Gaussian processes (GPs) are non-linear probabilistic models popular in many applications. However, na\"ive GP realizations require quadratic memory to store the covariance matrix and cubic computation to perform inference or evaluate the likelihood function. These bottlenecks have driven much investment in the development of approximate GP alternatives that scale to the large data sizes common in modern data-driven applications. We present in this manuscript MuyGPs, a novel efficient GP hyperparameter estimation method. MuyGPs builds upon prior methods that take advantage of the nearest neighbors structure of the data, and uses leave-one-out cross-validation to optimize covariance (kernel) hyperparameters without realizing a possibly expensive likelihood. We describe our model and methods in detail, and compare our implementations against the state-of-the-art competitors in a benchmark spatial statistics problem. We show that our method outperforms all known competitors both in terms of time-to-solution and the root mean squared error of the predictions.

The Semantic Web ontology language OWL 2 DL comes with a variety of language features that enable sophisticated and practically useful modeling. However, the use of these features has been severely restricted in order to retain decidability of the language. For example, OWL 2 DL does not allow a property to be both transitive and asymmetric, which would be desirable, e.g., for representing an ancestor relation. In this paper, we argue that the so-called global restrictions of OWL 2 DL preclude many useful forms of modeling, by providing a catalog of basic modeling patterns that would be available in OWL 2 DL if the global restrictions were discarded. We then report on the results of evaluating several state-of-the-art OWL 2 DL reasoners on problems that use combinations of features in a way that the global restrictions are violated. The systems turn out to rely heavily on the global restrictions and are thus largely incapable of coping with the modeling patterns. Next we show how off-the-shelf first-order logic theorem proving technology can be used to perform reasoning in the OWL 2 direct semantics, the semantics that underlies OWL 2 DL, but without requiring the global restrictions. Applying a naive proof-of-concept implementation of this approach to the test problems was successful in all cases. Based on our observations, we make suggestions for future lines of research on expressive description logic-style OWL reasoning.

OWL 2 has been standardized by the World Wide Web Consortium (W3C) as a family of ontology languages for the Semantic Web. The most expressive of these languages is OWL 2 Full, but to date no reasoner has been implemented for this language. Consistency and entailment checking are known to be undecidable for OWL 2 Full. We have translated a large fragment of the OWL 2 Full semantics into first-order logic, and used automated theorem proving systems to do reasoning based on this theory. The results are promising, and indicate that this approach can be applied in practice for effective OWL reasoning, beyond the capabilities of current Semantic Web reasoners. This is an extended version of a paper with the same title that has been published at CADE 2011, LNAI 6803, pp. 446-460. The extended version provides appendices with additional resources that were used in the reported evaluation.