauging) this reliability is, in general, NP-hard; in reasonable cases, there exist feasible algorithms for solving this problem. In this paper, we show that quantum computations techniques can drastically speed up the computation of reliability of given data.

Abstract—Since the 1960s, many algorithms have been designed to deal with interval uncertainty. In the last decade, there has been a lot of progress in extending these algorithms to the case when we have a combination of interval, probabilistic, and fuzzy uncertainty. We provide an overview of related algorithms, results, and remaining open problems. I. MAIN PROBLEM Why indirect measurements? In many real-life situations, we are interested in the value of a physical quantity y that is dif cult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure y indirectly. Speci cally, we nd some easier-to-measure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem). Then, to estimate y, we rst measure the values of the quantities x1,..., xn, and then we use the results ˜x1,..., ˜xn of these measurements to to compute an estimate ˜y for y as ˜y = f(˜x1,..., ˜xn). ˜x1 ˜x2

Int. J. Automation and Control, Vol. x, No. x, xxxx 1 Application-motivated combinations of fuzzy, interval, and probability approaches, with application to geoinformatics, bioinformatics,

Accepted (Day Month Year) Communicated by (xxxxxxxxxx) In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.

In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.

I would like to express my deep-felt gratitude to my mentor Dr. Vladik Kreinovich for his advice, encouragement, enduring patience and constant support. Dr. Kreinovich helped me to think, understand, and explain ideas in a critical, professional, and well-articulated manner. I also wish to thank the members of my committee, Dr. Aaron Velasco from the Geological Sciences Department, Dr. Scott Starks from the Electrical and Computer Engineering Department, and Dr. Luc Longpré from the Computer Science Department, for their help and advise. My special thanks: • to Dr. Benjamin C. Flores, and to the Louis Strokes Alliance for Minority Participation