We present a region-based geometric model for embedding normalized ℰℒℋ ontologies into a continuous vector space and formally prove that normalized ℰℒℋ has the strong faithfulness property on convex geometric models. This means that there is an embedding that precisely captures the original ontology. We first prove the statement assuming (possibly) non-convex regions, and then impose convexity on the regions, showing that the property still holds.
Knowledge Graphs (KGs) are a popular method for representing knowledge using triples of the form
(subject, predicate, object), called facts, corresponding to role assertions in DL ABoxes. Although KGs
contain a large number of facts, they are incomplete. This has sparked interest in using machine
learning methods to suggest plausible facts to add to the KG based on patterns found in the data using
KG embedding techniques, which aim to create representations of KGs in low-dimensional finite vector
spaces [
Embeddings that consider TBoxes formulated in description logic are a more recent phenomenon, here
referred as ontology embeddings, where the ontology can have both facts and a TBox [
We investigate whether ℰ ℒℋ has the strong faithfulness property over convex geometric models. We ifrst prove the statement for embeddings in low dimensions, considering a region-based representation for (possibly) non-convex regions. We investigate strong faithfulness on convex geometric models with a number of dimensions connected to an ontology’s signature and domain. We do so including embeddings for role inclusions, a problem that has not been well studied in the ℰ ℒℋ ontology embedding literature. Additionally, we consider model checking in convex geometric models.
Geometric models We go from the traditional model-theoretic interpretation of the ℰ ℒℋ description
logic to geometric interpretations, adapting definitions by Gutiérrez-Basulto and Schockaert [
Bourgaux et al. [12]. Here we present the basic ingredients of the notion of a geometric interpretation. The main idea of geometric models is that the sets on which we define our interpretation functions are continuous vector spaces, and that the interpretation of concepts and roles are defined over geometric regions.
Definition 1 (Isomorphism Preserving Linear Maps). Let be a natural number and : R ×
R ↦→
3. (R × { 0}) ∩ ({0} × R) = {02· }; where 0 denotes the vector (0, ..., 0) with zeros. For instance, the concatenation function is a linear map that satisfies Points 1, 2, and 3.
Definition 2 (Geometric Interpretation). Let be an isomorphism preserving linear map and a natural number. An -dimensional -geometric interpretation of ( , , ) assigns to each ∈ a region () ⊆ R, to each ∈ a region () ⊆ R2· , and to each ∈ a vector () ∈ R. We have that (⊥) := ∅, (⊤) := R, and for arbitrary ℰ ℒℋ concepts:
( ⊓ ) := () ∩ () and (∃.) := { ∈ R | ∃ ∈ () with (, ) ∈ ()}. An -dimensional -geometric interpretation satisfies • a concept assertion (), if () ∈ (), a role assertion (, ), if ( (), ()) ∈ (), • an ℰ ℒℋ instance query (IQ) (), if () ∈ (), • an ℰ ℒℋ CI ⊑ , if () ⊆ (), and an RI ⊑ , if () ⊆ ().
We write |= if satisfies an ℰ ℒℋ axiom . A geometric interpretation satisfies an ontology ,
in symbols |= , if it satisfies all axioms in . We define model faithfulness based on the work by [
We say that an ontology language has the strong faithfulness property over a class of geometric interpretations if for every satisfiable ontology in this language there is a geometric interpretation in that is both a strongly IQ-faithful and a strongly TBox-faithful model of .
We prove that normalized ℰ ℒℋ has the strong faithfulness property over a class of convex geometric
models. We introduce a new mapping from the domain of a classical interpretation ℐ to a vector space
and a new geometric interpretation ℐ based on this mapping. Our proofs now require us to fix the
isomorphism preserving linear map used in the definition of geometric interpretations (Definition 2).
We choose the concatenation function, denoted ⊕ , as done in Gutiérrez-Basulto and Schockaert [
Let ℐ = (∆ ℐ , · ℐ ) be a classical ℰ ℒℋ interpretation, and an ℰ ℒℋ ontology. We start by without loss of generality, a fixed ordering in our indexing system for positions in vectors, where indices 0 to | ()| −
1 correspond to the indices for individual names; | ()| to = | ()| + | ()| − 1 correspond to the indices for concept names; and to + (|()| · | ∆ ℐ ) | − 1 correspond to the indices for role names together with an element of ∆ ℐ . We write [], [], and [, ] to refer to the position in a vector corresponding to , , and together with an element , respectively (according to our indexing system). For example, [] = 0 means the value at the index corresponding to the individual name is 0. A vector is binary if ∈ {0, 1}d. We define using binary vectors. For all ∈ ∆ ℐ , ∈ , ∈ • ()[] = 1 if = ℐ , otherwise ()[] = 0, • ()[] = 1 if ∈ ℐ , otherwise ()[] = 0, and • ()[, ] = 1 if (, ) ∈ ℐ , otherwise ()[, ] = 0.
We introduce a definition for (possibly) non-convex geometric interpretations, using . Definition 5. defined as:
Let ℐ be a classical ℰ ℒℋ interpretation. The geometric interpretation of ℐ, denoted ℐ , is ℐ () := (ℐ ), for all ∈ , ℐ () := { () | ()[] = 1, ∈ ∆ ℐ }, for all ∈ , ℐ () := { () ⊕ () | ()[, ] = 1, , ∈ ∆ ℐ }, for all ∈ .
An intuitive way of thinking about our definition is that it maps domain elements to a subset of the vertex set of the d-dimensional unit hypercube, as in Fig. 1. ()[] = 1, and ()[] = 0, while ()[] = 0, ()[] = 1, and ()[] = 1.
Example 1. Consider , ∈ and ∈ . Let ℐ be an interpretation with , ∈ ∆ ℐ such that = ℐ , ∈ ℐ , and ∈ ℐ ∩ ℐ . We illustrate () and () in Fig. 1. In symbols, ()[] = 1, Theorem 1. For all ℰ ℒℋ axioms , ℐ |= if
ℐ |= .
The canonical models need to be finite because the definition of ℐ uses vectors in a finitelyifnite. The following holds for the canonical model just defined. dimensional space whose dimension depends on the size of ∆ and . We then employ finite canonical models for normalized ℰ ℒℋ since canonical models for arbitrary ℰ ℒℋ CIs are not guaranteed to be Theorem 2. Let be a normalized ℰ ℒℋ ontology. The following holds • for all ℰ ℒℋ IQs and CIs in normal form over sig(), ℐ |= if |= ; and • for all RIs over sig(), ℐ |= if |= .
Theorem 3. Let be an ℰ ℒℋ ontology and let ℐ be the canonical model of . The d-dimensional (possibly non-convex) ⊕ -geometric interpretation ℐ of ℐ is a strongly and IQ and TBox faithful model of .
Here, the vectors mapped by and the regions given by the non-convex geometric interpretation ℐ are the anchor points for the convex closure of these sets. The geometric interpretation obtained by taking the convex hull is denoted ℐ* .
Theorem 4. Let ℐ be a geometric interpretation as in Definition 5. If is an ℰ ℒℋ CI, an ℰ ℒℋ RI, or an ℰ ℒℋ IQ in normal form then ℐ |= if ℐ* |= .
Since normalized ℰ ℒℋ has a finite canonical model, we can take advantage of the previous theorem to generalize it to arbitrary interpretations by applying it to the canonical ℐ of . Theorem 5. Let be a normalized ℰ ℒℋ ontology and let ℐ be the canonical model of . The ddimensional convex ⊕ -geometric interpretation of ℐ is a strongly IQ and TBox faithful model of . Corollary 6. Normalized ℰ ℒℋ has the strong faithful property over ⊕ -geometric interpretations.
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