In this paper we discuss the relationships between conditional and preferential logics and neural network models, based on a multi-preferential semantics. We propose a concept-wise multipreference semantics, recently introduced for defeasible description logics to take into account preferences with respect to different concepts, as a tool for providing a semantic interpretation to neural network models. This approach has been explored both for unsupervised neural network models (Self-Organising Maps) and for supervised ones (Multilayer Perceptrons), and we expect that the same approach might be extended to other neural network models. It allows for logical properties of the network to be checked (by model checking) over an interpretation capturing the input-output behavior of the network. For Multilayer Perceptrons, the deep network itself can be regarded as a conditional knowledge base, in which synaptic connections correspond to weighted conditionals. The paper describes the general approach, through the cases of Self-Organising Maps and Multilayer Perceptrons, and discusses some open issues and perspectives.

In this paper we establish a link between preferential semantics for description logics and self-organising maps, which have been proposed as possible candidates to explain the psychological mechanisms underlying category generalisation. In particular, we show that a concept-wise multipreference semantics, which takes into account preferences with respect to different concepts and has been recently proposed for defeasible description logics, can be used to to provide a logical interpretation of SOMs. We also provide a logical interpretation of SOMs in terms of a fuzzy description logic as well as a probabilistic account.

In this paper we describe a concept-wise multi-preference semantics for description logic which has its root in the preferential approach for modeling defeasible reasoning in knowledge representation. We argue that this proposal, beside satisfying some desired properties, such as KLM postulates, and avoiding the drowning problem, also defines a plausible notion of semantics. We motivate the plausibility of the concept-wise multi-preference semantics by developing a logical semantics of self-organising maps, which have been proposed as possible candidates to explain the psychological mechanisms underlying category generalisation, in terms of multi-preference interpretations.

The paper describes a preferential approach for dealing with exceptions in KLM preferential logics, based on the rational closure. It is well known that the rational closure does not allow an independent handling of the inheritance of different defeasible properties of concepts. Several solutions have been proposed to face this problem and the lexicographic closure is the most notable one. In this work, we consider an alternative closure construction, called the Multi Preference closure (MP-closure), that has been first considered for reasoning with exceptions in DLs. Here, we reconstruct the notion of MP-closure in the propositional case and we show that it is a natural variant of Lehmann's lexicographic closure. Abandoning Maximal Entropy (an alternative route already considered but not explored by Lehmann) leads to a construction which exploits a different lexicographic ordering w.r.t. the lexicographic closure, and determines a preferential consequence relation rather than a rational consequence relation. We show that, building on the MP-closure semantics, rationality can be recovered, at least from the semantic point of view, resulting in a rational consequence relation which is stronger than the rational closure, but incomparable with the lexicographic closure. We also show that the MP-closure is stronger than the Relevant Closure.

Reasoning about exceptions in ontologies is nowadays one of the challenges the description logics community is facing. The paper describes a preferential approach for dealing with exceptions in Description Logics, based on the rational closure. The rational closure has the merit of providing a simple and efficient approach for reasoning with exceptions, but it does not allow independent handling of the inheritance of different defeasible properties of concepts. In this work we outline a possible solution to this problem by introducing a variant of the lexicographical closure, that we call skeptical closure, which requires to construct a single base. We develop a bi-preference semantics semantics for defining a characterization of the skeptical closure.

We define the notion of rational closure in the context of Description Logics extended with a tipicality operator. We start from ALC+T, an extension of ALC with a typicality operator T: intuitively allowing to express concepts of the form T(C), meant to select the "most normal" instances of a concept C. The semantics we consider is based on rational model. But we further restrict the semantics to minimal models, that is to say, to models that minimise the rank of domain elements. We show that this semantics captures exactly a notion of rational closure which is a natural extension to Description Logics of Lehmann and Magidor's original one. We also extend the notion of rational closure to the Abox component. We provide an ExpTime algorithm for computing the rational closure of an Abox and we show that it is sound and complete with respect to the minimal model semantics.