Resubmission of "Understanding the Expressive Capabilities of Knowledge Base Embeddings under Box Semantics" (NeSy 25) - Special Issue on NeSy 2025 Extended Papers - Mena Leemhuis & Oliver Kutz Dear editors, dear reviewers, This manuscript presents a significantly extended version of our paper "Understanding the Expressive Capabilities of Knowledge Base Embeddings under Box Semantics" presented at the 19th Conference on Neurosymbolic Learning and Reasoning (NeSy 2025) in Santa Cruz, California. Especially, we addressed the reviewers comments. The changes and additions comprise in detail the following: - We extended the preliminaries by adding a discussion on related work, both for ontology embeddings in general and for geometric ontology embeddings specifically. - In this context, also the box embeddings have been introduced in more depth, especially, a new definition (Definition 1) has been added to provide formal foundations and appropriate generalizations. There, the general principle of box embedding methods is stated by determining basic aspects that make up the basic building block of box embeddings. The box interpretation of Definition 22 can then be considered as a special case of this embedding, namely as one case that shows a 'Tarskian-style' semantics. - This 'Tarskian-style semantics' is discussed in detail in the new section "Towards Trustworthy and Interpretable Box Embeddings". There, we extend the discussion on the exact semantics of box embeddings started by Bourgaux et al. (KR 2024) and in our NeSy 2025 paper. With the help of the general embedding method defined in Definition 1, we add several new definitions on what an embedding method needs to fulfill to exhibit what we call a "Pseudo-Tarski" semantics. We prove this property in Theorem 17. This discussion and the two subsections of this new section on completeness and faithfulness detail the lack of intuitive semantics of existing KBE-approaches, the missing completeness and faithfulness of these approaches and the implications these restrictions have to real-world use cases. Though, this section uses KBE and box embeddings as motivation, its results are generally applicable to other types of embeddings resp. embedding methods. We point out the importance of discussing the semantics of the embedding approaches in general and offer ways to measure the degree of 'classicality' of an approach. - In general, we strengthened the connection to real-world use cases by adding several examples, e.g., Examples 29 and 30 on real-world use cases interfering with Helly's property and Examples 31, 44 and 49 discussing the influence on Helly's property on ontologies without disjointness axioms or with empty Aboxes, scenarios that are occurring often in real-world ontologies. - To further strengthen this connection, we added a new section on the faithfulness of the embeddings and how this faithfulness is affected by Helly's property. Faithfulness refers to the ability of modeling not only a correct embedding but an embedding that is based on data regularities and not based on geometric restrictions of the embedding method. In this context, we showed in which cases Helly's property hinders a faithful representation (see Proposition 47). - Additionally, we expanded the section on Helly-satisfiability by introducing several examples, e.g., for the construction of the Helly-Abox closure (Example 38) and the Helly-companion (Example 40). In this context, we formalized several observations that we made in our NeSy conference paper here in full detail, e.g., regarding the insufficiency of the Abox-closure rule for showing Helly-satisfiability (Corollary 37, Proposition 38).