How Expressive Are Box Embeddings?

1. Concepts as Sets

Let $V$ be a universe of objects. A concept $p$ is a Boolean predicate $p:V\to\{0,1\}$, and it determines a set $[p]=\{x\in V:p(x)=1\}\subseteq V$. Thus, we can identify a concept with the set of objects satisfying it.

Logical conjunction then becomes set intersection: $$[p\land q]=[p]\cap[q].$$ This gives a simple question: can we represent arbitrary concepts by boxes while preserving their intersection structure?

2. Representing Concepts by Boxes

Suppose each concept $A\subseteq V$ is represented by an axis-aligned box $B_A\subseteq\mathbb R^d$. If the representation preserves conjunction exactly, we would like $$B_{A\cap B}=B_A\cap B_B.$$ Boxes appear especially attractive for this purpose because they are closed under intersection: the intersection of two axis-aligned boxes is again an axis-aligned box, possibly empty.

But closure under an operation does not yet tell us which abstract intersection structures can be represented by that operation.

3. The Smallest Obstruction

Consider three concepts $A,B,C$ such that every pair is compatible: $$A\cap B\neq\varnothing,\qquad A\cap C\neq\varnothing,\qquad B\cap C\neq\varnothing.$$ At the same time, suppose the three concepts are jointly incompatible: $$A\cap B\cap C=\varnothing.$$ There is nothing contradictory about this pattern for arbitrary sets.

Can three boxes realize the same pattern?

4. Boxes Cannot Represent This Pattern

Let $B_1,B_2,B_3\subseteq\mathbb R^d$ be axis-aligned boxes. If $$B_1\cap B_2\neq\varnothing,\qquad B_1\cap B_3\neq\varnothing,\qquad B_2\cap B_3\neq\varnothing,$$ then necessarily $$B_1\cap B_2\cap B_3\neq\varnothing.$$ To see why, write each box as a product of intervals. In every coordinate, the three corresponding intervals intersect pairwise. For intervals on the real line, pairwise intersection implies a common intersection. Choosing one common point in every coordinate gives a point belonging to all three boxes.

This is the Helly property of axis-aligned boxes: their Helly number is $2$. Consequently, no dimension $d$ can make boxes represent the intersection pattern above exactly.

5. Closure Is Not Expressiveness

We therefore have an important distinction: $$\boxed{\text{closed under intersection}\;\not\Rightarrow\;\text{able to represent every intersection structure}.}$$ Boxes possess a particularly simple geometric operation for conjunction, but their geometry also imposes additional constraints that arbitrary sets do not satisfy.

Thus, being able to compute an operation inside a representation family is different from being able to faithfully represent every abstract structure generated by that operation.

6. Representation Capacity

The natural question is therefore not merely whether boxes support intersection, but which intersection structures boxes can represent exactly. More generally, for a geometric family $\mathcal F$, we can ask: which abstract meet or intersection structures admit a faithful representation inside $\mathcal F$?

For boxes, the Helly property already gives a minimal obstruction. It also points toward a broader problem: $$\boxed{\text{logical expressiveness}\quad\text{vs.}\quad\text{geometric simplicity}.}$$ The next question is therefore: which abstract intersection structures are exactly representable by boxes?