🤖 AI Summary
This paper investigates the decidability of the widely applicable query containment problem in first-order logic, focusing on cases admitting structurally simple countermodels—characterized by bounded treewidth, cliquewidth, and a newly introduced width measure, partitionwidth. We introduce the notion of “width-bounded universal model sets” and develop a unified framework grounded in partitionwidth, systematically integrating model-theoretic methods, graph width theory, and existential rule techniques. Partitionwidth is employed as the central width parameter for the first time, subsuming and extending classical decidable classes such as Datalog± and guarded rules. We establish decidability for several classes of homomorphism-closed queries under finite-partitionwidth rule sets. Furthermore, we expose inherent limitations of finite-unification sets and propose principled repairs to restore decidability.
📝 Abstract
We propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose to employ Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite partitionwidth sets of rules subsume other known abstract decidable classes but - leveraging existing notions of stratification - also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite unification sets into our picture and suggest several options for remedy.
