The Dichotomous Intensional Expressive Power of the Nested Relational Calculus with Powerset

Most existing studies on the expressive power of query languages have focused on what queries can be expressed and what queries cannot be expressed in a query language. They do not tell us much about whether a query can be implemented efficiently in a query language. Yet, paradoxically, efficiency is of primary concern in computer science. In this paper, the efficiency of queries in NRC(powerset), a nested relational calculus with a powerset operation, is discussed. A dichotomy in the efficiency of these queries on a large general class of structures— which include long chains, deep trees, etc.—is discovered. In particular, it is shown that these queries are either already expressible in the usual nested relational calculus or require at least exponential space. This Dichotomy Theorem, when coupled with the Bounded Degree Property of the usual nested relational calculus proved earlier by Libkin and Wong, becomes a powerful general tool in studying the intensional expressive power of query languages. The Bounded Degree Property makes it easy to prove that a query is inexpressible in the usual nested relational calculus. Then, if the query is expressible in NRC(powerset), subject to the conditions of the Dichotomy Theorem, the query must take at least