Undecidability of the unification and admissibility problems for modal and description logics

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ)

A Description Logic with Transitive and Converse Roles, Role Hierarchies and Qualifying Number Restrictions

As widely argued [HG97; Sat96], transitive roles play an important role in the adequate representation of aggregated objects: they allow these objects to be described by referring to their parts without specifying a level of decomposition. In [HG97], the Description Logic (DL) ALCHR+ is presented, which extends ALC with transitive roles and a role hierarchy. It is argued in [Sat98] that ALCHR+ is well-suited to the representation of aggregated objects in applications that require various part-whole relations to be distinguished, some of which are transitive. However, ALCHR+ allows neither the description of parts by means of the whole to which they belong, or vice versa. To overcome this limitation, we present the DL SHI which allows the use of, for example, has part as well as is part of. To achieve this, ALCHR+ was extended with inverse roles. It could be argued that, instead of defining yet another DL, one could make use of the results presented in [DL96] and use ALC extended with role expressions which include transitive closure and inverse operators. The reason for not proceeding like this is the fact that transitive roles can be implemented more efficiently than the transitive closure of roles (see [HG97]), although they lead to the same complexity class (ExpTime-hard) when added, together with role hierarchies, to ALC. Furthermore, it is still an open question whether the transitive closure of roles together with inverse roles necessitates the use of the cut rule [DM98], and this rule leads to an algorithm with very bad behaviour. We will present an algorithm for SHI without such a rule. Furthermore, we enrich the language with functional restrictions and, finally, with qualifying number restrictions. We give sound and complete decision proceduresfor the resulting logics that are derived from the initial algorithm for SHI. The structure of this report is as follows: In Section 2, we introduce the DL SI and present a tableaux algorithm for satisfiability (and subsumption) of SI-concepts—in another report [HST98] we prove that this algorithm can be refined to run in polynomial space. In Section 3 we add role hierarchies to SI and show how the algorithm can be modified to handle this extension appropriately. Please note that this logic, namely SHI, allows for the internalisation of general concept inclusion axioms, one of the most general form of terminological axioms. In Section 4 we augment SHI with functional restrictions and, using the so-called pairwise-blocking technique, the algorithm can be adapted to this extension as well. Finally, in Section 5, we show that standard techniques for handling qualifying number restrictions [HB91;BBH96] together with the techniques described in previous sections can be used to decide satisfiability and subsumption for SHIQ, namely ALC extended with transitive and inverse roles, role hierarchies, and qualifying number restrictions. Although Section 5 heavily depends on the previous sections, we have made it self-contained, i.e. it contains all necessary definitions and proofs from scratch, for a better readability. Building on the previous sections, Section 6 presents an algorithm that decides the satisfiability of SHIQ-ABoxes

Practical Reasoning for Very Expressive Description Logics

Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSPACE. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the worst-case complexity of the problem, exhibit good performance with real-life problems

Reasoning with Individuals for the Description Logic SHIQ

While there has been a great deal of work on the development of reasoning algorithms for expressive description logics, in most cases only Tbox reasoning is considered. In this paper we present an algorithm for combined Tbox and Abox reasoning in the SHIQ description logic. This algorithm is of particular interest as it can be used to decide the problem of (database) conjunctive query containment w.r.t. a schema. Moreover, the realisation of an efficient implementation should be relatively straightforward as it can be based on an existing highly optimised implementation of the Tbox algorithm in the FaCT system.Comment: To appear at CADE-1

An Abstract Tableau Calculus for the Description Logic SHOI Using UnrestrictedBlocking and Rewriting

Abstract This paper presents an abstract tableau calculus for the description logic SHOI. SHOI is the extension of ALC with singleton concepts, role inverse, transitive roles and role inclusion axioms. The presented tableau calculus is inspired by a recently introduced tableau synthesis framework. Termination is achieved by a variation of the unrestricted blocking mechanism that immediately rewrites terms with respect to the conjectured equalities. This approach leads to reduced search space for decision procedures based on the calculus. We also discuss restrictions of the application of the blocking rule by means of additional side conditions and/or additional premises.

Reasoning with Very Expressive Fuzzy Description Logics

It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN, as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN