S. K. M. Wong, L. S. Wang, and Y. Y. Yao, On modeling uncertainty with interval structures'', Computational Intelligence, 1995, Vol. 11, No. 2, pp. 406-426.

Traditional interval computations deal with the cases when we are interested in the values of some quantity $x$ and we do not know it precisely; instead, we know two numerical bounds $x^-\le x^+$ that contain $x$: $x^-\le x\le x^+$. The set of all numbers $x$ within this bounds is an {\it interval} $[x^-,x^+]$; inside the computer, a computer can be represented as a pair of numbers $(x^-,x^+)$ with $x^-\le x^+$.

In a more complicated situation, we want to know a set $S$. This set can be the shape of a geometric object (e.g., in graphics or in manufacturing), or the set of situations that satisfy a certain property (e.g., in expert systems, when we are interested, e.g., in the set of situations for which a patient has to have a surgery). Often, we do not have the complete information about what elements belong to the set $S$: about some elements, we are sure that they belong to $S$, about some, we are sure that they don't; about some other elements, we do not know. The resulting information about $S$ is that $S^-\subseteq S\subseteq S^+$, where $S^-$ denotes the set of all elements that are known to belong to $S$, and $S^+$ is the set of all elements that {\it may} belong to $S$ (i.e., that are either known to belong to $S$, or for which the answer to the question $s\in S$?'' is I don't know''). This information can be thus represented as a pair $(S^-,S^+)$, with $S^-\subseteq S^+$; this pair of {\it sets} can be viewed as a set analogue of an {\it interval}.

For uncertain knowledge, the queries that we can ask may contain logical connectives and'', or'', not'', etc. (of the type describe the situations in which the patient has the fever and the headache''). In other words, we have basic'' queries $Q_1$,..., $Q_m$, and we can form more complicated queries by using logical connectives. The resulting set of possible queries ${\cal Q}$ is a Boolean algebra.

Ideally, for each query $Q$, we would like to know the set of situations $t(Q)$ in which $Q$ is true. Since our knowledge is incomplete, in general, we will only produce a {\it pair} $(t^-(Q),t^+(Q))$ with the property that the unknown set $t(q)$ is in between $t^-(Q)$ and $t^+(Q)$. Therefore, our goal is, for every $Q$, to produce $t^-(Q)$ and $t^+(Q)$.

First, the authors notice that it is sufficient to produce $t^-(Q)$ (the set of all situations $s$ for which know that $Q$ is true), because $t^+(Q)$ can be described as a complement to $t^-(\neg Q)$ (i.e., as the set of all situations in which we do not know that $Q$ is false).

Second, for every two queries $Q$ and $Q'$, the set of all $s$ for which we know that both $Q$ and $Q'$ are true is exactly the intersection of the set of all $s$ for which $Q$ is true and the set of all $s$ for which $Q'$ is true: $t^-(Q\wedge Q')=t^-(Q)\cap t^-(Q')$. In other words, $t^-$ is a {\it $\wedge-$homomorphism} from the Boolean algebra of queries into a Boolean algebra of subsets. The authors define an {\it interval structure} as such a homomorphism, and illustrate this definition on two types of incomplete information: rough sets and expert-system style incomplete information.

A rough set, crudely speaking, describes the case when we have subdivided the set $A$ of all situations into finitely many mutually disjoint subsets $S_1,...,S_p$, and for each of these subsets $S_i$ and for each of the basic queries $Q_j$, we know whether $Q_j$ is true for all elements of $S_j$ or not. Then, for each $Q$, $t^-(Q)$ is the union of all subsets $S_j$ for which $Q$ is known to be true.

In expert-system style incomplete information, for each query $Q$, we describe the set of situations $s^-(Q)$ for which the expert is sure that $Q$ is true. The expert is not a perfect logician, so it can happen that the statement $Q(s)$ was not part of the expert's belief, but it can be deduced from the other beliefs of this expert. Therefore, the goal is to generate the set $t^-(Q)$ of all situations in which $Q(s)$ can be deduced from the expert's beliefs. The function $t^-$ is also an interval structure.

For each query $Q$, in addition to the set of situations in which $Q$ is true, we may also want to describe, e.g., the {\it probability} $p(Q)$ of $Q$. If we know the probability $p(s)$ of each situation $s$, then we can conclude that the desired probability $p(Q)$ belongs to the interval $[p(t^-(Q)),p(t^+(Q))]$. The authors show that intervals generated by Dempster-Shafer formalism can be thus interpreted.