Knowledge graph embeddings are now a widely adopted approach to knowledge representation in which entities and relationships are embedded in vector spaces. In this chapter, we introduce the reader to the concept of knowledge graph embeddings by explaining what they are, how they can be generated and how they can be evaluated. We summarize the state-of-the-art in this field by describing the approaches that have been introduced to represent knowledge in the vector space. In relation to knowledge representation, we consider the problem of explainability, and discuss models and methods for explaining predictions obtained via knowledge graph embeddings.

Data science is a cornerstone of current business practices. A major obstacle to its adoption is that most data analysis techniques are beyond the reach of typical end-users. Spreadsheets are a prime example of this phenomenon: despite being central in all sorts of data processing pipelines, the functionality necessary for processing and analyzing spreadsheets is hidden behind the high wall of spreadsheet formulas, which most end-users can neither write nor understand [Chambers and Scaffidi, 2010]. As a result, spreadsheets are often manipulated and analyzed manually. This increases the chance of making mistakes and prevents scaling beyond small data sets. Lowering the barrier to entry for specifying and solving data science tasks would help ameliorating these issues. Making data science tools more accessible would lower the cost of designing data processing pipelines and taking datadriven decisions. At the same time, accessible data science tools can prevent non-experts from relying on fragile heuristics and improvised solutions. The question we ask is then: is it possible to enable nontechnical end-users to specify and solve data science tasks that match their needs?

In this work we describe preferential Description Logics of typicality, a nonmonotonic extension of standard Description Logics by means of a typicality operator T allowing to extend a knowledge base with inclusions of the form T(C) D, whose intuitive meaning is that "normally/typically Cs are also Ds". This extension is based on a minimal model semantics corresponding to a notion of rational closure, built upon preferential models. We recall the basic concepts underlying preferential Description Logics. We also present two extensions of the preferential semantics: on the one hand, we consider probabilistic extensions, based on a distributed semantics that is suitable for tackling the problem of commonsense concept combination, on the other hand, we consider other strengthening of the rational closure semantics and construction to avoid the so called "blocking of property inheritance problem".

I know thou wilt say "ay," And I will take thy word. Yet if thou swear'st Thou mayst prove false. At lovers' perjuries, They say, Jove laughs.

Existential rules are a positive fragment of first-order logic that generalizes function-free Horn rules by allowing existentially quantified variables in rule heads. This family of languages has recently attracted significant interest in the context of ontology-mediated query answering. Forward chaining, also known as the chase, is a fundamental tool for computing universal models of knowledge bases, which consist of existential rules and facts. Several chase variants have been defined, which differ on the way they handle redundancies. A set of existential rules is bounded if it ensures the existence of a bound on the depth of the chase, independently from any set of facts. Deciding if a set of rules is bounded is an undecidable problem for all chase variants. Nevertheless, when computing universal models, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound remains unknown or even very large. Hence, we investigate the decidability of the k-boundedness problem, which asks whether the depth of the chase for a given set of rules is bounded by an integer k. We identify a general property which, when satisfied by a chase variant, leads to the decidability of k-boundedness. We then show that the main chase variants satisfy this property, namely the oblivious, semi-oblivious (aka Skolem), and restricted chase, as well as their breadth-first versions. This paper is under consideration for publication in Theory and Practice of Logic Programming.

We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3) logic for reasoning about the behavior of machine learning systems.

Intuitively, locally finiteness means that the refinement operator is computable, completeness means we can generate, by refinement of a, any element of G related to a given element g 1 by the order relation, and properness means that a refinement operator does not generate elements which are equivalent to the element being refined. When a refinement operator is locally finite, complete and proper, we say that it is ideal. Notice that all the subsumption relations presented above satisfy the reflexive 2 and transitive 3 properties. Therefore, the pair (G,), where G is the set of all DLGs given a set of labels L, and is any of the subsumption relations defined above is a quasi-ordered set. Thus, this opens the door to defining refinement operators for DLGs. Intuitively, a downward refinement operator for DLGs will generate refinements of a given DLG by either adding vertices, edges, or by making some of the labels more specific, thus making the graph more specific. In the following subsections, we will introduce a collection of refinement operators for connected DLGs, and discuss their theoretical properties. A summary of these operators is shown in Table 1, where we show that under the object-identity constraint, all the refinement operators presented in this document are ideal. If we do not impose object-identity, then the operators are locally complete and complete, but not proper.

Steen's (2018) Hintikka set properties for Church's type theory based on primitive equality are reduced to the Hintikka set properties of Benzm\"uller, Brown and Kohlhase (2004) which are based on the logical connectives negation, disjunction and universal quantification.

To a multi-agent logic of knowledge or belief we can add public announcements to model publicly observed information change, or action models to model information change that is differently observed by different agents, but also modalities representing quantification over such information change, such as quantifiers over announcements or quantifiers over actions models. Such additions may result in more complex or undecidable logics, and create a very open landscape of relative expressivity. The survey [88] of such logics focused on open problems. Some such open problems have since then been resolved, and yet others have come to the fore. In this updated survey we review what is known about such logics with quantification over information change, including digressions into what are known as relation changing modal(but often not epistemic) logics. Again we focus on open problems.

What kind of programming language would be most appropriate to serve the needs of integrative, multi-paradigm, multi-software-system approaches to AGI? This question is broached via exploring the more particular question of how to create a more scalable and usable version of the "Atomese" programming language that forms a key component of the OpenCog AGI design (an "Atomese 2.0") . It is tentatively proposed that the core of Atomese 2.0 should be a very flexible framework of rewriting rules for rewriting a metagraph (where the rules themselves are represented within the same metagraph, and some of the intermediate data created and used during the rule-interpretation process may be represented in the same metagraph). This framework should support concurrent rewriting of the metagraph according to rules that are labeled with various sorts of uncertainty-quantifications, and that are labeled with various sorts of types associated with various type systems. A gradual typing approach should be used to enable mixture of rules and other metagraph nodes/links associated with various type systems, and untyped metagraph nodes/links not associated with any type system. This must be done in a way that allows reasonable efficiency and scalability, including in concurrent and distributed processing contexts, in the case where a large percentage of of processing time is occupied with evaluating static pattern-matching queries on specific subgraphs of a large metagraph (including a rich variety of queries such as matches against nodes representing variables, and matches against whole subgraphs, etc.).