From Interval Arithmetic to Interval Constraints ∗

Two definitions of division by an interval containing zero are compared: a functional one and a relational one. We show that the relational definition also provides interval inverses for other functions that do not have point inverses, such as max and the absolute-value function. Applying the same approach to the ≤ relation introduces the “companion functions ” of relations. By regarding the arithmetic operations +, −, ∗, and / as ternary relations, we obtain the interval versions of the operations. This opens the way for regarding arithmetic problems such as evaluating expressions and solving equations as Constraint Satisfaction Problems (csps). These have a useful computational theory, which is, however, influenced by their predominantly discrete applications. We generalize the conventional formulation to better accommodate real-valued variables, and state the main results. When these results are applied to numerical csps we relate the interval evaluation of an arithmetic expression to the family of solving algorithms of csps. The key to our method of bringing interval arithmetic and interval constraints under a common denominator are companion functions. These functions form an alternative characterization of n-ary relations and appear to be a new contribution to the mathematical theory of relations