Datalog is a powerful rule-based language with numerous applications in databases and knowledge representation. Explaining why a fact belongs to the output of a Datalog program over a database is an essential task towards explainable and transparent data-intensive applications. A standard way of explaining a fact is the so-called why-provenance, which provides witnesses in the form of subsets of the input database that as a whole can be used to derive that fact. While why-provenance for Datalog has been extensively studied in the literature, the analogous notion for Datalog with negation remains unexplored. We extend why-provenance to Datalog with negation under the standard well-founded and stable model semantics, inherited from Logic Programming, by building on justification theory. We then perform a thorough data complexity analysis of the underlying explainability problem and show that it is in general intractable for both well-founded and stable model semantics; in particular, it is NP-complete, which is the best that we can hope for since the problem is already NP-complete for positive Datalog.

Abstract We propose a stable model semantics for higher-order logic programs. Our semantics is developed using Approximation Fixpoint Theory (AFT), a powerful formalism that has successfully been used to give meaning to diverse non-monotonic formalisms. The proposed semantics generalizes the classical two-valued stable model semantics of Gelfond and Lifschitz as well as the three-valued one of Przymusinski, retaining their desirable properties. Due to the use of AFT, we also get for free alternative semantics for higher-order logic programs, namely supported model, Kripke-Kleene, and well-founded. Additionally, we define a broad class of stratified higher-order logic programs and demonstrate that they have a unique two-valued higher-order stable model which coincides with the well-founded semantics of such programs. We provide a number of examples in different application domains, which demonstrate that higher-order logic programming under the stable model semantics is a powerful and versatile formalism, which can potentially form the basis of novel ASP systems.

Neural networks constitute a class of functions that are typically non-surjective, with high-dimensional fibers and complicated image. We prove two main results concerning the geometry of the loss landscape of a neural network. First, we provide an explicit effective bound on the sizes of the hidden layers so that the loss landscape has no spurious valleys, which guarantees the success of gradient descent methods. Second, we present a novel method for analyzing whether a given neural network architecture with monomial activation function can represent a target function of interest. The core of our analysis method is the study of a specific set of error values, and its behavior depending on different training datasets.

Common knowledge and only knowing capture two intuitive and natural notions that have proven to be useful in a variety of settings, for example to reason about coordination or agreement between agents, or to analyse the knowledge of knowledge-based agents. While these two epistemic operators have been extensively studied in isolation, the approaches made to encode their complex interplay failed to capture some essential properties of only knowing. We propose a novel solution by defining a notion of μ-biworld for countable ordinals μ, which approximates not only the worlds that an agent deems possible, but also those deemed impossible. This approach allows us to define a multi-agent epistemic logic with common knowledge and only knowing operators, and a three-valued model semantics for it. Moreover, we show that we only really need biworlds of depth at most ω²+1. Based on this observation, we define a Kripke semantics on a canonical Kripke structure and show that this semantics coincides with the model semantics. Finally, we discuss issues arising when combining negative introspection or truthfulness with only knowing and show how positive introspection can be integrated into our logic.