Domain independence and the relational calculus

Abstract

Several alternative semantics (or interpretations) of the relational (domain) calculus are studied here. It is shown that they all have the same expressive power, i.e., the selection of any of the semantics neither gains nor loses expressive power. Since the domain is potentially infinite, the answer to a relational calculus query is sometimes infinite (and hence not a relation). The following approaches which guarantee the finiteness of answers to queries are studied here:output-restricted unlimited interpretation, domain independent queries, output-restricted finite andcountable invention, andlimited interpretation. Of particular interest is the output-restricted unlimited interpretation—although the output is restricted to the active domain of the input and query, the quantified variables range over the infinite underlying domain. While this is close to the intuitive interpretation given to calculus formulas, the naive approach to evaluating queries under this semantics calls for the impossible task of examining infinitely many values. We describe here a constructiion which, given a queryQ under the output-restricted unlimited interpretation, yields a domain independent queryQ′, with length no more than exponential in the length ofQ, such thatQ andQ′ (under their respective semantics) express the same function.