From single to double use expressions, with applications to parametric interval linear systems: On computational complexity of fuzzy and interval computations

In many practical problems, we need to estimate the range of a given expression f(x₁, ..., x_n) when each input x_i belongs to a known interval [x_i, x_i] - or when each input x_i is described by a known fuzzy set. It is known that this problem is easy to solve when we have a Single Use Expression, i.e., an expression in which each variable x_i occurs only once. In this paper, we show that for similarly defined Double Use Expressions, the corresponding range estimation problem is NP-hard. Similar problems are analyzed for the problem of solving linear systems under interval (and fuzzy) uncertainty.