If we measure a number, we get an interval. What if we measure a function or an operator?", Reliable Computing (1996) [7 citations — 5 self]
Abstract:
Assume that we measure a physical quantity x with a measuring device whose accuracy is ffi (i.e., whose producers guarantee that the difference x \Gamma ~ x between the actual value x and the measured value ~ x does not exceed ffi). If the result of this measurement is ~ x, then possible values of x form an interval [~x \Gamma ffi; ~ x + ffi]. Suppose now that we know that a physical quantity y is a function of the physical quantity x (in other words, we know that y = f(x) for some function f(x)), but we do not know f. How to determine f? We can measure only finitely many values, with finite precision, so, after finitely many measurements, we get a set of possible functions f(x). This set can be called a function interval (function intervals were first analyzed by R. Moore himself). The situation can become even more complicated. For example, if we analyze how physical fields evolve, then in addition to functions, we must describe operators, i.e., mappings that transform a function (current value f(~x) of a physical field) into a function (predicted future value of this field). Again, since
