First-Order Rewritability and Complexity of Two-Dimensional Temporal Ontology-Mediated Queries

Abstract

Aiming at ontology-based data access to temporal data, we design two-dimensional temporal ontology and query languages by combining logics from the (extended) DL-Lite family with linear temporal logic LTL over discrete time (Z,<). Our main concern is first-order rewritability of ontology-mediated queries (OMQs) that consist of a 2D ontology and a positive temporal instance query. Our target languages for FO-rewritings are two-sorted FO(<) -- first-order logic with sorts for time instants ordered by the built-in precedence relation < and for the domain of individuals---its extension FO(<, ≡) with the standard congruence predicates t ≡ 0 (mod n), for any fixed n > 1, and FO(RPR) that admits relational primitive recursion. In terms of circuit complexity, FO(<, ≡)- and FO(RPR)-rewritability guarantee answering OMQs in uniform AC0 and NC1, respectively. We proceed in three steps. First, we define a hierarchy of 2D DL-Lite/LTL ontology languages and investigate the FO-rewritability of OMQs with atomic queries by constructing projections onto 1D LTL OMQs and employing recent results on the FO-rewritability of propositional LTL OMQs. As the projections involve deciding consistency of ontologies and data, we also consider the consistency problem for our languages. While the undecidability of consistency for 2D ontology languages with expressive Boolean role inclusions might be expected, we also show that, rather surprisingly, the restriction to Krom and Horn role inclusions leads to decidability (and ExpSpace-completeness), even if one admits full Booleans on concepts. As a final step, we lift some of the rewritability results for atomic OMQs to OMQs with expressive positive temporal instance queries. The lifting results are based on an in-depth study of the canonical models and only concern Horn ontologies.

Deciding FO-rewritability of Regular Languages and Ontology-Mediated Queries in Linear Temporal Logic

“…First, as well known, the interaction between temporal and description logic operators tends to dramatically increase the complexity of OMQ answering, which makes the uniform classification of OMQs according to their rewritability type much harder. Some initial steps in this direction have been made by Borgwardt et al (2019), Artale et al (2022). Second, even without the description logic constructs, pure one-dimensional temporal OMQs give rise to the complexity classes and target languages for FO-rewritings that have not occurred in the OBDA context so far.…”

“…There have been numerous attempts to extend ontology and query languages with constructors that are capable of representing events over temporal data; consult Lutz et al (2008), Artale et al (2017) for surveys and Gutiérrez-Basulto and Jung (2017), Borgwardt et al (2019), Walega et al (2020bWalega et al ( , 2020a, Artale et al (2022) for more recent developments. However, so far the focus has only been on the uniform complexity of reasoning with arbitrary ontologies and queries in a given language rather than on determining the data complexity and FO-rewritability of individual temporal OMQs.…”