Spectra of Cardinality Queries over Description Logic Knowledge Bases

TL;DR

This paper investigates the spectra of counting queries over Description Logic knowledge bases, identifying their possible shapes and providing effective representations, especially for atomic queries in $ ext{ALCIF}$ and its sublogics.

Contribution

It characterizes the spectra of atomic counting queries over $ ext{ALCIF}$ ontologies and introduces effective representations for these spectra, advancing finite model reasoning techniques.

Findings

Spectra are subsets of natural numbers closed under addition for $ ext{ALCIF}$.

Simpler spectra shapes like $[m, ext{infinity}]$ occur in sublogics.

Computing the spectra representations is $ ext{FP}^{ ext{NP}[ ext{log}]}$-complete.

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.