The Data Complexity of Description Logic Ontologies

TL;DR

This paper investigates the data complexity of query evaluation in description logic ontologies, providing complexity classifications and decidability results for various DLs and query types.

Contribution

It offers a non-uniform complexity analysis per ontology, establishing PTime/coNP-dichotomies and undecidability results for specific description logics.

Findings

PTime/coNP-dichotomy for ALCFI depth-one ontologies

Dichotomy for ALC and ALCI unrestricted depth ontologies

Undecidability of PTime evaluation for ALCF ontologies

Abstract

We analyze the data complexity of ontology-mediated querying where the ontologies are formulated in a description logic (DL) of the ALC family and queries are conjunctive queries, positive existential queries, or acyclic conjunctive queries. Our approach is non-uniform in the sense that we aim to understand the complexity of each single ontology instead of for all ontologies formulated in a certain language. While doing so, we quantify over the queries and are interested, for example, in the question whether all queries can be evaluated in polynomial time w.r.t. a given ontology. Our results include a PTime/coNP-dichotomy for ontologies of depth one in the description logic ALCFI, the same dichotomy for ALC- and ALCI-ontologies of unrestricted depth, and the non-existence of such a dichotomy for ALCF-ontologies. For the latter DL, we additionally show that it is undecidable whether a given ontology admits PTime query evaluation. We also consider the connection between PTime query evaluation and rewritability into (monadic) Datalog.