Pinpointing in the Description Logic $\mathcal {EL}^+$

Abstract

Axiom pinpointing has been introduced in description logics (DLs) to help the user understand the reasons why consequences hold by computing minimal subsets of the knowledge base that have the consequence in question. Until now, the pinpointing approach has only been applied to the DL ALC and some of its extensions. This paper considers axiom pinpointing in the less expressive DL EL + , for which subsumption can be decided in polynomial time. More precisely, we consider an extension of the pinpointing problem where the knowledge base is divided into a static part, which is always present, and a refutable part, of which subsets are taken. We describe an extension of the subsumption algorithm for EL + that can be used to compute all minimal subsets of (the refutable part of) a given TBox that imply a certain subsumption relationship. The worst-case complexity of this algorithm turns out to be exponential. This is not surprising since we can show that a given TBox may have exponentially many such minimal subsets. However, we can also show that the problem is not even output polynomial, i.e., unless P=NP, there cannot be an algorithm computing all such minimal sets that is polynomial in the size of its input and output. In addition, we show that finding out whether there is such a minimal subset within a given cardinality bound is an NP-complete problem. In contrast to these negative results, we also show that one such minimal subset can be computed in polynomial time. Finally, we provide some encouraging experimental results regarding the performance of a practical algorithm that computes one (small, but not necessarily minimal) subset that has a given subsumption relation as consequence.

A Generic Approach for Correcting Access Restrictions to a Consequence

“…Hence, we search for a change set of minimum cardinality. It follows from [5] that this problem is NP-complete.…”

Axiom Pinpointing in Lightweight Description Logics via Horn-SAT Encoding and Conflict Analysis

“…In this paper we build on previous work from the literature of EL + reasoning [4,6,7] and of SAT/SMT [19,33,12,17,23], and we propose a simple and novel approach for (concept subsumption and) axiom pinpointing in EL + (and hence in its sub-logics EL and ELH). In a nutshell, the idea is to generate polynomial-size Horn propositional formulas representing part or all the deduction steps performed by the classi cation algorithms of [4,6], and to manipulate them by exploiting the functionalities of modern con ict-driven SAT/SMT solvers like Boolean Contraint Propagation (BCP) [19], con ict analysis under assumptions [19,12], and all-SMT [17].…”

“…less expressive but tractable) description logics, like EL and its family [1,4,6,16,20,2]. In particular, the logic EL + [4,6,7] extends EL and is of particular relevance due to its algorithmic properties and due to its capability of expressing several important and widely-used bio-medical ontologies, such as Snomed-CT [31,30,32], NCI [29], GeneOntology [9] and the majority of Galen [21]. In fact in EL + not only standard logic problems such as concept subsumption (e.g., is Amputation-of-Finger a subconcept of Amputation-of-Arm in the ontology Snomed-CT?…”

“…To this extent, the problems of concept subsumption and axiom pinpointing in EL + have been thoroughly investigated, and e cient algorithm for these two functionalities have been implemented and tested with success on large ontologies, including Snomed-CT (see e.g. [4,6,7]). …”

“…In particular, we show that from an ontology T it is possible to generate in polynomial time Horn propositional formulas φ T , φ one for an amount of times up-to-linear in the size of the rst nMinA found; (iv) the same task of (iii) can also be computed by iteratively applying process (ii) on an up-to-linear sequence of increasingly-smaller formulas φ one T ,φ one S1 ,...φ one S k ; (v) all MinAs can be enumerated by means of AllSMT techniques on φ all T (po) , using step (iii) as a subroutine. It is worth noticing that (i) and (ii) are instantaneous even with huge φ T , φ one T and φ all T (po) , and that (v) requires building a polynomial-size formula φ all T (po) , in contrast to the exponential-size formula required by the AllMinA process of [6].We have implemented a prototype tool and performed a preliminary empirical evaluation on the available ontologies, whose results con rm the potential of our novel approach.Content. In 2 we provide the necessary background on EL + reasoning and…”