Several inconsistency-tolerant semantics have been introduced for querying inconsistent description logic knowledge bases. The first contribution of this paper is a practical approach for computing the query answers under three well-known such semantics, namely the AR, IAR and brave semantics, in the lightweight description logic DL-LiteR. We show that query answering under the intractable AR semantics can be performed efficiently by using IAR and brave semantics as tractable approximations and encoding the AR entailment problem as a propositional satisfiability (SAT) problem. The second issue tackled in this work is explaining why a tuple is a (non-)answer to a query under these semantics. We define explanations for positive and negative answers under the brave, AR and IAR semantics. We then study the computational properties of explanations in DL-LiteR. For each type of explanation, we analyze the data complexity of recognizing (preferred) explanations and deciding if a given assertion is relevant or necessary. We establish tight connections between intractable explanation problems and variants of SAT, enabling us to generate explanations by exploiting solvers for Boolean satisfaction and optimization problems. Finally, we empirically study the efficiency of our query answering and explanation framework using a benchmark we built upon the well-established LUBM benchmark.

Concept Induction refers to the problem of creating complex Description Logic class descriptions (i.e., TBox axioms) from instance examples (i.e., ABox data). In this paper we look particularly at the case where both a set of positive and a set of negative instances are given, and complex class expressions are sought under which the positive but not the negative examples fall. Concept induction has found applications in ontology engineering, but existing algorithms have fundamental performance issues in some scenarios, mainly because a high number of invokations of an external Description Logic reasoner is usually required. In this paper we present a new algorithm for this problem which drastically reduces the number of reasoner invokations needed. While this comes at the expense of a more limited traversal of the search space, we show that our approach improves execution times by up to several orders of magnitude, while output correctness, measured in the amount of correct coverage of the input instances, remains reasonably high in many cases. Our approach thus should provide a strong alternative to existing systems, in particular in settings where other systems are prohibitively slow.

Classification of description logic (DL) ontologies is a key computational problem in modern data management applications, so considerable effort has been devoted to the development and optimisation of practical reasoning calculi. Consequence-based calculi combine ideas from hypertableau and resolution in a way that has proved very effective in practice. However, existing consequence-based calculi can handle either Horn DLs (which do not support disjunction) or DLs without number restrictions. In this paper, we overcome this important limitation and present the first consequence-based calculus for deciding concept subsumption in the DL ALCHIQ+. Our calculus runs in exponential time assuming unary coding of numbers, and on ELH ontologies it runs in polynomial time. The extension to disjunctions and number restrictions is technically involved: we capture the relevant consequences using first-order clauses, and our inference rules adapt paramodulation techniques from first-order theorem proving. By using a well-known preprocessing step, the calculus can also decide concept subsumptions in SRIQ---a rich DL that covers all features of OWL 2 DL apart from nominals and datatypes. We have implemented our calculus in a new reasoner called Sequoia. We present the architecture of our reasoner and discuss several novel and important implementation techniques such as clause indexing and redundancy elimination. Finally, we present the results of an extensive performance evaluation, which revealed Sequoia to be competitive with existing reasoners. Thus, the calculus and the techniques we present in this paper provide an important addition to the repertoire of practical implementation techniques for description logic reasoning.

We propose a nonmonotonic Description Logic of typicality able to account for the phenomenon of concept combination of prototypical concepts. The proposed logic relies on the logic of typicality ALC TR, whose semantics is based on the notion of rational closure, as well as on the distributed semantics of probabilistic Description Logics, and is equipped with a cognitive heuristic used by humans for concept composition. We first extend the logic of typicality ALC TR by typicality inclusions whose intuitive meaning is that "there is probability p about the fact that typical Cs are Ds". As in the distributed semantics, we define different scenarios containing only some typicality inclusions, each one having a suitable probability. We then focus on those scenarios whose probabilities belong to a given and fixed range, and we exploit such scenarios in order to ascribe typical properties to a concept C obtained as the combination of two prototypical concepts. We also show that reasoning in the proposed Description Logic is EXPTIME-complete as for the underlying ALC.

This paper presents a novel SHOI tableau calculus which incorporates algebraic reasoning for deciding ontology consistency. Numerical restrictions imposed by nominals, existential and universal restrictions are encoded into a set of linear inequalities. Column generation and branch-and-price algorithms are used to solve these inequalities. Our preliminary experiments indicate that this calculus performs better on SHOI ontologies than standard tableau methods.

When modeling real world domains we have to deal with information that is incomplete or that comes from sources with different trust levels. This motivates the need for managing uncertainty in the Semantic Web. To this purpose, we introduced a probabilistic semantics, named DISPONTE, in order to combine description logics with probability theory. The probability of a query can be then computed from the set of its explanations by building a Binary Decision Diagram (BDD). The set of explanations can be found using the tableau algorithm, which has to handle non-determinism. Prolog, with its efficient handling of non-determinism, is suitable for implementing the tableau algorithm. TRILL and TRILLP are systems offering a Prolog implementation of the tableau algorithm. TRILLP builds a pinpointing formula, that compactly represents the set of explanations and can be directly translated into a BDD. Both reasoners were shown to outperform state-of-the-art DL reasoners. In this paper, we present an improvement of TRILLP, named TORNADO, in which the BDD is directly built during the construction of the tableau, further speeding up the overall inference process. An experimental comparison shows the effectiveness of TORNADO. All systems can be tried online in the TRILL on SWISH web application at http://trill.ml.unife.it/.

We study the problem of finite ontology mediated query answering (FOMQA), the variant of OMQA where the represented world is assumed to be finite, and thus only finite models of the ontology are considered. We adopt the most typical setting with unions of conjunctive queries and ontologies expressed in description logics (DLs). The study of FOMQA is relevant in settings that are not finitely controllable. This is the case not only for DLs without the finite model property, but also for those allowing transitive role declarations. When transitive roles are allowed, evaluating queries is challenging: FOMQA is undecidable for SHOIF and only known to be decidable for the Horn fragment of ALCIF. We show decidability of FOMQA for three proper fragments of SOIF: SOI, SOF, and SIF. Our approach is to characterise models relevant for deciding finite query entailment. Relying on a certain regularity of these models, we develop automata-based decision procedures with optimal complexity bounds.

Answer Set Programming (ASP) is a well-known declarative problem solving approach based on nonmonotonic logic programs, which has been successfully applied to a wide range of applications in artificial intelligence and beyond. To address the needs of modern applications, HEX-programs were introduced as an extension of ASP with external atoms for accessing information outside programs via an API style bi-directional interface mechanism. To evaluate such programs, conflict-driving learning algorithms for SAT and ASP solving have been extended in order to capture the semantics of external atoms. However, a drawback of the state-of-the-art approach is that external atoms are only evaluated under complete assignments (i.e., input to the external source) while in practice, their values often can be determined already based on partial assignments alone (i.e., from incomplete input to the external source). This prevents early backtracking in case of conflicts, and hinders more efficient evaluation of HEX-programs. We thus extend the notion of external atoms to allow for three-valued evaluation under partial assignments, while the two-valued semantics of the overall HEX-formalism remains unchanged. This paves the way for three enhancements: first, to evaluate external sources at any point during model search, which can trigger learning knowledge about the source behavior and/or early backtracking in the spirit of theory propagation in SAT modulo theories (SMT). Second, to optimize the knowledge learned in terms of so-called nogoods, which roughly speaking are impossible input-output configurations. Shrinking nogoods to their relevant input part leads to more effective search space pruning. And third, to make a necessary minimality check of candidate answer sets more efficient by exploiting early external evaluation calls. As this check usually accounts for a large share of the total runtime, optimization is here particularly important. We further present an experimental evaluation of an implementation of a novel HEX-algorithm that incorporates these enhancements using a benchmark suite. Our results demonstrate a clear efficiency gain over the state-of-the-art HEX-solver for the benchmarks, and provide insights regarding the most effective combinations of solver configurations.

We present a consequence-based calculus for concept subsumption and classification in the description logic ALCHOIQ, which extends ALC with role hierarchies, inverse roles, number restrictions, and nominals. By using standard transformations, our calculus extends to SROIQ, which covers all of OWL 2 DL except for datatypes. A key feature of our calculus is its pay-as-you-go behaviour: unlike existing algorithms, our calculus is worst-case optimal for all the well-known proper fragments of ALCHOIQ, albeit not for the full logic.

We study query answering in the description logic SQ supporting qualified number restrictions on both transitive and non-transitive roles. Our main contributions are a tree-like model property for SQ-knowledge bases and, building upon this, an optimal automata-based algorithm for answering positive existential regular path queries in 2EXPTIME.