Abstract: In a traditional database system, the result of a query is a set of values (those values that satisfy the query). In other data servers, such as a system with queries based on image content, or many text retrieval systems, the result of a query is a sorted list. For example, in the case of a system with queries based on image content, the query might ask for objects that are a particular shade of red, and the result of the query would be a sorted list of objects in the database, sorted by how well the color of the object matches that given in the query. A multimedia system must somehow synthesize both types of queries (those whose result is a set and those whose result is a sorted list) in a consistent manner. In this paper we discuss the solution adopted by Garlic, a multimedia information system being developed at the IBM Almaden Research Center. This solution is based on “graded” (or “fuzzy”) sets. Issues of efficient query evaluation in a multimedia system are very different from those in a traditional database system. This is because the multimedia system receives answers to subqueries from various subsystems, which can be accessed only in limited ways. For the important class of queries that are conjunctions of atomic queries (where each atomic query might be evaluated by a different subsystem), the naive algorithm must retrieve a number of elements that is linear in the database size. In contrast, in this paper an algorithm is given, which has been implemented in Garlic, such that if the conjuncts are independent, then with arbitrarily high probability, the total number of elements retrieved in evaluating the query is sublinear in the database size (in the case of two conjuncts, it is of the order of the square root of the database size). It is also shown that for such queries, the algorithm is optimal. The matching upper and lower bounds are robust, in the sense that they hold under almost any reasonable rule (including the standard min rule of fuzzy logic) for evaluating the conjunction. Finally, we find a query that is provably hard, in the sense that the naive linear algorithm is essentially optimal.