Query inseparability for ALC ontologies

Abstract

We investigate the problem whether two ALC ontologies are indistinguishable (or inseparable) by means of queries in a given signature, which is fundamental for ontology engineering tasks such as ontology versioning, modularisation, update, and forgetting. We consider both knowledge base (KB) and TBox inseparability. For KBs, we give modeltheoretic criteria in terms of (finite partial) homomorphisms and products and prove that this problem is undecidable for conjunctive queries (CQs), but 2ExpTime-complete for unions of CQs (UCQs). The same results hold if (U)CQs are replaced by rooted (U)CQs, where every variable is connected to an answer variable. We also show that inseparability by CQs is still undecidable if one KB is given in the lightweight DL EL and if no restrictions are imposed on the signature of the CQs. We also consider the problem whether two ALC TBoxes give the same answers to any query over any ABox in a given signature and show that, for CQs, this problem is undecidable, too. We then develop model-theoretic criteria for HornALC TBoxes and show using tree automata that, in contrast, inseparability becomes decidable and 2ExpTime-complete, even ExpTime-complete when restricted to (unions of) rooted CQs. module is extracted from an ontology. In ontology update or revision, the difference between the answers to queries over the updated or revised ontology and the original one should be minimised when constructing update or revision operators. Similarly, in forgetting, it is the answers to queries which should be preserved under appropriate forgetting operators. Thus, in the context of query answering, the fundamental relationship between ontologies is not whether they are logically equivalent (have the same models), but whether they give the same answers to any relevant query. To illustrate, consider the following simple TBox T = {Book ∃author.¬Book} saying that every book has an author who is not a book. Clearly, T is not logically equivalent to the TBoxwhich only states that every book has an author. However, if one takes as the query language the popular classes of conjunctive queries (CQs) or unions of CQs (UCQs), then no matter what the data is, every query will have the same answers independently of whether one uses T or T . Intuitively, the reason is that the 'positive' information given by T coincides with the 'positive' information given by T . If the main purpose of the ontology is answering UCQs, it is thus more important to know that T can be safely replaced by T without affecting the answers to UCQs than to establish that T and T are not logically equivalent.In most ontology engineering applications for ontology-based data access, the relevant class Q of queries can be further restricted to those given in a finite signature of relevant concept and role names. For example, to establish that a subset M of an ontology O is a module of O, one should not require that M and O give the same answers to all queries in Q, but only to those that are in the signature of M. Similarly, in the versioning ...

Conservative Extensions in Horn Description Logics with Inverse Roles

“…In this way, we establish decidability and a 2ExpTime upper bound for query entailment in the expressive Horn DL Horn-ALCHIF, and thus also for query inseparability and query conservative extensions. Together with known lower bounds (Botoeva et al, 2019), this yields 2ExpTime-completeness for all DLs between ELI and Horn-ALCHIF as well as between Horn-ALC and Horn-ALCHIF. To be more precise, the complexity is single exponential in the size of T 1 and double exponential only in the size of T 2 .…”

“…We now develop the announced characterization of CQ entailment. Known characterizations of entailment in Horn DLs without inverse roles (Lutz & Wolter, 2010;Botoeva et al, 2019) suggest that a first natural candidate characterization would be:…”

“…CQ entailment has been studied for various DLs (Kontchakov, Pulina, Sattler, Schneider, Selmer, Wolter, & Zakharyaschev, 2009;Lutz & Wolter, 2010;Konev et al, 2012;Botoeva, Lutz, Ryzhikov, Wolter, & Zakharyaschev, 2019), also in the KB version (Botoeva, Kontchakov, Ryzhikov, Wolter, & Zakharyaschev, 2016b;Botoeva et al, 2019), and also for OBDA specifications that involve mappings between data sources and the ABox (Bienvenu & Rosati, 2015), see also the survey by Botoeva, Konev, Lutz, Ryzhikov, Wolter, and Zakharyaschev (2016a). Nevertheless, there is still a notable gap in our understanding of this notion: query entailment between TBoxes is poorly understood in Horn DLs with inverse roles, which are considered a crucial feature in many applications.…”

Reasoning Web: Logical Foundation of Knowledge Graph Construction and Query Answering

“…The following result was first proved in [71] for EL. We state it here also for Horn-ALC [15] as this will be needed later on. Theorem 33.…”

“…The undecidability proof for CQ (and rCQ) entailment and inseparability of ALC KBs (Theorem 26) can also be lifted to the TBox case; see [15] for details. In contrast to the KB case, decidability of UCQ entailment and inseparability of ALC TBoxes remains open, as well as for the rUCQ versions.…”