Semiring Provenance for Lightweight Description Logics

We investigate semiring provenance--a successful framework originally defined in the relational database setting--for description logics. In this context, the ontology axioms are annotated with elements of a commutative semiring and these annotations are propagated to the ontology consequences in a way that reflects how they are derived. We define a provenance semantics for a language that encompasses several lightweight description logics and show its relationships with semantics that have been defined for ontologies annotated with a specific kind of annotation (such as fuzzy degrees). We show that under some restrictions on the semiring, the semantics satisfies desirable properties (such as extending the semiring provenance defined for databases). We then focus on the well-known why-provenance, which allows to compute the semiring provenance for every additively and multiplicatively idempotent commutative semiring, and for which we study the complexity of problems related to the provenance of an axiom or a conjunctive query answer. Finally, we consider two more restricted cases which correspond to the so-called positive Boolean provenance and lineage in the database setting. For these cases, we exhibit relationships with well-known notions related to explanations in description logics and complete our complexity analysis. As a side contribution, we provide conditions on an ELHI_bot ontology that guarantee tractable reasoning.

Answering Fuzzy Queries over Fuzzy DL-Lite Ontologies

A prominent problem in knowledge representation is how to answer queries taking into account also the implicit consequences of an ontology representing domain knowledge. While this problem has been widely studied within the realm of description logic ontologies, it has been surprisingly neglected within the context of vague or imprecise knowledge, particularly from the point of view of mathematical fuzzy logic. In this paper we study the problem of answering conjunctive queries and threshold queries w.r.t. ontologies in fuzzy DL-Lite. Specifically, we show through a rewriting approach that threshold query answering w.r.t. consistent ontologies remains in $AC_0$ in data complexity, but that conjunctive query answering is highly dependent on the selected triangular norm, which has an impact on the underlying semantics. For the idempodent G\"odel t-norm, we provide an effective method based on a reduction to the classical case. This paper is under consideration in Theory and Practice of Logic Programming (TPLP).

Union and Intersection of all Justifications

We present new algorithm for computing the union and intersection of all justifications for a given ontological consequence without first computing the set of all justifications. Through an empirical evaluation, we show that our approach works well in practice for expressive DLs. In particular, the union of all justifications can be computed much faster than with existing justification-enumeration approaches. We further discuss how to use these results to repair ontologies efficiently.

Reasoning with Contextual Knowledge and Influence Diagrams

Influence diagrams (IDs) are well-known formalisms extending Bayesian networks to model decision situations under uncertainty. Although they are convenient as a decision theoretic tool, their knowledge representation ability is limited in capturing other crucial notions such as logical consistency. We complement IDs with the light-weight description logic (DL) EL to overcome such limitations. We consider a setup where DL axioms hold in some contexts, yet the actual context is uncertain. The framework benefits from the convenience of using DL as a domain knowledge representation language and the modelling strength of IDs to deal with decisions over contexts in the presence of contextual uncertainty. We define related reasoning problems and study their computational complexity.

Axiom Pinpointing

Axiom pinpointing refers to the task of finding the specific axioms in an ontology which are responsible for a consequence to follow. This task has been studied, under different names, in many research areas, leading to a reformulation and reinvention of techniques. In this work, we present a general overview to axiom pinpointing, providing the basic notions, different approaches for solving it, and some variations and applications which have been considered in the literature. This should serve as a starting point for researchers interested in related problems, with an ample bibliography for delving deeper into the details.

Provenance for the Description Logic ELHr

We address the problem of handling provenance information in ELHr ontologies. We consider a setting recently introduced for ontology-based data access, based on semirings and extending classical data provenance, in which ontology axioms are annotated with provenance tokens. A consequence inherits the provenance of the axioms involved in deriving it, yielding a provenance polynomial as an annotation. We analyse the semantics for the ELHr case and show that the presence of conjunctions poses various difficulties for handling provenance, some of which are mitigated by assuming multiplicative idempotency of the semiring. Under this assumption, we study three problems: ontology completion with provenance, computing the set of relevant axioms for a consequence, and query answering.

Probabilistic Temporal Logic over Finite Traces (Technical Report)

Temporal logics over finite traces have recently gained attention due to their use in real-world applications, in particular in business process modelling and planning. In real life, processes contain some degree of uncertainty that is impossible to handle with classical logics. We propose a new probabilistic temporal logic over finite traces based on superposition semantics, where all possible evolutions are possible, until observed. We study the properties of the logic and provide automata-based mechanisms for deriving probabilistic inferences from its formulas. We ground the approach in the context of declarative process modelling, showing how the temporal patterns used in Declare can be lifted to our setting, and discussing how probabilistic inferences can be exploited to provide key offline and runtime reasoning tasks, and how to discover probabilistic Declare patterns from event data by minor adjustments to existing discovery algorithms.

A Decidable Very Expressive Description Logic for Databases (Extended Version)

We introduce $\mathcal{DLR}^+$, an extension of the n-ary propositionally closed description logic $\mathcal{DLR}$ to deal with attribute-labelled tuples (generalising the positional notation), projections of relations, and global and local objectification of relations, able to express inclusion, functional, key, and external uniqueness dependencies. The logic is equipped with both TBox and ABox axioms. We show how a simple syntactic restriction on the appearance of projections sharing common attributes in a $\mathcal{DLR}^+$ knowledge base makes reasoning in the language decidable with the same computational complexity as $\mathcal{DLR}$. The obtained $\mathcal{DLR}^\pm$ n-ary description logic is able to encode more thoroughly conceptual data models such as EER, UML, and ORM.

Towards Statistical Reasoning in Description Logics over Finite Domains (Full Version)

We present a probabilistic extension of the description logic $\mathcal{ALC}$ for reasoning about statistical knowledge. We consider conditional statements over proportions of the domain and are interested in the probabilistic-logical consequences of these proportions. After introducing some general reasoning problems and analyzing their properties, we present first algorithms and complexity results for reasoning in some fragments of Statistical $\mathcal{ALC}$.

Probabilistic Reasoning in the Description Logic ALCP with the Principle of Maximum Entropy (Full Version)

A central question for knowledge representation is how to encode and handle uncertain knowledge adequately. We introduce the probabilistic description logic ALCP that is designed for representing context-dependent knowledge, where the actual context taking place is uncertain. ALCP allows the expression of logical dependencies on the domain and probabilistic dependencies on the possible contexts. In order to draw probabilistic conclusions, we employ the principle of maximum entropy. We provide reasoning algorithms for this logic, and show that it satisfies several desirable properties of probabilistic logics.