Exploring Non-Regular Extensions of Propositional Dynamic Logic with Description-Logics Features

TL;DR

This paper explores the effects of non-regular path expressions on the decidability of description logics, showing that certain extensions lead to undecidability in satisfiability and query entailment.

Contribution

It introduces and analyzes non-regular extensions of ALC, demonstrating undecidability results with simple added features and specific non-regular languages.

Findings

Decidability is lost when adding the Self operator to ALCvpl.

Adding nominals to ALCvpl results in undecidability.

Undecidability holds even with a single non-regular language r#s# in query entailment.

Abstract

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.