The Complexity of Width Minimization for Existential Positive Queries

Existential positive queries are logical sentences built from conjunction, disjunction, and existential quantification, and are also known as select-project-join-union queries in database theory, where they are recognized as a basic and fundamental class of queries. It is known that the number of variables needed to express an existential positive query is the crucial parameter determining the complexity of evaluating it on a database, and is hence a natural measure from the perspective of query optimization and rewriting. In this article, we study the complexity of the natural decision problem associated to this measure, which we call the expressibility problem: Given an existential positive query and a number k, can the query be expressed using k (or fewer) variables? We precisely determine the complexity of the expressibility problem, showing that it is complete for the level Πp2 of the polynomial hierarchy. Moreover, we prove that the expressibility problem is undecidable in positive logic (that is, existential positive logic plus universal quantification), thus establishing existential positive logic as a maximal syntactic fragment where expressibility is decidable.