Expressive probabilistic description logics

AbstractThe work in this paper is directed towards sophisticated formalisms for reasoning under probabilistic uncertainty in ontologies in the Semantic Web. Ontologies play a central role in the development of the Semantic Web, since they provide a precise definition of shared terms in web resources. They are expressed in the standardized web ontology language OWL, which consists of the three increasingly expressive sublanguages OWL Lite, OWL DL, and OWL Full. The sublanguages OWL Lite and OWL DL have a formal semantics and a reasoning support through a mapping to the expressive description logics SHIF(D) and SHOIN(D), respectively. In this paper, we present the expressive probabilistic description logics P-SHIF(D) and P-SHOIN(D), which are probabilistic extensions of these description logics. They allow for expressing rich terminological probabilistic knowledge about concepts and roles as well as assertional probabilistic knowledge about instances of concepts and roles. They are semantically based on the notion of probabilistic lexicographic entailment from probabilistic default reasoning, which naturally interprets this terminological and assertional probabilistic knowledge as knowledge about random and concrete instances, respectively. As an important additional feature, they also allow for expressing terminological default knowledge, which is semantically interpreted as in Lehmann's lexicographic entailment in default reasoning from conditional knowledge bases. Another important feature of this extension of SHIF(D) and SHOIN(D) by probabilistic uncertainty is that it can be applied to other classical description logics as well. We then present sound and complete algorithms for the main reasoning problems in the new probabilistic description logics, which are based on reductions to reasoning in their classical counterparts, and to solving linear optimization problems. In particular, this shows the important result that reasoning in the new probabilistic description logics is decidable/computable. Furthermore, we also analyze the computational complexity of the main reasoning problems in the new probabilistic description logics in the general as well as restricted cases

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

Reasoning under fuzzy vagueness and probabilistic uncertainty in the Semantic Web

Combining data from many different sources or from sources that are not entirely trusted brings challenges to the automated processing of such data. Knowledge presented in natural language is another challenge for computing. In the semantic web, many applications such as personal agents need to be able to manage multiple kinds of uncertainty. There are two main approaches to modeling uncertainty in the literature - fuzzy and probabilistic. These approaches model semantically different types of uncertainty. This paper focuses on approaches that combine both fuzzy and probabilistic reasoning in one framework to provide automated agents the capability to deal with both types of uncertainty

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.Comment: Paper currently under review. 102 page

Synergies between machine learning and reasoning - An introduction by the Kay R. Amel group

This paper proposes a tentative and original survey of meeting points between Knowledge Representation and Reasoning (KRR) and Machine Learning (ML), two areas which have been developed quite separately in the last four decades. First, some common concerns are identified and discussed such as the types of representation used, the roles of knowledge and data, the lack or the excess of information, or the need for explanations and causal understanding. Then, the survey is organised in seven sections covering most of the territory where KRR and ML meet. We start with a section dealing with prototypical approaches from the literature on learning and reasoning: Inductive Logic Programming, Statistical Relational Learning, and Neurosymbolic AI, where ideas from rule-based reasoning are combined with ML. Then we focus on the use of various forms of background knowledge in learning, ranging from additional regularisation terms in loss functions, to the problem of aligning symbolic and vector space representations, or the use of knowledge graphs for learning. Then, the next section describes how KRR notions may benefit to learning tasks. For instance, constraints can be used as in declarative data mining for influencing the learned patterns; or semantic features are exploited in low-shot learning to compensate for the lack of data; or yet we can take advantage of analogies for learning purposes. Conversely, another section investigates how ML methods may serve KRR goals. For instance, one may learn special kinds of rules such as default rules, fuzzy rules or threshold rules, or special types of information such as constraints, or preferences. The section also covers formal concept analysis and rough sets-based methods. Yet another section reviews various interactions between Automated Reasoning and ML, such as the use of ML methods in SAT solving to make reasoning faster. Then a section deals with works related to model accountability, including explainability and interpretability, fairness and robustness. Finally, a section covers works on handling imperfect or incomplete data, including the problem of learning from uncertain or coarse data, the use of belief functions for regression, a revision-based view of the EM algorithm, the use of possibility theory in statistics, or the learning of imprecise models. This paper thus aims at a better mutual understanding of research in KRR and ML, and how they can cooperate. The paper is completed by an abundant bibliography

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Proceedings of the 11th Workshop on Nonmonotonic Reasoning

These are the proceedings of the 11th Nonmonotonic Reasoning Workshop. The aim of this series is to bring together active researchers in the broad area of nonmonotonic reasoning, including belief revision, reasoning about actions, planning, logic programming, argumentation, causality, probabilistic and possibilistic approaches to KR, and other related topics. As part of the program of the 11th workshop, we have assessed the status of the field and discussed issues such as: Significant recent achievements in the theory and automation of NMR; Critical short and long term goals for NMR; Emerging new research directions in NMR; Practical applications of NMR; Significance of NMR to knowledge representation and AI in general

Optimization Models Using Fuzzy Sets and Possibility Theory

Optimization is of central concern to a number of disciplines. Operations Research and Decision Theory are often considered to be identical with optimization. But also in other areas such as engineering design, regional policy, logistics and many others, the search for optimal solutions is one of the prime goals. The methods and models which have been used over the last decades in these areas have primarily been "hard" or "crisp", i.e. the solutions were considered to be either feasible or unfeasible, either above a certain aspiration level or below. This dichotomous structure of methods very often forced the modeler to approximate real problem situations of the more-or-less type by yes-or-no-type models, the solutions of which might turn out not to be the solutions to the real problems. This is particularly true if the problem under consideration includes vaguely defined relationships, human evaluations, uncertainty due to inconsistent or incomplete evidence, if natural language has to be modeled or if state variables can only be described approximately. Until recently, everything which was not known with certainty, i.e. which was not known to be either true or false or which was not known to either happen with certainty or to be impossible to occur, was modeled by means of probabilities. This holds in particular for uncertainties concerning the occurrence of events. probability theory was used irrespective of whether its axioms (such as, for instance, the law of large numbers) were satisfied or not, or whether the "events" could really be described unequivocally and crisply. In the meantime one has become aware of the fact that uncertainties concerning the occurrence as well as concerning the description of events ought to be modeled in a much more differentiated way. New concepts and theories have been developed to do this: the theory of evidence, possibility theory, the theory of fuzzy sets have been advanced to a stage of remarkable maturity and have already been applied successfully in numerous cases and in many areas. Unluckily, the progress in these areas has been so fast in the last years that it has not been documented in a way which makes these results easily accessible and understandable for newcomers to these areas: text-books have not been able to keep up with the speed of new developments; edited volumes have been published which are very useful for specialists in these areas, but which are of very little use to nonspecialists because they assume too much of a background in fuzzy set theory. To a certain degree the same is true for the existing professional journals in the area of fuzzy set theory. Altogether this volume is a very important and appreciable contribution to the literature on fuzzy set theory