Spectra of Cardinality Queries over Description Logic Knowledge Bases
Quentin Manière, Marcin Przybyłko
TL;DR
The paper studies spectra of counting queries over Description Logic knowledge bases, introducing spectra as the set of possible counts across all models and focusing on atomic CCQs. It identifies a broad class of queries (connected and individual-free) whose spectra are subsemigroups of $\mathbb{N}^{\infty}$ closed under addition and provides an effective representation $(S, M, \alpha)$ for these spectra; it also proves full realizability results for concept cardinality queries in $\mathcal{ALCIF}$ and analyzes spectra in sublogics (EL, DL-Lite variants) via cycle-reversion techniques. The main technical contributions include precise shape characterizations of spectra across several DL fragments, constructions that realize these shapes (including cycle-reversion for Horn-like logics), and data-complexity results showing ${\mathsf{FP}}^{\mathsf{NP}[\log]}$- or ${\mathsf{FP}}^{\mathsf{NP}[1]}$-completeness in various settings, with ${\mathsf{FP}}$ for DL-Lite core. The results establish a principled framework to represent and compute counting-query spectra, facilitating efficient reasoning about aggregates in ontology-mediated query answering and guiding future extensions to other DLs and semantics (e.g., bag semantics). Overall, the work advances the understanding of how spectral properties interact with description-logic expressivity and provides concrete computational procedures to obtain effective spectra representations.
Abstract
Recent works have explored the use of counting queries coupled with Description Logic ontologies. The answer to such a query in a model of a knowledge base is either an integer or $\infty$, and its spectrum is the set of its answers over all models. While it is unclear how to compute and manipulate such a set in general, we identify a class of counting queries whose spectra can be effectively represented. Focusing on atomic counting queries, we pinpoint the possible shapes of a spectrum over $\mathcal{ALCIF}$ ontologies: they are essentially the subsets of $\mathbb{N} \cup \{ \infty \}$ closed under addition. For most sublogics of $\mathcal{ALCIF}$, we show that possible spectra enjoy simpler shapes, being $[ m, \infty ]$ or variations thereof. To obtain our results, we refine constructions used for finite model reasoning and notably rely on a cycle-reversion technique for the Horn fragment of $\mathcal{ALCIF}$. We also study the data complexity of computing the proposed effective representation and establish the $\mathsf{FP}^{\mathsf{NP}[\log]}$-completeness of this task under several settings.