BoxLitE: A Faithful Knowledge Base Embedding Based on Convex Optimization

# BoxLitE: A Faithful Knowledge Base Embedding Based on Convex Optimization
Source: [https://arxiv.org/html/2605.23937](https://arxiv.org/html/2605.23937)
Hesham Morgan2Ana Ozaki3Aleksandar Pavlović4Emanuel Sallinger2 \\affiliations1Institute of Statistical Mathematics, Japan 2TU Wien, Austria 3University of Oslo, Norway 4University of Applied Sciences Campus Vienna, Austria \\emailsbruno@ism\.ac\.jp, hesham\.morgan@tuwien\.ac\.at, anaoz@uio\.no, aleksandar\.pavlovic@hcw\.ac\.at, sallinger@dbai\.tuwien\.ac\.at

###### Abstract

Knowledge base \(KB\) embeddings aim at combining the capability of classical knowledge graph embeddings to generalize the information present in facts, the ABox, with conceptual knowledge represented in an ontology language, the TBox\. Several authors have recently explored the idea of mapping concepts to*convex regions*in a vector space\. This is useful to represent hierarchies, typically present in TBoxes, since more general concepts can be mapped to larger regions, containing those regions associated with more specific concepts\. However, the power of convexity is rarely leveraged during the actual learning tasks\. Here, we introduce BoxLitE, a KB embedding model for DL\-LiteHthat allows for convex optimization\. We show that for any satisfiable DL\-LiteHKB, there is a BoxLitE embedding that is a weakly faithful model\. As a proof of concept, we show how to formulate the KB embedding task as a convex optimization problem and how to obtain embeddings with such desirable faithfulness properties\.

## 1Introduction

Knowledge base \(KB\) embeddings combine the capability of knowledge graph embeddings to perform inductive reasoning for link prediction with deductive reasoning, using logic expressions present in an ontology\(Bourgauxet al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib64)\)\. Several authors have recently explored the idea of mapping concepts in a KB to regions in a vector space\(Gutiérrez\-Basulto and Schockaert,[2018](https://arxiv.org/html/2605.23937#bib.bib44); Pavlović and Sallinger,[2023a](https://arxiv.org/html/2605.23937#bib.bib76); Pavlovićet al\.,[2025](https://arxiv.org/html/2605.23937#bib.bib49); Morganet al\.,[2026](https://arxiv.org/html/2605.23937#bib.bib78)\)\. Region\-based embeddings are important for KBs as they can naturally represent hierarchies: more general concepts can be mapped to larger regions, containing those regions associated with more specific concepts\.

Although concepts are usually mapped to*convex regions*in KBEs, e\.g\., balls, boxes, and cones\(Kulmanovet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib63); Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75); Pavlović and Sallinger,[2023b](https://arxiv.org/html/2605.23937#bib.bib48); Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37)\), the power of convexity is rarely leveraged during the actual learning tasks\. While being convex is not synonymous with “easy”, convexity brings a number of theoretical and practical benefits, in particular: all local minima must be global\. Under convexity, there are a number of efficient algorithms for many classes of convex problems such as linear programs and second\-order cone programs \(SOCPs\)\(Nesterov and Nemirovskii,[1994](https://arxiv.org/html/2605.23937#bib.bib26); Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25)\)\.

For KBs expressed in description logic, most of the literature work on region\-based embeddings focuses on theℰ​ℒ\{\\cal E\\\!L\}ontology language\(Yanget al\.,[2025](https://arxiv.org/html/2605.23937#bib.bib18); Jackermeieret al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib70); Lacerdaet al\.,[2024a](https://arxiv.org/html/2605.23937#bib.bib67); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75); Kulmanovet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib63); Mondalet al\.,[2021](https://arxiv.org/html/2605.23937#bib.bib46); Penget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib51); Lacerdaet al\.,[2024b](https://arxiv.org/html/2605.23937#bib.bib74)\), while some consider𝒜​ℒ​𝒞\\mathcal\{ALC\}\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37); Leemhuiset al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib40)\)\. However, in practice, ontologies present in large\-scale KBs tend to use features in the DL\-LiteHontology language\(Artaleet al\.,[2009](https://arxiv.org/html/2605.23937#bib.bib1)\), due to its simple but versatile expressivity, featuring low computational complexity\(Artaleet al\.,[2009](https://arxiv.org/html/2605.23937#bib.bib1)\)\. Despite its broad use, only a few works investigate logics in the DL\-Lite family in a KB embedding context\(Liet al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib73); Imeneset al\.,[2023](https://arxiv.org/html/2605.23937#bib.bib41); Bourgauxet al\.,[2021](https://arxiv.org/html/2605.23937#bib.bib47)\)and none provide a region\-based embedding implementation\.

In this paper, we introduce BoxLitE, a KB embedding model for DL\-LiteHontologies that allows for convex optimization\. In particular, we study the notion ofweakly faithful models\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37); Bourgauxet al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib64)\)in the optimization task\. This notion states that axioms that hold in the embedding are*consistent*with the KB and*entailments of the KB are satisfied*in the embedding\. In detail,

- •we select DL\-LiteHanontology languagethat allows to exploit theadvantages of convex optimization;
- •wedesign, implement, and evaluateBoxLitE, a KB embedding approach that allows for convex optimization;
- •we prove that every satisfiable DL\-LiteHhas a BoxLitE embedding that is aweakly faithful model;
- •we introduce a novel and efficient form ofnegative sampling based on existential concepts;
- •in contrast to commonly used unconstrained optimization approaches, our BoxLitE approach can enforce theTBox axioms using convex constraintswhile leaving the ABox terms in the objective function\.

Main Contribution\.Our main contribution lies on theoretical grounds, establishing the existence of a faithful model for DL\-LiteHand the proposal of a novel approach that computes such a model\. BoxLitE’s implementation and the empirical results serve as a proof of concept that our theoretical results translate into practical settings\. One of the motivations for this work is to try to solve link prediction using theoretically sounds techniques from a convex optimization perspective\. We would like to check how much performance can we get from a purely convex approach\. In this way, our work stands in contrast to previous approaches that relied on nonconvexity, yielding more complicated optimization problems\.

There are three primary sources of nonconvexity in contemporary KB embedding models \(see more details in Section[7](https://arxiv.org/html/2605.23937#S7)\): the way*negative sampling*is used by most works\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42); Sunet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib34); Yanget al\.,[2015](https://arxiv.org/html/2605.23937#bib.bib55); Trouillonet al\.,[2016](https://arxiv.org/html/2605.23937#bib.bib43); Kazemi and Poole,[2018](https://arxiv.org/html/2605.23937#bib.bib56); Balazevicet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib57); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75); Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Pavlović and Sallinger,[2024b](https://arxiv.org/html/2605.23937#bib.bib50)\); the*design of loss terms*involving operations that do not preserve convexity; and the choice of the*ontology language*, in particular, languages that allow conjunction on the left\-hand side of concept inclusions, such asℰ​ℒ\{\\cal E\\\!L\}and𝒜​ℒ​𝒞\\mathcal\{ALC\}, are likely to lead to nonconvexity\. This motivated us to consider DL\-LiteH\(Artaleet al\.,[2009](https://arxiv.org/html/2605.23937#bib.bib1)\), which is a simple but practically useful and well\-known ontology language without conjunction on the left\-hand side\.

On the algorithmic side, ADAM\(Kingma and Ba,[2015](https://arxiv.org/html/2605.23937#bib.bib12)\), the method of choice of several works, is often used in theoretically unsound ways\. For example, ADAM requires the objective function to be differentiable; however, it is not uncommon to see this requirement being ignored\. For instance, in\(Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75), Sections 4\.3, 4\.5\)and\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21), Section 4 and Appendix F\.2\), the authors describe nondifferentiable objective functions that are optimized via ADAM111In\(Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\), for example, loss terms for concepts assertions are expressed using the 2\-norm and the regularization term considered therein use a maximum of functions\. Both correspond to terms that are not differentiable in general, e\.g\., the 2\-norm function is not differentiable at the origin, which can be checked directly by the definition of Fréchet differentiability\.\. The rationale for that is not explained in the papers, but a reasonable guess seems to be that these functions are typically differentiable almost everywhere \(in a measure theory sense\), so there may be an underlying belief that iterates are unlikely to reach a point of nondifferentiability and the algorithm may still work in practice\. Besides, widely used tools such as PyTorch may attempt to differentiate nondifferentiable functions for the user by, e\.g\., selecting subgradients/supgradients when available\(PyTorch,[2026](https://arxiv.org/html/2605.23937#bib.bib65)\)\. Unfortunately, there are well\-known examples in the optimization literature showing that when a gradient method is applied directly to a nondifferentiable function, it may fail to find an optimal solution, even if the function is differentiable at the points generated by the method, e\.g\., see\(Beck,[2017](https://arxiv.org/html/2605.23937#bib.bib16), Section 8\.1\.2\)\. In the optimization community, there are works that explore the convergence properties of simple SGD methods when applied to nondifferentiable functions\(Bolte and Pauwels,[2020](https://arxiv.org/html/2605.23937#bib.bib71); Daviset al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib72)\), but, as far as we know, similar results have not been proved for ADAM\.

When performance is good, one may be tempted to overlook these issues, but when it is not, it may be hard to know what is to blame\. Is it the model, the parameter choice, the optimization algorithm, or a combination of those? Our proposed approach*does*include nondifferentiable terms as well, but, in contrast, our method of choice for solving the underlying optimization problem is suitable for handling nondifferentiability \(Section[7](https://arxiv.org/html/2605.23937#S7)\)\. In this sense, our approach is conceptually sound from an optimization point of view\.

Organization\. Our paper is organized as follows\. Section[2](https://arxiv.org/html/2605.23937#S2)provides basic definitions\. Section[3](https://arxiv.org/html/2605.23937#S3)defines the semantics of our BoxLitE approach\. Section[4](https://arxiv.org/html/2605.23937#S4)studies BoxLitE’s faithfulness properties\. Section[5](https://arxiv.org/html/2605.23937#S5)formulates BoxLitE’s convex optimization problem and shows faithfulness properties that are ensured by the problem formulation\. Section[6](https://arxiv.org/html/2605.23937#S6)discusses BoxLitE’s proof of concept implementation and experiments\. Section[8](https://arxiv.org/html/2605.23937#S8)provides a discussion on optimization in KB embeddings and concludes our paper\. Omitted proofs and additional details are given in the supplemental material\.

## 2Basic Definitions: DL\-Lite Ontologies

Let𝖭𝖢\{\\sf N\_\{C\}\},𝖭𝖱\{\\sf N\_\{R\}\}, and𝖭𝖨\{\\sf N\_\{I\}\}be*finite*, non\-empty, mutually disjoint sets of*concept*,*role*, and*individual*names, respectively\. We denote by𝖭𝖱−\{\\sf N\_\{R\}^\{\-\}\}the set𝖭𝖱∪\{R−∣R∈𝖭𝖱\}\{\\sf N\_\{R\}\}\\cup\\\{R^\{\-\}\\mid R\\in\{\\sf N\_\{R\}\}\\\}, by𝖭𝖢∃\{\\sf N\_\{C\}^\{\\exists\}\}the set𝖭𝖢∪\{∃R∣R∈𝖭𝖱−\}\{\\sf N\_\{C\}\}\\cup\\\{\\exists R\\mid R\\in\{\\sf N\_\{R\}^\{\-\}\}\\\}, and by𝖭𝖢​¬∃\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\neg\}the set𝖭𝖢∃∪\{¬E∣E∈𝖭𝖢∃\}\{\\sf N\_\{C\}^\{\\exists\}\}\\cup\\\{\\neg E\\mid E\\in\{\\sf N\_\{C\}^\{\\exists\}\}\\\}\. DL\-LiteHrole and concept inclusions are of the formS⊑TS\\sqsubseteq TandB⊑CB\\sqsubseteq C, resp\., whereS,T∈𝖭𝖱−S,T\\in\{\\sf N\_\{R\}^\{\-\}\}are roles222A*role*is a role name or the inverse of a role name\.andBB,CCare concepts built as follows:S::=R∣R−,B::=A∣∃S,C::=B∣¬B,S::=R\\mid R^\{\-\},\\ B::=A\\mid\\exists S,\\ C::=B\\mid\\neg B,withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}\. A DL\-LiteH*TBox*\(we may also use the less technical term*ontology*\) is a \(finite\) set of DL\-LiteHconcept and role inclusions\.*Assertions*are of the formD​\(a\)D\(a\)\(called*concept assertions*\) orR​\(a,b\)R\(a,b\)\(called*role assertions*\), whereD∈𝖭𝖢∃D\\in\{\\sf N\_\{C\}^\{\\exists\}\},R∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}, anda,b∈𝖭𝖨a,b\\in\{\\sf N\_\{I\}\}\. A DL\-LiteH*knowledge base*\(KB\) is a pair\(𝒯,𝒜\)\(\\mathcal\{T\},\\mathcal\{A\}\)where𝒯\\mathcal\{T\}is a DL\-LiteHTBox and𝒜\\mathcal\{A\}is a \(finite\) set of assertions, called*ABox*\. Following the KBE literature \(e\.g\.\(Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\)\) and to simplify our presentation, we assume that DL\-LiteHTBoxes are in*named form*, meaning that they contain only concept inclusions where one of the concepts is a concept name\. The semantics is given as usual by interpretationsℐ=\(Δℐ,⋅ℐ\)\\mathcal\{I\}=\(\\Delta^\{\\mathcal\{I\}\},\\cdot^\{\\mathcal\{I\}\}\)\(see supp\. material\)\. We call a DL\-LiteH*axiom*an expression that is a role inclusion \(RI\), a concept inclusion \(CI\), or an assertion\. We writeℐ⊧α\\mathcal\{I\}\\models\\alphaifℐ\\mathcal\{I\}satisfies an axiomα\\alpha\. An interpretationℐ\\mathcal\{I\}satisfies a KB𝒦=\(𝒯,𝒜\)\\mathcal\{K\}=\(\\mathcal\{T\},\\mathcal\{A\}\), writtenℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}, if it satisfies the axioms in𝒯\\mathcal\{T\}and𝒜\\mathcal\{A\}\. We say that𝒦\\mathcal\{K\}is*satisfiable*if such interpretationℐ\\mathcal\{I\}exists\. Also,𝒦\\mathcal\{K\}*entails*an axiomα\\alpha, written𝒦⊧α\\mathcal\{K\}\\models\\alpha, iff, for all interpretationsℐ\\mathcal\{I\}, ifℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}thenℐ⊧α\\mathcal\{I\}\\models\\alpha\. We say that an axiomα\\alphais*consistent*with a KB𝒦\\mathcal\{K\}if there is an interpretationℐ\\mathcal\{I\}such thatℐ⊧𝒦∪\{α\}\\mathcal\{I\}\\models\\mathcal\{K\}\\cup\\\{\\alpha\\\}\.

Canonical Model\.Our construction is based on previous definitions of canonical models for DL\-LiteH\(e\.g\.,\(Kontchakovet al\.,[2010](https://arxiv.org/html/2605.23937#bib.bib17)\)\)\. Though here, we ensure that*inclusions*hold in the model*only if*they are entailed by the ontology\. Before we provide the definition of the canonical model, we introduce the following notions\. Assume𝒦=\(𝒯,𝒜\)\\mathcal\{K\}=\(\\mathcal\{T\},\\mathcal\{A\}\)is a satisfiable DL\-LiteHKB\. We say that a conceptCCis*satisfiable w\.r\.t\.𝒦\\mathcal\{K\}*if there is an interpretationℐ\\mathcal\{I\}such thatℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}andCℐ≠∅C^\{\\mathcal\{I\}\}\\neq\\emptyset\. We write concepts of the formB⊓CB\\sqcap C\(with the*conjunction*operator\) only to falsify concept inclusions of the formB⊑¬CB\\sqsubseteq\\neg C\. In an interpretationℐ\\mathcal\{I\}, the meaning of\(B⊓C\)ℐ\(B\\sqcap C\)^\{\\mathcal\{I\}\}isBℐ∩CℐB^\{\\mathcal\{I\}\}\\cap C^\{\\mathcal\{I\}\}\. Let𝖭𝖢⊓∃\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\sqcap\}be the set𝖭𝖢∃∪\{D⊓E∣D,E∈𝖭𝖢∃\}\{\\sf N\_\{C\}^\{\\exists\}\}\\cup\\\{D\\sqcap E\\mid D,E\\in\{\\sf N\_\{C\}^\{\\exists\}\}\\\}\. AssumeΔ𝒦:=\{cD∣D∈𝖭𝖢⊓∃,D​is satisfiable w\.r\.t\.​𝒦\}\\Delta\_\{\\mathcal\{K\}\}:=\\\{c\_\{D\}\\mid D\\in\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\sqcap\},\\ D\\text\{ is satisfiable w\.r\.t\. \}\\mathcal\{K\}\\\}is disjoint from𝖭𝖨\{\\sf N\_\{I\}\}\. Given a roleSS, we writeS¯\\overline\{S\}as the result of switching between a role name and its inverse:S¯=S−\\overline\{S\}=S^\{\-\}ifS∈𝖭𝖱S\\in\{\\sf N\_\{R\}\}andS¯=R\\overline\{S\}=RifS=R−S=R^\{\-\}withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}\.

###### Definition 1\(Canonical Model\)\.

The canonical modelℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}for a satisfiable DL\-LiteHKB𝒦=\(𝒯,𝒜\)\\mathcal\{K\}=\(\\mathcal\{T\},\\mathcal\{A\}\)is:

- •Δℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\},aℐ𝒦:=aa^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=a, for alla∈𝖭𝖨a\\in\{\\sf N\_\{I\}\},
- •Aℐ𝒦:=\{a∈𝖭𝖨∣𝒦⊧A​\(a\)\}∪\{cD∈Δ𝒦∣𝒦⊧D⊑A\}A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\\\{a\\in\{\\sf N\_\{I\}\}\\mid\\mathcal\{K\}\\models A\(a\)\\\}\\cup\\\\ \\\{c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}\\mid\\mathcal\{K\}\\models D\\sqsubseteq A\\\}for allA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\},
- •Rℐ𝒦:=\{\(a,b\)∈𝖭𝖨×𝖭𝖨∣𝒦⊧R​\(a,b\)\}∪\{\(a,c∃S\)∈𝖭𝖨×Δ𝒦∣𝒦⊧∃S¯​\(a\),𝒦⊧S¯⊑R\}∪\{\(c∃S,a\)∈Δ𝒦×𝖭𝖨∣𝒦⊧∃S¯​\(a\),𝒦⊧S⊑R\}∪\{\(c∃S,c∃S¯\)∈Δ𝒦×Δ𝒦∣𝒦⊧S⊑R\}∪\{\(cD,c∃S\)∈Δ𝒦×Δ𝒦∣𝒦⊧D⊑∃S¯,𝒦⊧S¯⊑R\}∪\{\(c∃S,cD\)∈Δ𝒦×Δ𝒦∣𝒦⊧D⊑∃S¯,𝒦⊧S⊑R\}R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\\\{\(a,b\)\\in\{\\sf N\_\{I\}\}\\times\{\\sf N\_\{I\}\}\\mid\\mathcal\{K\}\\models R\(a,b\)\\\}\\cup\\\\ \\\{\(a,c\_\{\\exists S\}\)\\in\{\\sf N\_\{I\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\),\\ \\mathcal\{K\}\\models\\overline\{S\}\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},a\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\{\\sf N\_\{I\}\}\\mid\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\),\\ \\mathcal\{K\}\\models S\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},c\_\{\\exists\\overline\{S\}\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\mathcal\{K\}\\models S\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{D\},c\_\{\\exists S\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\{\\mid\}\\mathcal\{K\}\\models D\\sqsubseteq\\exists\\overline\{S\},\\ \\mathcal\{K\}\\models\\overline\{S\}\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},c\_\{D\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\mathcal\{K\}\\models D\\sqsubseteq\\exists\\overline\{S\},\\ \\mathcal\{K\}\\models S\\sqsubseteq R\\\}, for allR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}\.

###### Theorem 1\.

Let𝒦\\mathcal\{K\}be a satisfiable DL\-LiteHKB and letℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}be the canonical model of𝒦\\mathcal\{K\}\. Then, for all DL\-LiteHaxiomsα\\alpha, we have thatℐ𝒦⊧α\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\alphaiff𝒦⊧α\\mathcal\{K\}\\models\\alpha\.

## 3BoxLitE Semantics

Here we introduce a semantics for DL\-LiteHinspired by box\-based geometric models\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21)\)and suitable for defining a KB embedding model that can be optimized using convex optimization\. We define a geometric model that uses axis\-aligned hyper\-rectangles, called*boxes*\.

Letϵ\\epsilonandsΩ\{s\_\{\\Omega\}\}be fixed but arbitrary333In our work and, in particular, in the implementation, we take0<ϵ<ϵ𝑚𝑎𝑥0<\\epsilon<\\epsilon\_\{\\mathit\{max\}\}, whereϵ𝑚𝑎𝑥=0\.5\\epsilon\_\{\\mathit\{max\}\}=0\.5andϵ𝑚𝑎𝑥≤sΩ/8\\epsilon\_\{\\mathit\{max\}\}\\leq\{s\_\{\\Omega\}\}/8\. Makingϵ\\epsilonandsΩ\{s\_\{\\Omega\}\}learnable parameters would not enhance the representation capabilities of our model but allow for infinitely many equivalent solutions of our learned embeddings which only differ in scale\.positive constants withϵ≤sΩ\\epsilon\\leq\{s\_\{\\Omega\}\}\. Letϵ\\boldsymbol\{\\epsilon\}and𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}denote thedd\-dimensional vectors whose value in each dimension is equal toϵ\\epsilonandsΩ\{s\_\{\\Omega\}\}respectively\. Henceforth, we denote elementwise comparison operators with≤d\\mathrel\{\{\\leq\_\{d\}\}\}and≥d\\mathrel\{\{\\geq\_\{d\}\}\}\. Usingϵ\\boldsymbol\{\\epsilon\}and𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}, we define an axis\-aligned hyper\-rectangle, called the*universe box*:

Ω=\{𝐱∈ℝd∣−𝐬𝛀≤d𝐱≤d𝐬𝛀\}\{\\Omega\}=\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\-\{\\mathbf\{s\_\{\\Omega\}\}\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\{\\mathbf\{s\_\{\\Omega\}\}\}\\\}in ourdd\-dimensional Euclidean space, whered\>0d\>0\. The universe box is useful to establish basic properties of boxes \(Theorem[2](https://arxiv.org/html/2605.23937#Thmtheorem2)\) that are needed in our faithfulness proofs\.

###### Definition 2\.

A*box*is an element of the set𝖡𝗈𝗑\{\\sf Box\}, defined as:

𝖡𝗈𝗑≔\{\\displaystyle\{\\sf Box\}\\coloneqq\\\{\{𝐱∈ℝd∣𝐋\+ϵ≤d𝐱≤d𝐔−ϵ\}∣\\displaystyle\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\\mathbf\{L\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\}\-\\boldsymbol\{\\epsilon\}\\\}\\mid0≤d\(𝐔−𝐋\)≤d2𝐬𝛀,𝐋,𝐔∈ℝd\}\.\\displaystyle\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\(\\mathbf\{U\}\-\\mathbf\{L\}\)\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\},\\ \\mathbf\{L\},\\mathbf\{U\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}\.For a non\-empty box, the*lower*and*upper*bounds are denoted𝐋\\mathbf\{L\}and𝐔\\mathbf\{U\}\(resp\.\) of𝐗\\mathbf\{X\}\. If𝐗\\mathbf\{X\}is the empty box then we set𝐋:=𝐔:=𝟎\\mathbf\{L\}:=\\mathbf\{U\}:=\\mathbf\{0\}\. We denote with𝐱​\[i\]\\mathbf\{x\}\[i\]theii\-th dimension of a vector𝐱\\mathbf\{x\}\. If𝐗\\mathbf\{X\}is a box with lower and upper bounds𝐋\\mathbf\{L\}and𝐔\\mathbf\{U\}, resp\., then𝐗​\[i\]\\mathbf\{X\}\[i\]is the pair\(𝐋​\[i\],𝐔​\[i\]\)\(\\mathbf\{L\}\[i\],\\mathbf\{U\}\[i\]\)\.

Box Definition Intuition\.The definition of𝖡𝗈𝗑\{\\sf Box\}allows boxes to have a width of≤2​𝐬𝛀\\leq 2\{\\mathbf\{s\_\{\\Omega\}\}\}and to span outside of the universeΩ\{\\Omega\}, while we require vectors associated with individual names to be withinΩ\{\\Omega\}\(as we will see in Definition[3](https://arxiv.org/html/2605.23937#Thmdefinition3)\)\. This design is motivated \(in the spirit ofAbboudet al\.\([2020](https://arxiv.org/html/2605.23937#bib.bib21)\)\) by associating any individual name with a position and a bump vector, where the embeddings of a pair of individual names\(a,b\)\(a,b\)is retrieved by translating \(“*bumping*”\) the position ofaawith the bump ofbband vice versa\. Thus, if both the position and bump vectors are within the universe box, i\.e\., bounded by𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}, then the embeddings of any pair of individuals is bounded by2​𝐬𝛀2\{\\mathbf\{s\_\{\\Omega\}\}\}\. As we associate any concept and role name with a set of boxes that shall contain the embeddings of individual tuples, it is sufficient that the widths of boxes are bounded by2​𝐬𝛀2\{\\mathbf\{s\_\{\\Omega\}\}\}\. The intuition for*ϵ\\boldsymbol\{\\epsilon\}*is that most convex optimization tools do not allow strict inequalities or handle them less efficiently\. However, we need to be able to define empty boxes for concepts/roles that are unsatisfiable\. So we employ in[Definition2](https://arxiv.org/html/2605.23937#Thmdefinition2)*nonstrict inequalities*and add a small positive*ϵ\\boldsymbol\{\\epsilon\}*to emulate strict ones\.

###### Definition 3\.

A*box interpretation*η\\etais a function that maps:

- •each individual namea∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}to two vectorsη​\(a\)=\(𝗉𝗈𝗌​\(a\),𝖻𝗎𝗆𝗉​\(a\)\)\\eta\(a\)=\(\{\{\\sf pos\}\(a\)\},\{\{\\sf bump\}\(a\)\}\), namely, a position𝗉𝗈𝗌​\(e\)∈Ω\{\{\\sf pos\}\(e\)\}\\in\{\\Omega\}and a bump𝖻𝗎𝗆𝗉​\(e\)∈Ω\{\{\\sf bump\}\(e\)\}\\in\{\\Omega\};
- •each concept nameA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}to a boxη​\(A\)∈𝖡𝗈𝗑\{\{\\eta\}\(A\)\}\\in\{\\sf Box\};
- •each role nameR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}to three boxesη\(R\)=\(𝖧𝖾𝖺𝖽\(R\),𝖳𝖺𝗂𝗅\(R\),𝖡𝗎𝗆𝗉\(R\)\{\{\\eta\}\(R\)\}=\(\{\{\\sf Head\}\(R\)\},\{\{\\sf Tail\}\(R\)\},\{\{\\sf Bump\}\(R\)\}\), which we callRR’s head𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\}, tail𝖳𝖺𝗂𝗅​\(R\)\{\{\\sf Tail\}\(R\)\}and bump box𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}\.

We extend the mapping functionη\\etato arbitrary DL\-LiteHconcept and role expressions as follows:

η​\(¬C\)\\displaystyle\{\{\\eta\}\(\\neg\{C\}\)\}≔η​\(C\)¯,η​\(R−\)≔\(𝖳𝖺𝗂𝗅​\(R\),𝖧𝖾𝖺𝖽​\(R\),𝖡𝗎𝗆𝗉​\(R\)\),\\displaystyle\\coloneqq\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\},\\quad\{\{\\eta\}\(R^\{\-\}\)\}\\coloneqq\(\{\{\\sf Tail\}\(R\)\},\{\{\\sf Head\}\(R\)\},\{\{\\sf Bump\}\(R\)\}\),η​\(∃R\)\\displaystyle\{\{\\eta\}\(\\exists R\)\}≔\{𝐱∈ℝd∣𝐋𝐑𝐇−𝐔𝐑𝐁\+ϵ≤d𝐱≤d𝐔𝐑𝐇−𝐋𝐑𝐁−ϵ\}\\displaystyle\\coloneqq\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\\mathbf\{L^\{H\}\_\{R\}\}\-\\mathbf\{U^\{B\}\_\{R\}\}\+\\boldsymbol\{\\epsilon\}\{\\mathrel\{\{\\leq\_\{d\}\}\}\}\\mathbf\{x\}\{\\mathrel\{\{\\leq\_\{d\}\}\}\}\\mathbf\{U^\{H\}\_\{R\}\}\-\\mathbf\{L^\{B\}\_\{R\}\}\-\\boldsymbol\{\\epsilon\}\\\}where𝐋𝐑X\\mathbf\{L\}^\{X\}\_\{\\mathbf\{R\}\},𝐔𝐑X\\mathbf\{U\}^\{X\}\_\{\\mathbf\{R\}\}withX∈\{𝐇,𝐓,𝐁\}X\\in\\\{\\mathbf\{H\},\\mathbf\{T\},\\mathbf\{B\}\\\}are the lower and upper bounds of𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\},𝖳𝖺𝗂𝗅​\(R\)\{\{\\sf Tail\}\(R\)\}, and𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}; and whereη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}representsη​\(C\)\{\{\\eta\}\(C\)\}’s complement box\. Furthermore, for anyC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, the complement boxη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}is defined as follows:

η​\(C\)¯≔\{𝐱∈ℝd∣\(−𝐬𝛀−𝐋𝐂\+ϵ\)≤d𝐱≤d\(𝐬𝛀−𝐔𝐂−ϵ\)\}\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\\coloneqq\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}\+\\boldsymbol\{\\epsilon\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\-\\boldsymbol\{\\epsilon\}\)\\\}with𝐋𝐂\\mathbf\{L\_\{C\}\}and𝐔𝐂\\mathbf\{U\_\{C\}\}beingη​\(C\)\{\{\\eta\}\(C\)\}’s lower and upper bounds\.

Box Interpretation Intuition\.At first sight, an intuitive definition for the complement of a box would be the set complement\. However, we cannot use this notion as it leads to non\-convexity\. Thus, a different convex\-preserving definition of box complements is required that satisfies important properties of the usual complement, as shown in Theorem[2](https://arxiv.org/html/2605.23937#Thmtheorem2)\. Next, the box interpretation of inverse rolesη​\(R−\)\{\{\\eta\}\(R^\{\-\}\)\}swaps the head and tail boxes ofη​\(R\)\{\{\\eta\}\(R\)\}\. As for the existential boxes, we defineη​\(∃R\)\{\{\\eta\}\(\\exists R\)\}as𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\}enlarged by the boundaries of𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}, to ensure that it contains the embeddings of all subjects ofRRassertions, following∃R\\exists R’s semantics\.

###### Theorem 2\.

For any box interpretationη\{\\eta\}:

1. i\)for allC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\},η​\(C\)¯∈𝖡𝗈𝗑\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\\in\{\\sf Box\};
2. ii\)for allC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\},η​\(C\)¯¯=η​\(C\)\{\\overline\{\{\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\}\}\}=\{\{\\eta\}\(C\)\};
3. iii\)for allC,D∈𝖭𝖢∃C,D\\in\{\\sf N\_\{C\}^\{\\exists\}\}, ifη​\(C\)⊆η​\(D\)\{\{\\eta\}\(C\)\}\\subseteq\{\{\\eta\}\(D\)\}thenη​\(D\)¯⊆η​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}\\subseteq\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\.

Box Consistency\.A box interpretation is*box consistent*if for allC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}we have thatη​\(C\)∩η​\(C\)¯=∅\{\{\\eta\}\(C\)\}\\cap\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}=\\emptyset\.

###### Definition 4\.

A box interpretationη\{\\eta\}satisfies

- •R​\(a,b\)R\(a,b\)withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}anda,b∈𝖭𝖨a,b\\in\{\\sf N\_\{I\}\}iff i\)𝗉𝗈𝗌\(a\)\+𝖻𝗎𝗆𝗉\(b\)\\displaystyle i\)\\;\{\{\\sf pos\}\(a\)\}\+\{\{\\sf bump\}\(b\)\}∈𝖧𝖾𝖺𝖽​\(R\)\\displaystyle\\in\{\{\\sf Head\}\(R\)\}ii\)𝗉𝗈𝗌\(b\)\+𝖻𝗎𝗆𝗉\(a\)\\displaystyle ii\)\\;\{\{\\sf pos\}\(b\)\}\+\{\{\\sf bump\}\(a\)\}∈𝖳𝖺𝗂𝗅​\(R\)\\displaystyle\\in\{\{\\sf Tail\}\(R\)\}iii\)𝖻𝗎𝗆𝗉\(a\),𝖻𝗎𝗆𝗉\(b\)\\displaystyle iii\)\\;\{\{\\sf bump\}\(a\)\},\{\{\\sf bump\}\(b\)\}∈𝖡𝗎𝗆𝗉​\(R\)\\displaystyle\\in\{\{\\sf Bump\}\(R\)\}
- •R⊑SR\\sqsubseteq SwithR,S∈𝖭𝖱−R,S\\in\{\\sf N\_\{R\}^\{\-\}\}iff𝖡𝗎𝗆𝗉​\(R\)⊆𝖡𝗎𝗆𝗉​\(S\)\\;\{\{\\sf Bump\}\(R\)\}\\subseteq\{\{\\sf Bump\}\(S\)\} 𝖧𝖾𝖺𝖽​\(R\)⊆𝖧𝖾𝖺𝖽​\(S\)𝖳𝖺𝗂𝗅​\(R\)⊆𝖳𝖺𝗂𝗅​\(S\)\\displaystyle\{\{\\sf Head\}\(R\)\}\\subseteq\{\{\\sf Head\}\(S\)\}\\qquad\{\{\\sf Tail\}\(R\)\}\\subseteq\{\{\\sf Tail\}\(S\)\}
- •C⊑DC\\sqsubseteq D,C∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}andD∈𝖭𝖢​¬∃D\\in\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\neg\}iffη​\(C\)⊆η​\(D\)\\;\{\{\\eta\}\(C\)\}\\subseteq\{\{\\eta\}\(D\)\}\.

Regarding concept assertions withC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}anda∈𝖭𝖨a\\in\{\\sf N\_\{I\}\},η\\etasatisfiesC​\(a\)C\(a\)if𝗉𝗈𝗌​\(a\)∈η​\(C\)\{\{\\sf pos\}\(a\)\}\\in\{\{\\eta\}\(C\)\}andη\\etafalsifiesC​\(a\)C\(a\)if𝗉𝗈𝗌​\(a\)∈η​\(C\)¯\{\{\\sf pos\}\(a\)\}\\in\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\. Otherwise the status ofC​\(a\)C\(a\)is*unknown*444Geometric models for languages that have negation normally need to allow for the ‘unknown’ truth status\. This happens in the Cone Semantics\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37)\), tailored for𝒜​ℒ​𝒞\\mathcal\{ALC\}, which has negation\. Since DL\-LiteHallows forB⊑¬CB\\sqsubseteq\\neg Cwe use complement boxes and the ‘unknown’ truth state\.\.

We writeη⊧α\{\\eta\}\\models\\alphato indicate thatη\{\\eta\}satisfies an axiomα\\alpha, and writeη⊧̸α\{\\eta\}\\not\\models\\alphaotherwise\. Also, ifη⊧α\{\\eta\}\\models\\alphafor every axiomα\\alphain a KB𝒦\\mathcal\{K\}then we say thatη\{\\eta\}satisfies𝒦\\mathcal\{K\}or, equivalently, thatη\{\\eta\}is a model of𝒦\\mathcal\{K\}, in symbols,η⊧𝒦\{\\eta\}\\models\\mathcal\{K\}\.

The most intricate part of Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4)corresponds to role assertions, so we explain this in more details\. Itemsi\)i\)andii\)ii\)are as inAbboudet al\.\([2020](https://arxiv.org/html/2605.23937#bib.bib21)\)\. Itemiii\)iii\)is helpful to embed existential concepts\. Recall thatη​\(∃R\)\{\{\\eta\}\(\\exists R\)\}is𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\}enlarged by the boundaries of𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}\. Thus, ifR​\(a,b\)R\(a,b\)is satisfied in a box interpretation,𝗉𝗈𝗌​\(a\)\{\{\\sf pos\}\(a\)\}may lie at most the size of𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}away from𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\}\.

With this, we have defined BoxLitE’s semantics in terms of box interpretations\. In contrast to\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21)\), we\(i\)\(i\)define*complement boxes*, which allow the formulation of convex constraints that guarantee the satisfaction of concept disjointness axioms in our embedding solutions and\(i​i\)\(ii\)associate roles with additional*bump*boxes that constrain the maximal bump of an individual that is accepted by a role embedding\. The latter is important to distinguish between axioms of the formR⊑SR\\sqsubseteq Sand\{∃R⊑∃S,∃R−⊑∃S−\}\\\{\\exists R\\sqsubseteq\\exists S,\\exists R^\{\-\}\\sqsubseteq\\exists S^\{\-\}\\\}\. In the next section, we will define faithfulness properties of geometric models\. Furthermore, we will analyse which faithfulness properties for DL\-LiteHcan be satisfied by BoxLitE’s box interpretations\.

## 4Model Faithfulness

In this section, we study model faithfulness\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37)\), which is a property that is useful to show that geometric models correctly represent the knowledge present in KBs\. For KB completion, one is particularly interested in preserving the conceptual knowledge in TBoxes while allowing new assertions to hold\.

###### Definition 5\.

\(Adapted\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37); Bourgauxet al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib64)\)\) Letℒ\\mathcal\{L\}be an ontology language and𝒦\\mathcal\{K\}a satisfiable KB inℒ\\mathcal\{L\}\. A box interpretationη\{\\eta\}is

- •*weakly KB faithful*forℒ\\mathcal\{L\}and𝒦\\mathcal\{K\}if for every KB axiomα\\alphainℒ\\mathcal\{L\},η⊧α\{\\eta\}\\models\\alphaimplies thatα\\alphais consistent with𝒦\\mathcal\{K\};
- •*KB entailed*forℒ\\mathcal\{L\}and𝒦\\mathcal\{K\}if for every KB axiomα\\alphainℒ\\mathcal\{L\}that is entailed by𝒦\\mathcal\{K\},η⊧α\{\\eta\}\\models\\alpha\.

We may omit “weakly” and just say “faithful”\. We may also omitℒ\\mathcal\{L\}and/or𝒦\\mathcal\{K\}if they are clear from the context\. If a boxη\\etainterpretation satisfies both of the properties we say thatη\\etais a KBfaithful model\. These notions can be adapted for the case in whichℒ\\mathcal\{L\}is a description logic with TBox and ABox axioms and for the case where we consider only TBox axioms \(or only ABox axioms\) for TBox \(or ABox\) faithful models\.

[Proposition1](https://arxiv.org/html/2605.23937#Thmproposition1)and[Proposition2](https://arxiv.org/html/2605.23937#Thmproposition2)give basic conditions for faithfulness in DL\-LiteH\.

###### Proposition 1\.

Let𝒯\\mathcal\{T\}be a DL\-LiteHTBox\. Ifη⊧𝒯\\eta\\models\\mathcal\{T\}thenη\\etais \(weakly\) TBox faithful\.

Proposition[1](https://arxiv.org/html/2605.23937#Thmproposition1)follows from the fact that, in DL\-LiteH, TBoxes are always consistent, that is, inconsistencies only happen when considering a TBox and an ABox\.

###### Proposition 2\.

Let𝒦\\mathcal\{K\}be a DL\-LiteHKB andη\\etaa box interpretation that is box consistent\. Ifη⊧𝒦\\eta\\models\\mathcal\{K\}thenη\\etais \(weakly\) KB faithful\.

We are now ready to state[Theorem3](https://arxiv.org/html/2605.23937#Thmtheorem3), which is the main result of this section\.

###### Theorem 3\.

There exists a suitablesΩs\_\{\\Omega\}such that every satisfiable DL\-LiteHKB𝒦\\mathcal\{K\}has a box interpretationηℐ𝒦\\eta\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}that is a KB faithful model and box consistent\.

###### Sketch\.

The proof strategy of[Theorem3](https://arxiv.org/html/2605.23937#Thmtheorem3)consists of first defining a mapping that translates classical finite interpretations into box interpretations\. Then, given a DL\-LiteHKB𝒦\\mathcal\{K\}, we construct the canonical modelℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}in[Definition1](https://arxiv.org/html/2605.23937#Thmdefinition1)and use this mapping to create a box interpretationηℐ𝒦\\eta\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\(usingsΩ=4s\_\{\\Omega\}=4\) that is a KB faithful model of𝒦\\mathcal\{K\}\. In fact, our proof provides a stronger guarantee: the embedding is KB entailed and strongly KB faithful\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37); Bourgauxet al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib64)\)\. ∎

Informally, Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3)’s proof proposes for every satisfiable DL\-LiteHKB𝒦\\mathcal\{K\}an algorithm for constructing a box interpretation \(i\.e\., a BoxLitE embedding\) that is KB faithful and box consistent\. Using this algorithm, we can compute an upper bound for the minimal number of dimensions that a box interpretation requires to satisfy certain faithfulness properties, leading to Corollaries[1](https://arxiv.org/html/2605.23937#Thmcorollary1)and[2](https://arxiv.org/html/2605.23937#Thmcorollary2)\.

###### Corollary 1\.

Let𝒦\\mathcal\{K\}be a DL\-LiteHKB with an empty ABox\. Then for anyd≥dmind\\geq d\_\{\\min\}there is a box interpretationη\{\\eta\}with dimensionalityddsuch thatdmin=\|𝖭𝖢\|\+3​\|𝖭𝖱\|d\_\{\\min\}=\|\{\\sf N\_\{C\}\}\|\+3\|\{\\sf N\_\{R\}\}\|andη\{\\eta\}is a TBox faithful model of𝒦\\mathcal\{K\}\.

###### Corollary 2\.

Let𝒦\\mathcal\{K\}be a satisfiable DL\-LiteHKB\. Then for anyd≥dmind\\geq d\_\{\\min\}there is a box interpretationη\{\\eta\}with dimensionalityddsuch that:dmin=\|𝖭𝖢\|\+\|𝖭𝖱\|​\(2\+\|𝖭𝖨\|\+2​\|𝖭𝖱\|\)d\_\{\\min\}=\|\{\\sf N\_\{C\}\}\|\+\|\{\\sf N\_\{R\}\}\|\(2\+\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\)andη\{\\eta\}is a KB faithful model of𝒦\\mathcal\{K\}\.

With this we have finished the theoretical analysis of BoxLitE’s faithfulness properties\. What now remains to show is how our BoxLitE approach can be translated into a convex optimization problem that*ensures*certain faithfulness properties\. We will investigate this in the next section\.

## 5BoxLitE’s Convex Optimization Problem

Here, we formulate the representation of KBs with BoxLitE as a constrained convex optimization problem\(Boyd and Vandenberghe,[2004](https://arxiv.org/html/2605.23937#bib.bib22); Rockafellar,[1997](https://arxiv.org/html/2605.23937#bib.bib23)\)and investigate how the formulation relates to faithfulness properties\. In Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1), we describe how TBox axioms are translated to constraints\. In Section[5\.2](https://arxiv.org/html/2605.23937#S5.SS2), we discuss the objective and scoring functions\. In Section[5\.3](https://arxiv.org/html/2605.23937#S5.SS3), we establish that our problem formulation ensures the KB faithful and TBox entailed properties for DL\-LiteH\. In summary, the TBox corresponds to a part of the constraints of the convex optimization problem and the ABox corresponds to a part of the objective function of the problem\. In this way, a solver can handle both the ABox and TBox simultaneously by minimizing the objective function subject to the constraints\.

### 5\.1From TBox Axioms to Constraints

Given a box interpretationη\\eta, we concatenate all ofη\\eta’s parameters into a single vectorzzofn:=\(2​\|𝖭𝖨\|\+2​\|𝖭𝖢\|\+6​\|𝖭𝖱\|\)​dn:=\(2\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{C\}\}\|\+6\|\{\\sf N\_\{R\}\}\|\)ddimensions555For each individual in𝖭𝖨\{\\sf N\_\{I\}\}we have two vectors, one for the position and one for the bump\. For each concept in𝖭𝖢\{\\sf N\_\{C\}\}we also have two vectors, one for the upper corner and one for the lower corner of the box\. For each role in𝖭𝖱\{\\sf N\_\{R\}\}we have 3 boxes: head, tail, and bump; each box needs the upper and lower corner vectors\.\. We call this vectorz∈ℝnz\\in\\mathbb\{R\}^\{n\}an*embedding solution*\. Conversely, given such az∈ℝnz\\in\\mathbb\{R\}^\{n\}, we can reconstructη\\eta\. However, not allzzwill lead to anη\\etathat satisfies Definition[3](https://arxiv.org/html/2605.23937#Thmdefinition3)\. Therefore, we devise constraints thatzzmust satisfy, such that the correspondingη\\etahas desirable properties\.

In particular, given an arbitrary DL\-LiteHTBox𝒯\\mathcal\{T\}, we want to ensure that any feasible solution for a box interpretation \(i\) is box consistent, i\.e\., for allC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}it holds thatη​\(C\)∩η​\(¬C\)≠∅\{\{\\eta\}\(C\)\}\\cap\{\{\\eta\}\(\\neg C\)\}\\neq\\emptyset, \(ii\) ensures that any box width is positive and bounded by2​𝐬𝛀2\{\\mathbf\{s\_\{\\Omega\}\}\}, \(iii\) ensures that any individual embedding is within the universe box, and \(iv\) guarantees that any TBox axiom is satisfied in the embedding solution\. In the following, we define convex constraints for Points \(i\)–\(iv\)\.

Box Consistency Constraints\.Intuitively, Point \(i\) ensures that the set of feasible solutions only contains those solutions that do not predict any contradictions\. However, enforcing box consistency directly is non\-convex, due to the need of computing intersection boxes \(see Section[7](https://arxiv.org/html/2605.23937#S7)\)\. Thus, we need to define a convex alternative that guarantees non\-overlap ofη​\(C\)\{\{\\eta\}\(C\)\}andη​\(¬C\)\{\{\\eta\}\(\\neg C\)\}\. We ensure Point \(i\) by reserving one dimensioniCi\_\{C\}perC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and defining one constraint per dimensioniCi\_\{C\}, formally described in \([1](https://arxiv.org/html/2605.23937#S5.E1)\)\. By reserving one dimensioniCi\_\{C\}perC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, the minimal number of dimensions of our model depends on the TBox\. In particular, our model requires at least\|𝖭𝖢\|\+2​\|𝖭𝖱\|\|\{\\sf N\_\{C\}\}\|\+2\|\{\\sf N\_\{R\}\}\|dimensions666We have2​\|𝖭𝖱\|2\|\{\\sf N\_\{R\}\}\|because of∃R\\exists Rand∃R−\\exists R^\{\-\}for eachR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}\.for a TBox with\|𝖭𝖢\|\|\{\\sf N\_\{C\}\}\|concept names and\|𝖭𝖱\|\|\{\\sf N\_\{R\}\}\|role names\.

𝐋𝐂​\[iC\]\+𝐔𝐂​\[iC\]2≤−sΩ2\\frac\{\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]\+\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\}\{2\}\\leq\-\\frac\{\{s\_\{\\Omega\}\}\}\{2\}\(1\)The inequality in \([1](https://arxiv.org/html/2605.23937#S5.E1)\) is enough to ensure box consistency \(see Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6)in the appendix\)\.

Box Width Constraints\.As mentioned in Point \(ii\), our model assumes that the width of any box with lower and upper bounds𝐋\\mathbf\{L\}and𝐔\\mathbf\{U\}is positive and bounded by2​𝐬𝛀2\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}, i\.e\.,𝟎≤d𝐔−𝐋≤d2​𝐬𝛀\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\}\-\\mathbf\{L\}\\mathrel\{\{\\leq\_\{d\}\}\}2\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\. The following inequalities ensure this requirement:

𝟎≤d𝐔𝐂−𝐋𝐂≤d2​𝐬𝛀\\displaystyle\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\\mathrel\{\{\\leq\_\{d\}\}\}2\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\(2\)𝟎≤d𝐔𝐑X−𝐋𝐑X≤d2​𝐬𝛀,\\displaystyle\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\}^\{X\}\_\{\\mathbf\{R\}\}\-\\mathbf\{L\}^\{X\}\_\{\\mathbf\{R\}\}\\mathrel\{\{\\leq\_\{d\}\}\}2\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\},whereC∈𝖭𝖢C\\in\{\\sf N\_\{C\}\},R∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}, andX∈\{𝐇,𝐓,𝐁\}X\\in\\\{\\mathbf\{H\},\\mathbf\{T\},\\mathbf\{B\}\\\}\.

Universe Constraints\.As mentioned in Point \(iii\), our model assumes that positions and bumps of individual embeddings are within the universe boxΩ\{\\Omega\}\(see[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3)\)\. The following inequalities ensure this property:

−𝐬𝛀\+ϵ≤d𝗉𝗈𝗌​\(a\)≤d𝐬𝛀−ϵ\\displaystyle\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\{\{\\sf pos\}\(a\)\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\boldsymbol\{\\epsilon\}\(3\)−𝐬𝛀\+ϵ≤d𝖻𝗎𝗆𝗉​\(a\)≤d𝐬𝛀−ϵ,\\displaystyle\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\{\{\\sf bump\}\(a\)\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\boldsymbol\{\\epsilon\},wherea∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}\.

TBox Axiom Constraints\.Finally, to guarantee the satisfaction of TBox axioms in feasible solutions \(Point \(iv\)\), first we need to define the boundaries of concept and role embeddings\. Based on the definitions in Section[3](https://arxiv.org/html/2605.23937#S3), the boundaries ofη​\(¬C\)\\eta\(\\neg\{C\}\),η​\(∃R\)\\eta\(\\exists R\), andη​\(R−\)\\eta\(R^\{\-\}\)are defined as follows:

𝐋¬𝐂\\displaystyle\\mathbf\{L\_\{\\neg\{C\}\}\}≔−𝐬𝛀−𝐋𝐂,𝐔¬𝐂≔𝐬𝛀−𝐔𝐂,\\displaystyle\\coloneqq\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\},\\,\\,\\mathbf\{U\_\{\\neg\{C\}\}\}\\coloneqq\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\},𝐋∃𝐑\\displaystyle\\mathbf\{L\_\{\\exists R\}\}≔𝐋𝐑𝐇−𝐔𝐑𝐁,𝐔∃𝐑≔𝐔𝐑𝐇−𝐋𝐑𝐁,\\displaystyle\\coloneqq\\mathbf\{L^\{H\}\_\{R\}\}\-\\mathbf\{U^\{B\}\_\{R\}\},\\,\\,\\mathbf\{U\_\{\\exists R\}\}\\coloneqq\\mathbf\{U^\{H\}\_\{R\}\}\-\\mathbf\{L^\{B\}\_\{R\}\},𝖧𝖾𝖺𝖽​\(R−\)\\displaystyle\{\{\\sf Head\}\(R^\{\-\}\)\}≔𝖳𝖺𝗂𝗅​\(R\),𝖳𝖺𝗂𝗅​\(R−\)≔𝖧𝖾𝖺𝖽​\(R\),\\displaystyle\\coloneqq\{\{\\sf Tail\}\(R\)\},\\,\\,\{\{\\sf Tail\}\(R^\{\-\}\)\}\\coloneqq\{\{\\sf Head\}\(R\)\},𝖡𝗎𝗆𝗉​\(R−\)\\displaystyle\{\{\\sf Bump\}\(R^\{\-\}\)\}≔𝖡𝗎𝗆𝗉​\(R\)\.\\displaystyle\\coloneqq\{\{\\sf Bump\}\(R\)\}\.
LetB∈𝖭𝖢∃B\\in\{\\sf N\_\{C\}^\{\\exists\}\},C∈𝖭𝖢​¬∃C\\in\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\neg\}, andR,S∈𝖭𝖱−R,S\\in\{\\sf N\_\{R\}^\{\-\}\}\. To satisfy concept and role inclusions within any embedding solutionzz\(Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4)\), we employ the following linear inequalities:

- •B⊑CB\\sqsubseteq Ccorresponds to𝐋𝐂≤d𝐋𝐁\\mathbf\{L\_\{C\}\}\\leq\_\{d\}\\mathbf\{L\_\{B\}\}and𝐔𝐁≤d𝐔𝐂\\mathbf\{U\_\{B\}\}\\leq\_\{d\}\\mathbf\{U\_\{C\}\},
- •R⊑SR\\sqsubseteq Scorresponds to𝐋𝐒X≤𝐋𝐑X,𝐔𝐑X≤𝐔𝐒X\\mathbf\{L\}^\{X\}\_\{\\mathbf\{S\}\}\\leq\\mathbf\{L\}^\{X\}\_\{\\mathbf\{R\}\},\\mathbf\{U\}^\{X\}\_\{\\mathbf\{R\}\}\\leq\\mathbf\{U\}^\{X\}\_\{\\mathbf\{S\}\}withX∈\{𝐇,𝐓,𝐁\}X\\in\\\{\\mathbf\{H\},\\mathbf\{T\},\\mathbf\{B\}\\\}\.

Problem Size of the Translation\.We point out that the size of BoxLitE’s problem formulation increases linearly w\.r\.t\. the size of the KB𝒦\\mathcal\{K\}and\|𝖭𝖢∪𝖭𝖱∪𝖭𝖨\|\|\{\\sf N\_\{C\}\}\\cup\{\\sf N\_\{R\}\}\\cup\{\\sf N\_\{I\}\}\|\. Let𝒯\\mathcal\{T\}be𝒦\\mathcal\{K\}’s TBox, the number of inequalities, as described in this section, gives usO​\(\(\|𝖭𝖨\|\+\|𝖭𝖢\|\+\|𝖭𝖱\|\+\|𝒯\|\)​d\)O\(\(\|\{\\sf N\_\{I\}\}\|\+\|\{\\sf N\_\{C\}\}\|\+\|\{\\sf N\_\{R\}\}\|\+\|\\mathcal\{T\}\|\)d\)constraints\.

### 5\.2Optimization: Ranking Assertions

When ranking assertions, we would like the distance between the position of an individual w\.r\.t to a box associated with a concept name to affect the score of the corresponding assertion: the smaller the distance to the elements of the box, the higher the score\. The same intuition also applies for roles, but needs to take into account the way role assertions are satisfied in the embedding model \(Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4)\)\.

To express this intuition, we employ a*signed distance*function\. The signed distance assigns nonpositive values to the assertions satisfied in the embedding, and positive values to those that are violated\. Moreover, it reflects*how*an assertion is satisfied \(or violated\): for concept assertions, individuals with embeddings deeper inside the box have a smaller negative value than those closer to the border, while individuals with embeddings outside the box have positive values, and those values are higher the farther away the embeddings of the individuals are from the box\.

Signed Distance\.We move on to the formal definition of signed distance to a set\. First, the*Euclidean distance to a setS⊆ℝnS\\subseteq\\mathbb\{R\}^\{n\}*is defined as the function𝖽𝗂𝗌𝗍e:ℝn→ℝ\\mathop\{\\mathsf\{dist\}\_\{e\}\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}such that𝖽𝗂𝗌𝗍e\(y,S\)≔inf\{‖x−y‖2∣x∈S\},∀y∈ℝn\.\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,S\)\\coloneqq\\inf\\\{\\\|x\-y\\\|\_\{2\}\\mid x\\in S\\\},\\ \\forall y\\in\\mathbb\{R\}^\{n\}\.LetScS^\{c\}beℝn∖S\\mathbb\{R\}^\{n\}\\setminus S\. Then, the*signed distance*\(also called*oriented distance*\) toSSis

𝗌𝖽𝗂𝗌𝗍\(y,S\)≔\{𝖽𝗂𝗌𝗍e\(y,S\)if​y∉S−𝖽𝗂𝗌𝗍e\(y,Sc\)if​y∈S,\\mathop\{\\mathsf\{sdist\}\}\(y,S\)\\coloneqq\\begin\{cases\}\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,S\)&\\text\{ if \}y\\not\\in S\\\\ \-\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,S^\{c\}\)&\\text\{ if \}y\\in S,\\end\{cases\}\(4\)e\.g\., see Chapter 7 ofDelfour and Zolésio\([2011](https://arxiv.org/html/2605.23937#bib.bib15)\)\. Giveny∈ℝny\\in\\mathbb\{R\}^\{n\}, we denote byy\+∈ℝny^\{\+\}\\in\\mathbb\{R\}^\{n\}the vector that corresponds to replacing the negative components ofyyby0\. E\.g\.,\(1,−2,3,−4\)\+=\(1,0,3,0\)\(1,\-2,3,\-4\)^\{\+\}=\(1,0,3,0\)\.

###### Proposition 3\.

The signed distance toℝ−n\\mathbb\{R\}^\{n\}\_\{\-\}satisfies

𝗌𝖽𝗂𝗌𝗍\(y,ℝ−n\)=\{‖y\+‖2if​y∉ℝ−nmaxi∈\{1,…,n\}⁡yiif​y∈ℝ−n,\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=\\begin\{cases\}\\\|y^\{\+\}\\\|\_\{2\}&\\text\{ if \}y\\not\\in\\mathbb\{R\}^\{n\}\_\{\-\}\\\\ \\max\_\{i\\in\\\{1,\\ldots,n\\\}\}y\_\{i\}&\\text\{ if \}y\\in\\mathbb\{R\}^\{n\}\_\{\-\},\\end\{cases\}\(5\)whereℝ−n=\{y∈ℝn∣yi≤0,1≤i≤n\}\\mathbb\{R\}^\{n\}\_\{\-\}=\\\{y\\in\\mathbb\{R\}^\{n\}\\mid y\_\{i\}\\leq 0,1\\leq i\\leq n\\\}\.

Let𝐁\\mathbf\{B\}be a box with bounds𝐋\\mathbf\{L\}and𝐔\\mathbf\{U\}\. We define𝖽𝗂𝗌𝗍​\(𝐁,𝐱\)\{\\sf dist\}\(\\mathbf\{B\},\\mathbf\{x\}\)as the signed distance of the concatenated vector\(𝐋\+ϵ−𝐱\)⊕\(𝐱−𝐔\+ϵ\)\(\\mathbf\{L\}\+\\boldsymbol\{\\epsilon\}\-\\mathbf\{x\}\)\\oplus\(\\mathbf\{x\}\-\\mathbf\{U\}\+\\boldsymbol\{\\epsilon\}\)toℝ−2​d\\mathbb\{R\}^\{2d\}\_\{\-\}\. That is,

𝖽𝗂𝗌𝗍​\(𝐁,𝐱\)≔𝗌𝖽𝗂𝗌𝗍\(\(𝐋\+ϵ−𝐱\)⊕\(𝐱−𝐔\+ϵ\),ℝ−2​d\)\.\{\\sf dist\}\(\\mathbf\{B\},\\mathbf\{x\}\)\\coloneqq\\mathop\{\\mathsf\{sdist\}\}\(\(\\mathbf\{L\}\+\\boldsymbol\{\\epsilon\}\-\\mathbf\{x\}\)\\oplus\(\\mathbf\{x\}\-\\mathbf\{U\}\+\\boldsymbol\{\\epsilon\}\),\\mathbb\{R\}^\{2d\}\_\{\-\}\)\.We now define the loss terms using𝖽𝗂𝗌𝗍\{\\sf dist\}\.

Concept Assertion Loss\.We define the loss for concept assertionsD​\(a\)D\(a\)as follows:

ℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍​\(D,a\)≔𝖽𝗂𝗌𝗍​\(η​\(D\),𝗉𝗈𝗌​\(a\)\)\.\\mathcal\{L\}\_\{\\sf concept\}\(D,a\)\\coloneqq\{\\sf dist\}\(\{\{\\eta\}\(D\)\},\{\{\\sf pos\}\(a\)\}\)\.
Intuitively, minimizing the lossℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍​\(D,a\)\\mathcal\{L\}\_\{\\sf concept\}\(D,a\)of a concept assertionD​\(a\)D\(a\)pushes an individual’s position embedding𝗉𝗈𝗌​\(a\)\{\{\\sf pos\}\(a\)\}intoDD’s concept embeddingη​\(D\)\{\{\\eta\}\(D\)\}\.

Role Assertion Loss\.Similarly, the lossℒ𝗋𝗈𝗅𝖾​\(S,a,b\)\\mathcal\{L\}\_\{\\sf role\}\(S,a,b\)of a role assertionS​\(a,b\)S\(a,b\)pushes the translated individual embedding𝗉𝗈𝗌​\(a\)\+𝖻𝗎𝗆𝗉​\(b\)\{\{\\sf pos\}\(a\)\}\+\{\{\\sf bump\}\(b\)\}and, respectively,𝗉𝗈𝗌​\(b\)\+𝖻𝗎𝗆𝗉​\(a\)\{\{\\sf pos\}\(b\)\}\+\{\{\\sf bump\}\(a\)\}intoSS’s head box𝖧𝖾𝖺𝖽​\(S\)\{\{\\sf Head\}\(S\)\}andSS’s tail box𝖳𝖺𝗂𝗅​\(S\)\{\{\\sf Tail\}\(S\)\}\. Furthermore, it pushes the bumps ofaaandbbintoSS’s bump box𝖡𝗎𝗆𝗉​\(S\)\{\{\\sf Bump\}\(S\)\}\. Based on the distance function𝖽𝗂𝗌𝗍\{\\sf dist\}, we define the loss for role assertionsS​\(a,b\)S\(a,b\)as follows:

ℒ𝗋𝗈𝗅𝖾\(S,a,b\)≔max\(\\displaystyle\\mathcal\{L\}\_\{\\sf role\}\(S,a,b\)\\coloneqq\\max\\big\(𝖽𝗂𝗌𝗍​\(𝖧𝖾𝖺𝖽​\(S\),𝗉𝗈𝗌​\(a\)\+𝖻𝗎𝗆𝗉​\(b\)\),\\displaystyle\{\\sf dist\}\(\{\{\\sf Head\}\(S\)\},\{\{\\sf pos\}\(a\)\}\+\{\{\\sf bump\}\(b\)\}\),𝖽𝗂𝗌𝗍​\(𝖳𝖺𝗂𝗅​\(S\),𝗉𝗈𝗌​\(b\)\+𝖻𝗎𝗆𝗉​\(a\)\),\\displaystyle\{\\sf dist\}\(\{\{\\sf Tail\}\(S\)\},\{\{\\sf pos\}\(b\)\}\+\{\{\\sf bump\}\(a\)\}\),𝖽𝗂𝗌𝗍​\(𝖡𝗎𝗆𝗉​\(S\),𝖻𝗎𝗆𝗉​\(a\)\),\\displaystyle\{\\sf dist\}\(\{\{\\sf Bump\}\(S\)\},\{\{\\sf bump\}\(a\)\}\),𝖽𝗂𝗌𝗍\(𝖡𝗎𝗆𝗉\(S\),𝖻𝗎𝗆𝗉\(b\)\)\)\.\\displaystyle\{\\sf dist\}\(\{\{\\sf Bump\}\(S\)\},\{\{\\sf bump\}\(b\)\}\)\\big\)\.
In practice, KBs typically contain only positive assertions and no explicit negative ones\. Therefore, if we were to minimise only the loss terms of the assertions, BoxLitE would tend to assign high scores to all potential assertions, not distinguishing between true and false assertions\. A common way to address this issue in KB embedding models is*negative sampling*\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42); Sunet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib34); Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21)\)\. In that approach, given a role assertionR​\(h,t\)R\(h,t\)contained in the KB, one constructs a*corrupted*assertion by exchanging either the headhhor tailttof the role assertion by a randomly chosen individual from𝖭𝖨\{\\sf N\_\{I\}\}\. Since the number of assertions that hold is usually much smaller than the number of all possible assertions that can be made, most corrupted assertions tend to be false\. KB embedding models are then trained to assign high scores to ABox assertions in the training data and low scores to corrupted ones\. However, negative sampling leads to nonconvex loss terms, which are often incompatible with convex optimization \(see Section[7](https://arxiv.org/html/2605.23937#S7)\); and it is computationally expensive, as competitive performance often requires sampling hundreds or even thousands of negative assertions per ABox assertion\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Lu and Hu,[2020](https://arxiv.org/html/2605.23937#bib.bib33); Pavlović and Sallinger,[2023b](https://arxiv.org/html/2605.23937#bib.bib48)\)\. Next, we adapt negative sampling to concepts in𝖭𝖢∃\{\\sf N\_\{C\}^\{\\exists\}\}, leading to convex regularization terms that are fast to compute and push individual embeddings into complement boxes as required\.

Negative Concept Regularization\.Our approach builds on three observations: \(1\) directly pushing individual embeddings outside a box \(i\.e\., into its geometric complement\) leads to nonconvex optimization terms; \(2\) BoxLitE introduces convex boxes representing the complement of concept embeddings, which we can use to regularize the scores; and \(3\) BoxLitE does not offer convex representations for the complement of role embeddings, yet a role’s domain and range can be expressed as existential concept embeddings\. Based on these observations, we introduce a negative concept regularization term that for any potential concept assertionD​\(a\)∉𝒜D\(a\)\\not\\in\\mathcal\{A\}withD∈𝖭𝖢∃D\\in\{\\sf N\_\{C\}^\{\\exists\}\}anda∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}pushes𝗉𝗈𝗌​\(a\)\{\{\\sf pos\}\(a\)\}intoη​\(D\)¯\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}, reducing the score ofD​\(a\)D\(a\)\. This regularization term keeps plausibility scores for arbitrary assertions low, while the assertion loss terms selectively increase the scores of ABox assertions explicitly contained in the KB:

ℒ𝗇𝖾𝗀𝖺𝗍𝗂𝗏𝖾​\(D,a\)≔ℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍​\(¬D,a\)\.\\displaystyle\\mathcal\{L\}\_\{\\sf negative\}\(D,a\)\\hskip 2\.0pt\\coloneqq\\mathcal\{L\}\_\{\\mathsf\{concept\}\}\(\\neg D,a\)\.\\hskip 2\.0pt\\hskip 2\.0pt\(6\)Box Width Regularization\.A second strategy to keep scores for arbitrary assertions within a reasonable range is to regularize the box size of concept and role embeddings\. Specifically, for any concept nameD∈𝖭𝖢D\\in\{\\sf N\_\{C\}\}we regularize the size of its concept boxη​\(D\)\{\{\\eta\}\(D\)\}; and for any role nameS∈𝖭𝖱S\\in\{\\sf N\_\{R\}\}, we regularize the size of its head𝖧𝖾𝖺𝖽​\(S\)\{\{\\sf Head\}\(S\)\}, tail𝖳𝖺𝗂𝗅​\(S\)\{\{\\sf Tail\}\(S\)\}, and bump box𝖡𝗎𝗆𝗉​\(S\)\{\{\\sf Bump\}\(S\)\}\. Formally, we define the box regularization termℛ𝗐𝗂𝖽𝗍𝗁​\(𝐁\)\\mathcal\{R\}\_\{\\sf width\}\(\\mathbf\{B\}\)of a box𝐁\\mathbf\{B\}with bounds𝐔\\mathbf\{U\}and𝐋\\mathbf\{L\}, as:

ℛ𝗐𝗂𝖽𝗍𝗁​\(𝐁\)≔‖𝐔−𝐋‖2\.\\displaystyle\\mathcal\{R\}\_\{\\sf width\}\(\\mathbf\{B\}\)\\coloneqq\\\|\\mathbf\{U\}\-\\mathbf\{L\}\\\|\_\{2\}\.\(7\)Objective Function\.We now assemble these loss and regularization terms into our objective function:

max⁡\(maxD​\(a\)∈𝒜⁡ℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍​\(D,a\),maxS​\(a,b\)∈𝒜⁡ℒ𝗋𝗈𝗅𝖾​\(S,a,b\)\)\+λ1maxD​\(a\)∈NC∃∖𝒜⁡ℒ𝗇𝖾𝗀𝖺𝗍𝗂𝗏𝖾​\(D,a\)\+λ2\(∑D∈𝖭𝖢ℛ𝗐𝗂𝖽𝗍𝗁​\(η​\(D\)\)\+∑S∈𝖭𝖱ℛ𝗐𝗂𝖽𝗍𝗁\(𝖧𝖾𝖺𝖽\(S\)\)\+ℛ𝗐𝗂𝖽𝗍𝗁\(𝖳𝖺𝗂𝗅\(S\)\)\)\+λ3∑S∈𝖭𝖱ℛ𝗐𝗂𝖽𝗍𝗁​\(𝖡𝗎𝗆𝗉​\(S\)\)\.\\begin\{split\}&\\hskip\-14\.0pt\\max\\left\(\\max\_\{D\(a\)\\in\\mathcal\{A\}\}\\mathcal\{L\}\_\{\\sf concept\}\(D,a\),\\max\_\{S\(a,b\)\\in\\mathcal\{A\}\}\\mathcal\{L\}\_\{\\sf role\}\(S,a,b\)\\right\)\\hskip 2\.0pt\+\\\\ \\lambda\_\{1\}&\\hskip\-7\.0pt\\max\_\{D\(a\)\\in N\_\{C\}^\{\\exists\}\\setminus\\mathcal\{A\}\}\\mathcal\{L\}\_\{\\sf negative\}\(D,a\)\\hskip 2\.0pt\\hskip 2\.0pt\+\\\\ \\lambda\_\{2\}\\Big\(&\\sum\_\{D\\in\{\\sf N\_\{C\}\}\}\\mathcal\{R\}\_\{\\sf width\}\(\{\{\\eta\}\(D\)\}\)\\hskip 2\.0pt\+\\\\ &\\sum\_\{S\\in\{\\sf N\_\{R\}\}\}\\mathcal\{R\}\_\{\\sf width\}\(\{\{\\sf Head\}\(S\)\}\)\+\\mathcal\{R\}\_\{\\sf width\}\(\{\{\\sf Tail\}\(S\)\}\)\\Big\)\\hskip 2\.0pt\+\\\\ \\lambda\_\{3\}&\\sum\_\{S\\in\{\\sf N\_\{R\}\}\}\\mathcal\{R\}\_\{\\sf width\}\(\{\{\\sf Bump\}\(S\)\}\)\.\\end\{split\}
Scores\.To rank the assertions, we define two scoring functions, one for concept and one for role assertions\.

The scores​\(D,a\)s\(D,a\)of concept assertionsD​\(a\)D\(a\)is:

s​\(D,a\)≔\\displaystyle s\(D,a\)\\coloneqq−ℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍​\(D,a\)\\displaystyle\-\\mathcal\{L\}\_\{\\sf concept\}\(D,a\)and the scores​\(S,a,b\)s\(S,a,b\)of role assertionsS​\(a,b\)S\(a,b\)is:

s​\(S,a,b\)≔\\displaystyle s\(S,a,b\)\\coloneqq−ℒ𝗋𝗈𝗅𝖾​\(S,a,b\)\.\\displaystyle\-\\mathcal\{L\}\_\{\\sf role\}\(S,a,b\)\.The scoring functions of the concept and role assertions are the negative of the corresponding loss functions\. The intuition for this is that solving the optimization problem, i\.e\., minimizing the concept and role assertion loss for ABox assertions, maximizes their score\.

### 5\.3DL\-LiteHKB Faithfulness

Translating TBox axioms to linear inequalities that are used as convex constraints in the optimization problem \(see Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\), guarantees any BoxLitE embedding solutionzzto be TBox entailed and KB faithful for DL\-LiteH\.

Let𝒞𝒦⊆ℝn\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\\subseteq\\mathbb\{R\}^\{n\}be the set ofzz’s such that the constraints of Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)are satisfied\. That is,z∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}if and only if:\(a\)\(a\)for each TBox axiom the corresponding inequalities are satisfied; and\(b\)\(b\)the box consistency and universe constraints are satisfied, see the supplemental material for details\. Also, letf𝝀:ℝn→ℝf\_\{\\boldsymbol\{\\lambda\}\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}be the function that mapszzto the objective value for a given choice of nonnegative hyperparameters𝝀=\(λ1,λ2,λ3\)∈ℝ\+3\\boldsymbol\{\\lambda\}=\(\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}\)\\in\\mathbb\{R\}^\{3\}\_\{\+\}\.

###### Theorem 4\.

Let𝒦\\mathcal\{K\}be a satisfiable DL\-LiteHKB\. For nonnegative𝛌\\boldsymbol\{\\lambda\}the following optimization problem is convex\.

minz∈ℝn⁡f𝝀​\(z\),subject to​z∈𝒞𝒦\.\\displaystyle\\min\_\{z\\in\\mathbb\{R\}^\{n\}\}\\;f\_\{\\boldsymbol\{\\lambda\}\}\(z\),\\quad\\textup\{subject to\}\\;z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\.\(8\)In particular,f𝛌f\_\{\\boldsymbol\{\\lambda\}\}is a convex function,𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}is a polyhedral set and the following items hold\.

1. i\)Forddas in Corollary[1](https://arxiv.org/html/2605.23937#Thmcorollary1), andsΩ\{s\_\{\\Omega\}\}as in Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3),𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}is nonempty\.
2. ii\)Any embedding solutionz∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}corresponds to a box consistent interpretation that is TBox faithful\.
3. iii\)Ifλ1=0\\lambda\_\{1\}=0and there is an optimal solutionz∗z^\{\*\}such thatf𝝀​\(z∗\)≤0f\_\{\\boldsymbol\{\\lambda\}\}\(z^\{\*\}\)\\leq 0then the box interpretation corresponding toz∗z^\{\*\}is KB faithful\.
4. iv\)Suppose thatλ1=λ2=λ3=0\\lambda\_\{1\}=\\lambda\_\{2\}=\\lambda\_\{3\}=0\. Forddas in Corollary[2](https://arxiv.org/html/2605.23937#Thmcorollary2), andsΩ\{s\_\{\\Omega\}\}as in Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3), there is an optimal solutionz∗z^\{\*\}s\.t\.f𝝀​\(z∗\)≤0f\_\{\\boldsymbol\{\\lambda\}\}\(z^\{\*\}\)\\leq 0holds\.

Informally, Theorem[4](https://arxiv.org/html/2605.23937#Thmtheorem4)states that we can find a box interpretation for a satisfiable𝒦\\mathcal\{K\}via convex optimization over a polyhedral set\. Anyz∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\(whether optimal or not\) corresponds to a box interpretationη\\etathat is TBox faithful and box consistent\. Itemi\)i\)gives a bound on the minimum requiredddto ensure that𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}is nonempty, but this estimate seems to be conservative\. Itemsiii\)iii\)andiv\)iv\)imply that if we wish to find a solution that is KB faithful, this can be done by setting the hyperparameters associated to regularization terms to0, settingddto be sufficiently large and solving \([8](https://arxiv.org/html/2605.23937#S5.E8)\)\.

Finally, it turns out that \([8](https://arxiv.org/html/2605.23937#S5.E8)\) can be formulated as a*second\-order cone program*\(SOCP\)\(Loboet al\.,[1998](https://arxiv.org/html/2605.23937#bib.bib24)\),\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25), Lecture 3\)\.

###### Theorem 5\.

For nonnegative𝛌\\boldsymbol\{\\lambda\}, the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) can be reformulated as an equivalent SOCP\.

Theorem[5](https://arxiv.org/html/2605.23937#Thmtheorem5)is important because it shows that \([8](https://arxiv.org/html/2605.23937#S5.E8)\) can be solved efficiently via high\-quality open\-source solvers such as SeDuMi\(Sturm,[1999](https://arxiv.org/html/2605.23937#bib.bib38)\)and SDPT3\(Tütüncüet al\.,[2003](https://arxiv.org/html/2605.23937#bib.bib39)\)or commercial solvers such as Gurobi\(Gurobi Optimization, LLC,[2023](https://arxiv.org/html/2605.23937#bib.bib32)\)and MOSEK\(ApS,[2026](https://arxiv.org/html/2605.23937#bib.bib69)\)\. The conversion of a problem as in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) to a SOCP that can be handled by the aforementioned solvers, although tedious, can be automated by modelling tools for convex optimization such as CVXPY\(Agrawalet al\.,[2018](https://arxiv.org/html/2605.23937#bib.bib11)\)\.

## 6Proof of Concept

Here, we present empirical evidence for the theoretical foundations established so far\. We evaluate BoxLitE’s performance on subsets of the Family dataset\(Imeneset al\.,[2023](https://arxiv.org/html/2605.23937#bib.bib41)\), providing first results for its scalability and reasoning capabilities\. In our experiments, we only consider KBs with satisfiable concepts\.

Reproducibility\.We implemented BoxLitE’s optimization problem in Python 3\.12 using CVXPY\(Diamond and Boyd,[2016](https://arxiv.org/html/2605.23937#bib.bib13); Agrawalet al\.,[2018](https://arxiv.org/html/2605.23937#bib.bib11)\)for modeling and MOSEK\(ApS,[2026](https://arxiv.org/html/2605.23937#bib.bib69)\)for solving it\. In our evaluation we include experiments with other KBE approaches using classical stochastic gradient descent \(SGD\)\. We use PyKEEN 1\.11\.1\(Aliet al\.,[2021](https://arxiv.org/html/2605.23937#bib.bib68)\)in these experiments\. We ran each of our experiments on an Apple Mac Mini Desktop Computer with M4 Chip with 10 Core CPU and 10 Core GPU: 16GB \(Shared Memory\)\. More details can be found in the supplemental material and the code is available at[https://github\.com/AleksVap/BoxLitE](https://github.com/AleksVap/BoxLitE)\. We focus on the following questions\.

1. \(Q1\)What is the reasoning performance and what is the effect of the regularization terms on the results?
2. \(Q2\)How does increasing the dataset size affect the prediction performance, compilation, and solving time?
3. \(Q3\)How well does our model compare with classical embedding models based on stochastic gradient descent?

Experimental Setup\.To answer each of these questions, we have created a set of datasets \(F\_v1\-4\) of varying sizes from the family dataset\(Imeneset al\.,[2023](https://arxiv.org/html/2605.23937#bib.bib41)\)\. We derived these datasets by samplingkkassertions of the family dataset’s ABox with forest fire sampling\(Leskovecet al\.,[2005](https://arxiv.org/html/2605.23937#bib.bib53)\), a popular sampling technique for large graphs\. Furthermore, since the family dataset solely provides role assertions in its ABox, we selected all concept inclusions in the family dataset that only include roles and extended the TBox by the disjointness axiom∃𝗁𝖺𝗌𝖥𝖺𝗍𝗁𝖾𝗋−⊑¬∃𝗁𝖺𝗌𝖬𝗈𝗍𝗁𝖾𝗋−\\exists\{\\sf hasFather^\{\-\}\}\\sqsubseteq\\neg\\exists\{\\sf hasMother^\{\-\}\}\. We list the TBox of the created datasets in Figure[1](https://arxiv.org/html/2605.23937#S6.F1)\.

Evaluation Setup\.To evaluate BoxLitE’s performance, we created a set of inferred role assertions by\(i\)\(i\)adding any role assertion that logically follows from each dataset and\(i​i\)\(ii\)removing any assertion that occurs in the ABox\. We randomly split this set into a validation set \(20%\), used for model selection, and a test set \(80%\), used for evaluating the performance of the selected model\. We use the standard evaluation setting for KB completion777The evaluation of a KB embedding model typically needs a set of true and corrupted role assertions\. True role assertionsR​\(a,b\)R\(a,b\)of the KB are corrupted by replacingaaorbbby anyc∈𝖭𝖨c\\in\{\\sf N\_\{I\}\}such that the corrupted assertion is not within the KB\. The performance of KB embedding models is typically measured using the filtered versions\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42)\)of the*mean reciprocal rank*\(MRR\) and H@k, the proportion of true assertions within the predicted assertions whose rank is at maximum k\.\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Pavlović and Sallinger,[2024a](https://arxiv.org/html/2605.23937#bib.bib77); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\)\.

Dataset Properties\.Table[1](https://arxiv.org/html/2605.23937#S6.T1)lists the number of assertions and individuals of the train, validation, and test sets of F\_v1\-4\. We sampled each of these datasets individually\. Thus, although datasets F\_v1\-4 gradually increase in size, they are different from each other\.

𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾−⊑𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾\\displaystyle\{\\sf relative^\{\-\}\}\\sqsubseteq\{\\sf relative\}𝗁𝖺𝗌𝖲𝗂𝖻𝗅𝗂𝗇𝗀⊑𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾\\displaystyle\{\\sf hasSibling\}\\sqsubseteq\{\\sf relative\}𝗁𝖺𝗌𝖢𝗁𝗂𝗅𝖽⊑𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾\\displaystyle\{\\sf hasChild\}\\sqsubseteq\{\\sf relative\}𝗁𝖺𝗌𝖯𝖺𝗋𝖾𝗇𝗍⊑𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾\\displaystyle\{\\sf hasParent\}\\sqsubseteq\{\\sf relative\}𝗁𝖺𝗌𝖥𝖺𝗍𝗁𝖾𝗋⊑𝗁𝖺𝗌𝖯𝖺𝗋𝖾𝗇𝗍\\displaystyle\{\\sf hasFather\}\\sqsubseteq\{\\sf hasParent\}𝗁𝖺𝗌𝖬𝗈𝗍𝗁𝖾𝗋⊑𝗁𝖺𝗌𝖯𝖺𝗋𝖾𝗇𝗍\\displaystyle\{\\sf hasMother\}\\sqsubseteq\{\\sf hasParent\}𝗌𝗉𝗈𝗎𝗌𝖾−⊑𝗌𝗉𝗈𝗎𝗌𝖾\\displaystyle\{\\sf spouse^\{\-\}\}\\sqsubseteq\{\\sf spouse\}𝗁𝖺𝗌𝖲𝗂𝖻𝗅𝗂𝗇𝗀−⊑𝗁𝖺𝗌𝖲𝗂𝖻𝗅𝗂𝗇𝗀\\displaystyle\{\\sf hasSibling^\{\-\}\}\\sqsubseteq\{\\sf hasSibling\}𝗌𝗉𝗈𝗎𝗌𝖾⊑𝗋𝖾𝗅𝖺𝗍𝗂𝗏𝖾\\displaystyle\{\\sf spouse\}\\sqsubseteq\{\\sf relative\}∃𝗁𝖺𝗌𝖥𝖺𝗍𝗁𝖾𝗋−⊑¬∃𝗁𝖺𝗌𝖬𝗈𝗍𝗁𝖾𝗋−\\displaystyle\\exists\{\\sf hasFather^\{\-\}\}\\sqsubseteq\\neg\\exists\{\\sf hasMother^\{\-\}\}Figure 1:TBox of datasets F\_v1\-4\.Table 1:Dataset properties: Number of train, validation, and testing assertions, and individuals\.Table 2:Test Results on F\_v1\-4\. Average of 3 runs for SGD models\. Standard deviation nearly0in all cases\.\(Q1\) Performance\.In Table[2](https://arxiv.org/html/2605.23937#S6.T2)we present the link prediction results on the test set for BoxLitE with the three regularization terms that appear in the objetive function \(Section[5\.2](https://arxiv.org/html/2605.23937#S5.SS2)\)\. To study the effect of each of these terms, we also performed experiments in which we remove them\. We denote by BoxLitEiithe version of BoxLitE without the regularization term multiplied byλi\\lambda\_\{i\}\. The removal of each of the regularization terms decreases the overall performance of the model, with the removal of the term associated withλ2\\lambda\_\{2\}being the one that most negatively impacts the results\.

\(Q2\) Scalability\.The prediction performance on the test set, reported in Table[2](https://arxiv.org/html/2605.23937#S6.T2), reduces slowly with increasing dataset sizes\. Regarding the time required for each instance, we recall that a problem modelled through CVXPY is first compiled and then sent to a solver of the user’s choice, which in our case is MOSEK\. Given a specific choice of hyperparametersλ1,λ2,λ3\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}, Table[3](https://arxiv.org/html/2605.23937#S6.T3)displays the compilation and solving time required for obtaining an optimal solution to the problem in Theorem[4](https://arxiv.org/html/2605.23937#Thmtheorem4), for each of the datasets F\_v1\-4\. In our implementation, we tested 354 hyperparameter configurations for each dataset\. While changing the hyperparameters requires solving the optimization problem again, it does not require a recompilation\. Overall, compilation does not take more than a couple of seconds and all solution times were less than3030seconds, which is quite reasonable considering that the final SOCP corresponding toF​\_​v​4F\\\_v4has around3\.53\.5million variables and2\.32\.3million constraints, which has size comparable to some of the instances that appear in Mittelmann’s benchmark of large SOCPs\(Mittelmann,[2026](https://arxiv.org/html/2605.23937#bib.bib66)\)\. Even more, the compilation time in Table[3](https://arxiv.org/html/2605.23937#S6.T3)grows linearly with the number of training axioms, while the solving time increases only sublinearly\.

Table 3:Time in seconds split by dataset for BoxLitE\.\(Q3\) Comparison\.Comparing BoxLitE with other KBEs in a direct way is tricky since KBEs in the literature consider other languages and are mostly optimized using SGD\. BoxLitE is the first KBE for DL\-LiteHontologies and the first that solves link prediction via convex optimization\. In the discussion, we include an argument for why it is challenging to design convex optimization approaches for languages with conjunctions, which appear in other papers\. To illustrate how our approach roughly compares with classical embedding models such as BoxE, RotatE, and ComplEx, based on SGD, we run those models on the ABox part of F\_v1\-4\. We see that the results of BoxLitE are better than RotatE and ComplEx, but still behind BoxE\. One exception is F\_v2 where our model performs better\. None of the SGD models had any rule injections to reflect the DL\-LiteHTBox axioms used in BoxLitE\. This is a disadvantage for the SGD models\. On the other hand, BoxLitE has negative sampling applied only to existential assertions, which is a disadvantage for our case\. A possible main reason for the performance gap is BoxLitE’s hyperparameter optimization \(HPO\)\. It currently relies on grid search, which ensures full control over the explored parameter space, vital for ablations\. Yet, grid search does not adaptively guide the hyperparameter search\. By contrast, the SGD approaches, implemented in PyKEEN\(Aliet al\.,[2021](https://arxiv.org/html/2605.23937#bib.bib68)\), make use of more efficient, model‑based optimization techniques\. Incorporating such adaptive HPO methods in future work could lead to more effective exploration of BoxLitE’s hyperparameter landscape and performance gains\.

## 7Discussion on Optimization in KBEs

In this section, we discuss some aspects related to differentiability and primary causes for nonconvexity in previous KB embedding works\. We also recall how we address each challenge in our approach\.

Nondifferentiability\.In our approach nondifferentiability is not an issue because in view of Theorem[5](https://arxiv.org/html/2605.23937#Thmtheorem5)the underlying optimization problem can be cast as a second\-order cone program\. Informally, the nondifferentiable part of the problem gets embedded into the conic constraints and our solver of choice \(MOSEK\)\(ApS,[2026](https://arxiv.org/html/2605.23937#bib.bib69)\)can handle this kind of problem without theoretical issues\.

Negative sampling\.Negative sampling as described, say, in\(Sunet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib34), Section 3\.3\)includes the minimization of terms of the form

−log⁡\(σ​\(γ−d​\(x\)\)\)−∑i=1nwi​\(log⁡\(σ​\(d​\(xi\)−γ\)\)\),\-\\log\(\\sigma\(\\gamma\-d\(x\)\)\)\-\\sum\_\{i=1\}^\{n\}w\_\{i\}\(\\log\(\\sigma\(d\(x\_\{i\}\)\-\\gamma\)\)\),whereσ\\sigmais a sigmoid function \(e\.g\.,1/\(1\+e−x\)1/\(1\+e^\{\-x\}\)\),wiw\_\{i\}are nonnegative weights,γ\\gammais a margin parameter, thexix\_\{i\}’s are “negative samples” andddis a distance\-like function which may include, for example, app\-norm term\. Generally speaking, a function of the form−log\(σ\(d\(z\)−γ\)\-\\log\(\\sigma\(d\(z\)\-\\gamma\)is neither convex \(nor concave\) nor differentiable everywhere as a function ofzz\. This can be seen by considering the 1\-dimensional case, whereddis the absolute value function andzzis scalar, so that we obtain the function−ln⁡\(σ​\(\|z\|−γ\)\)=ln⁡\(1\+eγ−\|z\|\)\-\\ln\(\\sigma\(\|z\|\-\\gamma\)\)=\\ln\(1\+e^\{\\gamma\-\|z\|\}\), which, albeit continuous, is nonconvex and nondifferentiable atz=0z=0\. In contrast, BoxLitE adopts negative sampling in a convex way \(specifically by introducing the negative concept regularization terms of Section[5\.2](https://arxiv.org/html/2605.23937#S5.SS2)\) that push individuals into complement boxes\.

Nonconvex loss terms\.The loss terms proposed in previous works are often constructed through operations that do not preserve convexity in general\. We briefly take a look at this issue in two works that are more closely related to our approach\. To be fair, none of the works described below contain claims that their optimization problems are convex\. For BoxE, it is not clear whether the distance function considered in\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21), Section 4\)is convex as a*function of the parameters that need to be optimized during learning*, since it includes division and multiplication by a width term that depends on the size of the boxes\. Division and multiplication, generally speaking, are not operations that preserve convexity\. In BoxEL, the authors consider loss terms that are quotients of volumes of boxes \(or approximations thereof\), e\.g\., see\(Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75), Section 4\.4\)\. Again, division does not preserve convexity in general\. In contrast, all of BoxLitE’s loss terms together with the distance function in our approach are convex\.

The logic fragment\.A final source of nonconvexity for certain approaches seems to be the choice of the description logic fragment itself\. Approaches based on minimizing loss terms together with logical languages that include, say, conjunction on the left\-hand side typically lead to nonconvexity\. Suppose that the conjunction of conceptsC,DC,Dis interpreted as the set intersection of the corresponding boxesη​\(C\),η​\(D\)\\eta\(C\),\\eta\(D\), as observed in the works for theℰ​ℒ\{\\cal E\\\!L\}ontology language\. This leads to the requirement that𝐋E​\[i\]≤max⁡\(𝐋C​\[i\],𝐋D​\[i\]\)\\mathbf\{L\}\_\{E\}\[i\]\\leq\\max\(\\mathbf\{L\}\_\{C\}\[i\],\\mathbf\{L\}\_\{D\}\[i\]\)andmin⁡\(𝐔C​\[i\],𝐔D​\[i\]\)≤𝐔E​\[i\]\\min\(\\mathbf\{U\}\_\{C\}\[i\],\\mathbf\{U\}\_\{D\}\[i\]\)\\leq\\mathbf\{U\}\_\{E\}\[i\]must hold for each embedding dimensionii, where𝐋X,𝐔X\\mathbf\{L\}\_\{X\},\\mathbf\{U\}\_\{X\}indicates the lower and upper bounds of the box associated to a conceptXX\. Because the set𝒮≔\{\(a,b,c\)∈ℝ3∣min⁡\(a,b\)≤c\}\\mathcal\{S\}\\coloneqq\\\{\(a,b,c\)\\in\\mathbb\{R\}^\{3\}\\mid\\min\(a,b\)\\leq c\\\}is not convex888It suffices to observe that\(1,0,0\),\(0,1,0\)∈𝒮\(1,0,0\),\(0,1,0\)\\in\\mathcal\{S\}, but0\.5​\(1,0,0\)\+0\.5​\(0,1,0\)=\(0\.5,0\.5,0\)∉𝒮0\.5\(1,0,0\)\+0\.5\(0,1,0\)=\(0\.5,0\.5,0\)\\not\\in\\mathcal\{S\}\., the aforementioned restrictions are not convex in general\. Furthermore, recalling that optimal sets of convex functions are convex, it is not possible to devise a convexf:ℝ3→ℝf:\\mathbb\{R\}^\{3\}\\to\\mathbb\{R\}such that “\(a,b,c\)∈𝒮⇔\(a,b,c\)\(a,b,c\)\\in\\mathcal\{S\}\\Leftrightarrow\(a,b,c\)is optimal forff” holds\. In particular, absent extenuating circumstances, if the bounds ofC,D,EC,D,Eare parameters to be optimized during learning, it is impossible to construct a nonnegative*convex*loss function that is zero if and only if “C⊓D⊑EC\\sqcap D\\sqsubseteq E” is satisfied\. Regarding*cone semantics*, although convex optimization is mentioned as one motivation for using cones in\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37)\), the authors do not explain how exactly convex optimization fits in the picture of their approach\. In another work, the authors show how to use axis\-aligned cones and pairs of unions of convex cones to solve certain multi\-label classification problems\(Leemhuiset al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib40)\)via SVMs\. The approach described in\(Leemhuiset al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib40)\)seems to be significantly different from the loss function minimization approach described in other papers\. Furthermore, the Propositional𝒜​ℒ​𝒞\\mathcal\{ALC\}language used in\(Leemhuiset al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib40)\)is different from DL\-LiteH, which we consider in this work, since the latter features roles and inverses\.

## 8Conclusion and Future Work

We propose BoxLitE, a geometric model that allows for convex optimization and ensures the satisfaction of TBox axioms by translating them to convex constraints\. In addition, we prove that for any DL\-LiteHKB, there is a faithful embedding solution that is a KB model\. We practically implement and evaluate BoxLitE’s convex problem formulation for KB embeddings in CVXPY and MOSEK\. The results reveal that MOSEK finds a TBox entailed and KB faithful embedding solution for a prototypical ontology\. Also, within a few seconds of solving time, MOSEK finds embedding solutions on subsets \(F\_v1 to F\_v4\) of a real\-world KB that achieve good link prediction results\.

In the future, we will investigate BoxLitE’s practical side more and consider more efficient methods for the HPO and evaluation parts\. Also, we want to study how to ensure other theoretical properties in convex KBE models\. As languages that allow for conjunctions on the left of inclusions often lead to nonconvexity, another interesting line lies in studying sound nonconvex KBE approaches that can ensure the construction of faithful models\. Finally, we would like to investigate, using nonconvex tools, the effect of pushing negative samples outside the concept box into an ‘unknown’ truth state, and how this affects prediction results compared to the effect observed in our convex setting\.

## Acknowledgements

This work was supported by the “Strategic Research Projects” grant from ROIS \(Research Organization of Information and Systems\)\. Bruno F\. Lourenço’s work was partially supported by the JSPS Grant\-in\-Aid for Early\-Career Scientists 23K16844\. Ana Ozaki was supported by the Research Council of Norway, projects \(316022, 322480\) and Integreat \- Norwegian Centre for knowledge\-driven machine learning \(332645\)\. Emanuel Sallinger’s and Hesham Morgan’s work on this paper was funded by the Vienna Science and Technology Fund \(WWTF\) \[Grant ID: 10\.47379/VRG18013, 10\.47379/ICT25032, 10\.47379/NXT22018, 10\.47379/ ICT2201, 10\.47379/DCDH001\], and by the Austrian Science Fund \(FWF\) 10\.55776/COE12\.

## AI Declaration

Gen AI tools were only used to help find grammatical mistakes and to aid in the process of debugging the code\.

## References

- R\. Abboud, İ\. İ\. Ceylan, T\. Lukasiewicz, and T\. Salvatori \(2020\)BoxE: A box embedding model for knowledge base completion\.InNeurIPS,H\. Larochelle, M\. Ranzato, R\. Hadsell, M\. Balcan, and H\. Lin \(Eds\.\),Cited by:[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p1.5),[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p5.3),[§1](https://arxiv.org/html/2605.23937#S1.p6.1),[§3](https://arxiv.org/html/2605.23937#S3.p1.1),[§3](https://arxiv.org/html/2605.23937#S3.p3.12.12),[§3](https://arxiv.org/html/2605.23937#S3.p6.10.10),[§3](https://arxiv.org/html/2605.23937#S3.p7.5.5),[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p8.5),[§6](https://arxiv.org/html/2605.23937#S6.p5.2),[§7](https://arxiv.org/html/2605.23937#S7.p4.1)\.
- A\. Agrawal, R\. Verschueren, S\. Diamond, and S\. Boyd \(2018\)A rewriting system for convex optimization problems\.Journal of Control and Decision5\(1\),pp\. 42–60\.Cited by:[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p5.1),[§6](https://arxiv.org/html/2605.23937#S6.p2.1)\.
- M\. Ali, M\. Berrendorf, C\. T\. Hoyt, L\. Vermue, S\. Sharifzadeh, V\. Tresp, and J\. Lehmann \(2021\)PyKEEN 1\.0: A Python Library for Training and Evaluating Knowledge Graph Embeddings\.Journal of Machine Learning Research22\(82\),pp\. 1–6\.External Links:[Link](http://jmlr.org/papers/v22/20-825.html)Cited by:[§G\.6](https://arxiv.org/html/2605.23937#A7.SS6.p1.4),[§6](https://arxiv.org/html/2605.23937#S6.p2.1),[§6](https://arxiv.org/html/2605.23937#S6.p9.2)\.
- M\. ApS \(2026\)MOSEK optimizer api for python 11\.1\.3\.External Links:[Link](https://docs.mosek.com/latest/pythonapi/index.html)Cited by:[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p5.1),[§6](https://arxiv.org/html/2605.23937#S6.p2.1),[§7](https://arxiv.org/html/2605.23937#S7.p2.1)\.
- A\. Artale, D\. Calvanese, R\. Kontchakov, and M\. Zakharyaschev \(2009\)The DL\-Lite family and relations\.J\. Artif\. Intell\. Res\.36,pp\. 1–69\.Cited by:[Appendix B](https://arxiv.org/html/2605.23937#A2.p1.1),[§1](https://arxiv.org/html/2605.23937#S1.p3.3),[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- I\. Balazevic, C\. Allen, and T\. Hospedales \(2019\)TuckER: tensor factorization for knowledge graph completion\.InEMNLP\-IJCNLP,pp\. 5185–5194\.External Links:[Link](https://aclanthology.org/D19-1522),[Document](https://dx.doi.org/10.18653/v1/D19-1522)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- A\. Beck \(2017\)First\-order methods in optimization\.SIAM\.Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p6.1)\.
- A\. Ben\-Tal and A\. Nemirovski \(2001\)Lectures on modern convex optimization: analysis, algorithms, and engineering applications\.SIAM\.Cited by:[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.2.p2.6),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.3.p3.4),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.5.p5.15),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.p1.4),[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p4.1)\.
- J\. Bolte and E\. Pauwels \(2020\)Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning\.Mathematical Programming188\(1\),pp\. 19–51\.Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p6.1)\.
- A\. Bordes, N\. Usunier, A\. García\-Durán, J\. Weston, and O\. Yakhnenko \(2013\)Translating embeddings for modeling multi\-relational data\.In27th Annual Conference on Neural Information Processing Systems,C\. J\. C\. Burges, L\. Bottou, Z\. Ghahramani, and K\. Q\. Weinberger \(Eds\.\),pp\. 2787–2795\.External Links:[Link](https://proceedings.neurips.cc/paper/2013/hash/1cecc7a77928ca8133fa24680a88d2f9-Abstract.html)Cited by:[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p5.3),[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p8.5),[footnote 7](https://arxiv.org/html/2605.23937#footnote7)\.
- C\. Bourgaux, R\. Guimarães, R\. Koudijs, V\. Lacerda, and A\. Ozaki \(2024\)Knowledge base embeddings: semantics and theoretical properties\.InKR,P\. Marquis, M\. Ortiz, and M\. Pagnucco \(Eds\.\),External Links:[Link](https://doi.org/10.24963/kr.2024/77),[Document](https://dx.doi.org/10.24963/KR.2024/77)Cited by:[Appendix D](https://arxiv.org/html/2605.23937#A4.29.p1.14),[§1](https://arxiv.org/html/2605.23937#S1.p1.1),[§1](https://arxiv.org/html/2605.23937#S1.p4.1),[§4](https://arxiv.org/html/2605.23937#S4.1.p1.6),[Definition 5](https://arxiv.org/html/2605.23937#Thmdefinition5.p1.4.4)\.
- C\. Bourgaux, A\. Ozaki, and J\. Z\. Pan \(2021\)Geometric models for \(temporally\) attributed description logics\.In\(DL,M\. Homola, V\. Ryzhikov, and R\. A\. Schmidt \(Eds\.\),CEUR Workshop Proceedings, Vol\.2954\.External Links:[Link](https://ceur-ws.org/Vol-2954/paper-7.pdf)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- S\. Boyd and L\. Vandenberghe \(2004\)Convex optimization\.Cambridge University Press,Cambridge, England\.Cited by:[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.2.p2.1),[§5](https://arxiv.org/html/2605.23937#S5.p1.1)\.
- D\. Davis, D\. Drusvyatskiy, S\. Kakade, and J\. D\. Lee \(2019\)Stochastic subgradient method converges on tame functions\.Foundations of Computational Mathematics20\(1\),pp\. 119–154\.External Links:ISSN 1615\-3383,[Link](http://dx.doi.org/10.1007/s10208-018-09409-5),[Document](https://dx.doi.org/10.1007/s10208-018-09409-5)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p6.1)\.
- M\. C\. Delfour and J\. \-P\. Zolésio \(2011\)Shapes and geometries: metrics, analysis, differential calculus, and optimization, second edition\.Society for Industrial and Applied Mathematics\.Cited by:[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.p4.8),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.3.p3.7),[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p3.11)\.
- S\. Diamond and S\. Boyd \(2016\)CVXPY: A Python\-embedded modeling language for convex optimization\.Journal of Machine Learning Research17\(83\),pp\. 1–5\.Cited by:[§6](https://arxiv.org/html/2605.23937#S6.p2.1)\.
- Gurobi Optimization, LLC \(2023\)Gurobi Optimizer Reference Manual\.External Links:[Link](https://www.gurobi.com/)Cited by:[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p5.1)\.
- V\. Gutiérrez\-Basulto and S\. Schockaert \(2018\)From knowledge graph embedding to ontology embedding? an analysis of the compatibility between vector space representations and rules\.InKR,M\. Thielscher, F\. Toni, and F\. Wolter \(Eds\.\),pp\. 379–388\.External Links:[Link](https://aaai.org/ocs/index.php/KR/KR18/paper/view/18013)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p1.1)\.
- J\. Hiriart\-Urruty and C\. Lemaréchal \(1993a\)Convex analysis and minimization algorithms i: fundamentals\.Springer Berlin Heidelberg\.Cited by:[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.7.p1.2),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.3.p3.7)\.
- J\. Hiriart\-Urruty and C\. Lemaréchal \(1993b\)Convex analysis and minimization algorithms ii: advanced theory and bundle methods\.Springer,Berlin Heidelberg\.Cited by:[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.10.p4.11),[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.7.p1.2)\.
- A\. Imenes, R\. Guimarães, and A\. Ozaki \(2023\)Marrying query rewriting and knowledge graph embeddings\.InRuleML\+RR,Berlin, Heidelberg,pp\. 126–140\.External Links:ISBN 978\-3\-031\-45071\-6,[Link](https://doi.org/10.1007/978-3-031-45072-3_9),[Document](https://dx.doi.org/10.1007/978-3-031-45072-3%5F9)Cited by:[§G\.2](https://arxiv.org/html/2605.23937#A7.SS2.p1.4),[§1](https://arxiv.org/html/2605.23937#S1.p3.3),[§6](https://arxiv.org/html/2605.23937#S6.p1.1),[§6](https://arxiv.org/html/2605.23937#S6.p4.2)\.
- M\. Jackermeier, J\. Chen, and I\. Horrocks \(2024\)Dual box embeddings for the description logic EL\+\+\.InProceedings of the ACM Web Conference,WWW,pp\. 2250–2258\.External Links:ISBN 9798400701719,[Link](https://doi.org/10.1145/3589334.3645648),[Document](https://dx.doi.org/10.1145/3589334.3645648)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- S\. M\. Kazemi and D\. Poole \(2018\)SimplE embedding for link prediction in knowledge graphs\.InNeurIPS,S\. Bengio, H\. M\. Wallach, H\. Larochelle, K\. Grauman, N\. Cesa\-Bianchi, and R\. Garnett \(Eds\.\),pp\. 4289–4300\.External Links:[Link](https://proceedings.neurips.cc/paper/2018/hash/b2ab001909a8a6f04b51920306046ce5-Abstract.html)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- D\. P\. Kingma and J\. Ba \(2015\)Adam: A method for stochastic optimization\.InICLR,Y\. Bengio and Y\. LeCun \(Eds\.\),External Links:[Link](http://arxiv.org/abs/1412.6980)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p6.1)\.
- R\. Kontchakov, C\. Lutz, D\. Toman, F\. Wolter, and M\. Zakharyaschev \(2010\)The combined approach to query answering in dl\-lite\.InKR,F\. Lin, U\. Sattler, and M\. Truszczynski \(Eds\.\),External Links:[Link](http://aaai.org/ocs/index.php/KR/KR2010/paper/view/1282)Cited by:[Appendix C](https://arxiv.org/html/2605.23937#A3.p1.1),[§2](https://arxiv.org/html/2605.23937#S2.p2.24)\.
- M\. Kulmanov, W\. Liu\-Wei, Y\. Yan, and R\. Hoehndorf \(2019\)EL embeddings: geometric construction of models for the description logic EL\+\+\.InIJCAI,S\. Kraus \(Ed\.\),pp\. 6103–6109\.External Links:[Link](https://doi.org/10.24963/ijcai.2019/845),[Document](https://dx.doi.org/10.24963/IJCAI.2019/845)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- V\. Lacerda, A\. Ozaki, and R\. Guimarães \(2024a\)FaithEL: strongly tbox faithful knowledge base embeddings for*E*ℒ\\mathscr\{L\}\.InRuleML\+RR,S\. Kirrane, M\. Simkus, A\. Soylu, and D\. Roman \(Eds\.\),Lecture Notes in Computer Science, Vol\.15183,pp\. 191–199\.External Links:[Link](https://doi.org/10.1007/978-3-031-72407-7%5C_14),[Document](https://dx.doi.org/10.1007/978-3-031-72407-7%5F14)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- V\. Lacerda, A\. Ozaki, and R\. Guimarães \(2024b\)Strong faithfulness for ELH ontology embeddings\.TGDK2\(3\),pp\. 2:1–2:29\.External Links:[Link](https://doi.org/10.4230/TGDK.2.3.2),[Document](https://dx.doi.org/10.4230/TGDK.2.3.2)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- M\. Leemhuis, Ö\. L\. Özçep, and D\. Wolter \(2022\)Learning with cone\-based geometric models and orthologics\.Annals of Mathematics and Artificial Intelligence90\(11–12\),pp\. 1159–1195\.Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3),[§7](https://arxiv.org/html/2605.23937#S7.p5.16)\.
- J\. Leskovec, J\. M\. Kleinberg, and C\. Faloutsos \(2005\)Graphs over time: densification laws, shrinking diameters and possible explanations\.InACM SIGKDD,R\. Grossman, R\. J\. Bayardo, and K\. P\. Bennett \(Eds\.\),pp\. 177–187\.External Links:[Link](https://doi.org/10.1145/1081870.1081893),[Document](https://dx.doi.org/10.1145/1081870.1081893)Cited by:[§G\.2](https://arxiv.org/html/2605.23937#A7.SS2.p1.4),[§6](https://arxiv.org/html/2605.23937#S6.p4.2)\.
- W\. Li, X\. Zheng, H\. Gao, Q\. Ji, and G\. Qi \(2022\)Cosine\-based embedding for completing lightweight schematic knowledge in dl\-litecore\.Applied Sciences12\(20\)\.External Links:[Link](https://www.mdpi.com/2076-3417/12/20/10690),ISSN 2076\-3417,[Document](https://dx.doi.org/10.3390/app122010690)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- M\. S\. Lobo, L\. Vandenberghe, S\. Boyd, and H\. Lebret \(1998\)Applications of second\-order cone programming\.Linear Algebra and its Applications284\(1–3\),pp\. 193–228\.Cited by:[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.2.p2.6),[§E\.2](https://arxiv.org/html/2605.23937#A5.SS2.SSS0.Px1.p1.4),[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p4.1)\.
- H\. Lu and H\. Hu \(2020\)DensE: an enhanced non\-abelian group representation for knowledge graph embedding\.CoRRabs/2008\.04548\.External Links:2008\.04548,[Link](https://arxiv.org/abs/2008.04548)Cited by:[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p8.5)\.
- H\. Luo, X\. Wang, and B\. Lukens \(2018\)Variational analysis on the signed distance functions\.Journal of Optimization Theory and Applications180\(3\),pp\. 751–774\.Cited by:[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.1.p1.1),[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.p4.8)\.
- Ö\. Lütfü Özçep, M\. Leemhuis, and D\. Wolter \(2020\)Cone semantics for logics with negation\.InIJCAI,C\. Bessiere \(Ed\.\),pp\. 1820–1826\.Note:Main trackExternal Links:[Document](https://dx.doi.org/10.24963/ijcai.2020/252),[Link](https://doi.org/10.24963/ijcai.2020/252)Cited by:[Appendix D](https://arxiv.org/html/2605.23937#A4.29.p1.14),[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p3.3),[§1](https://arxiv.org/html/2605.23937#S1.p4.1),[§4](https://arxiv.org/html/2605.23937#S4.1.p1.6),[§4](https://arxiv.org/html/2605.23937#S4.p1.1),[§7](https://arxiv.org/html/2605.23937#S7.p5.16),[Definition 5](https://arxiv.org/html/2605.23937#Thmdefinition5.p1.4.4),[footnote 4](https://arxiv.org/html/2605.23937#footnote4)\.
- H\. Mittelmann \(2026\)Large second order cone benchmark\.Note:[https://plato\.asu\.edu/ftp/socp\.html](https://plato.asu.edu/ftp/socp.html)\[Online; accessed 17\-February\-2026\]Cited by:[§6](https://arxiv.org/html/2605.23937#S6.p8.5)\.
- S\. Mondal, S\. Bhatia, and R\. Mutharaju \(2021\)EmEL\+\+: embeddings for EL\+\+ description logic\.InAAAI\-MAKE,A\. Martin, K\. Hinkelmann, H\. Fill, A\. Gerber, D\. Lenat, R\. Stolle, and F\. van Harmelen \(Eds\.\),CEUR Workshop Proceedings, Vol\.2846\.External Links:[Link](https://ceur-ws.org/Vol-2846/paper19.pdf)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- H\. Morgan, A\. Pavlović, A\. Ozaki, B\. F\. Lourenço, and E\. Sallinger \(2026\)Convexity for trustworthy knowledge graph embeddings: from interpretability to reliability\.InProceedings of the 19th Forschungsforum der österreichischen Fachhochschulen \(FHK 2026\), Graz, Austria, April 15 \- April 16, 2026,Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p1.1)\.
- Y\. Nesterov and A\. Nemirovskii \(1994\)Interior\-point polynomial algorithms in convex programming\.SIAM\.Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p2.1)\.
- A\. Pavlović, E\. Sallinger, and S\. Schockaert \(2025\)Faithful Differentiable Reasoning with Reshuffled Region\-based Embeddings\.InProceedings of the 22nd International Conference on Principles of Knowledge Representation and Reasoning,pp\. 489–499\.External Links:[Document](https://dx.doi.org/10.24963/kr.2025/48),[Link](https://doi.org/10.24963/kr.2025/48)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p1.1)\.
- A\. Pavlović and E\. Sallinger \(2023a\)Building bridges: knowledge graph embeddings respecting logical rules\.InProceedings of the 15th Alberto Mendelzon International Workshop on Foundations of Data Management \(AMW 2023\), Santiago de Chile, Chile, May 22\-26, 2023,B\. Kimelfeld, M\. V\. Martinez, and R\. Angles \(Eds\.\),CEUR Workshop Proceedings, Vol\.3409\.External Links:[Link](https://ceur-ws.org/Vol-3409/paper9.pdf)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p1.1)\.
- A\. Pavlović and E\. Sallinger \(2023b\)ExpressivE: A spatio\-functional embedding for knowledge graph completion\.Inthe 11th International Conference on Learning Representations \(ICLR 2023\),External Links:[Link](https://openreview.net/pdf?id=xkev3%5C_np08z)Cited by:[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p1.5),[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p8.5)\.
- A\. Pavlović and E\. Sallinger \(2024a\)Raising the efficiency of knowledge graph embeddings while respecting logical rules\.InProceedings of the 16th Alberto Mendelzon International Workshop on Foundations of Data Management \(AMW 2024\), Mexico City, Mexico, September 30 \- October 4, 2024,G\. Montoya, E\. Sallinger, and G\. Vargas\-Solar \(Eds\.\),CEUR Workshop Proceedings\.Cited by:[§6](https://arxiv.org/html/2605.23937#S6.p5.2)\.
- A\. Pavlović and E\. Sallinger \(2024b\)SpeedE: Euclidean geometric knowledge graph embedding strikes back\.InFindings of the Association for Computational Linguistics: NAACL 2024,K\. Duh, H\. Gomez, and S\. Bethard \(Eds\.\),Mexico City, Mexico,pp\. 69–92\.External Links:[Link](https://aclanthology.org/2024.findings-naacl.6/),[Document](https://dx.doi.org/10.18653/v1/2024.findings-naacl.6)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- X\. Peng, Z\. Tang, M\. Kulmanov, K\. Niu, and R\. Hoehndorf \(2022\)Description logic EL\+\+ embeddings with intersectional closure\.CoRRabs/2202\.14018\.External Links:[Link](https://arxiv.org/abs/2202.14018)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.
- PyTorch \(2026\)Pytorch \- autograd mechanics\.Note:[https://docs\.pytorch\.org/docs/stable/notes/autograd\.html\#gradients\-for\-non\-differentiable\-functions](https://docs.pytorch.org/docs/stable/notes/autograd.html#gradients-for-non-differentiable-functions)\[Online; accessed 17\-February\-2026\]Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p6.1)\.
- R\. T\. Rockafellar \(1997\)Convex Analysis\.Princeton University Press\.Cited by:[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.7.p1.2),[§E\.1](https://arxiv.org/html/2605.23937#A5.SS1.p4.8),[§5](https://arxiv.org/html/2605.23937#S5.p1.1)\.
- J\. F\. Sturm \(1999\)Using SeDuMi 1\.02, a MATLAB toolbox for optimization over symmetric cones\.Optimization Methods and Software11\(1\-4\),pp\. 625–653\.Cited by:[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p5.1)\.
- Z\. Sun, Z\. Deng, J\. Nie, and J\. Tang \(2019\)RotatE: knowledge graph embedding by relational rotation in complex space\.InICLR,External Links:[Link](https://openreview.net/forum?id=HkgEQnRqYQ)Cited by:[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p5.3),[§5\.2](https://arxiv.org/html/2605.23937#S5.SS2.p8.5),[§7](https://arxiv.org/html/2605.23937#S7.p3.14)\.
- T\. Trouillon, J\. Welbl, S\. Riedel, É\. Gaussier, and G\. Bouchard \(2016\)Complex embeddings for simple link prediction\.InICML,M\. Balcan and K\. Q\. Weinberger \(Eds\.\),JMLR Workshop and Conference Proceedings, Vol\.48,pp\. 2071–2080\.External Links:[Link](http://proceedings.mlr.press/v48/trouillon16.html)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- R\. H\. Tütüncü, K\. C\. Toh, and M\. J\. Todd \(2003\)Solving semidefinite\-quadratic\-linear programs using SDPT3\.Mathematical Programming95\(2\),pp\. 189–217\.Cited by:[§5\.3](https://arxiv.org/html/2605.23937#S5.SS3.p5.1)\.
- B\. Xiong, N\. Potyka, T\. Tran, M\. Nayyeri, and S\. Staab \(2022\)Faithful embeddings for*E*ℒ\\mathscr\{L\}\+\+\{\}^\{\\mbox\{\+\+\}\}knowledge bases\.InISWC,U\. Sattler, A\. Hogan, C\. M\. Keet, V\. Presutti, J\. P\. A\. Almeida, H\. Takeda, P\. Monnin, G\. Pirrò, and C\. d’Amato \(Eds\.\),Lecture Notes in Computer Science, Vol\.13489,pp\. 22–38\.External Links:[Link](https://doi.org/10.1007/978-3-031-19433-7%5C_2),[Document](https://dx.doi.org/10.1007/978-3-031-19433-7%5F2)Cited by:[Appendix B](https://arxiv.org/html/2605.23937#A2.p2.18),[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p1.5),[§G\.4](https://arxiv.org/html/2605.23937#A7.SS4.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p2.1),[§1](https://arxiv.org/html/2605.23937#S1.p3.3),[§1](https://arxiv.org/html/2605.23937#S1.p5.3),[§1](https://arxiv.org/html/2605.23937#S1.p6.1),[§2](https://arxiv.org/html/2605.23937#S2.p1.53),[§6](https://arxiv.org/html/2605.23937#S6.p5.2),[§7](https://arxiv.org/html/2605.23937#S7.p4.1),[footnote 1](https://arxiv.org/html/2605.23937#footnote1)\.
- B\. Yang, W\. Yih, X\. He, J\. Gao, and L\. Deng \(2015\)Embedding entities and relations for learning and inference in knowledge bases\.InICLR,Y\. Bengio and Y\. LeCun \(Eds\.\),External Links:[Link](http://arxiv.org/abs/1412.6575)Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p5.3)\.
- H\. Yang, J\. Chen, and U\. Sattler \(2025\)TransBox: el\+\+\-closed ontology embedding\.InProceedings of the ACM on Web Conference,WWW,pp\. 22–34\.External Links:ISBN 9798400712746Cited by:[§1](https://arxiv.org/html/2605.23937#S1.p3.3)\.

## Appendix ASupplemental Material

This supplemental material contains additional information on the experimental setup, theoretical and empirical results, and complete proofs for each corollary, proposition, and theorem\. Specifically,[AppendixB](https://arxiv.org/html/2605.23937#A2)introduces the semantics of DL\-LiteHfor the convenience of the reader\. Next,[AppendixC](https://arxiv.org/html/2605.23937#A3)provides all proofs for the theoretical results in[Section2](https://arxiv.org/html/2605.23937#S2)\. Afterwards,[AppendixD](https://arxiv.org/html/2605.23937#A4)provides proofs for the theoretical results of[Section4](https://arxiv.org/html/2605.23937#S4)\. Moreover,[AppendixE](https://arxiv.org/html/2605.23937#A5)formulates in detail BoxLitE’s convex optimization problem and provides the proofs for each theoretical result of[Section5](https://arxiv.org/html/2605.23937#S5)\. Then,[AppendixF](https://arxiv.org/html/2605.23937#A6)provides additional information on the sizes of the optimization problem in the experiments\.[AppendixG](https://arxiv.org/html/2605.23937#A7)provides additional information on the experimental setup, including implementation details and a discussion on the creation of the datasets \(F\_v1\-4\), the training setup, hyperparameter optimization, evaluation protocol, and used metrics\.

## Appendix BBasic Definitions: DL\-LiteHSemantics

For the convenience of the reader, here we provide the definition of the semantics for DL\-LiteH, which is standard and can be found in references such as\(Artaleet al\.,[2009](https://arxiv.org/html/2605.23937#bib.bib1)\)\.

An*interpretation*ℐ\\mathcal\{I\}is a pair\(Δℐ,⋅ℐ\)\(\\Delta^\{\\mathcal\{I\}\},\\cdot^\{\\mathcal\{I\}\}\), whereΔℐ\\Delta^\{\\mathcal\{I\}\}is a non\-empty set, called the*domain*ofℐ\\mathcal\{I\}, and⋅ℐ\\cdot^\{\\mathcal\{I\}\}is a function that assigns a subsetAℐ⊆ΔℐA^\{\\mathcal\{I\}\}\\subseteq\\Delta^\{\\mathcal\{I\}\}of the domain to eachA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}, a binary relationRℐ⊆Δℐ×ΔℐR^\{\\mathcal\{I\}\}\\subseteq\\Delta^\{\\mathcal\{I\}\}\\times\\Delta^\{\\mathcal\{I\}\}over the domain to eachR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}, and an elementaℐ∈Δℐa^\{\\mathcal\{I\}\}\\in\\Delta^\{\\mathcal\{I\}\}to eacha∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}\. We extend⋅ℐ\\cdot^\{\\mathcal\{I\}\}to role and concept expressions as follows:

\(¬B\)ℐ:=\\displaystyle\(\\neg B\)^\{\\mathcal\{I\}\}:=\{\}Δℐ∖Bℐ;\\displaystyle\\Delta^\{\\mathcal\{I\}\}\\setminus B^\{\\mathcal\{I\}\};\(R−\)ℐ:=\\displaystyle\(R^\{\-\}\)^\{\\mathcal\{I\}\}:=\{\}\{\(e,d\)∣\(d,e\)∈Rℐ\};\\displaystyle\\\{\(e,d\)\\mid\(d,e\)\\in R^\{\\mathcal\{I\}\}\\\};\(∃S\)ℐ:=\\displaystyle\(\\exists S\)^\{\\mathcal\{I\}\}:=\{\}\{d∣∃e∈Δℐ​such that​\(d,e\)∈Sℐ\}\.\\displaystyle\\\{d\\mid\\exists e\\in\\Delta^\{\\mathcal\{I\}\}\\text\{ such that \}\(d,e\)\\in S^\{\\mathcal\{I\}\}\\\}\.We say that an interpretationℐ\\mathcal\{I\}*satisfies*

- •a role inclusionS⊑TS\\sqsubseteq TiffSℐ⊆TℐS^\{\\mathcal\{I\}\}\\subseteq T^\{\\mathcal\{I\}\};
- •a concept inclusionB⊑CB\\sqsubseteq CiffBℐ⊆CℐB^\{\\mathcal\{I\}\}\\subseteq C^\{\\mathcal\{I\}\};
- •a role assertionR​\(a,b\)R\(a,b\)iff\(aℐ,bℐ\)∈Rℐ\(a^\{\\mathcal\{I\}\},b^\{\\mathcal\{I\}\}\)\\in R^\{\\mathcal\{I\}\}; and
- •a concept assertionD​\(a\)D\(a\)iffaℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}\.

In this work𝖭𝖢\{\\sf N\_\{C\}\},𝖭𝖱\{\\sf N\_\{R\}\}, and𝖭𝖨\{\\sf N\_\{I\}\}are all*finite sets*, considered to be the relevant symbols to express KBs\. While in the Description Logic literature these sets are often assumed to be countably infinite, in the KB embedding literature, they are usually assumed to be finite, e\.g\.,\(Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\)and that is what we adopt here\. In what follows, we denote the size of a finite setVVby\|V\|\|V\|\.

## Appendix CProofs for Section[2](https://arxiv.org/html/2605.23937#S2)

Here, we provide proofs for the results in[Section2](https://arxiv.org/html/2605.23937#S2)\. The canonical interpretations found in the literature, e\.g\.,\(Kontchakovet al\.,[2010](https://arxiv.org/html/2605.23937#bib.bib17)\)are usually designed for query answering and may not satisfy the \(concept/role\) inclusions that are entailed by the TBox or falsify those inclusions that are not entailed\. Since satisfying the TBox is important in our work for establishing faithfulness results later, we provided our own definition of the canonical model \([Definition1](https://arxiv.org/html/2605.23937#Thmdefinition1)\) and now provide the full proof of[Theorem1](https://arxiv.org/html/2605.23937#Thmtheorem1)\.

See[1](https://arxiv.org/html/2605.23937#Thmtheorem1)

###### Proof\.

In the following, assumeA,B∈𝖭𝖢A,B\\in\{\\sf N\_\{C\}\}andR,S∈𝖭𝖱R,S\\in\{\\sf N\_\{R\}\}\.

###### Claim 1\.

ℐ𝒦⊧A⊑B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq Biff𝒦⊧A⊑B\\mathcal\{K\}\\models A\\sqsubseteq B\.

###### Proof\.

Assume𝒦⊧A⊑B\\mathcal\{K\}\\models A\\sqsubseteq B\.We make a case distinction based on the elements inΔℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\.

- •a∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}: Assumea∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thata∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. By assumption,𝒦⊧A⊑B\\mathcal\{K\}\\models A\\sqsubseteq B, so𝒦⊧B​\(a\)\\mathcal\{K\}\\models B\(a\)\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},a∈Bℐ𝒦a\\in B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Sinceaawas an arbitrary element of𝖭𝖨\{\\sf N\_\{I\}\}this holds for all elements of this kind\.
- •cD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}: AssumecD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatcD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. By assumption,𝒦⊧A⊑B\\mathcal\{K\}\\models A\\sqsubseteq B, so𝒦⊧D⊑B\\mathcal\{K\}\\models D\\sqsubseteq B\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},cD∈Bℐ𝒦c\_\{D\}\\in B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. SincecDc\_\{D\}was an arbitrary element ofΔ𝒦\\Delta\_\{\\mathcal\{K\}\}this holds for all elements of this kind\.

We have thus shown thatℐ𝒦⊧A⊑B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq B\.

Now, assume𝒦⊧̸A⊑B\\mathcal\{K\}\\not\\models A\\sqsubseteq B\.We show thatℐ𝒦⊧̸A⊑B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models A\\sqsubseteq B\. If𝒦⊧̸A⊑B\\mathcal\{K\}\\not\\models A\\sqsubseteq Bthen there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦\\mathcal\{K\}withAℐA^\{\\mathcal\{I\}\}non\-empty\. This means thatAAis satisfiable w\.r\.t\.𝒦\\mathcal\{K\}and thuscA∈Δ𝒦c\_\{A\}\\in\\Delta\_\{\\mathcal\{K\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatcA∈Aℐ𝒦c\_\{A\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}since𝒦⊧A⊑A\\mathcal\{K\}\\models A\\sqsubseteq Aholds trivially\. We now argue thatcA∉Bℐ𝒦c\_\{A\}\\notin B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, an element of the formcDc\_\{D\}is inBℐ𝒦B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧D⊑B\\mathcal\{K\}\\models D\\sqsubseteq B\. By assumption𝒦⊧̸A⊑B\\mathcal\{K\}\\not\\models A\\sqsubseteq B\. SocAc\_\{A\}is not inBℐ𝒦B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. ∎

###### Claim 2\.

ℐ𝒦⊧A⊑¬B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\neg Biff𝒦⊧A⊑¬B\\mathcal\{K\}\\models A\\sqsubseteq\\neg B\.

###### Proof\.

Assume𝒦⊧A⊑¬B\\mathcal\{K\}\\models A\\sqsubseteq\\neg B\.We make a case distinction based on the elements inΔℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\.

- •a∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}: Assumea∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thata∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. By assumption,𝒦⊧A⊑¬B\\mathcal\{K\}\\models A\\sqsubseteq\\neg B\. Also, by assumption𝒦\\mathcal\{K\}is satisfiable, meaning that𝒦⊧̸B​\(a\)\\mathcal\{K\}\\not\\models B\(a\)\. Then, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},a∉Bℐ𝒦a\\notin B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, that is,a∈\(¬B\)ℐ𝒦a\\in\(\\neg B\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Asaawas an arbitrary element of𝖭𝖨\{\\sf N\_\{I\}\}this holds for all elements of this kind\.
- •cD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}: AssumecD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatcD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. By assumption,𝒦⊧A⊑¬B\\mathcal\{K\}\\models A\\sqsubseteq\\neg B, so𝒦⊧D⊑¬B\\mathcal\{K\}\\models D\\sqsubseteq\\neg B\. AscD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}, by definition ofΔ𝒦\\Delta\_\{\\mathcal\{K\}\},DDis satisfiable w\.r\.t𝒦\\mathcal\{K\}\. So𝒦⊧̸D⊑B\\mathcal\{K\}\\not\\models D\\sqsubseteq B\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},cD∈\(¬B\)ℐ𝒦c\_\{D\}\\in\(\\neg B\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. SincecDc\_\{D\}was an arbitrary element ofΔ𝒦\\Delta\_\{\\mathcal\{K\}\}this holds for all elements of this kind\.

We have thus shown thatℐ𝒦⊧A⊑¬B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\neg B\.

Now, assume𝒦⊧̸A⊑¬B\\mathcal\{K\}\\not\\models A\\sqsubseteq\\neg B\.We show thatℐ𝒦⊧̸A⊑¬B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models A\\sqsubseteq\\neg B\. If𝒦⊧̸A⊑¬B\\mathcal\{K\}\\not\\models A\\sqsubseteq\\neg Bthen there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦\\mathcal\{K\}withAℐ∩BℐA^\{\\mathcal\{I\}\}\\cap B^\{\\mathcal\{I\}\}non\-empty\. This means thatA⊓BA\\sqcap Bis satisfiable w\.r\.t\.𝒦\\mathcal\{K\}and thuscA⊓B∈Δ𝒦c\_\{A\\sqcap B\}\\in\\Delta\_\{\\mathcal\{K\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatcA⊓B∈Aℐ𝒦c\_\{A\\sqcap B\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}andcA⊓B∈Bℐ𝒦c\_\{A\\sqcap B\}\\in B^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}since𝒦⊧A⊓B⊑A\\mathcal\{K\}\\models A\\sqcap B\\sqsubseteq Aand𝒦⊧A⊓B⊑B\\mathcal\{K\}\\models A\\sqcap B\\sqsubseteq B\. Soℐ𝒦⊧̸A⊑¬B\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models A\\sqsubseteq\\neg B\. ∎

###### Claim 3\.

ℐ𝒦⊧R⊑S\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models R\\sqsubseteq Siff𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S\.

###### Proof\.

Assume𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S\.We make a case distinction based on the elements inΔℐ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}and how they can be related in the extension of a role name in the definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}\.

- •\(a,b\)∈𝖭𝖨×𝖭𝖨\(a,b\)\\in\{\\sf N\_\{I\}\}\\times\{\\sf N\_\{I\}\}: Assume\(a,b\)∈Rℐ𝒦\(a,b\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. We first argue that in this case𝒦⊧R​\(a,b\)\\mathcal\{K\}\\models R\(a,b\)\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(a,b\)∈Rℐ𝒦​iff​𝒦⊧R​\(a,b\)\(a,b\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\\text\{ iff \}\\mathcal\{K\}\\models R\(a,b\)\. Since by assumption𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq Swe have that𝒦⊧S​\(a,b\)\\mathcal\{K\}\\models S\(a,b\), so\(a,b\)∈Sℐ𝒦\(a,b\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Since\(a,b\)\(a,b\)was an arbitrary pair in𝖭𝖨×𝖭𝖨\{\\sf N\_\{I\}\}\\times\{\\sf N\_\{I\}\}, this holds for all such kinds of pairs\.
- •\(a,c∃R′\)∈𝖭𝖨×Δ𝒦\(a,c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in\{\\sf N\_\{I\}\}\\times\\Delta\_\{\\mathcal\{K\}\}: Assume\(a,c∃R′\)∈Rℐ𝒦\(a,c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧∃R′¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{R^\{\\prime\}\}\(a\)and𝒦⊧R′¯⊑R\\mathcal\{K\}\\models\\overline\{R^\{\\prime\}\}\\sqsubseteq R\. By assumption𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S\. So𝒦⊧R′¯⊑S\\mathcal\{K\}\\models\\overline\{R^\{\\prime\}\}\\sqsubseteq S\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that\(a,c∃R′\)∈Sℐ𝒦\(a,c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Since\(a,c∃R′\)\(a,c\_\{\\exists\{R^\{\\prime\}\}\}\)was an arbitrary pair of this format in𝖭𝖨×Δ𝒦\{\\sf N\_\{I\}\}\\times\\Delta\_\{\\mathcal\{K\}\}, this holds for all such kinds of pairs\.
- •\(c∃R′,a\)∈Δ𝒦×𝖭𝖨\(c\_\{\\exists R^\{\\prime\}\},a\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\{\\sf N\_\{I\}\}: Assume\(c∃R′,a\)∈Rℐ𝒦\(c\_\{\\exists R^\{\\prime\}\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},𝒦⊧∃R′¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{R^\{\\prime\}\}\(a\)and𝒦⊧R′⊑R\\mathcal\{K\}\\models\{R^\{\\prime\}\}\\sqsubseteq R\. By assumption𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S\. So𝒦⊧R′⊑S\\mathcal\{K\}\\models R^\{\\prime\}\\sqsubseteq S\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that\(c∃R′,a\)∈Sℐ𝒦\(c\_\{\\exists R^\{\\prime\}\},a\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Since\(c∃R′,a\)\(c\_\{\\exists\{R^\{\\prime\}\}\},a\)was an arbitrary pair of this format in𝖭𝖨×Δ𝒦\{\\sf N\_\{I\}\}\\times\\Delta\_\{\\mathcal\{K\}\}, this argument can be applied for all such kinds of pairs\.
- •\(c∃R′,c∃R′¯\)∈Δ𝒦×Δ𝒦\(c\_\{\\exists R^\{\\prime\}\},c\_\{\\exists\\overline\{R^\{\\prime\}\}\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}: Assume\(c∃R′,c∃R′¯\)∈Rℐ𝒦\(c\_\{\\exists R^\{\\prime\}\},c\_\{\\exists\\overline\{R^\{\\prime\}\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧R′⊑R\\mathcal\{K\}\\models R^\{\\prime\}\\sqsubseteq R\. By assumption,𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S, so𝒦⊧R′⊑S\\mathcal\{K\}\\models R^\{\\prime\}\\sqsubseteq S\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(c∃R′,c∃R′¯\)∈Sℐ𝒦\(c\_\{\\exists R^\{\\prime\}\},c\_\{\\exists\\overline\{R^\{\\prime\}\}\}\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\.
- •\(cD,c∃R′\)∈Δ𝒦×Δ𝒦\(c\_\{D\},c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}: Assume\(cD,c∃R′\)∈Rℐ𝒦\(c\_\{D\},c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧D⊑∃R′¯\\mathcal\{K\}\\models D\\sqsubseteq\\exists\\overline\{R^\{\\prime\}\}and𝒦⊧R′¯⊑R\\mathcal\{K\}\\models\\overline\{R^\{\\prime\}\}\\sqsubseteq R\. By assumption,𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S, so𝒦⊧R′¯⊑S\\mathcal\{K\}\\models\\overline\{R^\{\\prime\}\}\\sqsubseteq S\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(cD,c∃R′\)∈Sℐ𝒦\(c\_\{D\},c\_\{\\exists\{R^\{\\prime\}\}\}\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\.
- •\(c∃R′,cD\)∈Δ𝒦×Δ𝒦\(c\_\{\\exists R^\{\\prime\}\},c\_\{D\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}: Assume\(c∃R′,cD\)∈Rℐ𝒦\(c\_\{\\exists\{R^\{\\prime\}\}\},c\_\{D\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧D⊑∃R′¯\\mathcal\{K\}\\models D\\sqsubseteq\\exists\\overline\{R^\{\\prime\}\}and𝒦⊧R′⊑R\\mathcal\{K\}\\models R^\{\\prime\}\\sqsubseteq R\. By assumption,𝒦⊧R⊑S\\mathcal\{K\}\\models R\\sqsubseteq S, so𝒦⊧R′⊑S\\mathcal\{K\}\\models R^\{\\prime\}\\sqsubseteq S\. Then, again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(c∃R′,cD\)∈Sℐ𝒦\(c\_\{\\exists\{R^\{\\prime\}\}\},c\_\{D\}\)\\in S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\.

We have thus shown thatℐ𝒦⊧R⊑S\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models R\\sqsubseteq S\.

Now, assume𝒦⊧̸R⊑S\\mathcal\{K\}\\not\\models R\\sqsubseteq S\.We show thatℐ𝒦⊧̸R⊑S\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models R\\sqsubseteq S\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that\{\(c∃S,c∃S¯\)∈Δ𝒦×Δ𝒦∣𝒦⊧S⊑R\}⊆Rℐ𝒦\\\{\(c\_\{\\exists S\},c\_\{\\exists\\overline\{S\}\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\mathcal\{K\}\\models S\\sqsubseteq R\\\}\\subseteq R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By takingS=RS=R\(and since trivially𝒦⊧R⊑R\\mathcal\{K\}\\models R\\sqsubseteq R\), we have in particular that\(c∃R,c∃R¯\)∈Rℐ𝒦\(c\_\{\\exists R\},c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. We now argue that\(c∃R,c∃R¯\)∉Sℐ𝒦\(c\_\{\\exists R\},c\_\{\\exists\\overline\{R\}\}\)\\notin S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, a pair of the form\(c∃S′,c∃S′¯\)\(c\_\{\\exists S^\{\\prime\}\},c\_\{\\exists\\overline\{S^\{\\prime\}\}\}\)is inSℐ𝒦S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧S′⊑S\\mathcal\{K\}\\models S^\{\\prime\}\\sqsubseteq S\. By assumption𝒦⊧̸R⊑S\\mathcal\{K\}\\not\\models R\\sqsubseteq S\. So\(c∃R,c∃R¯\)∉Sℐ𝒦\(c\_\{\\exists R\},c\_\{\\exists\\overline\{R\}\}\)\\notin S^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. ∎

###### Claim 4\.

ℐ𝒦⊧∃R⊑A\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R\\sqsubseteq Aiff𝒦⊧∃R⊑A\\mathcal\{K\}\\models\\exists R\\sqsubseteq A\.

###### Proof\.

Assume𝒦⊧∃R⊑A\\mathcal\{K\}\\models\\exists R\\sqsubseteq A\.We make a case distinction based on the elements inΔℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\.

- •a∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}: Assumea∈\(∃R\)ℐ𝒦a\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In this case, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, either \(1\) there isb∈𝖭𝖨b\\in\{\\sf N\_\{I\}\}such that\(a,b\)∈Rℐ𝒦\(a,b\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}or \(2\) there isc∃R¯∈Δ𝒦c\_\{\\exists\\overline\{R\}\}\\in\\Delta\_\{\\mathcal\{K\}\}such that\(a,c∃R¯\)∈Rℐ𝒦\(a,c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In case \(1\), by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(a,b\)∈Rℐ𝒦\(a,b\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}implies that𝒦⊧R​\(a,b\)\\mathcal\{K\}\\models R\(a,b\)\. Together with the assumption that𝒦⊧∃R⊑A\\mathcal\{K\}\\models\\exists R\\sqsubseteq A, this means that𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. Again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thata∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In case \(2\), by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(a,c∃R¯\)∈Rℐ𝒦\(a,c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}implies that𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)\. By assumption𝒦⊧∃R⊑A\\mathcal\{K\}\\models\\exists R\\sqsubseteq A, which means that𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. Again by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thata∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Sinceaawas an arbitrary element in𝖭𝖨\{\\sf N\_\{I\}\}, this argument can be applied for all elements of this kind\.
- •cD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}: AssumecD∈\(∃R\)ℐ𝒦c\_\{D\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. We first show that𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. IfcD∈\(∃R\)ℐ𝒦c\_\{D\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}then, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, either \(1\) there isa∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}such that\(cD,a\)∈Rℐ𝒦\(c\_\{D\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}or \(2\) there iscD′∈Δ𝒦c\_\{D^\{\\prime\}\}\\in\\Delta\_\{\\mathcal\{K\}\}such that\(cD,cD′\)∈Rℐ𝒦\(c\_\{D\},c\_\{D^\{\\prime\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In case \(1\), by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(cD,a\)∈Rℐ𝒦\(c\_\{D\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}means thatDDis of the form∃S\\exists S,𝒦⊧S⊑R\\mathcal\{K\}\\models S\\sqsubseteq R, and𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\)\. So𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. In case \(2\), by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, there are three possibilities: - \(a\)DDis of the form∃S\\exists SandD′D^\{\\prime\}is of the form∃S¯\\exists\\overline\{S\}; - \(b\)D′D^\{\\prime\}is of the form∃S\\exists Sand𝒦⊧D⊑∃S¯\\mathcal\{K\}\\models D\\sqsubseteq\\exists\\overline\{S\}; - \(c\)DDis of the form∃S\\exists Sand𝒦⊧D′⊑∃S¯\\mathcal\{K\}\\models D^\{\\prime\}\\sqsubseteq\\exists\\overline\{S\}\. In the sub\-cases \(a\) and \(c\) we also have that𝒦⊧S⊑R\\mathcal\{K\}\\models S\\sqsubseteq R\. So𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. In the sub\-case \(b\), we have that𝒦⊧S¯⊑R\\mathcal\{K\}\\models\\overline\{S\}\\sqsubseteq R\. Then again𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. By assumption𝒦⊧∃R⊑A\\mathcal\{K\}\\models\\exists R\\sqsubseteq A, which means that𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. Then,cD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}\. SincecDc\_\{D\}was an arbitrary element inΔ𝒦\\Delta\_\{\\mathcal\{K\}\}, this argument can be applied for all elements of this kind\.

We have thus shown that, for all elementsddinΔℐ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, ifd∈\(∃R\)ℐ𝒦d\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}thend∈Aℐ𝒦d\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Soℐ𝒦⊧∃R⊑A\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R\\sqsubseteq A\.

Now, assume𝒦⊧̸∃R⊑A\\mathcal\{K\}\\not\\models\\exists R\\sqsubseteq A\.If𝒦⊧̸∃R⊑A\\mathcal\{K\}\\not\\models\\exists R\\sqsubseteq Athen there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦\\mathcal\{K\}with\(∃R\)ℐ\(\\exists R\)^\{\\mathcal\{I\}\}non\-empty\. This means that∃R\\exists Ris satisfiable w\.r\.t\.𝒦\\mathcal\{K\}and thusc∃R∈Δ𝒦c\_\{\\exists R\}\\in\\Delta\_\{\\mathcal\{K\}\}\. We show thatℐ𝒦⊧̸∃R⊑A\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models\\exists R\\sqsubseteq Aby showing thatc∃R∈\(∃R\)ℐ𝒦c\_\{\\exists R\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}butc∃R∉Aℐ𝒦c\_\{\\exists R\}\\not\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By the definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(c∃S,c∃S¯\)∈Rℐ𝒦\(c\_\{\\exists S\},c\_\{\\exists\\overline\{S\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}if𝒦⊧S⊑R\\mathcal\{K\}\\models S\\sqsubseteq R, which is trivially the case forS=RS=R\. Soc∃R∈\(∃R\)ℐ𝒦c\_\{\\exists R\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. We now argue thatc∃R∉Aℐ𝒦c\_\{\\exists R\}\\not\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, an element of the formcDc\_\{D\}is inAℐ𝒦A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. By assumption𝒦⊧̸∃R⊑A\\mathcal\{K\}\\not\\models\\exists R\\sqsubseteq A\. Soc∃Rc\_\{\\exists R\}is not inAℐ𝒦A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. ∎

###### Claim 5\.

ℐ𝒦⊧A⊑∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\exists Riff𝒦⊧A⊑∃R\\mathcal\{K\}\\models A\\sqsubseteq\\exists R\.

###### Proof\.

Assume𝒦⊧A⊑∃R\\mathcal\{K\}\\models A\\sqsubseteq\\exists R\. We make a case distinction based on the elements inΔℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\.

- •a∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}: Assumea∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In this case, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. By assumption,𝒦⊧A⊑∃R\\mathcal\{K\}\\models A\\sqsubseteq\\exists R\. So𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)\. Then, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},\(a,c∃R¯\)∈Rℐ𝒦\(a,c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Thus,a∈\(∃R\)ℐ𝒦a\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, as required\. Sinceaawas an arbitrary element in𝖭𝖨\{\\sf N\_\{I\}\}, this argument can be applied for all elements of this kind\.
- •cD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}: AssumecD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In this case, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. By assumption,𝒦⊧A⊑∃R\\mathcal\{K\}\\models A\\sqsubseteq\\exists R, so𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. Then, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that\(cD,c∃S¯\)∈Rℐ𝒦\(c\_\{D\},c\_\{\\exists\\overline\{S\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}forS∈𝖭𝖱−S\\in\{\\sf N\_\{R\}^\{\-\}\}such that𝒦⊧S⊑R\\mathcal\{K\}\\models S\\sqsubseteq R\. TakingS=RS=Rthis trivially holds, so\(cD,c∃R¯\)∈Rℐ𝒦\(c\_\{D\},c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. This means thatcD∈\(∃R\)ℐ𝒦c\_\{D\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, as required\. SincecDc\_\{D\}was an arbitrary element inΔ𝒦\\Delta\_\{\\mathcal\{K\}\}, this argument can be applied for all elements of this kind\.

We have thus shown that, for all elementsddinΔℐ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, ifd∈Aℐ𝒦d\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}thend∈\(∃R\)ℐ𝒦d\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Soℐ𝒦⊧A⊑∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\exists R\.

Now, assume𝒦⊧̸A⊑∃R\\mathcal\{K\}\\not\\models A\\sqsubseteq\\exists R\.If𝒦⊧̸A⊑∃R\\mathcal\{K\}\\not\\models A\\sqsubseteq\\exists Rthen there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦\\mathcal\{K\}withAℐA^\{\\mathcal\{I\}\}non\-empty\. This means thatAAis satisfiable w\.r\.t\.𝒦\\mathcal\{K\}and thuscA∈Δ𝒦c\_\{A\}\\in\\Delta\_\{\\mathcal\{K\}\}\. We show thatℐ𝒦⊧̸A⊑∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models A\\sqsubseteq\\exists Rby showing thatcA∈Aℐ𝒦c\_\{A\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}butcA∉\(∃R\)ℐ𝒦c\_\{A\}\\not\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, an element of the formcDc\_\{D\}is inAℐ𝒦A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A, which is trivially the case forD=AD=A\. SocA∈Aℐ𝒦c\_\{A\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. We now argue thatcA∉\(∃R\)ℐ𝒦c\_\{A\}\\not\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\},cA∈\(∃R\)ℐ𝒦c\_\{A\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}iff𝒦⊧A⊑∃S\\mathcal\{K\}\\models A\\sqsubseteq\\exists Sand𝒦⊧S⊑R\\mathcal\{K\}\\models S\\sqsubseteq R\. By assumption𝒦⊧̸A⊑∃R\\mathcal\{K\}\\not\\models A\\sqsubseteq\\exists R\. So either𝒦⊧̸A⊑∃S\\mathcal\{K\}\\not\\models A\\sqsubseteq\\exists Sor𝒦⊧̸S⊑R\\mathcal\{K\}\\not\\models S\\sqsubseteq R\. Thus,cA∉\(∃R\)ℐ𝒦c\_\{A\}\\notin\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. ∎

###### Claim 6\.

ℐ𝒦⊧A⊑¬∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\neg\\exists Riff𝒦⊧A⊑¬∃R\\mathcal\{K\}\\models A\\sqsubseteq\\neg\\exists R\.

###### Proof\.

Assume𝒦⊧A⊑¬∃R\\mathcal\{K\}\\models A\\sqsubseteq\\neg\\exists R\. We make a case distinction based on the elements inΔℐ𝒦:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\.

- •a∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}: Assumea∈Aℐ𝒦a\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In this case, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. By assumption,𝒦⊧A⊑¬∃R\\mathcal\{K\}\\models A\\sqsubseteq\\neg\\exists R\. As𝒦\\mathcal\{K\}is satisfiable,𝒦⊧̸∃R​\(a\)\\mathcal\{K\}\\not\\models\\exists R\(a\)\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, to show thatℐ𝒦⊧̸∃R​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models\\exists R\(a\), we need to rule out the following two cases, whereR′∈𝖭𝖱R^\{\\prime\}\\in\{\\sf N\_\{R\}\}\. - –\(a,b\)∈R′⁣ℐ𝒦\(a,b\)\\in R^\{\\prime\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}andR′=RR^\{\\prime\}=R\. Since this would imply𝒦⊧R′​\(a,b\)\\mathcal\{K\}\\models R^\{\\prime\}\(a,b\)and𝒦⊧̸∃R​\(a\)\\mathcal\{K\}\\not\\models\\exists R\(a\), this cannot happen\. - –\(a,c∃S\)∈R′⁣ℐ𝒦\(a,c\_\{\\exists S\}\)\\in R^\{\\prime\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}andR′=RR^\{\\prime\}=R\. Since𝒦⊧̸∃R​\(a\)\\mathcal\{K\}\\not\\models\\exists R\(a\), there is no roleSSsuch that\(a,c∃S\)∈R′⁣ℐ𝒦\(a,c\_\{\\exists S\}\)\\in R^\{\\prime\\mathcal\{I\}\_\{\\mathcal\{K\}\}\},𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\),𝒦⊧S¯⊑R\\mathcal\{K\}\\models\\overline\{S\}\\sqsubseteq R\. So this case cannot happen because otherwise we would have that𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)\. - –\(c∃S,a\)∈R′⁣ℐ𝒦\(c\_\{\\exists S\},a\)\\in R^\{\\prime\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}andR′=R−R^\{\\prime\}=R^\{\-\}\. Since𝒦⊧̸∃R​\(a\)\\mathcal\{K\}\\not\\models\\exists R\(a\), there is no roleSSsuch that\(c∃S,a\)∈R′⁣ℐ𝒦\(c\_\{\\exists S\},a\)\\in R^\{\\prime\\mathcal\{I\}\_\{\\mathcal\{K\}\}\},𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\), and𝒦⊧S⊑R\\mathcal\{K\}\\models\{S\}\\sqsubseteq R\. So this case cannot happen because otherwise𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)would hold\. Then, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatℐ𝒦⊧̸∃R​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models\\exists R\(a\)\. Sinceaawas an arbitrary element in𝖭𝖨\{\\sf N\_\{I\}\}, this argument can be applied for all elements of this kind\.
- •cD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}: AssumecD∈Aℐ𝒦c\_\{D\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. In this case, by definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have that𝒦⊧D⊑A\\mathcal\{K\}\\models D\\sqsubseteq A\. By assumption,𝒦⊧A⊑¬∃R\\mathcal\{K\}\\models A\\sqsubseteq\\neg\\exists R, so𝒦⊧D⊑¬∃R\\mathcal\{K\}\\models D\\sqsubseteq\\neg\\exists R\. In the second item of the argument of Claim[4](https://arxiv.org/html/2605.23937#Thmclaim4), we have shown that ifcD∈\(∃R\)ℐ𝒦c\_\{D\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}then𝒦⊧D⊑∃R\\mathcal\{K\}\\models D\\sqsubseteq\\exists R\. AscD∈Δ𝒦c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}, we have thatDDis satisfiable\. So𝒦⊧̸D⊑∃R\\mathcal\{K\}\\not\\models D\\sqsubseteq\\exists R\. Then, by contrapositive, we have thatcD∉\(∃R\)ℐ𝒦c\_\{D\}\\notin\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\.

We have thus shown that, for all elementsddinΔℐ𝒦\\Delta^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}, ifd∈Aℐ𝒦d\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}thend∈\(¬∃R\)ℐ𝒦d\\in\(\\neg\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\. Soℐ𝒦⊧A⊑¬∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\neg\\exists R\.

Now, assume𝒦⊧̸A⊑¬∃R\\mathcal\{K\}\\not\\models A\\sqsubseteq\\neg\\exists R\.If𝒦⊧̸A⊑¬∃R\\mathcal\{K\}\\not\\models A\\sqsubseteq\\neg\\exists Rthen there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦\\mathcal\{K\}withAℐ∩\(∃R\)ℐA^\{\\mathcal\{I\}\}\\cap\(\\exists R\)^\{\\mathcal\{I\}\}non\-empty\. This means thatA⊓∃RA\\sqcap\\exists Ris satisfiable w\.r\.t\.𝒦\\mathcal\{K\}and thuscA⊓∃R∈Δ𝒦c\_\{A\\sqcap\\exists R\}\\in\\Delta\_\{\\mathcal\{K\}\}\. By definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}, we have thatcA⊓∃R∈Aℐ𝒦c\_\{A\\sqcap\\exists R\}\\in A^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}andcA⊓∃R∈\(∃R\)ℐ𝒦c\_\{A\\sqcap\\exists R\}\\in\(\\exists R\)^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}since𝒦⊧A⊓∃R⊑A\\mathcal\{K\}\\models A\\sqcap\\exists R\\sqsubseteq Aand𝒦⊧A⊓∃R⊑∃R\\mathcal\{K\}\\models A\\sqcap\\exists R\\sqsubseteq\\exists R\(in more details, we have that\(cA⊓∃R,c∃R−\)∈Rℐ𝒦\(c\_\{A\\sqcap\\exists R\},c\_\{\\exists R^\{\-\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\)\. Soℐ𝒦⊧̸A⊑¬∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\not\\models A\\sqsubseteq\\neg\\exists R\. ∎

###### Claim 7\.

ℐ𝒦⊧∃R⊑¬A\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R\\sqsubseteq\\neg Aiff𝒦⊧∃R⊑¬A\\mathcal\{K\}\\models\\exists R\\sqsubseteq\\neg A\.

###### Proof\.

Since𝒦⊧∃R⊑¬A\\mathcal\{K\}\\models\\exists R\\sqsubseteq\\neg Aiff𝒦⊧A⊑¬∃R\\mathcal\{K\}\\models A\\sqsubseteq\\neg\\exists Randℐ𝒦⊧∃R⊑¬A\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R\\sqsubseteq\\neg Aiffℐ𝒦⊧A⊑¬∃R\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\\sqsubseteq\\neg\\exists R, this claim follows from Claim[6](https://arxiv.org/html/2605.23937#Thmclaim6)\. ∎

Regarding the assertions, we have thatℐ𝒦⊧R​\(a,b\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models R\(a,b\)iff𝒦⊧R​\(a,b\)\\mathcal\{K\}\\models R\(a,b\)directly follows from the definition ofℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}\(Definition[1](https://arxiv.org/html/2605.23937#Thmdefinition1)\)\. The same holds forℐ𝒦⊧A​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models A\(a\)iff𝒦⊧A​\(a\)\\mathcal\{K\}\\models A\(a\)\. It remains to argue about assertions of the form∃R′​\(a\)\\exists R^\{\\prime\}\(a\)\(whereR′R^\{\\prime\}is a role name or its inverse\)\.

###### Claim 8\.

ℐ𝒦⊧∃R′​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R^\{\\prime\}\(a\)iff𝒦⊧∃R′​\(a\)\\mathcal\{K\}\\models\\exists R^\{\\prime\}\(a\)\.

###### Proof\.

We first show that if𝒦⊧∃R′​\(a\)\\mathcal\{K\}\\models\\exists R^\{\\prime\}\(a\)thenℐ𝒦⊧∃R′​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R^\{\\prime\}\(a\)\. We make a case distinction\.

- •R′=RR^\{\\prime\}=R\. If𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)andR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}then\(a,c∃R¯\)∈Rℐ𝒦\(a,c\_\{\\exists\\overline\{R\}\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\(takeS¯=R\\overline\{S\}=Rin Definition[1](https://arxiv.org/html/2605.23937#Thmdefinition1)\)\. Soℐ𝒦⊧∃R​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R\(a\)\.
- •R′=R−R^\{\\prime\}=R^\{\-\}\. If𝒦⊧∃R−​\(a\)\\mathcal\{K\}\\models\\exists R^\{\-\}\(a\)andR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}then\(c∃R,a\)∈Rℐ𝒦\(c\_\{\\exists\{R\}\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\(takeS¯=R−\\overline\{S\}=R^\{\-\}in Definition[1](https://arxiv.org/html/2605.23937#Thmdefinition1)\)\. Soℐ𝒦⊧∃R−​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R^\{\-\}\(a\)\.

Conversely, assumeℐ𝒦⊧∃R′​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R^\{\\prime\}\(a\)\. There are two cases\.

- •R′=RR^\{\\prime\}=Rand\(a,c∃S\)∈Rℐ𝒦\(a,c\_\{\\exists S\}\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}for someSSsuch that𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\)and𝒦⊧S¯⊑R\\mathcal\{K\}\\models\\overline\{S\}\\sqsubseteq R\. In this case,𝒦⊧∃R​\(a\)\\mathcal\{K\}\\models\\exists R\(a\)\. Then𝒦⊧∃R′​\(a\)\\mathcal\{K\}\\models\\exists R^\{\\prime\}\(a\)since hereR′=RR^\{\\prime\}=R\.
- •R′=R−R^\{\\prime\}=R^\{\-\}and\(c∃S,a\)∈Rℐ𝒦\(c\_\{\\exists S\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}for someSSsuch that𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\)and𝒦⊧S⊑R\\mathcal\{K\}\\models\{S\}\\sqsubseteq R\. In this case, note that\(c∃S,a\)∈Rℐ𝒦\(c\_\{\\exists S\},a\)\\in R^\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}meansℐ𝒦⊧∃R−​\(a\)\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\exists R^\{\-\}\(a\)\. As𝒦⊧∃S¯​\(a\)\\mathcal\{K\}\\models\\exists\\overline\{S\}\(a\)and𝒦⊧S⊑R\\mathcal\{K\}\\models\{S\}\\sqsubseteq Rwe have that𝒦⊧∃R−​\(a\)\\mathcal\{K\}\\models\\exists R^\{\-\}\(a\)so𝒦⊧∃R′​\(a\)\\mathcal\{K\}\\models\\exists R^\{\\prime\}\(a\)since hereR′=R−R^\{\\prime\}=R^\{\-\}\.

∎

We have completed now our proof with all DL\-LiteHaxioms\. ∎

###### Corollary 3\.

Let𝒦\\mathcal\{K\}be a satisfiable DL\-LiteHKB and letℐ𝒦′\\mathcal\{I\}^\{\\prime\}\_\{\\mathcal\{K\}\}be the result of modifying the canonical modelℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}of𝒦\\mathcal\{K\}by settingΔ𝒦\\Delta\_\{\\mathcal\{K\}\}as\{c∃R,c∃R−∣R∈𝖭𝖱\}\\\{c\_\{\\exists R\},c\_\{\\exists R^\{\-\}\}\\mid R\\in\{\\sf N\_\{R\}\}\\\}\. Then, for all DL\-LiteHaxiomsα\\alpha, we have that if𝒦⊧α\\mathcal\{K\}\\models\\alphathenℐ𝒦⊧α\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\alpha\.

###### Proof\.

The proof is the same as in[Theorem1](https://arxiv.org/html/2605.23937#Thmtheorem1), except that since here we only want one direction of the theorem \(showing that the interpretation satisfies𝒦\\mathcal\{K\}\) we do not require all the elements ofΔ𝒦\\Delta\_\{\\mathcal\{K\}\}used in[Theorem1](https://arxiv.org/html/2605.23937#Thmtheorem1)\. ∎

See[2](https://arxiv.org/html/2605.23937#Thmtheorem2)

###### Proof\.

\(Condition[i\)](https://arxiv.org/html/2605.23937#S3.I2.i1)\) What is to be shown is that for all concept embeddingsη​\(C\)∈𝖡𝗈𝗑\{\{\\eta\}\(C\)\}\\in\{\\sf Box\}withC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, there is a complementη​\(C\)¯∈𝖡𝗈𝗑\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\\in\{\\sf Box\}\. By the definition of the complement of boxes, we know thatη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}is:

\{𝐱∣\(−𝐬𝛀−𝐋𝐂\+ϵ\)≤d𝐱≤d\(𝐬𝛀−𝐔𝐂−ϵ\),𝐱∈ℝd\}\.\\\{\\mathbf\{x\}\\mid\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}\+\\boldsymbol\{\\epsilon\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\-\\boldsymbol\{\\epsilon\}\),\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}\.
Given[Definition2](https://arxiv.org/html/2605.23937#Thmdefinition2)and[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3), the bounds of the complement boxη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}are𝐋¬𝐂=−𝐬𝛀−𝐋𝐂\\mathbf\{L\_\{\\neg C\}\}=\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}and𝐔¬𝐂=𝐬𝛀−𝐔𝐂\\mathbf\{U\_\{\\neg C\}\}=\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\. The negation of a boxη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}is again a box, since: \(i\) we can compute the width ofη​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}as𝐔¬𝐂−𝐋¬𝐂=\(𝐬𝛀−𝐔𝐂\)−\(−𝐬𝛀−𝐋𝐂\)=2​𝐬𝛀−\(𝐔𝐂−𝐋𝐂\)\\mathbf\{U\_\{\\neg C\}\}\-\\mathbf\{L\_\{\\neg C\}\}=\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\)\-\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}\)=2\{\\mathbf\{s\_\{\\Omega\}\}\}\-\(\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\)and \(ii\) by the definition of the set𝖡𝗈𝗑\{\\sf Box\}of boxes we know that𝟎≤d\(𝐔𝐂−𝐋𝐂\)≤d2​𝐬𝛀\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\(\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}\. From \(i\) and \(ii\) we have that𝟎≤d2​𝐬𝛀−\(𝐔𝐂−𝐋𝐂\)≤d2​𝐬𝛀\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}\-\(\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}, proving thatη​\(C\)¯∈𝖡𝗈𝗑\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\\in\{\\sf Box\}\. Thus, we have shown that for all concept embeddingsη​\(C\)∈𝖡𝗈𝗑\{\{\\eta\}\(C\)\}\\in\{\\sf Box\}withC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, it holds thatη​\(C\)¯∈𝖡𝗈𝗑\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\\in\{\\sf Box\}\.

\(Condition[ii\)](https://arxiv.org/html/2605.23937#S3.I2.i2)\) We show thatη​\(C\)¯¯=η​\(C\)\{\\overline\{\{\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\}\}\}=\{\{\\eta\}\(C\)\}for allη​\(C\)∈𝖡𝗈𝗑\{\{\\eta\}\(C\)\}\\in\{\\sf Box\}withC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}\. We have thatη​\(C\)¯¯\{\\overline\{\{\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\}\}\}is

=\{𝐱∣𝐋𝐂\+ϵ≤d𝐱≤d𝐔𝐂−ϵ,𝐱∈ℝd\}¯¯\\displaystyle=\{\\overline\{\{\{\\overline\{\{\\\{\\mathbf\{x\}\\mid\\mathbf\{L\_\{C\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{C\}\}\-\\boldsymbol\{\\epsilon\},\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}\}\}\}\}\}\}=\{𝐱∣\(−𝐬𝛀−𝐋𝐂\+ϵ\)≤d𝐱≤d\(𝐬𝛀−𝐔𝐂−ϵ\),𝐱∈ℝd\}¯\\displaystyle=\{\\overline\{\{\\\{\\mathbf\{x\}\\mid\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}\+\\boldsymbol\{\\epsilon\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\-\\boldsymbol\{\\epsilon\}\),\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}\}\}\}=\{𝐱∣\(−𝐬𝛀−\(−𝐬𝛀−𝐋𝐂\)\+ϵ\)≤d𝐱≤d\\displaystyle=\\\{\\mathbf\{x\}\\mid\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\(\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}\)\+\\boldsymbol\{\\epsilon\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\(𝐬𝛀−\(𝐬𝛀−𝐔𝐂\)−ϵ\),𝐱∈ℝd\}\\displaystyle\\qquad\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\(\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{U\_\{C\}\}\)\-\\boldsymbol\{\\epsilon\}\),\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}=\{𝐱∣𝐋𝐂\+ϵ≤d𝐱≤d𝐔𝐂−ϵ,𝐱∈ℝd\}\\displaystyle=\\\{\\mathbf\{x\}\\mid\\mathbf\{L\_\{C\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{C\}\}\-\\boldsymbol\{\\epsilon\},\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\\}=η​\(C\)\.\\displaystyle=\{\{\\eta\}\(C\)\}\.\(Condition[iii\)](https://arxiv.org/html/2605.23937#S3.I2.i3)\) We show that ifη​\(C\)⊆η​\(D\)\{\{\\eta\}\(C\)\}\\subseteq\{\{\\eta\}\(D\)\}thenη​\(D\)¯⊆η​\(C\)¯\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}\\subseteq\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}, for allη​\(C\),η​\(D\)∈𝖡𝗈𝗑\{\{\\eta\}\(C\)\},\{\{\\eta\}\(D\)\}\\in\{\\sf Box\}withC,D∈𝖭𝖢∃C,D\\in\{\\sf N\_\{C\}^\{\\exists\}\}:

η​\(C\)⊆η​\(D\)\\displaystyle\{\{\\eta\}\(C\)\}\\subseteq\{\{\\eta\}\(D\)\}⇔\(𝐋𝐃≤d𝐋𝐂\)∧\(𝐔𝐂≤d𝐔𝐃\)\\displaystyle\\Leftrightarrow\(\\mathbf\{L\_\{D\}\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{L\_\{C\}\}\)\\land\(\\mathbf\{U\_\{C\}\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{D\}\}\)⇔\(\(𝐬𝛀\+𝐋𝐃\)≤d\(𝐬𝛀\+𝐋𝐂\)\)∧\(\(−𝐬𝛀\+𝐔𝐂\)≤d\\displaystyle\\Leftrightarrow\(\(\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\mathbf\{L\_\{D\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\(\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\mathbf\{L\_\{C\}\}\)\)\\land\(\(\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\mathbf\{U\_\{C\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\(−𝐬𝛀\+𝐔𝐃\)\)\\displaystyle\\qquad\(\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\+\\mathbf\{U\_\{D\}\}\)\)⇔\(\(−𝐬𝛀−𝐋𝐂\)≤d\(−𝐬𝛀−𝐋𝐃\)\)∧\(\(𝐬𝛀−𝐔𝐃\)≤d\\displaystyle\\Leftrightarrow\(\(\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\mathbf\{L\_\{C\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\(\-\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\mathbf\{L\_\{D\}\}\)\)\\land\(\(\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\mathbf\{U\_\{D\}\}\)\\mathrel\{\{\\leq\_\{d\}\}\}\(𝐬𝛀−𝐔𝐂\)\)\\displaystyle\\qquad\(\\mathbf\{\{\\mathbf\{s\_\{\\Omega\}\}\}\}\-\\mathbf\{U\_\{C\}\}\)\)⇔\(𝐋η​\(𝐂\)¯≤d𝐋η​\(𝐃\)¯\)∧\(𝐔η​\(𝐃\)¯≤d𝐔η​\(𝐂\)¯\)\\displaystyle\\Leftrightarrow\(\\mathbf\{L\_\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{L\_\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}\}\)\\land\(\\mathbf\{U\_\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\}\)⇔η​\(D\)¯⊆η​\(C\)¯\.\\displaystyle\\Leftrightarrow\{\\overline\{\{\{\{\\eta\}\(D\)\}\}\}\}\\subseteq\{\\overline\{\{\{\{\\eta\}\(C\)\}\}\}\}\.∎

## Appendix DProofs for Section[4](https://arxiv.org/html/2605.23937#S4)

We start by proving[Proposition1](https://arxiv.org/html/2605.23937#Thmproposition1)and[Proposition2](https://arxiv.org/html/2605.23937#Thmproposition2)stated in the main text of Section[4](https://arxiv.org/html/2605.23937#S4)\.

See[1](https://arxiv.org/html/2605.23937#Thmproposition1)

###### Proof\.

Letη\\etabe a box interpretation for a DL\-LiteHKB𝒦\\mathcal\{K\}with empty ABox and assumeη⊧𝒦\\eta\\models\\mathcal\{K\}\.

###### Claim 9\.

If𝒦\\mathcal\{K\}is a DL\-LiteHKB with an empty ABox then𝒦\\mathcal\{K\}is satisfiable\.

###### Proof\.

Letℐ\\mathcal\{I\}be an interpretation such that for allA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}and allR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}we have thatAℐ=Rℐ=∅A^\{\\mathcal\{I\}\}=R^\{\\mathcal\{I\}\}=\\emptyset\. Then, trivially,Cℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}andRℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}whereC,DC,Dare arbitrary DL\-LiteHconcepts andR,SR,Sare arbitrary roles with symbols in𝖭𝖢\{\\sf N\_\{C\}\}and𝖭𝖱\{\\sf N\_\{R\}\}\. This means that,ℐ⊧α\\mathcal\{I\}\\models\\alphafor all \(concept/role\) inclusionsα\\alphawith symbols in𝖭𝖢∪𝖭𝖱\{\\sf N\_\{C\}\}\\cup\{\\sf N\_\{R\}\}\. If𝒦\\mathcal\{K\}has an empty ABox, all axioms in𝒦\\mathcal\{K\}are inclusions and thus all are satisfied byℐ\\mathcal\{I\}\. In other words,ℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}, so𝒦\\mathcal\{K\}is satisfiable\. ∎

###### Claim 10\.

Assume𝒦\\mathcal\{K\}is a satisfiable DL\-LiteHKB with an empty ABox\. If𝒦∪\{α\}\\mathcal\{K\}\\cup\\\{\\alpha\\\}is unsatisfiable thenα\\alphais an assertion\.

###### Proof\.

Suppose𝒦∪\{α\}\\mathcal\{K\}\\cup\\\{\\alpha\\\}is unsatisfiable and𝒦\\mathcal\{K\}is a satisfiable DL\-LiteHKB with an empty ABox\. By the proof of Claim[9](https://arxiv.org/html/2605.23937#Thmclaim9), any interpretationℐ\\mathcal\{I\}withAℐ=Rℐ=∅A^\{\\mathcal\{I\}\}=R^\{\\mathcal\{I\}\}=\\emptyset, for allA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}and allR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}, satisfies not only𝒦\\mathcal\{K\}but also any extension of𝒦\\mathcal\{K\}with an inclusion\. Thus, if𝒦∪\{α\}\\mathcal\{K\}\\cup\\\{\\alpha\\\}is unsatisfiableα\\alphacannot be an inclusion\. That is,α\\alphamust be an assertion\. ∎

By Claim[9](https://arxiv.org/html/2605.23937#Thmclaim9),𝒦\\mathcal\{K\}is satisfiable\. By Claim[10](https://arxiv.org/html/2605.23937#Thmclaim10),𝒦∪\{α\}\\mathcal\{K\}\\cup\\\{\\alpha\\\}is unsatisfiable only ifα\\alphais an assertion\. Then the lemma follows since we only require TBox faithfulness, soα\\alphain Definition[5](https://arxiv.org/html/2605.23937#Thmdefinition5)ranges only over inclusions, not assertions\. ∎

See[2](https://arxiv.org/html/2605.23937#Thmproposition2)

###### Proof\.

Letη\\etabe a consistent box interpretation for a DL\-LiteHKB𝒦\\mathcal\{K\}and assumeη⊧𝒦\\eta\\models\\mathcal\{K\}\. Let𝒦′\\mathcal\{K\}^\{\\prime\}be\{α∣η⊧α\}\\\{\\alpha\\mid\\eta\\models\\alpha\\\}\. As𝖭𝖨,𝖭𝖢,𝖭𝖱\{\\sf N\_\{I\}\},\{\\sf N\_\{C\}\},\{\\sf N\_\{R\}\}are finite, there are finitely many axioms, so𝒦′\\mathcal\{K\}^\{\\prime\}is finite\. Asη\\etais box consistent, there is an interpretationℐ\\mathcal\{I\}that satisfies𝒦′\\mathcal\{K\}^\{\\prime\}\. Indeed,ℐ\\mathcal\{I\}can be constructed fromη\\etaas follows\.

- •LetΔℐ:=𝖭𝖨∪Δ𝒦\\Delta^\{\\mathcal\{I\}\}:=\{\\sf N\_\{I\}\}\\cup\\Delta\_\{\\mathcal\{K\}\}\(whereΔ𝒦\\Delta\_\{\\mathcal\{K\}\}is as in[Definition1](https://arxiv.org/html/2605.23937#Thmdefinition1)\)\.
- •For alla∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}, we have thataℐ:=aa^\{\\mathcal\{I\}\}:=a\. Moreover,
- •Aℐ:=\{a∈𝖭𝖨∣η⊧A​\(a\)\}∪\{cD∈Δ𝒦∣η⊧D⊑A\}A^\{\\mathcal\{I\}\}:=\\\{a\\in\{\\sf N\_\{I\}\}\\mid\\eta\\models A\(a\)\\\}\\cup\\\{c\_\{D\}\\in\\Delta\_\{\\mathcal\{K\}\}\\mid\\eta\\models D\\sqsubseteq A\\\}, for allA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}, and
- •Rℐ:=\{\(a,b\)∈𝖭𝖨×𝖭𝖨∣η⊧R​\(a,b\)\}∪\{\(a,c∃S\)∈𝖭𝖨×Δ𝒦∣η⊧∃S¯​\(a\),η⊧S¯⊑R\}∪\{\(c∃S,a\)∈Δ𝒦×𝖭𝖨∣η⊧∃S¯​\(a\),η⊧S⊑R\}∪\{\(c∃S,c∃S¯\)∈Δ𝒦×Δ𝒦∣η⊧S⊑R\}∪\{\(cD,c∃S\)∈Δ𝒦×Δ𝒦∣η⊧D⊑∃S¯,η⊧S¯⊑R\}∪\{\(c∃S,cD\)∈Δ𝒦×Δ𝒦∣η⊧D⊑∃S¯,η⊧S⊑R\}R^\{\\mathcal\{I\}\}:=\\\{\(a,b\)\\in\{\\sf N\_\{I\}\}\\times\{\\sf N\_\{I\}\}\\mid\\eta\\models R\(a,b\)\\\}\\cup\\\\ \\\{\(a,c\_\{\\exists S\}\)\\in\{\\sf N\_\{I\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\eta\\models\\exists\\overline\{S\}\(a\),\\ \\eta\\models\\overline\{S\}\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},a\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\{\\sf N\_\{I\}\}\\mid\\eta\\models\\exists\\overline\{S\}\(a\),\\ \\eta\\models S\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},c\_\{\\exists\\overline\{S\}\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\eta\\models S\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{D\},c\_\{\\exists S\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\{\\mid\}\\eta\\models D\\sqsubseteq\\exists\\overline\{S\},\\ \\eta\\models\\overline\{S\}\\sqsubseteq R\\\}\\cup\\\\ \\\{\(c\_\{\\exists S\},c\_\{D\}\)\\in\\Delta\_\{\\mathcal\{K\}\}\\times\\Delta\_\{\\mathcal\{K\}\}\\mid\\eta\\models D\\sqsubseteq\\exists\\overline\{S\},\\ \\eta\\models S\\sqsubseteq R\\\}, for allR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}\.

We can see thatℐ\\mathcal\{I\}is defined usingη\\etain the same wayℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}is defined using𝒦\\mathcal\{K\}in[Definition1](https://arxiv.org/html/2605.23937#Thmdefinition1)\(box consistency ofη\\etaensures thatℐ\\mathcal\{I\}is well\-defined\)\. One can then employ the same argument as in[Theorem1](https://arxiv.org/html/2605.23937#Thmtheorem1)to show thatη⊧α\\eta\\models\\alphaiffℐ⊧α\\mathcal\{I\}\\models\\alpha\. This means thatℐ⊧𝒦′\\mathcal\{I\}\\models\\mathcal\{K\}^\{\\prime\}\.

Ifη⊧α\\eta\\models\\alphathenα∈𝒦′\\alpha\\in\\mathcal\{K\}^\{\\prime\}\. Asℐ⊧𝒦′\\mathcal\{I\}\\models\\mathcal\{K\}^\{\\prime\}, we have thatℐ⊧α\\mathcal\{I\}\\models\\alpha\. By assumptionη⊧𝒦\\eta\\models\\mathcal\{K\}\. Then, by definition,𝒦′\\mathcal\{K\}^\{\\prime\}contains all axioms in𝒦\\mathcal\{K\}\. Asℐ⊧𝒦′\\mathcal\{I\}\\models\\mathcal\{K\}^\{\\prime\}, we have thatℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}\. This means thatℐ⊧𝒦∪\{α\}\\mathcal\{I\}\\models\\mathcal\{K\}\\cup\\\{\\alpha\\\}\. So𝒦∪\{α\}\\mathcal\{K\}\\cup\\\{\\alpha\\\}is satisfiable\. ∎

Table 4:Parameters for boxesI=I\_\{=\},I⊂I\_\{\\subset\},I⊃I\_\{\\supset\},I⊅I\_\{\\not\\supset\},I∩I\_\{\\cap\}, andΩ\{\\Omega\}in dimensioniCi\_\{C\}\.Table 5:Parameters for𝒫𝖢\\mathcal\{P\}\_\{\\mathsf\{C\}\}and𝒫𝖢¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}in dimensioniCi\_\{C\}\.Table 6:Parameters for boxesII,𝒮⊂\\mathcal\{S\}\_\{\\subset\},𝒮⊂¬\\mathcal\{S\}^\{\\neg\}\_\{\\subset\},ℬ𝖱,𝖨\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\},ℬ𝖱,𝖨¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}, andΩ\{\\Omega\}in dimensioniR,ai\_\{R,a\}\.Table 7:Parameters for point embeddings𝒫𝖱\\mathcal\{P\}\_\{\\mathsf\{R\}\},𝒫𝖱¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\},ℬ𝖱\\mathcal\{B\}\_\{\\mathsf\{R\}\}, andℬ𝖱¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}in dimensioniR,ai\_\{R,a\}\.iR,ai\_\{R,a\}−𝐬𝛀\-\{\\mathbf\{s\_\{\\Omega\}\}\}𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}II𝒮⊂\\mathcal\{S\}\_\{\\subset\}𝒮⊂¬\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}𝒫𝖱\\mathcal\{P\}\_\{\\mathsf\{R\}\}𝒫𝖱¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}ℬ𝖱¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}ℬ𝖱¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}

Figure 2:Visualization of the parameters ofII,𝒮⊂\\mathcal\{S\}\_\{\\subset\},𝒮⊂¬\\mathcal\{S\}^\{\\neg\}\_\{\\subset\},𝒫𝖱\\mathcal\{P\}\_\{\\mathsf\{R\}\},𝒫𝖱¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\},ℬ𝖱\\mathcal\{B\}\_\{\\mathsf\{R\}\},ℬ𝖱¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}, andΩ\{\\Omega\}in dimensioniR,ai\_\{R,a\}\.iCi\_\{C\}−𝐬𝛀\-\{\\mathbf\{s\_\{\\Omega\}\}\}𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}I=I\_\{=\}I⊂I\_\{\\subset\}I⊅I\_\{\\not\\supset\}I⊃I\_\{\\supset\}I∩I\_\{\\cap\}𝒫𝖢\\mathcal\{P\}\_\{\\mathsf\{C\}\}𝒫𝖢¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}

Figure 3:Visualization of the parameters ofI=I\_\{=\},I⊂I\_\{\\subset\},I⊃I\_\{\\supset\},I⊅I\_\{\\not\\supset\},I∩I\_\{\\cap\},𝒫𝖢\\mathcal\{P\}\_\{\\mathsf\{C\}\},𝒫𝖢¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}andΩ\{\\Omega\}in dimensioniCi\_\{C\}\.As already hinted in the main text, the proof strategy of[Theorem3](https://arxiv.org/html/2605.23937#Thmtheorem3)consists of first creating a mapping between finite interpretations and box interpretations and then using the canonical model for a KB to establish KB faithfulness\.[Definition6](https://arxiv.org/html/2605.23937#Thmdefinition6)describes this mapping\. The values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4),[5](https://arxiv.org/html/2605.23937#A4.T5),[6](https://arxiv.org/html/2605.23937#A4.T6), and[7](https://arxiv.org/html/2605.23937#A4.T7)satisfy the constraints in[Fig\.2](https://arxiv.org/html/2605.23937#A4.F2)and[Fig\.3](https://arxiv.org/html/2605.23937#A4.F3)\. Any choice of values that satisfy[Fig\.2](https://arxiv.org/html/2605.23937#A4.F2)and[Fig\.3](https://arxiv.org/html/2605.23937#A4.F3)could be used\.

###### Definition 6\.

Given an interpretationℐ\\mathcal\{I\}with finite domain, we define a box interpretationηℐ\\eta\_\{\\mathcal\{I\}\}in add\-dimensional Euclidean space, whered=\|𝖭𝖢∃\|\+\|𝖭𝖱\|⋅\|Δℐ\|d=\|\{\\sf N\_\{C\}^\{\\exists\}\}\|\+\|\{\\sf N\_\{R\}\}\|\\cdot\|\\Delta^\{\\mathcal\{I\}\}\|, as follows\. To each conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, we associate a dimension in the vector space and denote its index byiCi\_\{C\}\. Similarly, to each pair\(R,c\)\(R,c\)in𝖭𝖱×Δℐ\{\\sf N\_\{R\}\}\\times\\Delta^\{\\mathcal\{I\}\}, we associate a dimensioniR,ci\_\{R,c\}\. In our construction, we use constants defined in Tables[4](https://arxiv.org/html/2605.23937#A4.T4),[5](https://arxiv.org/html/2605.23937#A4.T5),[6](https://arxiv.org/html/2605.23937#A4.T6), and[7](https://arxiv.org/html/2605.23937#A4.T7)\. For eacha∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}, letηℐ​\(a\):=\{𝗉𝗈𝗌ℐ​\(a\),𝖻𝗎𝗆𝗉ℐ​\(a\)\}\\eta\_\{\\mathcal\{I\}\}\(a\):=\\\{\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\),\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\\}where:

- •for everyC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\},𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}ifaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}, and𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}ifaℐ∉Cℐa^\{\\mathcal\{I\}\}\\notin C^\{\\mathcal\{I\}\}; also𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]=ℬ𝖢\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{B\}\_\{\\mathsf\{C\}\}\(ifaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}or not\);
- •for every pair\(R,c\)\(R,c\)withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\},𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iR,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}if\(c,aℐ\)∈Rℐ\(c,a^\{\\mathcal\{I\}\}\)\\in R^\{\\mathcal\{I\}\}, otherwise𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iR,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\},𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}ifaℐ=ca^\{\\mathcal\{I\}\}=c, otherwise \(that is,aℐ≠ca^\{\\mathcal\{I\}\}\\neq c\)𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]=𝒫𝖱¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\.

For eachD∈𝖭𝖢∃D\\in\{\\sf N\_\{C\}^\{\\exists\}\}with∅=Dℐ\\emptyset=D^\{\\mathcal\{I\}\}, letηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(D\)be the box with all dimensions set toI0I\_\{0\}, and, for eachD∈𝖭𝖢​¬∃D\\in\{\{\\sf N\}\}^\{\\exists\}\_\{\\sf C\\neg\}with∅≠Dℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}, letηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(D\)be the box where:

- •for everyC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, - –ηℐ​\(D\)​\[iC\]:=I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]:=I\_\{=\}iff∅≠Dℐ=Cℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}=C^\{\\mathcal\{I\}\}, - –ηℐ​\(D\)​\[iC\]:=I⊂\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]:=I\_\{\\subset\}iff∅≠Dℐ⊂Cℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}\\subset C^\{\\mathcal\{I\}\}, - –ηℐ​\(D\)​\[iC\]:=I⊃\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]:=I\_\{\\supset\}iffDℐ⊃Cℐ≠∅D^\{\\mathcal\{I\}\}\\supset C^\{\\mathcal\{I\}\}\\neq\\emptyset, - –ηℐ​\(D\)​\[iC\]:=I⊅\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]:=I\_\{\\not\\supset\}iff∅≠Cℐ\\emptyset\\neq C^\{\\mathcal\{I\}\},∅≠Dℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}, and∅=Cℐ∩Dℐ\\emptyset=C^\{\\mathcal\{I\}\}\\cap D^\{\\mathcal\{I\}\}, - –elseηℐ​\(D\)​\[iC\]:=I∩\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]:=I\_\{\\cap\};
- •for every pair\(R,c\)\(R,c\)withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\},ηℐ​\(D\)​\[iR,c\]:=I\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{R,c\}\]:=I\.

For eachS∈𝖭𝖱S\\in\{\\sf N\_\{R\}\}, letηℐ​\(S\)\\eta\_\{\\mathcal\{I\}\}\(S\)be the boxes𝖧𝖾𝖺𝖽ℐ​\(S\)\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\},𝖳𝖺𝗂𝗅ℐ​\(S\)\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\},𝖡𝗎𝗆𝗉ℐ​\(S\)\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}:

- •for everyC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\},𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iC\]:=ηℐ​\(∃S\)​\[iC\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]:=\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)\[i\_\{C\}\],𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iC\]:=ηℐ​\(∃S−\)​\[iC\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]:=\\eta\_\{\\mathcal\{I\}\}\(\\exists S^\{\-\}\)\[i\_\{C\}\],𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iC\]:=I0\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]:=I\_\{0\};
- •for every pair\(R,c\)\(R,c\)withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}, - –𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iR,c\]:=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{S\}\_\{\\subset\}and𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iR,c\]:=ℬ𝖱,𝖨\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}if for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}implies\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}, otherwise𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iR,c\]:=𝒮⊂¬\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}and𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iR,c\]:=ℬ𝖱,𝖨¬\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}, - –𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iR,c\]:=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{S\}\_\{\\subset\}if for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(e,c\)∈Sℐ\(e,c\)\\in S^\{\\mathcal\{I\}\}implies\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}, otherwise𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iR,c\]:=𝒮⊂¬\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{R,c\}\]:=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\.

We first argue thatηℐ\\eta\_\{\\mathcal\{I\}\}is well\-defined\. That is,ηℐ\\eta\_\{\\mathcal\{I\}\}is indeed a box consistent interpretation\. We also show that it satisfies additional properties that are used in other proofs\.

###### Theorem 6\.

Letℐ\\mathcal\{I\}be an interpretation with finite domain, and letϵ\\epsilonbe an arbitrary value with0<ϵ≤ϵ𝑚𝑎𝑥0<\\epsilon\\leq\\epsilon\_\{\\mathit\{max\}\}\. Then,ηℐ\{\\eta\}\_\{\\mathcal\{I\}\}constructed fromℐ\\mathcal\{I\}, as in[Definition6](https://arxiv.org/html/2605.23937#Thmdefinition6), is a box interpretation that satisfies the following additional property\.

- •For any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, it holds that: 𝐋𝐂​\[iC\]\+𝐔𝐂​\[iC\]2≤−sΩ2,\\displaystyle\\frac\{\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]\+\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\}\{2\}\\leq\-\\frac\{\{s\_\{\\Omega\}\}\}\{2\},which implies thatηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is box consistent\.

###### Proof\.

Letℐ\\mathcal\{I\}be an interpretation with finite domain and letϵ\\epsilonbe an arbitrary value with0<ϵ≤ϵ𝑚𝑎𝑥0<\\epsilon\\leq\\epsilon\_\{\\mathit\{max\}\}, and recall that in Section[3](https://arxiv.org/html/2605.23937#S3)we assume thatϵ𝑚𝑎𝑥\\epsilon\_\{\\mathit\{max\}\}is bounded by0\.50\.5throughout this paper\. Also,Ω​\[iC\]=\(−4,4\)\{\\Omega\}\[i\_\{C\}\]=\(\-4,4\)\([Table6](https://arxiv.org/html/2605.23937#A4.T6)\) impliessΩ=4s\_\{\\Omega\}=4\. We first prove thatηℐ\\eta\_\{\\mathcal\{I\}\}as described in Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)is well\-defined, i\.e\., that it is a box interpretation \(c\.f\.[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3)\)\. We start by proving the conditions related to individual names in[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3), namely

- •each individual namea∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}is mapped to two vectorsη​\(a\)=\(𝗉𝗈𝗌​\(a\),𝖻𝗎𝗆𝗉​\(a\)\)\\eta\(a\)=\(\{\{\\sf pos\}\(a\)\},\{\{\\sf bump\}\(a\)\}\), namely, a position𝗉𝗈𝗌​\(e\)∈Ω\{\{\\sf pos\}\(e\)\}\\in\{\\Omega\}and a bump𝖻𝗎𝗆𝗉​\(e\)∈Ω\{\{\\sf bump\}\(e\)\}\\in\{\\Omega\}\.

Indeed[Definition6](https://arxiv.org/html/2605.23937#Thmdefinition6)maps eacha∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}to two vectors:𝗉𝗈𝗌ℐ​\(a\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)and𝖻𝗎𝗆𝗉ℐ​\(a\)\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\. To complete this item, we need to show that𝗉𝗈𝗌ℐ​\(a\),𝖻𝗎𝗆𝗉ℐ​\(a\)∈Ω\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\),\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\in\{\\Omega\}, which we do in[11](https://arxiv.org/html/2605.23937#Thmclaim11)\.

###### Claim 11\.

For any individuala∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}:

−𝐬𝛀\+ϵ≤d𝗉𝗈𝗌ℐ​\(a\)≤d𝐬𝛀−ϵ\\displaystyle\-\{\\mathbf\{s\_\{\\Omega\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\mathrel\{\{\\leq\_\{d\}\}\}\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\boldsymbol\{\\epsilon\}−𝐬𝛀\+ϵ≤d𝖻𝗎𝗆𝗉ℐ​\(a\)≤d𝐬𝛀−ϵ\\displaystyle\-\{\\mathbf\{s\_\{\\Omega\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\mathrel\{\{\\leq\_\{d\}\}\}\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\boldsymbol\{\\epsilon\}

###### Proof\.

For anya∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}and for any dimensioniiwith0≤i≤d0\\leq i\\leq d, Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)assigns \(i\)𝗉𝗈𝗌ℐ​\(a\)​\[i\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\]either to𝒫𝖢,𝒫𝖢¬,𝒫𝖱,\\mathcal\{P\}\_\{\\mathsf\{C\}\},\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\},\\mathcal\{P\}\_\{\\mathsf\{R\}\},or𝒫𝖱¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. Furthermore, by Tables[4](https://arxiv.org/html/2605.23937#A4.T4),[5](https://arxiv.org/html/2605.23937#A4.T5), and[7](https://arxiv.org/html/2605.23937#A4.T7)it holds that \(ii\)−sΩ\+ϵ𝑚𝑎𝑥≤𝒫𝖢,𝒫𝖢¬,𝒫𝖱,𝒫𝖱¬≤sΩ−ϵ𝑚𝑎𝑥\-\{s\_\{\\Omega\}\}\+\\epsilon\_\{\\mathit\{max\}\}\\leq\\mathcal\{P\}\_\{\\mathsf\{C\}\},\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\},\\mathcal\{P\}\_\{\\mathsf\{R\}\},\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\\leq\{s\_\{\\Omega\}\}\-\\epsilon\_\{\\mathit\{max\}\}\. By \(i\) and \(ii\) for alla∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}it holds that−𝐬𝛀\+ϵ≤d𝗉𝗈𝗌ℐ​\(a\)≤d𝐬𝛀−ϵ\-\{\\mathbf\{s\_\{\\Omega\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\mathrel\{\{\\leq\_\{d\}\}\}\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\boldsymbol\{\\epsilon\}\.

For anya∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}and for any dimensioniiwith0≤i≤d0\\leq i\\leq d, Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)assigns \(i\)𝖻𝗎𝗆𝗉ℐ​\(a\)​\[i\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\]either toℬ𝖱\\mathcal\{B\}\_\{\\mathsf\{R\}\}orℬ𝖱¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. Also, by Tables[5](https://arxiv.org/html/2605.23937#A4.T5),[6](https://arxiv.org/html/2605.23937#A4.T6), and[7](https://arxiv.org/html/2605.23937#A4.T7)it holds that \(ii\)−sΩ\+ϵ𝑚𝑎𝑥≤ℬ𝖱,ℬ𝖱¬≤sΩ−ϵ𝑚𝑎𝑥\-\{s\_\{\\Omega\}\}\+\\epsilon\_\{\\mathit\{max\}\}\\leq\\mathcal\{B\}\_\{\\mathsf\{R\}\},\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}\\leq\{s\_\{\\Omega\}\}\-\\epsilon\_\{\\mathit\{max\}\}\. By \(i\) and \(ii\), for alla∈𝖭𝖨a\\in\{\\sf N\_\{I\}\},−𝐬𝛀\+ϵ≤d𝖻𝗎𝗆𝗉ℐ​\(a\)≤d𝐬𝛀−ϵ\-\{\\mathbf\{s\_\{\\Omega\}\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\mathrel\{\{\\leq\_\{d\}\}\}\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\boldsymbol\{\\epsilon\}\. ∎

We now proceed with the second item of[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3)\.

- •Each concept nameA∈𝖭𝖢A\\in\{\\sf N\_\{C\}\}is mapped toη​\(A\)∈𝖡𝗈𝗑\{\{\\eta\}\(A\)\}\\in\{\\sf Box\}\.

To show thatη​\(A\)∈𝖡𝗈𝗑\{\{\\eta\}\(A\)\}\\in\{\\sf Box\}we need[12](https://arxiv.org/html/2605.23937#Thmclaim12)to hold\.

###### Claim 12\.

For any conceptC∈𝖭𝖢C\\in\{\\sf N\_\{C\}\}, it holds that:

𝟎≤d𝐔𝐂−𝐋𝐂≤d2​𝐬𝛀\\displaystyle\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}\(in fact this holds for allC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}\)\.

###### Proof\.

For any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and for any dimensioniiwith0≤i≤d0\\leq i\\leq d,[Definition6](https://arxiv.org/html/2605.23937#Thmdefinition6)assignsηℐ​\(C\)​\[i\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\]either toI0,I=,I⊂,I⊃,I⊅,I∩,or​II\_\{0\},I\_\{=\},I\_\{\\subset\},I\_\{\\supset\},I\_\{\\not\\supset\},I\_\{\\cap\},\\text\{ or \}I\. Then, by Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[6](https://arxiv.org/html/2605.23937#A4.T6), for\(𝐋𝐂​\[i\],𝐔𝐂​\[i\]\):=ηℐ​\(C\)​\[i\]\(\\mathbf\{L\_\{C\}\}\[i\],\\mathbf\{U\_\{C\}\}\[i\]\):=\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\]it holds that0≤𝐔𝐂​\[i\]−𝐋𝐂​\[i\]≤2​sΩ0\\leq\\mathbf\{U\_\{C\}\}\[i\]\-\\mathbf\{L\_\{C\}\}\[i\]\\leq 2\{s\_\{\\Omega\}\}\. Thus,𝟎≤d𝐔𝐂−𝐋𝐂≤d2​𝐬𝛀\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\_\{C\}\}\-\\mathbf\{L\_\{C\}\}\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}\. ∎

The third item of[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3)is:

- •each role nameR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}is mapped to three boxesη\(R\)=\(𝖧𝖾𝖺𝖽\(R\),𝖳𝖺𝗂𝗅\(R\),𝖡𝗎𝗆𝗉\(R\)\{\{\\eta\}\(R\)\}=\(\{\{\\sf Head\}\(R\)\},\{\{\\sf Tail\}\(R\)\},\{\{\\sf Bump\}\(R\)\}\), which we callRR’s head𝖧𝖾𝖺𝖽​\(R\)\{\{\\sf Head\}\(R\)\}, tail𝖳𝖺𝗂𝗅​\(R\)\{\{\\sf Tail\}\(R\)\}and bump box𝖡𝗎𝗆𝗉​\(R\)\{\{\\sf Bump\}\(R\)\}\.

We show this in[13](https://arxiv.org/html/2605.23937#Thmclaim13)\.

###### Claim 13\.

For any roleR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}, it holds that:

𝟎≤d𝐔𝐑X−𝐋𝐑X≤d2​𝐬𝛀​𝗐𝗂𝗍𝗁​X∈\{𝐇,𝐓,𝐁\}\.\\displaystyle\\mathbf\{0\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U\}^\{X\}\_\{\\mathbf\{R\}\}\-\\mathbf\{L\}^\{X\}\_\{\\mathbf\{R\}\}\\mathrel\{\{\\leq\_\{d\}\}\}2\{\\mathbf\{s\_\{\\Omega\}\}\}\\;\{\\sf with\}\\;X\\in\\\{\\mathbf\{H\},\\mathbf\{T\},\\mathbf\{B\}\\\}\.

###### Proof\.

For anyR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}and for any dimensioniiwith0≤i≤d0\\leq i\\leq d, Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)assigns𝖧𝖾𝖺𝖽ℐ​\(R\)​\[i\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\]and𝖳𝖺𝗂𝗅ℐ​\(R\)​\[i\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\]either to𝒮⊂,𝒮⊂¬,I0,I=,I⊂,I⊃,I⊅,\\mathcal\{S\}\_\{\\subset\},\\mathcal\{S\}^\{\\neg\}\_\{\\subset\},\{I\_\{0\},\}I\_\{=\},I\_\{\\subset\},I\_\{\\supset\},I\_\{\\not\\supset\},orI∩I\_\{\\cap\}\. Then, by Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[6](https://arxiv.org/html/2605.23937#A4.T6), for\(𝐋𝐑𝐇​\[i\],𝐔𝐑𝐇​\[i\]\):=𝖧𝖾𝖺𝖽ℐ​\(R\)​\[i\]\(\\mathbf\{L^\{H\}\_\{R\}\}\[i\],\\mathbf\{U^\{H\}\_\{R\}\}\[i\]\):=\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\]it holds that0≤𝐔𝐑𝐇​\[i\]−𝐋𝐑𝐇​\[i\]≤2​sΩ0\\leq\\mathbf\{U^\{H\}\_\{R\}\}\[i\]\-\\mathbf\{L^\{H\}\_\{R\}\}\[i\]\\leq 2\{s\_\{\\Omega\}\}, and for\(𝐋𝐑𝐓​\[i\],𝐔𝐑𝐓​\[i\]\):=𝖳𝖺𝗂𝗅ℐ​\(R\)​\[i\]\(\\mathbf\{L^\{T\}\_\{R\}\}\[i\],\\mathbf\{U^\{T\}\_\{R\}\}\[i\]\):=\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\]it holds that0≤𝐔𝐑𝐓​\[i\]−𝐋𝐑𝐓​\[i\]≤2​sΩ0\\leq\\mathbf\{U^\{T\}\_\{R\}\}\[i\]\-\\mathbf\{L^\{T\}\_\{R\}\}\[i\]\\leq 2\{s\_\{\\Omega\}\}\.

For anyR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}and for any dimensioniiwith0≤i≤d0\\leq i\\leq d, Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)assigns𝖻𝗎𝗆𝗉ℐ​\(R\)​\[i\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(R\)\[i\]either toI0,ℬ𝖱,𝖨I\_\{0\},\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}, orℬ𝖱,𝖨¬\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. Then, by Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[6](https://arxiv.org/html/2605.23937#A4.T6), for\(𝐋𝐑𝐁​\[i\],𝐔𝐑𝐁​\[i\]\):=𝖻𝗎𝗆𝗉ℐ​\(R\)​\[i\]\(\\mathbf\{L^\{B\}\_\{R\}\}\[i\],\\mathbf\{U^\{B\}\_\{R\}\}\[i\]\):=\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(R\)\[i\]it holds that0≤𝐔𝐑𝐁​\[i\]−𝐋𝐑𝐁​\[i\]≤2​sΩ0\\leq\\mathbf\{U^\{B\}\_\{R\}\}\[i\]\-\\mathbf\{L^\{B\}\_\{R\}\}\[i\]\\leq 2\{s\_\{\\Omega\}\}\. ∎

The extension ofηℐ\\eta\_\{\\mathcal\{I\}\}to arbitrary DL\-LiteHconcept and role expressions is as in[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3), though, for presentation purposes we did not explicitly define the head and tail boxes for the dimensioniCi\_\{C\}\(this was defined in terms ofηℐ​\(∃S\)\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)andηℐ​\(∃S−\)\\eta\_\{\\mathcal\{I\}\}\(\\exists S^\{\-\}\)\)\. So here we need to argue thatηℐ​\(∃S\)\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)is equal to

\{𝐱∈ℝd∣𝐋𝐒𝐇−𝐔𝐒𝐁\+ϵ≤d𝐱≤d𝐔𝐒𝐇−𝐋𝐒𝐁−ϵ\}\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\\mathbf\{L^\{H\}\_\{S\}\}\-\\mathbf\{U^\{B\}\_\{S\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U^\{H\}\_\{S\}\}\-\\mathbf\{L^\{B\}\_\{S\}\}\-\\boldsymbol\{\\epsilon\}\\\}\(recall that𝐋∃𝐒=𝐋𝐒𝐇−𝐔𝐒𝐁\\mathbf\{L\_\{\\exists S\}\}=\\mathbf\{L^\{H\}\_\{S\}\}\-\\mathbf\{U^\{B\}\_\{S\}\}and𝐔∃𝐒=𝐔𝐒𝐇−𝐋𝐒𝐁\\mathbf\{U\_\{\\exists S\}\}=\\mathbf\{U^\{H\}\_\{S\}\}\-\\mathbf\{L^\{B\}\_\{S\}\}see[Definition3](https://arxiv.org/html/2605.23937#Thmdefinition3)\)\. Indeed, by the values in Table[6](https://arxiv.org/html/2605.23937#A4.T6), for the dimensions of the formiR,ci\_\{R,c\}we have that𝐋𝐒𝐇​\[iR,c\]−𝐔𝐒𝐁​\[iR,c\]=−2\\mathbf\{L^\{H\}\_\{S\}\}\[i\_\{R,c\}\]\-\\mathbf\{U^\{B\}\_\{S\}\}\[i\_\{R,c\}\]=\-2and𝐔𝐒𝐇​\[iR,c\]−𝐋𝐒𝐁​\[iR,c\]=2\\mathbf\{U^\{H\}\_\{S\}\}\[i\_\{R,c\}\]\-\\mathbf\{L^\{B\}\_\{S\}\}\[i\_\{R,c\}\]=2, which correspond toII, the value ofηℐ​\(∃S\)​\[iR,c\]\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)\[i\_\{R,c\}\]\. For the dimensions of the formiCi\_\{C\},𝐋𝐒𝐇​\[iC\]−𝐔𝐒𝐁​\[iC\]=𝐋𝐒𝐇​\[iC\]\\mathbf\{L^\{H\}\_\{S\}\}\[i\_\{C\}\]\-\\mathbf\{U^\{B\}\_\{S\}\}\[i\_\{C\}\]=\\mathbf\{L^\{H\}\_\{S\}\}\[i\_\{C\}\]and𝐔𝐒𝐇​\[iC\]−𝐋𝐒𝐁​\[iC\]=𝐔𝐒𝐇​\[iC\]\\mathbf\{U^\{H\}\_\{S\}\}\[i\_\{C\}\]\-\\mathbf\{L^\{B\}\_\{S\}\}\[i\_\{C\}\]=\\mathbf\{U^\{H\}\_\{S\}\}\[i\_\{C\}\]since in this case𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iC\]:=I0\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]:=I\_\{0\}, which is\(0,0\)\(0,0\), by the values in Table[4](https://arxiv.org/html/2605.23937#A4.T4)\. This is as required as in this case we have𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iC\]=ηℐ​\(∃S\)​\[iC\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)\[i\_\{C\}\]\. The argument thatηℐ​\(∃S−\)\\eta\_\{\\mathcal\{I\}\}\(\\exists S^\{\-\}\)is equal to

\{𝐱∈ℝd∣𝐋𝐒𝐓−𝐔𝐒𝐁\+ϵ≤d𝐱≤d𝐔𝐒𝐓−𝐋𝐒𝐁−ϵ\}\\\{\\mathbf\{x\}\\in\{\{\\mathbb\{R\}\}^\{d\}\}\\mid\\mathbf\{L^\{T\}\_\{S\}\}\-\\mathbf\{U^\{B\}\_\{S\}\}\+\\boldsymbol\{\\epsilon\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{x\}\\mathrel\{\{\\leq\_\{d\}\}\}\\mathbf\{U^\{T\}\_\{S\}\}\-\\mathbf\{L^\{B\}\_\{S\}\}\-\\boldsymbol\{\\epsilon\}\\\}is similar\. It remains to argue the following\.

###### Claim 14\.

For any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, it holds that:

𝐋𝐂​\[iC\]\+𝐔𝐂​\[iC\]2≤−sΩ2\\displaystyle\\frac\{\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]\+\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\}\{2\}\\leq\-\\frac\{\{s\_\{\\Omega\}\}\}\{2\}\(9\)and that this impliesηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is box consistent\.

###### Proof\.

For any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and dimensioniCi\_\{C\}, \(i\) Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)assignsηℐ​\(C\)​\[iC\]=I=\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{C\}\]=I\_\{=\}\. Then, by Table[4](https://arxiv.org/html/2605.23937#A4.T4)it holds that \(ii\) the lower bound ofI=I\_\{=\}is set to−4\-4and the upper bound ofI=I\_\{=\}is set to−0\.5\-0\.5\. By \(i\) and \(ii\), we have that \(iii\)𝐋𝐂​\[iC\]\+𝐔𝐂​\[iC\]2=−2\.25≤−sΩ/2=−2\\frac\{\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]\+\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\}\{2\}=\-2\.25\\leq\-\{s\_\{\\Omega\}\}/2=\-2\. From \(i\), \(ii\), and \(iii\), for any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}, it holds that𝐋𝐂​\[iC\]\+𝐔𝐂​\[iC\]2≤−sΩ2\\frac\{\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]\+\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\}\{2\}\\leq\-\\frac\{\{s\_\{\\Omega\}\}\}\{2\}, proving \(Equation \([9](https://arxiv.org/html/2605.23937#A4.E9)\)\)\.

Finally, we briefly check that \([9](https://arxiv.org/html/2605.23937#A4.E9)\) guarantees box consistency\. LetCCbe an arbitrary conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and assume that \([9](https://arxiv.org/html/2605.23937#A4.E9)\) holds\. Next, recall that𝐋¬𝐂=−𝐬𝛀−𝐋𝐂\\mathbf\{L\_\{\\neg C\}\}=\-\{\\mathbf\{s\_\{\\Omega\}\}\}\-\\mathbf\{L\_\{C\}\}holds \(see Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\)\. Now we have that:

𝐔𝐂​\[iC\]\\displaystyle\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]≤−sΩ−𝐋𝐂​\[iC\]=𝐋¬𝐂​\[iC\]\\displaystyle\\leq\-\{s\_\{\\Omega\}\}\-\\mathbf\{L\_\{C\}\}\[i\_\{C\}\]=\\mathbf\{L\_\{\\neg C\}\}\[i\_\{C\}\]\(10\)
In particular,

𝐔𝐂​\[iC\]−ϵ<𝐋¬𝐂​\[iC\]\+ϵ,\\mathbf\{U\_\{C\}\}\[i\_\{C\}\]\-\\epsilon<\\mathbf\{L\_\{\\neg C\}\}\[i\_\{C\}\]\+\\epsilon,which means that the boxes defined byCCand¬C\\neg Cdo not intersect in dimensioniCi\_\{C\}\. This implies thatηℐ​\(C\)∩ηℐ​\(¬C\)≠∅\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\\cap\{\\eta\}\_\{\\mathcal\{I\}\}\(\\neg C\)\\neq\\emptyset, proving thatηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is box consistent\. ∎

This finishes the proof of this theorem\. ∎

[Fig\.3](https://arxiv.org/html/2605.23937#A4.F3)and[Fig\.2](https://arxiv.org/html/2605.23937#A4.F2)help to visualize how the constants defined in Tables[4](https://arxiv.org/html/2605.23937#A4.T4),[5](https://arxiv.org/html/2605.23937#A4.T5),[6](https://arxiv.org/html/2605.23937#A4.T6), and[7](https://arxiv.org/html/2605.23937#A4.T7)relate to each other\. In the next lemmas, we argue thatℐ⊧α\\mathcal\{I\}\\models\\alphaiffηℐ⊧α\\eta\_\{\\mathcal\{I\}\}\\models\\alphawhenα\\alphais a concept assertion, role assertion, concept inclusion, or role inclusion\.

###### Lemma 1\.

Given an interpretationℐ\\mathcal\{I\}with finite domain, letηℐ\\eta\_\{\\mathcal\{I\}\}be as in Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\. For alla∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}and all conceptsD∈𝖭𝖢∃D\\in\{\\sf N\_\{C\}^\{\\exists\}\},ℐ⊧D​\(a\)\\mathcal\{I\}\\models D\(a\)iffηℐ⊧D​\(a\)\\eta\_\{\\mathcal\{I\}\}\\models D\(a\)\.

###### Proof\.

We need to show thataℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}iff𝗉𝗈𝗌ℐ​\(a\)∈ηℐ​\(D\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\. \(⇒\\Rightarrow\) Supposeaℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}\. To show that𝗉𝗈𝗌ℐ​\(a\)∈ηℐ​\(D\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\in\\eta\_\{\\mathcal\{I\}\}\(D\), we argue that𝗉𝗈𝗌ℐ​\(a\)​\[j\]∈ηℐ​\(D\)​\[j\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[j\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[j\], for every dimension1≤j≤d1\\leq j\\leq d\. We make a case distinction withC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}\.

- •DimensioniCi\_\{C\}withaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}\.In this case, \(i\)𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}\. By the parameters of the boxes of dimensioniCi\_\{C\}\(see Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5), and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\), it holds that \(ii\)𝒫𝖢∈I=∩I⊃∩I⊂∩I∩\\mathcal\{P\}\_\{\\mathsf\{C\}\}\\in I\_\{=\}\\cap I\_\{\\supset\}\\cap I\_\{\\subset\}\\cap I\_\{\\cap\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(D\)​\[iC\]\{\\eta\}\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]is either equal toI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\(since by the \(⇒\\Rightarrow\) assumptionDℐ≠∅D^\{\\mathcal\{I\}\}\\neq\\emptyset\)\. By the \(⇒\\Rightarrow\) assumption,aℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}and by the assumption of this caseaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have thatηℐ​\(D\)​\[iC\]\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]is notI⊅I\_\{\\not\\supset\}becauseaℐ∈Cℐ∩Dℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}\\cap D^\{\\mathcal\{I\}\}; so it holds that \(iii\)ηℐ​\(D\)​\[iC\]≠I⊅\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]\\neq I\_\{\\not\\supset\}\. By \(i\)\-\(iii\) it holds that𝗉𝗈𝗌ℐ​\(a\)​\[iC\]∈ηℐ​\(D\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]\.
- •DimensioniCi\_\{C\}withaℐ∉Cℐa^\{\\mathcal\{I\}\}\\not\\in C^\{\\mathcal\{I\}\}\.In this case, \(i\)𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}\. By the parameters of the boxes of dimensioniCi\_\{C\}\(see Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5), and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\), it holds that \(ii\)𝒫𝖢¬∈I⊅∩I⊃∩I∩\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}\\in I\_\{\\not\\supset\}\\cap I\_\{\\supset\}\\cap I\_\{\\cap\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(D\)​\[iC\]\{\\eta\}\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]is either equal toI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}, \(since by the \(⇒\\Rightarrow\) assumptionDℐ≠∅D^\{\\mathcal\{I\}\}\\neq\\emptyset\)\. By the \(⇒\\Rightarrow\) assumption,aℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}and by the assumption of this caseaℐ∉Cℐa^\{\\mathcal\{I\}\}\\not\\in C^\{\\mathcal\{I\}\}\. Also, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)we have thatηℐ​\(D\)​\[iC\]\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]is neitherI=I\_\{=\}norI⊂I\_\{\\subset\}because the former requiresCℐ=DℐC^\{\\mathcal\{I\}\}=D^\{\\mathcal\{I\}\}, the latter requiresDℐ⊂CℐD^\{\\mathcal\{I\}\}\\subset C^\{\\mathcal\{I\}\}, but in the case we consider here we haveaℐ∈Dℐ∖Cℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}\\setminus C^\{\\mathcal\{I\}\}; so \(iii\)ηℐ​\(D\)​\[iC\]≠I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]\\neq I\_\{=\}and \(iv\)ηℐ​\(D\)​\[iC\]≠I⊂\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]\\neq I\_\{\\subset\}\. By \(i\)\-\(iv\) it holds that𝗉𝗈𝗌ℐ​\(a\)​\[iC\]∈ηℐ​\(D\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{C\}\]\.

Finally, consider the dimensions of the formiR,ci\_\{R,c\}, whereR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\. In this case \(i\)ηℐ​\(D\)​\[iR,c\]=I\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{R,c\}\]=Ifor anyR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\. Furthermore, for any individual name in𝖭𝖨\{\\sf N\_\{I\}\}, in particularaa, we have that \(ii\)𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}or𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]=𝒫𝖱¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. By the parameters ofII,𝒫𝖱\\mathcal\{P\}\_\{\\mathsf\{R\}\}, and𝒫𝖱¬\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\(see Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7), and Figure[2](https://arxiv.org/html/2605.23937#A4.F2)\), it holds that \(iii\)𝒫𝖱∈I\\mathcal\{P\}\_\{\\mathsf\{R\}\}\\in Iand𝒫𝖱¬∈I\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\\in I\. By \(i\)\-\(iii\) it holds that𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]∈ηℐ​\(D\)​\[iR,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{R,c\}\]provided thatϵ≤ϵ𝑚𝑎𝑥\\epsilon\\leq\\epsilon\_\{\\mathit\{max\}\}\(see Table[7](https://arxiv.org/html/2605.23937#A4.T7)and Section[3](https://arxiv.org/html/2605.23937#S3)\)\. We have shown that𝗉𝗈𝗌ℐ​\(a\)​\[j\]∈ηℐ​\(D\)​\[j\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[j\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[j\]for any dimension1≤j≤d1\\leq j\\leq d\. Thus, we have shown that𝗉𝗈𝗌ℐ​\(a\)∈ηℐ​\(D\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\in\\eta\_\{\\mathcal\{I\}\}\(D\)ifaℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}\.

\(⇐\\Leftarrow\) Now suppose𝗉𝗈𝗌ℐ​\(a\)∈ηℐ​\(D\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\. This means, for dimensioniDi\_\{D\}in particular, that \(i\)𝗉𝗈𝗌ℐ​\(a\)​\[iD\]∈ηℐ​\(D\)​\[iD\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{D\}\]\\in\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{D\}\]\. By the construction of dimensioniDi\_\{D\}ofηℐ\\eta\_\{\\mathcal\{I\}\}it holds that \(ii\)ηℐ​\(D\)​\[iD\]=I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{D\}\]=I\_\{=\}\. From \(i\) and \(ii\),𝗉𝗈𝗌ℐ​\(a\)​\[iD\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{D\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}\. By the construction of dimensioniDi\_\{D\}𝗉𝗈𝗌ℐ​\(a\)​\[iD\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{D\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}can only be ifaℐ∈Dℐa^\{\\mathcal\{I\}\}\\in D^\{\\mathcal\{I\}\}\. ∎

###### Lemma 2\.

Given an interpretationℐ\\mathcal\{I\}with finite domain, letηℐ\\eta\_\{\\mathcal\{I\}\}be as in Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\. For alla,b∈𝖭𝖨a,b\\in\{\\sf N\_\{I\}\}and allR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\},ℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\)iffηℐ⊧R​\(a,b\)\\eta\_\{\\mathcal\{I\}\}\\models R\(a,b\)\.

###### Proof\.

\(⇒\\Rightarrow\) Assumeℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\)\. We start by showing that𝗉𝗈𝗌ℐ​\(a\)\+𝖻𝗎𝗆𝗉ℐ​\(b\)∈𝖧𝖾𝖺𝖽ℐ​\(R\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}by showing that this holds in every dimension\. For this we make a case distinction, whereC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\},S∈𝖭𝖱S\\in\{\\sf N\_\{R\}\}, andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\.

- •DimensioniCi\_\{C\}\.We want to show that𝗉𝗈𝗌ℐ​\(a\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]∈𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]=ηℐ​\(∃R\)​\[iC\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists R\)\[i\_\{C\}\]can be either equal toI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\(since by the \(⇒\\Rightarrow\) assumptionℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\), so\(∃R\)ℐ≠∅\(\\exists R\)^\{\\mathcal\{I\}\}\\neq\\emptyset\)\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}ifaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}, and𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}ifaℐ∉Cℐa^\{\\mathcal\{I\}\}\\notin C^\{\\mathcal\{I\}\}; also𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]=ℬ𝖢\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{B\}\_\{\\mathsf\{C\}\}\(ifaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}or not\)\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5), we have that all possible values for𝗉𝗈𝗌ℐ​\(a\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]are inI⊃∩I∩I\_\{\\supset\}\\cap I\_\{\\cap\}\. So we need to consider three cases, namely, \(i\) when𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]=I=\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{=\}, \(ii\) when𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]=I⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{\\subset\}, and \(iii\) when𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]=I⊅\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{\\not\\supset\}\. By the \(⇒\\Rightarrow\) assumption,ℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\), soaℐ∈\(∃R\)ℐa^\{\\mathcal\{I\}\}\\in\(\\exists R\)^\{\\mathcal\{I\}\}\. In case \(i\) and in case \(ii\), by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have that\(∃R\)ℐ=Cℐ\(\\exists R\)^\{\\mathcal\{I\}\}=C^\{\\mathcal\{I\}\}and\(∃R\)ℐ⊂Cℐ\(\\exists R\)^\{\\mathcal\{I\}\}\\subset C^\{\\mathcal\{I\}\}, respectively\. So, in both casesaℐ∈Cℐa^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}and, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5),𝗉𝗈𝗌ℐ​\(a\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]∈I=∩I⊂\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\\in I\_\{=\}\\cap I\_\{\\subset\}\. In case \(iii\),\(∃R\)ℐ∩Cℐ=∅\(\\exists R\)^\{\\mathcal\{I\}\}\\cap C^\{\\mathcal\{I\}\}=\\emptyset\. Soaℐ∉Cℐa^\{\\mathcal\{I\}\}\\notin C^\{\\mathcal\{I\}\}and, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(a\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5),𝗉𝗈𝗌ℐ​\(a\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]∈I⊅\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\\in I\_\{\\not\\supset\}\.
- •DimensioniS,ci\_\{S,c\}withaℐ=ca^\{\\mathcal\{I\}\}=cand for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\},\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), whenaℐ=ca^\{\\mathcal\{I\}\}=c, we have that𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}and𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}\. Also, in this case𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]is𝒮⊂\\mathcal\{S\}\_\{\\subset\}\. By the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]∈𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.
- •DimensioniS,ci\_\{S,c\}withaℐ=ca^\{\\mathcal\{I\}\}=cand there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}, with\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}but\(c,e\)∉Sℐ\(c,e\)\\notin S^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), whenaℐ=ca^\{\\mathcal\{I\}\}=c, we have that𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}and𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]=𝒮⊂¬\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also,𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. In both cases, by the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]∈𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.
- •DimensioniS,ci\_\{S,c\}withaℐ≠ca^\{\\mathcal\{I\}\}\\neq c\.In this case, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]=𝒫𝖱¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. It can be that𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}\_\{\\subset\}or𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]=𝒮⊂¬\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also, it can be that𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}\(depending onℐ\\mathcal\{I\}\) but in all cases, by the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(a\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]∈𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\], as required\.

We now show that𝗉𝗈𝗌ℐ​\(b\)\+𝖻𝗎𝗆𝗉ℐ​\(a\)∈𝖳𝖺𝗂𝗅ℐ​\(R\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}by showing that this holds in every dimension\. We make a similar case distinction as in the argument above\.

- •DimensioniCi\_\{C\}\.We want to show that𝗉𝗈𝗌ℐ​\(b\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]∈𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]=ηℐ​\(∃R−\)​\[iC\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists R^\{\-\}\)\[i\_\{C\}\]can be either equal toI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\(since by the \(⇒\\Rightarrow\) assumptionℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\), so\(∃R−\)ℐ≠∅\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}\\neq\\emptyset\)\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(b\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}ifbℐ∈Cℐb^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}, and𝗉𝗈𝗌ℐ​\(b\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}ifbℐ∉Cℐb^\{\\mathcal\{I\}\}\\notin C^\{\\mathcal\{I\}\}; also𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]=ℬ𝖢\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{B\}\_\{\\mathsf\{C\}\}\(ifbℐ∈Cℐb^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}or not\)\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5), we have that all possible values for𝗉𝗈𝗌ℐ​\(b\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]are inI⊃∩I∩I\_\{\\supset\}\\cap I\_\{\\cap\}\. So we need to consider three cases, namely, \(i\) when𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]=I=\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{=\}, \(ii\) when𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]=I⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{\\subset\}, and \(iii\) when𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]=I⊅\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{\\not\\supset\}\. By the \(⇒\\Rightarrow\) assumption,ℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\), sobℐ∈\(∃R−\)ℐb^\{\\mathcal\{I\}\}\\in\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}\. In case \(i\) and in case \(ii\), by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have that\(∃R−\)ℐ=Cℐ\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}=C^\{\\mathcal\{I\}\}and\(∃R−\)ℐ⊂Cℐ\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}\\subset C^\{\\mathcal\{I\}\}, respectively\. So, in both casesbℐ∈Cℐb^\{\\mathcal\{I\}\}\\in C^\{\\mathcal\{I\}\}and, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(b\)​\[iC\]=𝒫𝖢\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{P\}\_\{\\mathsf\{C\}\}\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5),𝗉𝗈𝗌ℐ​\(b\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]∈I=∩I⊂\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\\in I\_\{=\}\\cap I\_\{\\subset\}\. In case \(iii\),\(∃R−\)ℐ∩Cℐ=∅\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}\\cap C^\{\\mathcal\{I\}\}=\\emptyset\. Sobℐ∉Cℐb^\{\\mathcal\{I\}\}\\notin C^\{\\mathcal\{I\}\}and, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(b\)​\[iC\]=𝒫𝖢¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{C\}\}\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5),𝗉𝗈𝗌ℐ​\(b\)​\[iC\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]∈I⊅\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]\\in I\_\{\\not\\supset\}\.
- •DimensioniS,ci\_\{S,c\}withbℐ=cb^\{\\mathcal\{I\}\}=cand for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\},\(e,c\)∈Rℐ\(e,c\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), whenbℐ=cb^\{\\mathcal\{I\}\}=c, we have that𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}and𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. We also have that𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}becauseℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\)and in our particular case this also impliesℐ⊧R​\(b,a\)\\mathcal\{I\}\\models R\(b,a\)\. By the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]∈𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.
- •DimensioniS,ci\_\{S,c\}withbℐ=cb^\{\\mathcal\{I\}\}=cand there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}, with\(e,c\)∈Rℐ\(e,c\)\\in R^\{\\mathcal\{I\}\}but\(c,e\)∉Sℐ\(c,e\)\\notin S^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), whenbℐ=cb^\{\\mathcal\{I\}\}=c, we have that𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}and𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]=𝒮⊂¬\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also,𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. In both cases, by the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]∈𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.
- •DimensioniS,ci\_\{S,c\}withbℐ≠cb^\{\\mathcal\{I\}\}\\neq c\.In this case, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]=𝒫𝖱¬\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]=\\mathcal\{P\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. It can be that𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}\_\{\\subset\}or𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]=𝒮⊂¬\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also, it can be that𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}\(depending onℐ\\mathcal\{I\}\) but in all cases, by the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝗉𝗈𝗌ℐ​\(b\)​\[iS,c\]\+𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]∈𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iS,c\]\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.

It remains to show that𝖻𝗎𝗆𝗉ℐ​\(a\)∈𝖡𝗎𝗆𝗉ℐ​\(R\)\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}and𝖻𝗎𝗆𝗉ℐ​\(b\)∈𝖡𝗎𝗆𝗉ℐ​\(R\)\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\. We again make a case distinction\.

- •DimensioniCi\_\{C\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have that𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iC\]=I0\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=I\_\{0\}and𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]=𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]=ℬ𝖢\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]=\\mathcal\{B\}\_\{\\mathsf\{C\}\}\. By the values in Tables[4](https://arxiv.org/html/2605.23937#A4.T4)and[5](https://arxiv.org/html/2605.23937#A4.T5),𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iC\]=𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iC\]∈𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iC\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{C\}\]=\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{C\}\]\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]\.
- •DimensioniS,ci\_\{S,c\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have that𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}and the same for𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\. Also,𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iS,c\]=ℬ𝖱,𝖨\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}or𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iS,c\]=ℬ𝖱,𝖨¬\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. In all cases, by the values in Tables[6](https://arxiv.org/html/2605.23937#A4.T6)and[7](https://arxiv.org/html/2605.23937#A4.T7),𝖻𝗎𝗆𝗉ℐ​\(a\)​\[iS,c\]∈𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iS,c\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{S,c\}\]\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]and𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iS,c\]∈𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iS,c\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{S,c\}\]\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\.

We have thus shown that

𝗉𝗈𝗌ℐ​\(a\)\+𝖻𝗎𝗆𝗉ℐ​\(b\)\\displaystyle\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)∈𝖧𝖾𝖺𝖽ℐ​\(R\)\\displaystyle\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}𝗉𝗈𝗌ℐ​\(b\)\+𝖻𝗎𝗆𝗉ℐ​\(a\)\\displaystyle\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(b\)\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)∈𝖳𝖺𝗂𝗅ℐ​\(R\)\\displaystyle\\in\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}𝖻𝗎𝗆𝗉ℐ​\(a\)\\displaystyle\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(a\)∈𝖡𝗎𝗆𝗉ℐ​\(R\)\\displaystyle\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}𝖻𝗎𝗆𝗉ℐ​\(b\)\\displaystyle\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)∈𝖡𝗎𝗆𝗉ℐ​\(R\)\.\\displaystyle\\in\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\.Then, by Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4), we have thatηℐ⊧R​\(a,b\)\\eta\_\{\\mathcal\{I\}\}\\models R\(a,b\)\.

\(⇐\\Leftarrow\) Assumeηℐ⊧R​\(a,b\)\\eta\_\{\\mathcal\{I\}\}\\models R\(a,b\)\. Letc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}be the element such thataℐ=ca^\{\\mathcal\{I\}\}=c\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]=𝒫𝖱\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]=\\mathcal\{P\}\_\{\\mathsf\{R\}\}and𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iR,c\]:=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{R,c\}\]:=\\mathcal\{S\}\_\{\\subset\}\. Also, either𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iR,c\]=ℬ𝖱\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{R,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\}\}or𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iR,c\]=ℬ𝖱¬\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{R,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\}\}\. By Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4),𝗉𝗈𝗌ℐ​\(a\)\+𝖻𝗎𝗆𝗉ℐ​\(b\)∈𝖧𝖾𝖺𝖽ℐ​\(R\)\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\. This means, in particular, that for the dimensioniR,ci\_\{R,c\}

𝗉𝗈𝗌ℐ​\(a\)​\[iR,c\]\+𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iR,c\]∈𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iR,c\]\.\\mathsf\{pos\}\_\{\\mathcal\{I\}\}\(a\)\[i\_\{R,c\}\]\+\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{R,c\}\]\\in\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{R,c\}\]\.The only possible value of𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iR,c\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{R,c\}\]that satisfies the conditions above isℬ𝖱\\mathcal\{B\}\_\{\\mathsf\{R\}\}, which is assigned to𝖻𝗎𝗆𝗉ℐ​\(b\)​\[iR,c\]\\mathsf\{bump\}\_\{\\mathcal\{I\}\}\(b\)\[i\_\{R,c\}\]iff\(c,bℐ\)∈Rℐ\(c,b^\{\\mathcal\{I\}\}\)\\in R^\{\\mathcal\{I\}\}\. Sinceaℐ=ca^\{\\mathcal\{I\}\}=c, we have that\(aℐ,bℐ\)∈Rℐ\(a^\{\\mathcal\{I\}\},b^\{\\mathcal\{I\}\}\)\\in R^\{\\mathcal\{I\}\}\. In other words,ℐ⊧R​\(a,b\)\\mathcal\{I\}\\models R\(a,b\)\. ∎

###### Lemma 3\.

Given an interpretationℐ\\mathcal\{I\}with finite domain, letηℐ\\eta\_\{\\mathcal\{I\}\}be as in Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\. For all DL\-LiteHCIsC⊑DC\\sqsubseteq D,ℐ⊧C⊑D\\mathcal\{I\}\\models C\\sqsubseteq Diffηℐ⊧C⊑D\\eta\_\{\\mathcal\{I\}\}\\models C\\sqsubseteq D\.

###### Proof\.

We need to show thatCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}iffηℐ​\(C\)⊆ηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(C\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\. \(⇒\\Rightarrow\) SupposeCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. To show thatηℐ​\(C\)⊆ηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(C\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\), we argue thatηℐ​\(C\)​\[j\]⊆ηℐ​\(D\)​\[j\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[j\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[j\], for every dimension1≤j≤d1\\leq j\\leq d\. First assume∅=Dℐ\\emptyset=D^\{\\mathcal\{I\}\}\. Then, by the \(⇒\\Rightarrow\) assumption,∅=Cℐ\\emptyset=C^\{\\mathcal\{I\}\}\. By the box definition \(see Section[3](https://arxiv.org/html/2605.23937#S3)\), we have thatηℐ​\(C\)=ηℐ​\(D\)=∅\\eta\_\{\\mathcal\{I\}\}\(C\)=\\eta\_\{\\mathcal\{I\}\}\(D\)=\\emptyset\(whenϵ\>0\\epsilon\>0, which is assumed to be the case in this work\)\. Soηℐ​\(C\)⊆ηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(C\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\. We now make a case distinction with∅≠Dℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}andE∈𝖭𝖢∃E\\in\{\\sf N\_\{C\}^\{\\exists\}\}\.

- •DimensioniEi\_\{E\}with∅≠Dℐ=Eℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}=E^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(E\)​\[iE\]=I=\\eta\_\{\\mathcal\{I\}\}\(E\)\[i\_\{E\}\]=I\_\{=\}\. By the assumption in this case \(i\)Dℐ=EℐD^\{\\mathcal\{I\}\}=E^\{\\mathcal\{I\}\}, so \(ii\)ηℐ​\(D\)​\[iE\]=I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{=\}by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\. By the \(⇒\\Rightarrow\) assumption,Cℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. We have two cases\. - –IfCℐ=DℐC^\{\\mathcal\{I\}\}=D^\{\\mathcal\{I\}\}then, by \(i\),Cℐ=EℐC^\{\\mathcal\{I\}\}=E^\{\\mathcal\{I\}\}\. So, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), we have thatηℐ​\(C\)​\[iE\]=I=\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=I\_\{=\}, which impliesηℐ​\(C\)​\[iE\]=ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]and thenηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\], as required\. - –Otherwise,Cℐ⊂DℐC^\{\\mathcal\{I\}\}\\subset D^\{\\mathcal\{I\}\}\. Then, by \(i\),Cℐ⊂EℐC^\{\\mathcal\{I\}\}\\subset E^\{\\mathcal\{I\}\}\. IfCℐ≠∅C^\{\\mathcal\{I\}\}\\neq\\emptysetthen by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(C\)​\[iE\]=I⊂\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=I\_\{\\subset\}\. By \(ii\),ηℐ​\(D\)​\[iE\]=I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{=\}\. SinceI⊂⊆I=I\_\{\\subset\}\\subseteq I\_\{=\}\(see Table[4](https://arxiv.org/html/2605.23937#A4.T4)and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\),ηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]\. Otherwise \(whenCℐ=∅C^\{\\mathcal\{I\}\}=\\emptyset\) thenηℐ​\(C\)=∅\\eta\_\{\\mathcal\{I\}\}\(C\)=\\emptysetand we are done\.
- •DimensioniEi\_\{E\}with∅≠Dℐ⊂Eℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}\\subset E^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)and the assumption in this case \(i\)ηℐ​\(D\)​\[iE\]=I⊂\{\\eta\}\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{\\subset\}\. By the \(⇒\\Rightarrow\) assumptionCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}and by the assumption in this caseDℐ⊂EℐD^\{\\mathcal\{I\}\}\\subset E^\{\\mathcal\{I\}\}, soCℐ⊂EℐC^\{\\mathcal\{I\}\}\\subset E^\{\\mathcal\{I\}\}\. IfCℐ≠∅C^\{\\mathcal\{I\}\}\\neq\\emptysetthen by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), \(ii\)ηℐ​\(C\)​\[iE\]=I⊂\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=I\_\{\\subset\}\. Then, by \(i\) and \(ii\) it holds thatηℐ​\(C\)​\[iE\]=ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]soηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]\. Otherwise \(whenCℐ=∅C^\{\\mathcal\{I\}\}=\\emptyset\),ηℐ​\(C\)=∅\\eta\_\{\\mathcal\{I\}\}\(C\)=\\emptysetand we are done\.
- •DimensioniEi\_\{E\}withDℐ⊃Eℐ≠∅D^\{\\mathcal\{I\}\}\\supset E^\{\\mathcal\{I\}\}\\neq\\emptyset\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)and the assumption in this case \(i\)ηℐ​\(D\)​\[iE\]=I⊃\{\\eta\}\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{\\supset\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)ηℐ​\(C\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]is either equal to \(ii\)∅\\emptyset,I=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\. By the chosen parameters ofI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, andI⊅I\_\{\\not\\supset\}\(see Table[4](https://arxiv.org/html/2605.23937#A4.T4)and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\) it holds that \(iii\)I=∪I⊃∪I⊂∪I∩∪I⊅⊆I⊃I\_\{=\}\\cup I\_\{\\supset\}\\cup I\_\{\\subset\}\\cup I\_\{\\cap\}\\cup I\_\{\\not\\supset\}\\subseteq I\_\{\\supset\}\. Then, by \(i\)\-\(iii\), any possible value ofηℐ​\(C\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]leads toηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\{\\eta\}\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\], as required\.
- •DimensioniEi\_\{E\}with∅≠Eℐ\\emptyset\\neq E^\{\\mathcal\{I\}\},∅≠Dℐ\\emptyset\\neq D^\{\\mathcal\{I\}\}, and∅=Eℐ∩Dℐ\\emptyset=E^\{\\mathcal\{I\}\}\\cap D^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)and the assumption in this case \(i\)ηℐ​\(D\)​\[iE\]=I⊅\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{\\not\\supset\}\. By the \(⇒\\Rightarrow\) assumptionCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}, so∅=Eℐ∩Cℐ\\emptyset=E^\{\\mathcal\{I\}\}\\cap C^\{\\mathcal\{I\}\}\. If∅≠Cℐ\\emptyset\\neq C^\{\\mathcal\{I\}\}then by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\(ii\)ηℐ​\(C\)​\[iE\]=I⊅\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=I\_\{\\not\\supset\}\. By \(i\) and \(ii\)ηℐ​\(C\)​\[iE\]=ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\], soηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]\. The case whereCℐ=∅C^\{\\mathcal\{I\}\}=\\emptysetis as argued above\.
- •DimensioniEi\_\{E\}with none of the above\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)and the assumption in this case \(i\)ηℐ​\(D\)​\[iE\]=I∩\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]=I\_\{\\cap\}\. Again by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(C\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]is either equal to \(ii\)∅\\emptyset,I=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\. By the chosen parameters ofI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, andI⊅I\_\{\\not\\supset\}\(see Table[4](https://arxiv.org/html/2605.23937#A4.T4)and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\) it holds that \(iii\)I⊂∪I∩∪I⊅⊆I∩I\_\{\\subset\}\\cup I\_\{\\cap\}\\cup I\_\{\\not\\supset\}\\subseteq I\_\{\\cap\}\(also∅⊆I∩\\emptyset\\subseteq I\_\{\\cap\}\)\. Ifηℐ​\(C\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]is∅\\emptyset,I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}then by \(i\)\-\(iii\)ηℐ​\(C\)​\[iE\]⊆ηℐ​\(D\)​\[iE\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{E\}\]\. Otherwiseηℐ​\(C\)​\[iE\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{E\}\]is eitherI=I\_\{=\}orI⊃I\_\{\\supset\}\. The former case happens iffCℐ=Eℐ≠∅C^\{\\mathcal\{I\}\}=E^\{\\mathcal\{I\}\}\\neq\\emptyset\. However, by the \(⇒\\Rightarrow\) assumption, we have thatCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. SoCℐ=Eℐ≠∅C^\{\\mathcal\{I\}\}=E^\{\\mathcal\{I\}\}\\neq\\emptysetimplies∅≠Eℐ⊆Dℐ\\emptyset\\neq E^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}, which cannot happen by the “none of the above” assumption in this case\. The latter case happens iffCℐ⊃Eℐ≠∅C^\{\\mathcal\{I\}\}\\supset E^\{\\mathcal\{I\}\}\\neq\\emptyset\. Again by the \(⇒\\Rightarrow\) assumption, we have thatCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. SoCℐ⊃Eℐ≠∅C^\{\\mathcal\{I\}\}\\supset E^\{\\mathcal\{I\}\}\\neq\\emptysetimplies∅≠Eℐ⊆Dℐ\\emptyset\\neq E^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}, which cannot happen by the “none of the above” assumption\.

It remains to argue for the dimensions of the formiR,ci\_\{R,c\}, whereR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), for every pair\(R,c\)\(R,c\)withR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}andc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\},ηℐ​\(C\)​\[iR,c\]=ηℐ​\(D\)​\[iR,c\]=I\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{R,c\}\]=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{R,c\}\]=I\. So, for every dimension of this form,ηℐ​\(C\)​\[iR,c\]⊆ηℐ​\(D\)​\[iR,c\]\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{R,c\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{R,c\}\]\.

\(⇐\\Leftarrow\) Now supposeηℐ​\(C\)⊆ηℐ​\(D\)\\eta\_\{\\mathcal\{I\}\}\(C\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(D\)\. Ifηℐ​\(C\)=∅\\eta\_\{\\mathcal\{I\}\}\(C\)=\\emptysetthen, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), this happens iffCℐ=∅C^\{\\mathcal\{I\}\}=\\emptysetand we are done since this trivially impliesCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. Ifηℐ​\(D\)=∅\\eta\_\{\\mathcal\{I\}\}\(D\)=\\emptysetthen, by the \(⇐\\Leftarrow\) assumption,ηℐ​\(C\)=∅\\eta\_\{\\mathcal\{I\}\}\(C\)=\\emptyset\. Since, again by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), this happens iffCℐ=∅C^\{\\mathcal\{I\}\}=\\emptysetwe are done since this trivially impliesCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}\. We can then assume that bothηℐ​\(C\)≠∅\\eta\_\{\\mathcal\{I\}\}\(C\)\\neq\\emptysetandηℐ​\(D\)≠∅\\eta\_\{\\mathcal\{I\}\}\(D\)\\neq\\emptysethold, which means by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)thatCℐ≠∅C^\{\\mathcal\{I\}\}\\neq\\emptysetandDℐ≠∅D^\{\\mathcal\{I\}\}\\neq\\emptyset\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),ηℐ​\(D\)​\[iD\]=I=\\eta\_\{\\mathcal\{I\}\}\(D\)\[i\_\{D\}\]=I\_\{=\}\. By the \(⇐\\Leftarrow\) assumption, \(i\)ηℐ​\(C\)​\[iD\]⊆I=\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{D\}\]\\subseteq I\_\{=\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)ηℐ​\(C\)​\[iD\]\{\\eta\}\_\{\\mathcal\{I\}\}\(C\)\[i\_\{D\}\]is either equal toI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, orI⊅I\_\{\\not\\supset\}\. By the chosen parameters ofI=I\_\{=\},I⊃I\_\{\\supset\},I⊂I\_\{\\subset\},I∩I\_\{\\cap\}, andI⊅I\_\{\\not\\supset\}\(see Table[4](https://arxiv.org/html/2605.23937#A4.T4)and Figure[3](https://arxiv.org/html/2605.23937#A4.F3)\) the only options that are subsets ofI=I\_\{=\}areI=I\_\{=\}itself andI⊂I\_\{\\subset\}\. By \(i\), eitherηℐ​\(C\)​\[iD\]=I=\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{D\}\]=I\_\{=\}orηℐ​\(C\)​\[iD\]=I⊂\\eta\_\{\\mathcal\{I\}\}\(C\)\[i\_\{D\}\]=I\_\{\\subset\}\. The former case can only happen ifCℐ=DℐC^\{\\mathcal\{I\}\}=D^\{\\mathcal\{I\}\}, which means thatCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}, as required\. The latter case can only happen ifCℐ⊂DℐC^\{\\mathcal\{I\}\}\\subset D^\{\\mathcal\{I\}\}, which means thatCℐ⊆DℐC^\{\\mathcal\{I\}\}\\subseteq D^\{\\mathcal\{I\}\}, as required\. ∎

###### Lemma 4\.

Given an interpretationℐ\\mathcal\{I\}with finite domain, letηℐ\\eta\_\{\\mathcal\{I\}\}be as in Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\. For all DL\-LiteHRIsR⊑SR\\sqsubseteq S,ℐ⊧R⊑S\\mathcal\{I\}\\models R\\sqsubseteq Siffηℐ⊧R⊑S\\eta\_\{\\mathcal\{I\}\}\\models R\\sqsubseteq S\.

###### Proof\.

\(⇒\\Rightarrow\) AssumeRℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. To show thatηℐ​\(R\)⊆ηℐ​\(S\)\\eta\_\{\\mathcal\{I\}\}\(R\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(S\), we show that

𝖧𝖾𝖺𝖽ℐ​\(R\)\\displaystyle\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}⊆𝖧𝖾𝖺𝖽ℐ​\(S\),\\displaystyle\\subseteq\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\},𝖳𝖺𝗂𝗅ℐ​\(R\)\\displaystyle\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}⊆𝖳𝖺𝗂𝗅ℐ​\(S\),\\displaystyle\\subseteq\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\},𝖡𝗎𝗆𝗉ℐ​\(R\)\\displaystyle\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}⊆𝖡𝗎𝗆𝗉ℐ​\(S\)\.\\displaystyle\\subseteq\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\.We start with arguing that𝖧𝖾𝖺𝖽ℐ​\(R\)⊆𝖧𝖾𝖺𝖽ℐ​\(S\)\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\\subseteq\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\. We make the following case distinction\.

- •DimensioniCi\_\{C\}\.By assumption,Rℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. This implies\(∃R\)ℐ⊆\(∃S\)ℐ\(\\exists R\)^\{\\mathcal\{I\}\}\\subseteq\(\\exists S\)^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iC\]=ηℐ​\(∃R\)​\[iC\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists R\)\[i\_\{C\}\]and𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iC\]=ηℐ​\(∃S\)​\[iC\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)\[i\_\{C\}\]\. By Lemma[3](https://arxiv.org/html/2605.23937#Thmlemma3),ηℐ​\(∃R\)​\[iC\]⊆ηℐ​\(∃S\)​\[iC\]\\eta\_\{\\mathcal\{I\}\}\(\\exists R\)\[i\_\{C\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(\\exists S\)\[i\_\{C\}\]and we are done\.
- •DimensioniT,ci\_\{T,c\}and for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}implies\(c,e\)∈Tℐ\(c,e\)\\in T^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iT,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. By assumption,Rℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. Then, for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Tℐ\(c,e\)\\in T^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. So𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]=𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iT,c\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]and thus𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]⊆𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iT,c\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.
- •DimensioniT,ci\_\{T,c\}and there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}with\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}but\(c,e\)∉Tℐ\(c,e\)\\notin T^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iT,c\]=𝒮⊂¬\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also,𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}or𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]=𝒮⊂¬\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. In both cases, by the values in Table[6](https://arxiv.org/html/2605.23937#A4.T6), we have that𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iT,c\]⊆𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iT,c\]\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.

We now argue that𝖳𝖺𝗂𝗅ℐ​\(R\)⊆𝖳𝖺𝗂𝗅ℐ​\(S\)\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\\subseteq\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\.

- •DimensioniCi\_\{C\}\.By assumption,Rℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. This implies\(R−\)ℐ⊆\(S−\)ℐ\(R^\{\-\}\)^\{\\mathcal\{I\}\}\\subseteq\(S^\{\-\}\)^\{\\mathcal\{I\}\}, so\(∃R−\)ℐ⊆\(∃S−\)ℐ\(\\exists R^\{\-\}\)^\{\\mathcal\{I\}\}\\subseteq\(\\exists S^\{\-\}\)^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iC\]=ηℐ​\(∃R−\)​\[iC\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists R^\{\-\}\)\[i\_\{C\}\]and𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iC\]=ηℐ​\(∃S−\)​\[iC\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]=\\eta\_\{\\mathcal\{I\}\}\(\\exists S^\{\-\}\)\[i\_\{C\}\]\. Then, by Lemma[3](https://arxiv.org/html/2605.23937#Thmlemma3),ηℐ​\(∃R−\)​\[iC\]⊆ηℐ​\(∃S−\)​\[iC\]\\eta\_\{\\mathcal\{I\}\}\(\\exists R^\{\-\}\)\[i\_\{C\}\]\\subseteq\\eta\_\{\\mathcal\{I\}\}\(\\exists S^\{\-\}\)\[i\_\{C\}\]and we are done\.
- •DimensioniT,ci\_\{T,c\}and for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(e,c\)∈Sℐ\(e,c\)\\in S^\{\\mathcal\{I\}\}implies\(c,e\)∈Tℐ\(c,e\)\\in T^\{\\mathcal\{I\}\}\.In this case,𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iT,c\]=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. By assumption,Rℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. Then, for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(e,c\)∈Rℐ\(e,c\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Tℐ\(c,e\)\\in T^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. So𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]=𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iT,c\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]and thus𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]⊆𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iT,c\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.
- •DimensioniT,ci\_\{T,c\}and there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}with\(e,c\)∈Sℐ\(e,c\)\\in S^\{\\mathcal\{I\}\}but\(c,e\)∉Tℐ\(c,e\)\\notin T^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iT,c\]=𝒮⊂¬\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Also,𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]=𝒮⊂\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}\_\{\\subset\}or𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]=𝒮⊂¬\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. In both cases, by the values in Table[6](https://arxiv.org/html/2605.23937#A4.T6), we have that𝖳𝖺𝗂𝗅ℐ​\(R\)​\[iT,c\]⊆𝖳𝖺𝗂𝗅ℐ​\(S\)​\[iT,c\]\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Tail\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.

Finally, we argue that𝖡𝗎𝗆𝗉ℐ​\(R\)⊆𝖡𝗎𝗆𝗉ℐ​\(S\)\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\\subseteq\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\.

- •DimensioniCi\_\{C\}\.In this case, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iC\]=𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iC\]=I0\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]=\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]=I\_\{0\}\. Then, trivially,𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iC\]⊆𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iC\]\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{C\}\]\\subseteq\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{C\}\]\.
- •DimensioniS,ci\_\{S,c\}and for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}we have that\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Tℐ\(c,e\)\\in T^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iT,c\]=ℬ𝖱,𝖨\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. Also,𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]=ℬ𝖱,𝖨\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{B\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}or𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]=ℬ𝖱,𝖨¬\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. In both cases, by the values in Table[6](https://arxiv.org/html/2605.23937#A4.T6), we have that𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]⊆𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iT,c\]\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.
- •DimensioniT,ci\_\{T,c\}and there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}with\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}but\(c,e\)∉Tℐ\(c,e\)\\notin T^\{\\mathcal\{I\}\}\.By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]=ℬ𝖱,𝖨¬\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. By assumption, we have thatRℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. This implies that there ise∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\}with\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}but\(c,e\)∉Tℐ\(c,e\)\\notin T^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iT,c\]=ℬ𝖱,𝖨¬\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]=\\mathcal\{B\}^\{\\neg\}\_\{\\mathsf\{R\},\\mathsf\{I\}\}\. So𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]=𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iT,c\]\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]=\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]and thus𝖡𝗎𝗆𝗉ℐ​\(R\)​\[iT,c\]⊆𝖡𝗎𝗆𝗉ℐ​\(S\)​\[iT,c\]\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{T,c\}\]\\subseteq\{\{\\sf Bump\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{T,c\}\]\.

\(⇐\\Leftarrow\) Assumeηℐ​\(R\)⊆ηℐ​\(S\)\\eta\_\{\\mathcal\{I\}\}\(R\)\\subseteq\\eta\_\{\\mathcal\{I\}\}\(S\)\. Letccbe an arbitrary element ofΔℐ\\Delta^\{\\mathcal\{I\}\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6),𝖧𝖾𝖺𝖽ℐ​\(S\)​\[iS,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}\_\{\\subset\}\. By Definition[4](https://arxiv.org/html/2605.23937#Thmdefinition4)and the \(⇐\\Leftarrow\) assumption,𝖧𝖾𝖺𝖽ℐ​\(R\)⊆𝖧𝖾𝖺𝖽ℐ​\(S\)\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\\subseteq\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(S\)\}, so \(i\)𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]⊆𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]\\subseteq\\mathcal\{S\}\_\{\\subset\}\. By Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)again,𝖧𝖾𝖺𝖽ℐ​\(R\)\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}can be either𝒮⊂\\mathcal\{S\}\_\{\\subset\}or𝒮⊂¬\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\. Since𝒮⊂¬⊈𝒮⊂\\mathcal\{S\}^\{\\neg\}\_\{\\subset\}\\not\\subseteq\\mathcal\{S\}\_\{\\subset\}and \(i\) holds, we actually have that \(ii\)𝖧𝖾𝖺𝖽ℐ​\(R\)​\[iS,c\]=𝒮⊂\{\{\\sf Head\}\_\{\\mathcal\{I\}\}\(R\)\}\[i\_\{S,c\}\]=\\mathcal\{S\}\_\{\\subset\}and, by Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6), for alle∈Δℐe\\in\\Delta^\{\\mathcal\{I\}\},\(c,e\)∈Rℐ\(c,e\)\\in R^\{\\mathcal\{I\}\}implies\(c,e\)∈Sℐ\(c,e\)\\in S^\{\\mathcal\{I\}\}\. Sinceccwas an arbitrary element ofΔℐ\\Delta^\{\\mathcal\{I\}\}, \(iii\) this holds for all such elements\. Thus, by \(ii\)\-\(iii\),Rℐ⊆SℐR^\{\\mathcal\{I\}\}\\subseteq S^\{\\mathcal\{I\}\}\. ∎

See[3](https://arxiv.org/html/2605.23937#Thmtheorem3)

###### Proof\.

Given a satisfiable DL\-LiteHKB𝒦\\mathcal\{K\}, letℐ𝒦\\mathcal\{I\}\_\{\\mathcal\{K\}\}be the canonical model for𝒦\\mathcal\{K\}, as in Definition[1](https://arxiv.org/html/2605.23937#Thmdefinition1)\. By Theorem[1](https://arxiv.org/html/2605.23937#Thmtheorem1), for all DL\-LiteHaxiomsα\\alpha, we have that𝒦⊧α\\mathcal\{K\}\\models\\alphaiffℐ𝒦⊧α\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\alpha\. By Lemmas[1](https://arxiv.org/html/2605.23937#Thmlemma1),[2](https://arxiv.org/html/2605.23937#Thmlemma2),[3](https://arxiv.org/html/2605.23937#Thmlemma3), and[4](https://arxiv.org/html/2605.23937#Thmlemma4),ℐ𝒦⊧α\\mathcal\{I\}\_\{\\mathcal\{K\}\}\\models\\alphaiffηℐ𝒦⊧α\\eta\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\\models\\alpha\. Then,𝒦⊧α\\mathcal\{K\}\\models\\alphaiffηℐ𝒦⊧α\\eta\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}\\models\\alpha, which corresponds to the notion of strong KB faithfulness\(Lütfü Özçepet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib37); Bourgauxet al\.,[2024](https://arxiv.org/html/2605.23937#bib.bib64)\)\. Note thatΩ​\[iC\]=\(−4,4\)\{\\Omega\}\[i\_\{C\}\]=\(\-4,4\)\([Table6](https://arxiv.org/html/2605.23937#A4.T6)\) impliessΩ=4s\_\{\\Omega\}=4in our construction\. ∎

See[1](https://arxiv.org/html/2605.23937#Thmcorollary1)

###### Proof\.

By Claim[9](https://arxiv.org/html/2605.23937#Thmclaim9), any DL\-LiteHKB with an empty ABox is satisfiable\. Then, Corollary[1](https://arxiv.org/html/2605.23937#Thmcorollary1)follows from the proof of Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3)\. Specifically, if𝒦\\mathcal\{K\}’s ABox is empty, then the translation of a modelℐ\\mathcal\{I\}of𝒦\\mathcal\{K\}toηℐ\{\\eta\}\_\{\\mathcal\{I\}\}\(see Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\) reduces to \(1\) one dimensioniCi\_\{C\}for any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and \(2\) one dimensioniR,ci\_\{R,c\}for any pair of rolesR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}and elementsc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\. Thus, the constructed embedding has\|𝖭𝖢∃\|\+\|𝖭𝖱\|​\|Δℐ\|\|\{\\sf N\_\{C\}^\{\\exists\}\}\|\+\|\{\\sf N\_\{R\}\}\|\|\\Delta^\{\\mathcal\{I\}\}\|dimensions\. This dimensionality bound depends on\|Δℐ\|\|\\Delta^\{\\mathcal\{I\}\}\|\. Thus, translating different models of𝒦\\mathcal\{K\}leads to box interpretations with different dimensionalities\. This means that the selected model of𝒦\\mathcal\{K\}influences the dimensionality bounds for weak faithfulness, as we show next\.

For anyℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\},\(i\)\(i\)by Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6)it holds that the constructedηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is box consistent and\(i​i\)\(ii\)by Lemmas[1](https://arxiv.org/html/2605.23937#Thmlemma1)\-[4](https://arxiv.org/html/2605.23937#Thmlemma4), it holds thatηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is KB\-entailed for𝒦\\mathcal\{K\}\. By Points\(i\)\(i\)and\(i​i\)\(ii\)and Proposition[1](https://arxiv.org/html/2605.23937#Thmproposition1)it holds that anyηℐ\{\\eta\}\_\{\\mathcal\{I\}\}withℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}is a \(weakly\) TBox faithful model of𝒦\\mathcal\{K\}\. In particular, we can choose a modelℐ\\mathcal\{I\}with the smallest domain\|Δℐ\|=1\|\\Delta^\{\\mathcal\{I\}\}\|=1, which results inηℐ\{\\eta\}\_\{\\mathcal\{I\}\}having the following number of dimensions:

\|𝖭𝖢∃\|\+\|𝖭𝖱\|​\|Δℐ\|\\displaystyle\|\{\\sf N\_\{C\}^\{\\exists\}\}\|\+\|\{\\sf N\_\{R\}\}\|\|\\Delta^\{\\mathcal\{I\}\}\|=\\displaystyle=\|𝖭𝖢\|\+2​\|𝖭𝖱\|\+\|𝖭𝖱\|\\displaystyle\|\{\\sf N\_\{C\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\+\|\{\\sf N\_\{R\}\}\|=\\displaystyle=\|𝖭𝖢\|\+3​\|𝖭𝖱\|\\displaystyle\|\{\\sf N\_\{C\}\}\|\+3\|\{\\sf N\_\{R\}\}\|
One can extendηℐ𝒦\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}to a box interpretationηℐ𝒦′\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}^\{\\prime\}with more dimensions while keepingηℐ𝒦′\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}^\{\\prime\}a weakly TBox faithful model of𝒦\\mathcal\{K\}\. This can be done, for instance, by copying an arbitrary dimension ofηℐ𝒦\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}multiple times\. This keeps the extendedηℐ𝒦′\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}^\{\\prime\}a weakly TBox faithful model of𝒦\\mathcal\{K\}, while increasing its dimensionality arbitrarily\. Thus, we have shown that for anyd≥\|𝖭𝖢\|\+3​\|𝖭𝖱\|d\\geq\|\{\\sf N\_\{C\}\}\|\+3\|\{\\sf N\_\{R\}\}\|there is aη\{\\eta\}, s\.t\.,η\{\\eta\}is a weakly TBox faithful model of𝒦\\mathcal\{K\}\. ∎

See[2](https://arxiv.org/html/2605.23937#Thmcorollary2)

###### Proof\.

Analogous to the proof of Corollary[1](https://arxiv.org/html/2605.23937#Thmcorollary1), Corollary[2](https://arxiv.org/html/2605.23937#Thmcorollary2)essentially follows from the proof of Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3)\. Specifically, the translation of a modelℐ\\mathcal\{I\}of𝒦\\mathcal\{K\}toηℐ\{\\eta\}\_\{\\mathcal\{I\}\}\(see Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)\) reduces to \(1\) one dimensioniCi\_\{C\}for any conceptC∈𝖭𝖢∃C\\in\{\\sf N\_\{C\}^\{\\exists\}\}and \(2\) one dimensioniR,ci\_\{R,c\}for any pair of rolesR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}and elementsc∈Δℐc\\in\\Delta^\{\\mathcal\{I\}\}\. Thus, the constructed embedding has\|𝖭𝖢∃\|\+\|𝖭𝖱\|​\|Δℐ\|\|\{\\sf N\_\{C\}^\{\\exists\}\}\|\+\|\{\\sf N\_\{R\}\}\|\|\\Delta^\{\\mathcal\{I\}\}\|dimensions\. This dimensionality bound depends again on\|Δℐ\|\|\\Delta^\{\\mathcal\{I\}\}\|\. Thus, translating different models of𝒦\\mathcal\{K\}leads to box interpretations with different dimensionalities\. This means, in particular, that the selected model of𝒦\\mathcal\{K\}influences the dimensionality bounds for faithfulness, as we show next\.

For anyℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\},\(i\)\(i\)by Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6)it holds that the constructedηℐ\{\\eta\}\_\{\\mathcal\{I\}\}is box consistent and\(i​i\)\(ii\)by Lemmas[1](https://arxiv.org/html/2605.23937#Thmlemma1)\-[4](https://arxiv.org/html/2605.23937#Thmlemma4), it holds thatηℐ⊧𝒦\{\\eta\}\_\{\\mathcal\{I\}\}\\models\\mathcal\{K\}\. By Points\(i\)\(i\)and\(i​i\)\(ii\)and Proposition[2](https://arxiv.org/html/2605.23937#Thmproposition2)it holds that anyηℐ\{\\eta\}\_\{\\mathcal\{I\}\}withℐ⊧𝒦\\mathcal\{I\}\\models\\mathcal\{K\}is weakly KB faithful\. In particular, we know by Corollary[3](https://arxiv.org/html/2605.23937#Thmcorollary3)that𝒦\\mathcal\{K\}has a modelℐ\\mathcal\{I\}with\|Δℐ\|=\|𝖭𝖨\|\+2​\|𝖭𝖱\|\|\\Delta^\{\\mathcal\{I\}\}\|=\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{R\}\}\|, which results inηℐ\{\\eta\}\_\{\\mathcal\{I\}\}having the following number of dimensions:

\|𝖭𝖢∃\|\+\|𝖭𝖱\|​\|Δℐ\|\\displaystyle\|\{\\sf N\_\{C\}^\{\\exists\}\}\|\+\|\{\\sf N\_\{R\}\}\|\|\\Delta^\{\\mathcal\{I\}\}\|=\\displaystyle=\|𝖭𝖢\|\+2​\|𝖭𝖱\|\+\|𝖭𝖱\|​\(\|𝖭𝖨\|\+2​\|𝖭𝖱\|\)\\displaystyle\|\{\\sf N\_\{C\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\+\|\{\\sf N\_\{R\}\}\|\(\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\)=\\displaystyle=\|𝖭𝖢\|\+\|𝖭𝖱\|​\(2\+\|𝖭𝖨\|\+2​\|𝖭𝖱\|\)\\displaystyle\|\{\\sf N\_\{C\}\}\|\+\|\{\\sf N\_\{R\}\}\|\(2\+\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\)
As argued in[Corollary1](https://arxiv.org/html/2605.23937#Thmcorollary1), one can modifyηℐ𝒦\{\\eta\}\_\{\\mathcal\{I\}\_\{\\mathcal\{K\}\}\}so as to add more dimensions while keeping it a weakly KB faithful model of𝒦\\mathcal\{K\}\. Thus, we have shown that for anyd≥\|𝖭𝖢\|\+\|𝖭𝖱\|​\(2\+\|𝖭𝖨\|\+2​\|𝖭𝖱\|\)d\\geq\|\{\\sf N\_\{C\}\}\|\+\|\{\\sf N\_\{R\}\}\|\(2\+\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{R\}\}\|\)there is aη\{\\eta\}, s\.t\.,η\{\\eta\}is a weakly KB faithful model of𝒦\\mathcal\{K\}\. ∎

## Appendix EConvex Optimization Formulation for BoxLitE and Proofs for Section[5](https://arxiv.org/html/2605.23937#S5)

Here we provide the details for the signed distance function and the convex optimization problem in Equation \([8](https://arxiv.org/html/2605.23937#S5.E8)\)\.

### E\.1Signed distance function

The notation here is as described in Section[5\.2](https://arxiv.org/html/2605.23937#S5.SS2)\. See[3](https://arxiv.org/html/2605.23937#Thmproposition3)

###### Proof\.

This proposition also follows from Example 5\.1 in\(Luoet al\.,[2018](https://arxiv.org/html/2605.23937#bib.bib14)\), where the expression for𝗌𝖽𝗂𝗌𝗍\(y,ℝ\+n\)\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\+\}\)is described\. For the sake of self\-containment, we present a complete proof here\.

We consider two cases\. First, supposey∉ℝ−ny\\not\\in\\mathbb\{R\}^\{n\}\_\{\-\}, so, by definition,𝗌𝖽𝗂𝗌𝗍\(y,ℝ−n\)=𝖽𝗂𝗌𝗍e\(y,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)\. By direct computation,

𝖽𝗂𝗌𝗍e\(y,ℝ−n\)\\displaystyle\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=infx∈ℝ−n\[∑i=1n\(yi−xi\)2\]1/2\\displaystyle=\\inf\_\{x\\in\\mathbb\{R\}^\{n\}\_\{\-\}\}\\left\[\\sum\_\{i=1\}^\{n\}\(y\_\{i\}\-x\_\{i\}\)^\{2\}\\right\]^\{1/2\}=\[∑i=1ninfxi∈ℝ−\(yi−xi\)2\]1/2\\displaystyle=\\left\[\\sum\_\{i=1\}^\{n\}\\inf\_\{x\_\{i\}\\in\\mathbb\{R\}\_\{\-\}\}\(y\_\{i\}\-x\_\{i\}\)^\{2\}\\right\]^\{1/2\}=\[∑i=1nmax\(yi,0\)2\]1/2\\displaystyle=\\left\[\\sum\_\{i=1\}^\{n\}\\max\(y\_\{i\},0\)^\{2\}\\right\]^\{1/2\}=‖y\+‖2\.\\displaystyle=\\\|y^\{\+\}\\\|\_\{2\}\.This takes care of the first case\.

Next, suppose thaty∈ℝ−ny\\in\\mathbb\{R\}^\{n\}\_\{\-\}\. Let∂ℝ−n\\partial\\mathbb\{R\}^\{n\}\_\{\-\}denote the topological boundary ofℝ−n\\mathbb\{R\}^\{n\}\_\{\-\}, which corresponds to the elements ofℝ−n\\mathbb\{R\}^\{n\}\_\{\-\}that have at least one zero component\. Asy∈ℝ−ny\\in\\mathbb\{R\}^\{n\}\_\{\-\}by assumption, the distance betweenyyand\(ℝ−n\)c\(\\mathbb\{R\}^\{n\}\_\{\-\}\)^\{c\}is the same as the distance betweenyyand the boundary∂ℝ−n\\partial\\mathbb\{R\}^\{n\}\_\{\-\}, that is,𝖽𝗂𝗌𝗍e\(y,\(ℝ−n\)c\)=𝖽𝗂𝗌𝗍e\(y,∂ℝ−n\)\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\(\\mathbb\{R\}^\{n\}\_\{\-\}\)^\{c\}\)=\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\partial\\mathbb\{R\}^\{n\}\_\{\-\}\)\.

Let𝒫\\mathcal\{P\}denote the set of all subsets of\{1,…,n\}\\\{1,\\ldots,n\\\}except for∅\\emptyset\. Then,∂ℝ−n\\partial\\mathbb\{R\}^\{n\}\_\{\-\}is the union of sets of the formℝ𝒬n≔\{x∈ℝ−n∣xi=0,∀i∈𝒬\}\\mathbb\{R\}^\{n\}\_\{\\mathcal\{Q\}\}\\coloneqq\\\{x\\in\\mathbb\{R\}^\{n\}\_\{\-\}\\mid x\_\{i\}=0,\\forall i\\in\\mathcal\{Q\}\\\}, where𝒬∈𝒫\\mathcal\{Q\}\\in\\mathcal\{P\}\. With that,𝖽𝗂𝗌𝗍e\(y,\(ℝ−n\)c\)\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\(\\mathbb\{R\}^\{n\}\_\{\-\}\)^\{c\}\)is the minimum of the distances betweenyyand allℝ𝒬n\\mathbb\{R\}^\{n\}\_\{\\mathcal\{Q\}\}for𝒬∈𝒫\\mathcal\{Q\}\\in\\mathcal\{P\}\.

Given𝒬∈𝒫\\mathcal\{Q\}\\in\\mathcal\{P\}, we have that𝖽𝗂𝗌𝗍e\(y,ℝ𝒬n\)\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\\mathcal\{Q\}\}\)is equal to

infx∈ℝ𝒬n\[∑i∈𝒬\(yi−xi\)2\+∑i∈\{1,…,n\}∖𝒬\(yi−xi\)2\]1/2\\displaystyle\\inf\_\{x\\in\\mathbb\{R\}^\{n\}\_\{\\mathcal\{Q\}\}\}\\left\[\\sum\_\{i\\in\\mathcal\{Q\}\}\(y\_\{i\}\-x\_\{i\}\)^\{2\}\+\\sum\_\{i\\in\\\{1,\\ldots,n\\\}\\setminus\\mathcal\{Q\}\}\(y\_\{i\}\-x\_\{i\}\)^\{2\}\\right\]^\{1/2\}=\[∑i∈Qyi2\]1/2,\\displaystyle=\\left\[\\sum\_\{i\\in Q\}y\_\{i\}^\{2\}\\right\]^\{1/2\},where the equality follows becausexi=0x\_\{i\}=0fori∈Qi\\in Qand the infimum value for the second summation is zero sincexix\_\{i\}can be taken to be equal toyiy\_\{i\}wheniidoes not belong toQQ\.

Therefore, in order to minimize𝖽𝗂𝗌𝗍e\(y,ℝ𝒬n\)\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\\mathcal\{Q\}\}\)over all𝒬∈𝒫\\mathcal\{Q\}\\in\\mathcal\{P\}, it is enough to consider a singleton subset𝒬\\mathcal\{Q\}corresponding to a component ofyythat has the smallest absolute value\. Sincey∈ℝ−ny\\in\\mathbb\{R\}^\{n\}\_\{\-\}, the absolute value of a smallest component ofyyis given by\|maxi∈\{1,…,n\}⁡yi\|\|\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}y\_\{i\}\|\. Overall

−𝖽𝗂𝗌𝗍e\(y,\(ℝ−n\)c\)=−𝖽𝗂𝗌𝗍e\(y,∂ℝ−n\)=maxi∈\{1,…,n\}⁡yi\.\-\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\(\\mathbb\{R\}^\{n\}\_\{\-\}\)^\{c\}\)=\-\\mathop\{\\mathsf\{dist\}\_\{e\}\}\(y,\\partial\\mathbb\{R\}^\{n\}\_\{\-\}\)=\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}y\_\{i\}\.∎

As CVXPY does not implement signed distance functions natively, we implement𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\ \\cdot\\ ,\\mathbb\{R\}^\{n\}\_\{\-\}\)using the support function feature provided by CVXPY\. We now review some convex analysis concepts related to that\.

Given a setS⊆ℝnS\\subseteq\\mathbb\{R\}^\{n\}, we define the*support function ofSS*byσS​\(y\)≔sup\{⟨y,z⟩∣z∈S\}\\sigma\_\{S\}\(y\)\\coloneqq\\sup\\\{\\langle y,z\\rangle\\mid z\\in S\\\}, where⟨y,z⟩\\langle y,z\\rangleindicates the usual Euclidean dot product betweenyyandzz\.

The signed distance function to a convex set is convex, e\.g\., see Theorem 10\.1 in Chapter 7 of\(Delfour and Zolésio,[2011](https://arxiv.org/html/2605.23937#bib.bib15)\)or Section 3\.3 of\(Luoet al\.,[2018](https://arxiv.org/html/2605.23937#bib.bib14)\)\. In particular, sinceℝ−n\\mathbb\{R\}^\{n\}\_\{\-\}is a convex cone999A convex cone𝒦⊆ℝn\\mathcal\{K\}\\subseteq\\mathbb\{R\}^\{n\}is a convex set satisfyingα​y∈𝒦\\alpha y\\in\\mathcal\{K\}for allα≥0\\alpha\\geq 0and ally∈𝒦y\\in\\mathcal\{K\}\., the function𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{n\}\_\{\-\}\)is convex and positively homogeneous101010That is,𝗌𝖽𝗂𝗌𝗍\(α​y,ℝ−n\)=α​𝗌𝖽𝗂𝗌𝗍\(y,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\alpha y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=\\alpha\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)holds, for allα∈ℝ\+\\alpha\\in\\mathbb\{R\}\_\{\+\},y∈ℝny\\in\\mathbb\{R\}^\{n\}\.\. Furthermore,𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{n\}\_\{\-\}\)is finite everywhere\. Then, it follows from Corollary 13\.2\.2 in\(Rockafellar,[1997](https://arxiv.org/html/2605.23937#bib.bib23)\)that𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{n\}\_\{\-\}\)can be expressed as the support function of a certain convex set\. That is, there exists a convex setC⊆ℝnC\\subseteq\\mathbb\{R\}^\{n\}such that𝗌𝖽𝗂𝗌𝗍\(y,ℝ−n\)=σC​\(y\)\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=\\sigma\_\{C\}\(y\)holds for everyy∈ℝny\\in\\mathbb\{R\}^\{n\}\. We implement𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−n\)\\mathop\{\\mathsf\{sdist\}\}\(\\ \\cdot\\ ,\\mathbb\{R\}^\{n\}\_\{\-\}\)in our code by expressing it as the support function of a certain convex set\.

LetBn≔\{y∈ℝn∣‖y‖2≤1\}B\_\{n\}\\coloneqq\\\{y\\in\\mathbb\{R\}^\{n\}\\mid\\\|y\\\|\_\{2\}\\leq 1\\\}denote the unit ball inℝn\\mathbb\{R\}^\{n\}andPn≔\{y∈ℝn∣y≥0,y1\+⋯​yn≥1\}P\_\{n\}\\coloneqq\\\{y\\in\\mathbb\{R\}^\{n\}\\mid y\\geq 0,y\_\{1\}\+\\cdots y\_\{n\}\\geq 1\\\}\. In what follows, giveny,z∈ℝny,z\\in\\mathbb\{R\}^\{n\}, we usey≤zy\\leq zto indicate thatyi≤ziy\_\{i\}\\leq z\_\{i\}holds fori∈\{1,…,n\}i\\in\\\{1,\\ldots,n\\\}\. With that, the next proposition tell us precisely how to obtain the signed distance function toℝ−n\\mathbb\{R\}^\{n\}\_\{\-\}as the support function of a convex set\.

###### Proposition 4\.

LetCn=Bn∩PnC\_\{n\}=B\_\{n\}\\cap P\_\{n\}\. For everyy∈ℝny\\in\\mathbb\{R\}^\{n\}the following hold\.

𝗌𝖽𝗂𝗌𝗍\(y,ℝ−n\)\\displaystyle\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{n\}\_\{\-\}\)=σCn​\(y\)\\displaystyle=\\sigma\_\{C\_\{n\}\}\(y\)=infz∈ℝn\{‖z‖2\+maxi∈\{1,…,n\}⁡\(yi−zi\)∣y≤z\},\\displaystyle=\\inf\_\{z\\in\\mathbb\{R\}^\{n\}\}\\\{\\\|z\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}\)\\mid y\\leq z\\\},

###### Proof\.

The result is straightforward forn=1n=1, so henceforth we assume thatn≥2n\\geq 2\. Before we proceed, we need some extra convex analysis preliminaries, for more details see\(Rockafellar,[1997](https://arxiv.org/html/2605.23937#bib.bib23); Hiriart\-Urruty and Lemaréchal,[1993a](https://arxiv.org/html/2605.23937#bib.bib20),[b](https://arxiv.org/html/2605.23937#bib.bib19)\)\.

For a setS⊆ℝnS\\subseteq\\mathbb\{R\}^\{n\}denote its indicator function byδS:ℝn→ℝn∪\{\+∞\}\\delta\_\{S\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}^\{n\}\\cup\\\{\+\\infty\\\}\. By definition, we have

δS​\(y\)≔\{0if​y∈S\+∞if​y∉S,,σS​\(y\)≔supz∈S⟨z,y⟩\.\\delta\_\{S\}\(y\)\\coloneqq\\begin\{cases\}0&\\text\{ if \}y\\in S\\\\ \+\\infty&\\text\{ if \}y\\not\\in S,\\end\{cases\},\\qquad\\sigma\_\{S\}\(y\)\\coloneqq\\sup\_\{z\\in S\}\\,\\langle z,\\,y\\rangle\.Next, letf:ℝn→ℝ∪\{\+∞\}f:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}\\cup\\\{\+\\infty\\\}be a convex function\. Its conjugate function is defined as

f∗​\(s\)≔supx∈ℝn\(⟨s,x⟩−f​\(x\)\)\.f^\{\*\}\(s\)\\coloneqq\\sup\_\{x\\in\\mathbb\{R\}^\{n\}\}\(\\langle s,\\,x\\rangle\-f\(x\)\)\.We note thatδS∗=σS\\delta\_\{S\}^\{\*\}=\\sigma\_\{S\}, forS⊆ℝnS\\subseteq\\mathbb\{R\}^\{n\}\.

SinceCn=Pn∩BnC\_\{n\}=P\_\{n\}\\cap B\_\{n\}, we haveδBn\+δPn=δCn\\delta\_\{B\_\{n\}\}\+\\delta\_\{P\_\{n\}\}=\\delta\_\{C\_\{n\}\}\. Then, for everyy∈ℝny\\in\\mathbb\{R\}^\{n\}, we have

supx∈Cn⟨y,x⟩\\displaystyle\\sup\_\{x\\in C\_\{n\}\}\\,\\langle y,\\,x\\rangle=supx∈ℝn\(⟨y,x⟩−δBn​\(x\)−δPn​\(x\)\)\\displaystyle=\\sup\_\{x\\in\\mathbb\{R\}^\{n\}\}\\,\(\\langle y,\\,x\\rangle\-\\delta\_\{B\_\{n\}\}\(x\)\-\\delta\_\{P\_\{n\}\}\(x\)\)=\(δBn\+δPn\)∗​\(y\)\.\\displaystyle=\(\\delta\_\{B\_\{n\}\}\+\\delta\_\{P\_\{n\}\}\)^\{\*\}\(y\)\.
Sincen≥2n\\geq 2, the \(topological\) interiors ofPnP\_\{n\}andBnB\_\{n\}intersect\. For example,\(0\.6,0\.6,ϵ,…,ϵ\)\(0\.6,0\.6,\\epsilon,\\ldots,\\epsilon\)belongs to the interior of both sets for sufficiently smallϵ\>0\\epsilon\>0whenn≥3n\\geq 3\. Forn=2n=2, it is enough to take\(0\.6,0\.6\)\(0\.6,0\.6\)\. Under this condition, a theorem from convex analysis says that\(δBn\+δPn\)∗\(\\delta\_\{B\_\{n\}\}\+\\delta\_\{P\_\{n\}\}\)^\{\*\}is the exact infimal convolution betweenσBn\\sigma\_\{B\_\{n\}\}andσPn\\sigma\_\{P\_\{n\}\}, e\.g\., see Theorem 2\.3\.2 of Chapter X in\(Hiriart\-Urruty and Lemaréchal,[1993b](https://arxiv.org/html/2605.23937#bib.bib19)\)\. This means that

σCn​\(y\)=supx∈Cn⟨y,x⟩=infz∈ℝn\{σBn​\(z\)\+σPn​\(y−z\)\}\\sigma\_\{C\_\{n\}\}\(y\)=\\sup\_\{x\\in C\_\{n\}\}\\,\\langle y,\\,x\\rangle=\\inf\_\{z\\in\\mathbb\{R\}^\{n\}\}\\\{\\sigma\_\{B\_\{n\}\}\(z\)\+\\sigma\_\{P\_\{n\}\}\(y\-z\)\\\}\(11\)and the infimum is attained for everyyy\. Now, forz∈ℝnz\\in\\mathbb\{R\}^\{n\}we have

σBn​\(z\)=‖z‖2,\\sigma\_\{B\_\{n\}\}\(z\)=\\\|z\\\|\_\{2\},which follows from the Cauchy\-Schwarz inequality\. Defininge≔\(−1,−1,…,−1\)e\\coloneqq\(\-1,\-1,\\ldots,\-1\), we have forw∈ℝnw\\in\\mathbb\{R\}^\{n\}

σPn​\(w\)\\displaystyle\\sigma\_\{P\_\{n\}\}\(w\)=sup0≤u,1≤u1\+⋯​un⟨u,w⟩=inf0≤e​t−w,0≤t−t\\displaystyle=\\sup\_\{0\\leq u,1\\leq u\_\{1\}\+\\cdots u\_\{n\}\}\\langle u,\\,w\\rangle=\\inf\_\{0\\leq et\-w,0\\leq t\}\-t=\{maxi∈\{1,…,n\}⁡wiif​w∈ℝ−n\+∞otherwise,\\displaystyle=\\begin\{cases\}\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}w\_\{i\}&\\text\{ if \}w\\in\\mathbb\{R\}^\{n\}\_\{\-\}\\\\ \+\\infty&\\text\{ otherwise \}\\end\{cases\},
where the second equality follows from linear programming duality\. The third equality holds because the constraint0≤e​t−w0\\leq et\-wimplies that−t≥wi\-t\\geq w\_\{i\}for allii\. So minimizing−t\-tunder this constraint and the constraint thatt≥0t\\geq 0leads tomaxi∈\{1,…,n\}⁡wi\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}w\_\{i\}ifw∈ℝ−nw\\in\\mathbb\{R\}^\{n\}\_\{\-\}\. If some component ofwiw\_\{i\}is positive, then the problem is infeasible, so the infimum is\+∞\+\\infty\.

Plugging the expressions forσBn\\sigma\_\{B\_\{n\}\}andσPn\\sigma\_\{P\_\{n\}\}into Equation \([11](https://arxiv.org/html/2605.23937#A5.E11)\) leads to

σCn​\(y\)=infz∈ℝn\{‖z‖2\+maxi∈\{1,…,n\}⁡\(yi−zi\)∣y≤z\}\.\\sigma\_\{C\_\{n\}\}\(y\)=\\inf\_\{z\\in\\mathbb\{R\}^\{n\}\}\\\{\\\|z\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}\)\\mid y\\leq z\\\}\.
Therefore, in order to show that the proposition holds, it is enough to constructx∗∈Cnx^\{\*\}\\in C\_\{n\}andz∗∈ℝnz^\{\*\}\\in\\mathbb\{R\}^\{n\}satisfyingy≤z∗y\\leq z^\{\*\}and

⟨y,x∗⟩\\displaystyle\\langle y,\\,x^\{\*\}\\rangle=‖z∗‖2\+maxi∈\{1,…,n\}⁡\(yi−zi∗\)\\displaystyle=\\\|z^\{\*\}\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}^\{\*\}\)=\{‖y\+‖2if​y∉ℝ−nmaxi∈\{1,…,n\}⁡yiif​y∈ℝ−n\.\\displaystyle=\\begin\{cases\}\\\|y^\{\+\}\\\|\_\{2\}&\\text\{ if \}y\\not\\in\\mathbb\{R\}^\{n\}\_\{\-\}\\\\ \\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\{y\_\{i\}\}&\\text\{ if \}y\\in\\mathbb\{R\}^\{n\}\_\{\-\}\.\\end\{cases\}This can be done constructively case\-by\-case as follows\.

\(i\)\(i\)Supposey∉ℝ−ny\\not\\in\\mathbb\{R\}^\{n\}\_\{\-\}\.Then, at least one component ofyyis positive, soy\+≠0y^\{\+\}\\neq 0\. Letx∗≔y\+‖y\+‖2x^\{\*\}\\coloneqq\\frac\{y^\{\+\}\}\{\\\|y^\{\+\}\\\|\_\{2\}\}andz∗≔y\+z^\{\*\}\\coloneqq y^\{\+\}\.

Thenx∗x^\{\*\}belongs toCnC\_\{n\}, because‖x∗‖2=1\\\|x^\{\*\}\\\|\_\{2\}=1andx1∗\+⋯\+xn∗=‖x∗‖1≥‖x∗‖2=1x\_\{1\}^\{\*\}\+\\cdots\+x\_\{n\}^\{\*\}=\\\|x^\{\*\}\\\|\_\{1\}\\geq\\\|x^\{\*\}\\\|\_\{2\}=1\. We also havey−z∗=y−≤0y\-z^\{\*\}=y^\{\-\}\\leq 0, wherey−y^\{\-\}is the nonpositive part ofyy\. Also, since at least one component ofyyis positivey−z∗y\-z^\{\*\}is a nonpositive vector with at least one entry equal to zero\. Somaxi∈\{1,…,n\}⁡\(yi−zi∗\)=0\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}^\{\*\}\)=0\. Overall,

⟨y,x∗⟩=‖z∗‖2\+maxi∈\{1,…,n\}⁡\(yi−zi∗\)=‖y\+‖2\.\\langle y,\\,x^\{\*\}\\rangle=\\\|z^\{\*\}\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}^\{\*\}\)=\\\|y^\{\+\}\\\|\_\{2\}\.
\(i​i\)\(ii\)Supposey∈ℝ−ny\\in\\mathbb\{R\}^\{n\}\_\{\-\}\.Letjjbe an index ofyyassociated to its largest component111111There may be multiplejj’s, but any will work\.\. We letx∗∈ℝnx^\{\*\}\\in\\mathbb\{R\}^\{n\}be such thatxj∗≔1x^\{\*\}\_\{j\}\\coloneqq 1andxk∗≔0x^\{\*\}\_\{k\}\\coloneqq 0fork≠jk\\neq j\. We have‖x∗‖2=1\\\|x^\{\*\}\\\|\_\{2\}=1andx1∗\+⋯​xn∗=xj∗=1x^\{\*\}\_\{1\}\+\\cdots x^\{\*\}\_\{n\}=x^\{\*\}\_\{j\}=1, sox∗∈Cx^\{\*\}\\in C\. Finally, letz∗≔0z^\{\*\}\\coloneqq 0\. With that, sincey∈ℝ−ny\\in\\mathbb\{R\}^\{n\}\_\{\-\}, we havey−z∗=y≤0y\-z^\{\*\}=y\\leq 0and

yj=⟨y,x∗⟩=‖z∗‖2\+maxi∈\{1,…,n\}⁡\(yi−zi∗\)=maxi∈\{1,…,n\}⁡yi\.y\_\{j\}=\\langle y,\\,x^\{\*\}\\rangle=\\\|z^\{\*\}\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}\(y\_\{i\}\-z\_\{i\}^\{\*\}\)=\\max\_\{i\\in\\\{1,\\ldots,n\\\}\}y\_\{i\}\.
∎

### E\.2Convex Optimization

Let𝒦=\(𝒯,𝒜\)\\mathcal\{K\}=\(\\mathcal\{T\},\\mathcal\{A\}\)be a DL\-LiteHKB, letddbe the embedding dimension and let𝐬𝛀\{\\mathbf\{s\_\{\\Omega\}\}\}and theϵ\>0\\epsilon\>0be as in Section[3](https://arxiv.org/html/2605.23937#S3)\. Following Section[3](https://arxiv.org/html/2605.23937#S3), a given box interpretationη\\etaassociates to each individual namea∈𝖭𝖨a\\in\{\\sf N\_\{I\}\}, each concept nameC∈𝖭𝖢C\\in\{\\sf N\_\{C\}\}and each role nameR∈𝖭𝖱R\\in\{\\sf N\_\{R\}\}the following objects: two vectors\(𝗉𝗈𝗌​\(a\),𝖻𝗎𝗆𝗉​\(a\)\)∈Ω×Ω\(\{\{\\sf pos\}\(a\)\},\{\{\\sf bump\}\(a\)\}\)\\in\{\\Omega\}\\times\{\\Omega\}, a boxη​\(C\)\{\{\\eta\}\(C\)\}and three boxes\(𝖧𝖾𝖺𝖽​\(R\),𝖳𝖺𝗂𝗅​\(R\),𝖡𝗎𝗆𝗉​\(R\)\)\(\{\{\\sf Head\}\(R\)\},\{\{\\sf Tail\}\(R\)\},\{\{\\sf Bump\}\(R\)\}\), respectively\. Each box is parameterized by two vectors inℝd\\mathbb\{R\}^\{d\}representing lower and upper bounds\.

We recall that we concatenate all the parameters ofη\\etainto a single vectorzzof dimensionn:=\(2​\|𝖭𝖨\|\+2​\|𝖭𝖢\|\+6​\|𝖭𝖱\|\)​dn:=\(2\|\{\\sf N\_\{I\}\}\|\+2\|\{\\sf N\_\{C\}\}\|\+6\|\{\\sf N\_\{R\}\}\|\)d\. With that, let𝒞𝒦⊆ℝn\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\\subseteq\\mathbb\{R\}^\{n\}be the set ofzz’s such that the constraints defined in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)are satisfied\. That is,z∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}if and only if theη\\etacorresponding tozzis such that: for each TBox axiom the corresponding inequalities are satisfied; and the box consistency and universe constraints are satisfied\. Also, letf𝝀:ℝn→ℝf\_\{\\boldsymbol\{\\lambda\}\}:\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}be the function that mapszzto the objective value described in Section[5\.2](https://arxiv.org/html/2605.23937#S5.SS2)for a given choice of nonnegative hyperparameters𝝀=\(λ1,λ2,λ3\)∈ℝ\+3\\boldsymbol\{\\lambda\}=\(\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}\)\\in\\mathbb\{R\}^\{3\}\_\{\+\}\.

In what follows, we recall that a set inℝn\\mathbb\{R\}^\{n\}is said to be*polyhedral*if it can be written as the set of solutions of finitely many linear equalities/inequalities\.

See[4](https://arxiv.org/html/2605.23937#Thmtheorem4)

###### Proof\.

As described in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1), each inclusion in the TBox of the KB is translated into finitely many linear inequalities in terms of the parameters of the box interpretation, which are exactly the components ofzz\. Similarly, for each individual, concept and role, the box consistency and universe constraints in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)are translated to finitely many linear inequalities inzz\. Since𝖭𝖨\{\\sf N\_\{I\}\},𝖭𝖢\{\\sf N\_\{C\}\},𝖭𝖱\{\\sf N\_\{R\}\}, and the TBox are all finite,𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}is the intersection of solution sets of finitely many linear inequalities\. Thus,𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}is a polyhedral set inℝn\\mathbb\{R\}^\{n\}\.

Next, we move on to the convexity off𝝀f\_\{\\boldsymbol\{\\lambda\}\}\. First, we recall that the composition of a convex function with an affine function is still convex\. Also, sums of convex functions are convex\. Similarly, the maximum of convex functions is also convex\. See\(Boyd and Vandenberghe,[2004](https://arxiv.org/html/2605.23937#bib.bib22), Section 3\.2\)for a review of calculus rules for convex functions\.

First, the signed distance function𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)is convex, see Theorem 10\.1 in Chapter 7 of\(Delfour and Zolésio,[2011](https://arxiv.org/html/2605.23937#bib.bib15)\)or pg\.154 in\(Hiriart\-Urruty and Lemaréchal,[1993a](https://arxiv.org/html/2605.23937#bib.bib20)\)\. With that, the𝖽𝗂𝗌𝗍\{\\sf dist\}function is also convex, as it is the composition of a convex function with an affine function\. This implies that the concept assertion lossℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍\\mathcal\{L\}\_\{\\sf concept\}is convex as well as it is again a composition of a convex function with an affine function\. Similarly, both the role assertion lossℒ𝗋𝗈𝗅𝖾\\mathcal\{L\}\_\{\\sf role\}and the negative concept regularizationℒ𝗇𝖾𝗀𝖺𝗍𝗂𝗏𝖾\\mathcal\{L\}\_\{\\sf negative\}are convex, as they are the maximum of finitely many convex functions\. Finally, the box width regularizationℛ𝗐𝗂𝖽𝗍𝗁\\mathcal\{R\}\_\{\\sf width\}is also convex, since it is the composition of a convex function \(the22\-norm\) with a linear function\.

Overall the objective functionf𝝀f\_\{\\boldsymbol\{\\lambda\}\}is convex as it is obtained by taking sums and maximums of finitely many convex functions\.

So far, we have shown that the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) is a convex optimization problem with polyhedral constraints\. Next, we move on to the proofs of the items\.

i\)i\)Let~​𝒦\\tilde\{\}\\mathcal\{K\}be the KB which coincides with𝒦\\mathcal\{K\}except for the fact that its ABox is empty\. By itemi\)i\)of Corollary[1](https://arxiv.org/html/2605.23937#Thmcorollary1), for everyd≥dmind\\geq d\_\{\\min\}, there exists a box interpretationη\\etathat is weakly TBox faithful\. Because of the faithfulness of the embedding,η\\etasatisfies the constraints described in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\.

First we recall that the proof of Corollary[1](https://arxiv.org/html/2605.23937#Thmcorollary1)is done by constructingη\\etafollowing Definition[6](https://arxiv.org/html/2605.23937#Thmdefinition6)withsΩ=4\{s\_\{\\Omega\}\}=4\(as in the proof of Theorem[3](https://arxiv.org/html/2605.23937#Thmtheorem3)\), see also Table[6](https://arxiv.org/html/2605.23937#A4.T6)\. This allows us to invoke Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6), which ensures thatη\\etasatisfies \([1](https://arxiv.org/html/2605.23937#S5.E1)\) in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\. Furthermore, becauseη\\etais a box interpretation, the inequalities in \([2](https://arxiv.org/html/2605.23937#S5.E2)\) and \(LABEL:eq:indUn\) \(Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\) must be satisfied, see Definition[3](https://arxiv.org/html/2605.23937#Thmdefinition3)\. This is because Definition[3](https://arxiv.org/html/2605.23937#Thmdefinition3)imposes the same restrictions on the widths of the boxes\.

Finally, aggregating the parameters ofη\\etainto a single vectorz∈ℝnz\\in\\mathbb\{R\}^\{n\}in an appropriate order, we will havez∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}, which implies that𝒞𝒦≠∅\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\\neq\\emptyset\.

ii\)ii\)The proof of itemii\)ii\)follows from the definition of𝒞𝒦\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\. We recall thatz∈𝒞𝒦z\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}if and only if the corresponding inequalities in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)are satisfied\. As in the proof of Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6), the inequality in \([1](https://arxiv.org/html/2605.23937#S5.E1)\) \(Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)\) implies box consistency\. We conclude that the box interpretation corresponding to a givenz∈𝒞𝒦z\\in\{\{\\mathcal\{C\}\}\}\_\{\\mathcal\{K\}\}is box consistent and satisfies all the TBox axioms of the underlying KB\. By Proposition[1](https://arxiv.org/html/2605.23937#Thmproposition1), the box interpretation associated tozzis TBox faithful, which concludes the proof\.

iii\)iii\)Sincez∗z^\{\*\}is assumed to be an optimal solution, we havez∗∈𝒞𝒦z^\{\*\}\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\. By itemii\)ii\), the box interpretation corresponding toz∗z^\{\*\}satisfies all the TBox axioms and must be box consistent\. Becausef𝝀​\(z∗\)≤0f\_\{\\boldsymbol\{\\lambda\}\}\(z^\{\*\}\)\\leq 0holds,λ1\\lambda\_\{1\}is zero and the regularization terms associated toλ2,λ3\\lambda\_\{2\},\\lambda\_\{3\}are nonnegative, we conclude that the first max term of the objective function term is nonpositive\. In particular, all theℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍\\mathcal\{L\}\_\{\\sf concept\}andℒ𝗋𝗈𝗅𝖾\\mathcal\{L\}\_\{\\sf role\}terms inside the max must be nonpositive\. By the definition of the signed distance function𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\), this implies that the corresponding ABox axioms are satisfied as well\. By Proposition[2](https://arxiv.org/html/2605.23937#Thmproposition2), the box interpretation associated withz∗z^\{\*\}is KB faithful\.

iv\)iv\)By itemi\)i\)of Corollary[2](https://arxiv.org/html/2605.23937#Thmcorollary2), there exists a weakly KB faithful box interpretationη\\etaof𝒦\\mathcal\{K\}for anyd≥dmind\\geq d\_\{\\min\}\. As in the proof of itemi\)i\),η\\etasatisfies the constraints described in Section[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)forsΩ=4\{s\_\{\\Omega\}\}=4, which comes as consequence of Theorem[6](https://arxiv.org/html/2605.23937#Thmtheorem6), the proof of Corollary[2](https://arxiv.org/html/2605.23937#Thmcorollary2)and the definition of box interpretation in Definition[3](https://arxiv.org/html/2605.23937#Thmdefinition3)\.

Aggregating the parameters of the box interpretationη\\etainto a single vectorz∗∈ℝnz^\{\*\}\\in\\mathbb\{R\}^\{n\}in an appropriate order, we havez∗∈𝒞𝒦z^\{\*\}\\in\{\\mathcal\{C\}\}\_\{\\mathcal\{K\}\}\. Becauseη⊧𝒦\\eta\\models\\mathcal\{K\}, all the ABox axioms are satisfied, so the theℒ𝖼𝗈𝗇𝖼𝖾𝗉𝗍\\mathcal\{L\}\_\{\\sf concept\}andℒ𝗋𝗈𝗅𝖾\\mathcal\{L\}\_\{\\sf role\}terms in in first max term of the objective function must be nonpositive\. As a consequence, if the regularization parametersλ1,λ2,λ3\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}in the objective function term are zero as well, thenf𝝀​\(z∗\)≤0f\_\{\\boldsymbol\{\\lambda\}\}\(z^\{\*\}\)\\leq 0holds\. ∎

#### Second\-order cone representability\.

For those familiar with conic optimization, the proof of Theorem[5](https://arxiv.org/html/2605.23937#Thmtheorem5)is routine and can be summarized as follows\. We employ the notion of the*epigraph*of a functionf:ℝs→ℝ∪\{\+∞\}f:\\mathbb\{R\}^\{s\}\\to\\mathbb\{R\}\\cup\\\{\+\\infty\\\}, defined as the set\{\(x,t\)∈ℝs×ℝ∣f​\(x\)≤t\}\\\{\(x,t\)\\in\\mathbb\{R\}^\{s\}\\times\\mathbb\{R\}\\mid f\(x\)\\leq t\\\}\. With that, the epigraphs of the functions corresponding to each of the terms appearing inf𝝀f\_\{\\boldsymbol\{\\lambda\}\}are*second\-order cone representable*\(SOCr\(Loboet al\.,[1998](https://arxiv.org/html/2605.23937#bib.bib24)\), also called CQr in\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25), Lecture 3\)\)\. Furthermore, second\-order cone representability is preserved by adding functions, composition with affine functions and taking a maximum of finitely many SOCr functions, see\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25), Section 3\.3\)\. Overall, applying appropriate calculus rules, we see that the epigraph off𝝀f\_\{\\boldsymbol\{\\lambda\}\}is SOCr, which implies in particular that the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) has a SOCP formulation\. For the sake of self\-containment, we present a detailed proof\.

See[5](https://arxiv.org/html/2605.23937#Thmtheorem5)“Equivalent” in the statement of Theorem[5](https://arxiv.org/html/2605.23937#Thmtheorem5)means that the optimal value and optimal solutions from the former can be recovered from the optimal value and optimal solutions to the latter and vice\-versa\.

###### Proof\.

In its most general form, a SOCP can be written as

miny∈ℝs\\displaystyle\\min\_\{y\\in\\mathbb\{R\}^\{s\}\}cT​y\\displaystyle\\quad c^\{T\}y\(12\)subject toy∈𝒫\\displaystyle\\quad y\\in\\mathcal\{P\}bi\+Ai​y∈𝒦2ni,i=1​…,m,\\displaystyle\\quad b\_\{i\}\+A\_\{i\}y\\in\{\\mathcal\{K\}\_\{2\}^\{n\_\{i\}\}\},\\qquad i=1\\ldots,m,where each𝒦2ni≔\{\(t,z\)∈ℝ×ℝni−1∣t≥‖z‖2\}\{\\mathcal\{K\}\_\{2\}^\{n\_\{i\}\}\}\\coloneqq\\\{\(t,z\)\\in\\mathbb\{R\}\\times\\mathbb\{R\}^\{n\_\{i\}\-1\}\\mid t\\geq\\\|z\\\|\_\{2\}\\\}is the second\-order cone inℝni\\mathbb\{R\}^\{n\_\{i\}\},𝒫⊆ℝs\\mathcal\{P\}\\subseteq\\mathbb\{R\}^\{s\}is a polyhedral set described via finitely many linear equalities/inequalities121212Depending on the reference, the “standard form” of SOCPs may not include linear inequalities directly, but this can be bypassed easily since a linear inequality of the form “aT​y≥da^\{T\}y\\geq d” is equivalent to the SOC constraint “aT​y−d∈𝒦12a^\{T\}y\-d\\in\{\\mathcal\{K\}\_\{1\}^\{2\}\}”\., thebib\_\{i\}’s are vectors,AiA\_\{i\}’s are linear maps,c∈ℝsc\\in\\mathbb\{R\}^\{s\}is a fixed vector andcT​yc^\{T\}yindicates the Euclidean inner product betweenccandyyso thatcT​y=∑i=1sci​yic^\{T\}y=\\sum\_\{i=1\}^\{s\}c\_\{i\}y\_\{i\}holds\.

Comparing the SOCP in \([12](https://arxiv.org/html/2605.23937#A5.E12)\) with the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) and recalling that𝒞𝒦\\mathcal\{C\}\_\{\\mathcal\{K\}\}is polyhedral, we see that \([8](https://arxiv.org/html/2605.23937#S5.E8)\) is not a SOCP only because its objective function is nonlinear\. Here, we will use the common optimization trick of “dropping the objective to the constraints”, e\.g\., see\(Loboet al\.,[1998](https://arxiv.org/html/2605.23937#bib.bib24), Section 2\.5\)or\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25), Chapter 3\)\. The idea is as follows, ifg:ℝs→ℝg:\\mathbb\{R\}^\{s\}\\to\\mathbb\{R\}is a real function andS⊆ℝsS\\subseteq\\mathbb\{R\}^\{s\}, then the problem “miny⁡g​\(y\)​subject to​y∈S\\min\_\{y\}g\(y\)\\,\\,\\text\{subject to\}\\,\\,y\\in S” is equivalent to “mint,y⁡t​subject to​g​\(y\)≤t,y∈S\\min\_\{t,y\}t\\,\\,\\text\{ subject to \}\\,\\,g\(y\)\\leq t,y\\in S”\. Similarly, if there were several functionsgjg\_\{j\}we would have that the problem

miny​∑j=1ℓgj​\(y\)​subject to​y∈S\\min\_\{y\}\\sum\_\{j=1\}^\{\\ell\}g\_\{j\}\(y\)\\,\\,\\text\{subject to\}\\,\\,y\\in Sis equivalent to

mint1,…,tℓ,y​∑i=1ℓtj​subject to​gj​\(y\)≤tj​\(j=1,…,ℓ\),y∈S\\min\_\{t\_\{1\},\\ldots,t\_\{\\ell\},y\}\\sum\_\{i=1\}^\{\\ell\}t\_\{j\}\\,\\,\\text\{ subject to \}\\,\\,g\_\{j\}\(y\)\\leq t\_\{j\}\\,\\,\(j=1,\\ldots,\\ell\),\\,y\\in S\(13\)Here, the constraints “gj​\(y\)≤tjg\_\{j\}\(y\)\\leq t\_\{j\}” simply mean that\(y,tj\)\(y,t\_\{j\}\)belongs to the epigraph ofgjg\_\{j\}\. In particular, ifSSand the epigraphs ofgjg\_\{j\}can be represented via finitely many linear equalities/inequalities and SOC constraints, then the problem in \([13](https://arxiv.org/html/2605.23937#A5.E13)\) can reformulated as a SOCP since its objective function is linear\. Informally, we say that a function is SOCr if its epigraph can be represented via finitely many equalities/inequalities and SOC constraints \(adding auxiliary variables if necessary\)\.

This discussion provides a blueprint for proving that the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) can be reformulated as a SOCP\. The objective functionf𝝀f\_\{\\boldsymbol\{\\lambda\}\}is a sum of four terms: the loss terms associated to concept and role assertions which are aggregated with a max, the negative sampling component, and two terms for width regularization\. For each term that appear, we add one auxiliary variabletjt\_\{j\}, we “drop the objective function terms to the constraints” and we argue that each resulting constraint can be written in terms of linear equalities/inequalities and SOC constraints, i\.e\., the epigraphs of the functions are SOCr\. Naturally, ifggis SOCr then the composition ofggwith an affine function is SOCr, e\.g\., see\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25), Remark 3\.3\.1\)131313It is enough to observe that ifhhis an affine function of the formh​\(x\)=B​x\+dh\(x\)=Bx\+d, forBBa linear map andddis a vector, we can obtain a SOC representation of the epigraph ofg∘hg\\circ hby adding the linear constrainty=B​x\+dy=Bx\+dto a SOC representation of\{\(y,t\)∣g​\(y\)≤t\}\\\{\(y,t\)\\mid g\(y\)\\leq t\\\}\(the epigraph ofgg\)\.\.

The building blocks for the loss term and regularization terms inf𝝀f\_\{\\boldsymbol\{\\lambda\}\}boil down to two functions: the signed distance function𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)and the 2\-norm function∥⋅∥2\\\|\\cdot\\\|\_\{2\}\. All the four terms inf𝝀f\_\{\\boldsymbol\{\\lambda\}\}are obtained by composing copies of𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)and∥⋅∥2\\\|\\cdot\\\|\_\{2\}with affine functions and either adding them together or taking maximums\. In view of our discussion so far, it is enough to establish that the epigraphs of these two functions can be written in terms of linear equalities/inequalities and SOC constraints\. In other words, we need to show that both functions are SOCr\.

First, we will show that𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)is SOCr\. For that, we define the auxiliary functionψ:ℝ2​d×ℝ2​d→ℝ∪\{\+∞\}\\psi:\\mathbb\{R\}^\{2d\}\\times\\mathbb\{R\}^\{2d\}\\to\\mathbb\{R\}\\cup\\\{\+\\infty\\\}such that

ψ​\(y,z\)≔\{‖z‖2\+maxi∈\{1,…,2​d\}⁡\(yi−zi\)if​y≤z\+∞otherwise\.\\psi\(y,z\)\\coloneqq\\begin\{cases\}\\\|z\\\|\_\{2\}\+\\max\_\{i\\in\\\{1,\\ldots,2d\\\}\}\(y\_\{i\}\-z\_\{i\}\)&\\text\{if \}y\\leq z\\\\ \+\\infty&\\text\{otherwise\.\}\\end\{cases\}The functionψ​\(y,z\)\\psi\(y,z\)is SOCr becauseψ​\(y,z\)≤t\\psi\(y,z\)\\leq tholds fort∈ℝt\\in\\mathbb\{R\}if and only if there existst1,t2∈ℝt\_\{1\},t\_\{2\}\\in\\mathbb\{R\}such that

t1\+t2\\displaystyle t\_\{1\}\+t\_\{2\}≤t,\\displaystyle\\leq t,\(t1,‖z‖2\)\\displaystyle\(t\_\{1\},\\\|z\\\|\_\{2\}\)∈𝒦22​d\+1,\\displaystyle\\in\{\\mathcal\{K\}\_\{2\}^\{2d\+1\}\},yi−zi\\displaystyle y\_\{i\}\-z\_\{i\}≤t2,∀i∈\{1,…,2​d\}\\displaystyle\\leq t\_\{2\},\\quad\\forall i\\in\\\{1,\\ldots,2d\\\}y\\displaystyle y≤z\.\\displaystyle\\leq z\.Furthermore, Proposition[4](https://arxiv.org/html/2605.23937#Thmproposition4)implies that𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)is the partial minimization ofψ\\psiwith respect to the second argument, i\.e\.,

𝗌𝖽𝗂𝗌𝗍\(y,ℝ−2​d\)=infz∈ℝ2​dψ​\(y,z\)\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{2d\}\_\{\-\}\)=\\inf\_\{z\\in\\mathbb\{R\}^\{2d\}\}\\psi\(y,z\)holds for everyy∈ℝ2​dy\\in\\mathbb\{R\}^\{2d\}\. Furthermore, the proof of Proposition[4](https://arxiv.org/html/2605.23937#Thmproposition4)shows that for everyyy, there existszyz\_\{y\}such that𝗌𝖽𝗂𝗌𝗍\(y,ℝ−2​d\)=ψ​\(y,zy\)\\mathop\{\\mathsf\{sdist\}\}\(y,\\mathbb\{R\}^\{2d\}\_\{\-\}\)=\\psi\(y,z\_\{y\}\)holds, i\.e\., the infimum is achieved for everyyy\. The property of being SOCr is preserved by partial minimization assuming that for everyyythe infimum is achieved, e\.g\., see Section 3\.3 in\(Ben\-Tal and Nemirovski,[2001](https://arxiv.org/html/2605.23937#bib.bib25)\)\. Therefore,𝗌𝖽𝗂𝗌𝗍\(⋅,ℝ−2​d\)\\mathop\{\\mathsf\{sdist\}\}\(\\cdot,\\mathbb\{R\}^\{2d\}\_\{\-\}\)is SOCr as well\.

Next, the function∥⋅∥2\\\|\\cdot\\\|\_\{2\}is also SOCr, since‖y‖2≤t\\\|y\\\|\_\{2\}\\leq tholds fory∈ℝd,t∈ℝy\\in\\mathbb\{R\}^\{d\},t\\in\\mathbb\{R\}if and only if\(t,y\)∈𝒦2d\(t,y\)\\in\{\\mathcal\{K\}\_\{2\}^\{d\}\}\.

In conclusion, all the functions used to build the objective functionf𝝀f\_\{\\boldsymbol\{\\lambda\}\}are SOCr, so overall, the problem in \([8](https://arxiv.org/html/2605.23937#S5.E8)\) has a SOCP formulation\. ∎

## Appendix FSize of the final optimization problem

As mentioned in Section[6](https://arxiv.org/html/2605.23937#S6), a problem modelled through CVXPY is first compiled and then sent to a solver such as MOSEK\. Here we report in Table[8](https://arxiv.org/html/2605.23937#A6.T8)the number of variables and constraints of the final optimization problem that CVXPY outputs to MOSEK\.

Table 8:Problem sizes of the final optimization problem solved by MOSEK split by dataset\.
## Appendix GExperimental Details

This section describes our experimental setup, created benchmark datastes, and evaluation protocol in detail\. In particular, Section[G\.1](https://arxiv.org/html/2605.23937#A7.SS1)contains details about the implementation of BoxLitE\. Furthermore, Section[G\.2](https://arxiv.org/html/2605.23937#A7.SS2)discusses the properties of the created benchmark datasets F\_v1\-4\. Next, Section[G\.3](https://arxiv.org/html/2605.23937#A7.SS3), describes our experimental setup, including a detailed description of the learning setup, used hardware, and selected hyperparameters\. Continuing from that, Section[G\.4](https://arxiv.org/html/2605.23937#A7.SS4)discusses the evaluation protocol and metrics in detail\. Moreover, Section[G\.5](https://arxiv.org/html/2605.23937#A7.SS5)shows the runtime of BoxLitE and the SGD solvers over the various Family dataset subsets\. Finally, Section[G\.6](https://arxiv.org/html/2605.23937#A7.SS6)details the hyperparameters used for the SGD models we have shown in our experiments\.

### G\.1Implementation & Reproducibility

We implemented the BoxLitE \(convex\) second\-order cone optimization problem in Python 3\.12 using CVXPY for formulating the problem and MOSEK for optimizing it\. To make our findings reproducible, we include BoxLitE’s code, the created datasets F\_v1\-4, a ReadMe\.md file listing library dependencies, installation, and execution instructions as part of the supplemental material\. The seed used for the SGD models is69346934\. We will make all provided supplementary materials \(including BoxLitE’s code base and the created datasets\) publicly available upon acceptance of our paper\.

### G\.2Details on F\_v1\-4

This section contains details about the created benchmark datasets F\_v1\-4 of Section[6](https://arxiv.org/html/2605.23937#S6)\. In particular, we have created a set of datasets \(F\_v1\-4\) of varying sizes from the family dataset\(Imeneset al\.,[2023](https://arxiv.org/html/2605.23937#bib.bib41)\)\. We derived these datasets by sampling approximatelyk∈\{300,500,1000,3000\}k\\in\\\{300,500,1000,3000\\\}assertions of the family dataset’s ABox with forest fire sampling\(Leskovecet al\.,[2005](https://arxiv.org/html/2605.23937#bib.bib53)\), a popular sampling technique for large graphs, using a forward burning probability \(𝗉𝖿=0\.7\{\\sf pf\}=0\.7\) and a backward burning probability \(𝖻𝖿=0\{\\sf bf\}=0\)\. Furthermore, since the family dataset solely provides role assertions in its ABox, we selected any TBox inclusion of the family dataset that includes solely roles and extended the TBox by the disjointness axioms∃𝗁𝖺𝗌𝖥𝖺𝗍𝗁𝖾𝗋−⊑¬∃𝗁𝖺𝗌𝖬𝗈𝗍𝗁𝖾𝗋−\\exists\{\\sf hasFather^\{\-\}\}\\sqsubseteq\\neg\\exists\{\\sf hasMother^\{\-\}\}\. Figure[1](https://arxiv.org/html/2605.23937#S6.F1)lists the TBox of the created datasets F\_v1\-4\.

Next, we created a set of inferred assertions by\(i\)\(i\)adding any assertion that logically follows from each dataset and\(i​i\)\(ii\)removing any assertion that occurs in the ABox\. We randomly split this set of inferred assertions into a validation set \(20%\), used for model selection, and a test set \(80%\), used for evaluating the performance of the selected model\.

### G\.3Experimental Setup

Training Setup\.During the training phase, we optimized the loss described in[5\.1](https://arxiv.org/html/2605.23937#S5.SS1)using MOSEK on the train set\. After retrieving an embedding solution from MOSEK, we evaluated its performance on the validation set, which we used for selecting the best embedding solution \(see Section[G\.4](https://arxiv.org/html/2605.23937#A7.SS4)for more details on the evaluation protocol\)\. We now discuss BoxLitE’s hyperparameter optimization\.

Hyperparameter Optimization\.We setsΩ=1\{s\_\{\\Omega\}\}=1andϵ=10−2\\epsilon=10^\{\-2\}for all of our experiments\. For the benchmark results on datasets F\_v1\-4, we setd=32d=32and tuned the hyperparameters within the following ranges:λ1,λ2,λ3∈\{0,0\.001,0\.003,0\.1,0\.3,1,3\}\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}\\in\\\{0,0\.001,0\.003,0\.1,0\.3,1,3\\\}\. We list the best found hyperparameters for each of the datasets in Table[9](https://arxiv.org/html/2605.23937#A7.T9)\.

Table 9:Best found hyperparameters for BoxLitE\.Table 10:Best found hyperparameters for BoxLitE1\.Table 11:Best found hyperparameters for BoxLitE2\.Table 12:Best found hyperparameters for BoxLitE3\.
### G\.4Evaluation Protocol

We have evaluated BoxLitE, by following the standard evaluation setting for KB completion as described by\(Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Pavlović and Sallinger,[2023b](https://arxiv.org/html/2605.23937#bib.bib48); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\)\. In particular, this includes measuring the ranking quality of each role assertionR​\(a,b\)R\(a,b\)in the test set over any possible individual in the first position of the assertion, i\.e\.,R​\(a′,b\)R\(a^\{\\prime\},b\)for alla′∈𝖭𝖨a^\{\\prime\}\\in\{\\sf N\_\{I\}\}, and the second position of the assertion, i\.e\.,R​\(a,b′\)R\(a,b^\{\\prime\}\)for allb′∈𝖭𝖨b^\{\\prime\}\\in\{\\sf N\_\{I\}\}\. Furthermore, we used the standard metrics for KB completion, namely, the mean reciprocal rank \(MRR\) and hits at k \(H@k\)\.

As typically done in the literature\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42); Sunet al\.,[2019](https://arxiv.org/html/2605.23937#bib.bib34); Abboudet al\.,[2020](https://arxiv.org/html/2605.23937#bib.bib21); Pavlović and Sallinger,[2023b](https://arxiv.org/html/2605.23937#bib.bib48); Xionget al\.,[2022](https://arxiv.org/html/2605.23937#bib.bib75)\), we presented the filtered versions of these merics introduced by\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42)\)\. This means in particular that for hyperparameter tuning, we evaluated each of the found BoxLitE embedding solutions on the validation set and excluded any assertion from the ranking that occurs in the train or validation set \(apart from the validation assertion whose score shall be computed\)\. We selected those embedding solutions that reached the highest scores on the validation set\. We followed the filtered setting\(Bordeset al\.,[2013](https://arxiv.org/html/2605.23937#bib.bib42)\)also during the final evaluation, i\.e\., we evaluated the selected embedding solutions on the test set and excluded any assertion from the ranking that occurs in the train, validation, or test set \(apart from the test assertion whose score shall be computed\)\.

The intuition of the filtered setting is that we exclude assertions from the ranking that are during the current evaluation stage known to be true, as assigning a high score to these assertions does not indicate a wrong inference\. Specifically, during hyperparameter tuning on the validation set, the train and validation assertions are known to be true and thus need to be excluded; while during the final evaluation on the test set, the train, validation, and test assertions are known to be true and thus need to be excluded from the ranking\. Finally, we briefly review the definition of H@k and the MRR: H@k reflects the proportion of true assertions within the predicted assertions whose rank is at mostkk, whereas the MRR represents their average of inverse ranks \(1/rank1/\\textit\{rank\}\)\.

### G\.5Running Time

For each choice of the hyperparametersλ1,λ2,λ3\\lambda\_\{1\},\\lambda\_\{2\},\\lambda\_\{3\}, the evaluation of BoxLitE takes less than a minute for F\_v1 and F\_v2\. It takes 2 minutes and 38 seconds for F\_v3 and more than 20 minutes for F\_v4\. Improving our evaluation procedure would contribute for scalability\.

Table 13:Training time split by dataset for SGD methods\.Regarding SGD methods, we provide in Table[13](https://arxiv.org/html/2605.23937#A7.T13)the training time for the SGD methods with100100trials and500500epochs for each method\. We used the same dimensionality3232as in BoxLitE \(see[G\.3](https://arxiv.org/html/2605.23937#A7.SS3)\)\.

### G\.6SGD setup

The SGD models shown in our experiments were trained using PyKEEN’s implementation of BoxE, RotatE, and ComplEx\. We trained each model on the different Family subsets for 100 trials over 500 epochs\. The optimizer used is Adagrad, together with early stopping\. The frequency, patience, and relative delta of the stopper are1010,1010, and0\.010\.01, respectively\. Lastly, the embedding dimension, as in BoxLitE, is set to3232\. The remaining parameters are set to the defaults of the KGE implementations in PykEEN\(Aliet al\.,[2021](https://arxiv.org/html/2605.23937#bib.bib68)\)\.