To describe the state of the world, we need to describe the values of all physical quantities. In practice, due to inevitable measurement inaccuracy, we do not know the exact values of these quantities, we only know the sets of possible values for these quantities. On the class of such uncertainty-related sets, we can naturally define arithmetic operations that transform, e.g., uncertainty in a and b into uncertainty with which we know the sum a + b.
In many applications, it has been useful to reformulate the problem in purely algebraic terms, i.e., in terms of axioms that the basic operations must satisfy: there are useful applications of groups, rings, fields, etc. From this viewpoint, it is desirable to be able to describe the class of uncertainty-related sets with the corresponding arithmetic operations in algebraic terms. In this paper, we provide such a representation.
Our representation has the same complexity complexity as the usual algebraic description of a field (such as the field of real numbers).