Rebuttal for the paper with tracking Separability and its Approximations in Ontology-based Data Management

We would like to thank the reviewers for their detailed and constructive comments, which helped us to improve the paper. Before providing point-by-point notes on how we implemented the reviewers’ comments, we would like to highlight the main difference between the newly submitted version and the previous version. While we were following Reviewer#3’s suggestion about improving the presentation of the proof of Theorem 8, we noticed a sneaky error in the lower bound proof, actually inherited from the conference version of this paper. In particular, following the referenced paper [59], the Succint clique problem is known to be NExpTime -complete only when seeking for cliques of size | V | / 2, and not for cliques of arbitrary sizes (the size of the parameter k does not have to be provided in the input). Actually, if k is given and is in unary notation, then we could guess k many nodes, i.e. their binary representations, and check in polynomial time with the circuit that they form a clique in the succinctly represented graph. Thus, the succinct clique problem as described in the previous version of our paper would be in NP, and therefore not NExpTime -complete as we claimed. In the newly submitted version we solve this issue and show that, under GLAV mappings, both the separability and characterizability decision problems are actually Π p 3 -complete. We have done our best to write the new Π p 3 -completeness proof as clear as possible. Abstract. 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).