Worst-Case Optimal Querying of Very Expressive Description Logics with Path Expressions and Succinct Counting

Abstract

Among the most expressive knowledge representation formalisms are the description logics of the Z family. For well-behaved fragments of ZOIQ, entailment of positive two-way regular path queries is well known to be 2EXPTIME-complete under the proviso of unary encoding of numbers in cardinality constraints. We show that this assumption can be dropped without an increase in complexity and EXPTIME-completeness can be achieved when bounding the number of query atoms, using a novel reduction from query entailment to knowledge base satisfiability. These findings allow to strengthen other results regarding query entailment and query containment problems in very expressive description logics. Our results also carry over to GC2, the two-variable guarded fragment of first-order logic with counting quantifiers, for which hitherto only conjunctive query entailment has been investigated.

On Finite Entailment of Non-Local Queries in Description Logics

“…Our main finding is that finite entailment of UCQs with transitive closure in ALCOI + and ALCOQ + is 2EXPTIMEcomplete. As unrestricted entailment of positive RPQs is 2EXPTIME-complete for both logics (Calvanese, Eiter, and Ortiz 2014;Bednarczyk and Rudolph 2019), and the hardness already holds for CQs and both ALCI and ALCO (Lutz 2008;Ngo, Ortiz, and Simkus 2016), we conclude that the complexity of our setting is the same over arbitrary and finite models. As finite entailment of two-way RPQs in ALCIOF is undecidable (Rudolph 2016), our results are quite close to the undecidability frontier.…”

“…The single work studying extensions of UCQs (Rudolph 2016), brings only undecidability results for navigational queries and various expressive DLs. In contrast, for unrestricted entailment a large body of literature is available, providing multiple positive results for navigational queries and both expressive and lightweight DLs (Calvanese, Eiter, and Ortiz 2014;Stefanoni et al 2014;Bienvenu, Ortiz, and Simkus 2015;Jung et al 2017;Gutiérrez-Basulto, Ibáñez-García, and Jung 2018;Gogacz et al 2019;Bednarczyk and Rudolph 2019). The lack of research on finite entailment of non-local queries (e.g.…”

On Finite Entailment of Non-Local Queries in Description Logics

“…Our main finding is that finite entailment of UCQs with transitive closure over ALCOI + or ALCOQ + knowledge bases is 2EXPTIME-complete. As unrestricted entailment of positive regular path queries is 2EXPTIMEcomplete for both logics (Calvanese, Eiter, and Ortiz 2014;Bednarczyk and Rudolph 2019), and the hardness already holds for CQs and both ALCI and ALCO (Lutz 2008;Ngo, Ortiz, and Simkus 2016), we have that the complexity of our setting is the same over arbitrary and finite models. As finite entailment of two-way regular path queries over ALCIOF knowledge bases is undecidable (Rudolph 2016), our results are quite close to the undecidability frontier.…”

“…The single work studying extensions of UCQs (Rudolph 2016), brings only undecidability results for navigational queries and various expressive description logics. In contrast, for unrestricted entailment a large body of literature is available, providing multiple positive results for navigational queries and both expressive and lightweight DLs (Calvanese, Eiter, and Ortiz 2014;Stefanoni et al 2014;Bienvenu, Ortiz, and Simkus 2015;Jung et al 2017;Gutiérrez-Basulto, Ibáñez-García, and Jung 2018;Gogacz et al 2019;Bednarczyk and Rudolph 2019).…”

Finite Entailment of Local Queries in the Z Family of Description Logics

“…2EXPTIME) KBSat (resp. PEQ entailment) problems, see: (Calvanese, Eiter, and Ortiz 2009;Bednarczyk and Rudolph 2019). The proofs of these facts rely on the quasiforest model property, which admits automata-theoretic approach (a reduction to the so-called fully enriched automata).…”