We thank the reviewers and editors for the valuable feedback on our manuscript. We have considered your comments and suggestions, responded each of them in this document and made the appropriate changes in the manuscript. This submission is a resubmit of 797-1788 --------------------------------------------------------------- Response to Reviewer 1 This paper presents a new geometric embedding method based on order embedding for knowledge bases of Description Logic ALC. It includes two steps: (1) transforming the axioms/ontology to a DAG composed of subsubsumption relationships of sub-expressions (which include the named concepts and the other expressions like the concept conjunction and disjunction); (2) using an order embedding algorithm to embed the nodes in the DAG. The paper is well written and easy to follow with good technical quality. The idea is novel, and the codes look to be well maintained. ------ * In Line 29, Page 5, "The lattice construction is not compete since we consider a subset $\hat{C}$ from the infinite set ${C}$ of possible concept descriptions". Could the authors give more justification? What concept descriptions could be ignored? What problem will this lead to? ** Response: The lattice is constructed based on the existing axioms in the knowledge base and the elements of the lattice are the subexpressions in each axiom. Some deduction rules can generate new concept descriptions, which will not be considered for the lattice construction and therefore the lattice will ignore that information. We have added these explanation in the beginning of Section 3.1 to also motivate the use of saturation procedures that can include some of the missing concept descriptions and potentially improve the quality of the embeddings. ------ * In Theorem 2, the authors proves that the embedding (mapping function \(f_{\theta}\)) is a lattice preserving function of \((\hat{bf{C}}, \subseteq)\). But how about preservation of the semantics of the original ontology \(\mathcal{O}\)? The author can give some justification on whether the embedding is sound enough to keep the semantics of the original ontology with some specific settings in learning. If not, why, and what semantics cannot be preserved? ** Response: The embedding $f_\theta$ preserves a lattice structure. The lattice CatE preserves is generated from the objects and morphisms of the category that provides the semantics for a theory $T$. This category is shown to be compatible with classical semantics in the sense that the theory $T$ is category-theoretically unsatisfiable if and only if $T$ is set-theoretically unsatisfiable (\url{https://ceur-ws.org/Vol-2954/paper-22.pdf}). Therefore, by preserving the lattice structure, $f_\theta$ also preserves the categorical semantics, and therefore also classical semantics. We have added this explanation in the manuscript and also added a toy example to illustrate how the entities would behave in the embedding space. ------ * In the second step, beside order embedding algorithms, will authors consider some other hierarchy embedding methods such as poincare ball-based hyperbolic embedding methods and box/cone-based geometric embedding methods? Please justify (and implement and evaluate if possible) its potential extensions. ** Response: Hierarchical embedding methods such as such as hyperbolic embeddings are suitable for tree-form structures because trees have negative Ricci curvature (\url{https://openreview.net/pdf?id=BylEqnVFDB}). In the case of lattices, the curvature is not necessarily negative, therefore we believe hierachical methods are not suitable in this case.} ------ * ELEmbedding and Box2EL are for ontologies of DL EL++. How are axioms beyond DL EL++ processed? Do GO and FoodOn include axioms beyond DL ALC? If yes, how are they processed? In Table 1, please give the DL languages (expressivities) of the ontologies used for experiments. ** Response: To apply ELEmbeddings and Box2EL, we normalize the axioms to be compatible with $\mathcal{EL^{++}}$ normal forms; however, axioms beyond $\mathcal{EL}^{++}$ are ignored by these methods. Regarding the expressivity of ontologies, ORE1, GO and PPI are expressed in $\mathcal{EL}^{++}$ whereas FoodOn is expressed in $\mathcal{ALC}$. We have extended Table 1 to include this information. In the case of FoodOn, initially we used the $\mathcal{EL}^{++}$ version for baselines and $\mathcal{ALC}$ version for our method. We have extended Table 3 with the evaluation of our method in the $\mathcal{EL}^{++}$ version of FoodOn.} ------ * In Line 2, Page 6, "correspond" --> "corresponds". ** Response: Thank you. We fixed this typo. ------ ---------------------------------------------------- Response to Reviewer 2 The paper is interesting and the results overall show the benefit of the approach. The paper heavily leverages the work by C. Le Duc (reference 27, in this paper), which, in part, limits the theoretical novelties of this paper. That said, the numerical results are convincing and the code has been made available. ------ * I would recommend to split the long introduction by introducing a subsection towards the end where the authors describe the proposed approach and the benefits. That helps the readers to jump to the main part of in case they are familiar with the related work. ** Response: We have followed your suggestion and added a subsection at the end of the introduction to describe our work and contributions. ------ * At some level, there is a bit of a question if one should consider the approach category theory based, given that in the end the authors are really using the algebraic structure of a lattice. Sure, a poset is a category, but at some levels the use of this very general terminology in this specific setting might be counter productive, i.e. "scare" possible readers away. I am not suggesting the authors to make changes, just making a general comment. ** Response: Thank you for your comment. We acknoledwege that even though the foundation of our method is Category Theory (CT), we are using quite simple elements from CT. However, we also tried to highlight that we construct a lattice that serves as our central structure to embed.} ------ * Theorem 1 is trivial and does not really warrant a proof and could be condensed to a "Follows by checking the properties of a poset." ** Response: We agree and have changed it to a Lemma and moved it to Appendix ------ * Some minor suggestions: Page 5 line 34, why do you put "saturation" in quotes? It has been used in other previous parts without quotation. Page 5 line 36, 'inmediate' -> 'immediate' Page 6 line 4, 'per morphism' or 'for each morphism' Page 6 line 4 , 'Due to the large space complexity of Equation 5' -> 'Due to the large space complexity required when implementing Equation 5'. ** Response: Thank you, we have made the corresponding fixes in the manuscript.