On the decidability and axiomatization of query finiteness in deductive databases

A database query is finite if its result consists of a finite sets tuples. For queries formulated as sets of pure Horn rules, the problem of determining finiteness is, in general, undecidable. In this paper, we consider superfiniteness—a stronger kind of finiteness, which applies to Horn queries whose function symbols are replaced by the abstraction of infinite relations with finiteness constraints (abbr., FC's). We show that superfiniteness is not only decidable but also axiomatizable, and the axiomatization yields an effective decision procedure. Although there are finite queries that are not superfinite, we demonstrate that superfinite queries represent an interesting and nontrivial subclass within the class of all finite queries. The we turn to the issue of inference of finiteness constraints—an important practical problem that is instrumental in deciding if a query is evaluable by a bottom-up algorithm. Although it is not known whether FC-entailment is decidable for sets of function-free Horn rules, we show that super-entailment, a stronger form of entailment, is decidable. We also show how a decision procedure for super-entailment can be used to enhance tests for query finiteness.