Original manuscript tracking number: 854-1860. We would like to thank all reviewers and editors for their valuable feedback and insightful comments which has strengthen our work. We appreciate the possibility to revise and resubmit our manuscript 'DELE: Deductive EL++ Embeddings for Knowledge Base Completion', an extended version of NeSy 2024 conference paper 'Enhancing Geometric Ontology Embeddings for EL++ with Negative Sampling and Deductive Closure Filtering'. See below a point-by-point response to all reviewers' comments. The file 'dele_highlighted.pdf' contains the revised version of the manuscript with changes highlighted in blue. The file 'dele.pdf' contains the same revised version yet without highlighted text. ======================= Review # 2 ================================ Reviewer 2: 1) A minor note -- in page 13, line 18, it should be "decrease" and not "dicrease". Response: Thank you, we fixed this typo. ======================= Review # 3 ================================ Reviewer 3: In this new version, the authors have improved their paper in various ways taking into account the comments of the reviewers. It is appreciated that the authors expand on how they define the deductive closure in a way that it is finite. I believe the intension is to say that the deductive closure is finite because it is restricted to normalised axioms. Though, the writing of this part is still confusing. The recommendation is the following: 1) When you define EL ++ (before normalization), you have "concept descriptions are constructed with the grammar ⊥|⊤|A⊓B|∃r.A| {a}". Here the recommendation is that you (i) replace A and B by C,D and (ii) say that C,D are EL ++ concept descriptions and r is a role name. The reason for (i) is that you used A to denote a concept name and, in the description logic literature, it is usual to write A,B for concept names and C,D for complex concept expressions (see e.g. the book an Introduction to Description Logic page 10 definition 2.1 and then page 13 definition 2.2). The reason for (ii) is that you need to define from where the symbols in the grammar are coming from, so that one can see this is an inductive definition. Response: As suggested, we replaced concept description A, B by C, D in Section 2.1 'Description Logic EL++' while defining EL++. We also added the comment explaining that C, D are EL++ concept descriptions and r is a role name. Following the logic of this comment we changed some other notations in Section 2.1 to make sure that C or D are reserved for concept descriptions and A or B represent concept names. To further improve clarity of the text we replaced every occurence of 'relation' by 'role'. Reviewer 3: 2) When you talk about normalized axioms in Table 1, use A,B instead of C,D since C,D usually represents complex concepts. Response: We replaced every occurences of C and D in Table 1 by A and B as suggested. Reviewer 3: 3) Still about Table 1, make clear that A,B are indeed concept names and r, r1, r2, s are role names. Response: As suggested, we added clarifications of notations used in Table 1 to the table caption. Reviewer 3: 4) Why use both R and r in Table 1? Response: To make Table 1 easier to read and understand we decided unify all notations involving role names (descriptions of GCI2, GCI3, GCI3-BOT, RI0 and RI1) and used only r, r_1, r_2 and s to represent role names. We also replaced every occurence of 'R' to 'r' and 'S' to 's' related to roles in formulas. Since we used previously 'R' to represent roles in Eq. 1-18 and 'r' to represent a corresponding parameter of embedding functions in geometric models, we now use 'r' for role names and 'r_0' for corresponding parameters of embedding functions. Reviewer 3: 5) In section 2.3 we have e.g. "For example, all entailed axioms of type C ⊑ D will be a subset of the set of all possible axioms of GCI0 type having cardinality |C|2 where |C| is the cardinality of the set of all concept names." By changing the notation in Table 1 you would also need to change here and check elsewhere. Response: Since we work with normalized EL++ theories we changed all occurences of C and D concept names to A and B as suggested, to avoid confusion between complex concept descriptions and concept names (changes are made in sections 2.1, 2.3, 4, 4.1-4.3, 5, 6.2-6.4, 7, Appendix E, F). Reviewer 3: 6) Regarding related to work when you say "Its extension, FaithEL [22], interprets concepts and relations as subsets of unitary hypercubes. Although these geometric methods construct a “canonical” model of an ontology, they are not intended to predict new knowledge, which however is useful in real-world applications: knowledge bases may be incomplete, and ontology embedding methods designed for knowledge base completion would be able to find a tradeoff between the prediction of entailed and novel knowledge. " This is not totally correct. The idea of FaithEL [22] is to allow to predict new knowledge but only new assertions, not new TBox axioms. Not allowing new TBox axioms can be useful for some real world applications. Consider for example that you want to create an embedding of your TBox and ABox and consider that, in your data, every instance of a bird is also an instance of something that can fly. Then, it could be that an embedding model satisfies the concept inclusion Bird \subseteq CanFly. However, if this is not something that follows from your TBox (e.g. penguins are birds but do not fly) and your embedding method created strongly TBox faithful models then this wrong inclusion is avoided. Please change the explanation of this reference to clarify that it does allow the prediction of new knowledge, as long as it is consistent with the TBox part of the knowledge base. Response: Thank you. We clarified the description of FaithEL method in Section 3.2 'Geometric-Based Ontology Embeddings' as suggested (highlighting that it can predict new assertions consistent with the TBox part). Sincerely, The authors