Towards a General Description of Interval Multiplications: Algebraic Analysis and Its Relation to t-Norms

It is well known that interval computations are very important, both by themselves (as a method for processing data known with interval uncertainty) and as a way to process fuzzy data. In general, the problem of computing the range of a given function under interval uncertainty is computationally difficult (NP-hard). As a result, there exist different methods for estimating such a range: some methods require a longer computation time and lead to more accurate results, other methods lead to somewhat less accurate results but are much faster than the more accurate techniques. In particular, different methods exist for interval multiplication, i.e., for computing the range of a product of two numbers known with interval uncertainty. To select a method which is the best in a given situation, it is desired to be able to describe all possible methods. In this paper, we provide a description of all possible operations for interval multiplication; this description is based on the same ideas as a known description of t-norms in fuzzy logic.