Finite Open-World Query Answering with Number Restrictions (Extended Version)

TL;DR

This paper establishes the first decidability results for finite open-world query answering with both referential constraints and number restrictions, using new techniques to construct finite models for arbitrary signatures.

Contribution

It introduces novel methods to determine decidability of FQA combining referential constraints and number restrictions for any signature.

Findings

Decidability of FQA with unary inclusion and functional dependencies.

Finite implication closure suffices for decidability in these cases.

New techniques for constructing finite universal models.

Abstract

Open-world query answering is the problem of deciding, given a set of facts, conjunction of constraints, and query, whether the facts and constraints imply the query. This amounts to reasoning over all instances that include the facts and satisfy the constraints. We study finite open-world query answering (FQA), which assumes that the underlying world is finite and thus only considers the finite completions of the instance. The major known decidable cases of FQA derive from the following: the guarded fragment of first-order logic, which can express referential constraints (data in one place points to data in another) but cannot express number restrictions such as functional dependencies; and the guarded fragment with number restrictions but on a signature of arity only two. In this paper, we give the first decidability results for FQA that combine both referential constraints and number restrictions for arbitrary signatures: we show that, for unary inclusion dependencies and functional dependencies, the finiteness assumption of FQA can be lifted up to taking the finite implication closure of the dependencies. Our result relies on new techniques to construct finite universal models of such constraints, for any bound on the maximal query size.