Separability and Its Approximations in Ontology-based Data Management

Given two datasets, i.e., two sets of tuples of constants, representing positive and negative examples, logical separability is the reasoning task of finding a formula in a certain target query language that separates them. As already pointed out in previous works, this task turns out to be relevant in several application scenarios such as concept learning and generating referring expressions. Besides, if we think of the input datasets of positive and negative examples as composed of tuples of constants classified, respectively, positively and negatively by a black-box model, then the separating formula can be used to provide global post-hoc explanations of such a model. In this paper, we study the separability task in the context of Ontology-based Data Management (OBDM), in which a domain ontology provides a high-level, logic-based specification of a domain of interest, semantically linked through suitable mapping assertions to the data source layer of an information system. Since a formula that properly separates (proper separation) two input datasets does not always exist, our first contribution is to propose (best) approximations of the proper separation, called (minimally) complete and (maximally) sound separations. We do this by presenting a general framework for separability in OBDM. Then, in a scenario that uses by far the most popular languages for the OBDM paradigm, our second contribution is a comprehensive study of three natural computational problems associated with the framework, namely Verification (check whether a given formula is a proper, complete, or sound separation of two given datasets), Existence (check whether a proper, or best approximated separation of two given datasets exists at all), and Computation (compute any proper, or any best approximated separation of two given datasets).