Towards an Algebraic Description of Set Arithmetic

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). 1 Formulation of the Problem Need for intervals (and more general sets): a brief reminder. To describe the exact state of the world, we need to know the numerical values of the corresponding physical quantities. Some of the quantities can only take values from a given discrete set: e.g., the electric charge must be proportional to the smallest possible charge. However, for most physical quantities, all real numbers are possible values. In practice, we rarely know the exact value a of a physical characteristic; due to inevitable measurement uncertainty, we only know a set of possible values; see, e.g., [5]. In many cases, all we know is the measurement result ã and the upper bound ∆ on the absolute value of the measurement error ã − a. In this case, all we know about the actual (unknown) value a of this quantity is that it belongs to the interval [ã−∆, ã+∆].