In many real-life situations, we are interested in the physical quantities that are difficult or even impossible to measure directly. To estimate the value of such quantity y, we measure the values of auxiliary quantities x1,...,xn that are related to y by a known functional relation y=f(x1,...,xn), and we then use the results Xi of measuring xi to find the desired estimate Y=f(X1,..,Xn). Due to measurement errors, the measured values Xi are slightly different from the actual (unknown) values xi; as a result, our estimate Y is different from the actual value y=f(x1,...,xn) of the desired quantity.
When xi and y are numbers, then the measurement accuracy can be usually represented in interval terms, and interval computations can be used to estimate the resulting uncertainty in y. In some real-life problems, what we are interested in is more complex than a number. For example, we may be interested in the dependence of the one physical quantity x1 on another one x2: we may be interested in how the material strain depends on the applied stress, or in how the temperature depends on a point in 3-D space; in all such cases, what we are interested in is a function. We may be interested in even more complex structures: e.g., in quantum mechanics, measuring instruments are described by operators in a Hilbert space, so if we want to have a precise description of an actual (imperfect) measuring instrument, what we are interested in is an operator.
For many of such mathematical structures, researchers have developed ways to represent uncertainty, but usually, for each new structure, we have to perform a lot of complex analysis from scratch. It is desirable to come up with a general methodology that would automatically produce a natural description of validated uncertainty for all physically interesting situations (or at least for as many such situations as possible). In this paper, we produce the foundations for such a methodology; it turns out that this problem naturally leads to the technique of domains first introduced by D. Scott in the 1970s.