The embedding of description logic ontologies into low dimensional vector spaces is, e.g., in the context of knowledge graph embedding, an established option to do for example link prediction or concept membership prediction with the help of background information in form of an ontology. However, it is not only of interest to model expressive description logics such as ℒ but also to model temporal aspects, thus the evolution of concepts over time geometrically. Therefore, there is a need to model operators such as, e.g., eventually or next, thus temporal description logics geometrically. In this paper, an approach for embedding an expressive Boolean temporal description logic based on the embedding of concepts as closed convex cones is presented and it is proven that an ontology is satisfiable in the classical sense if and only if it is satisfiable in a geometric model based on the presented embedding. This model is a first step towards a learning approach able to model background information in form of an expressive temporal description logic ontology.
Though subsymbolic learning approaches gained
importance due to good-quality results in the last years, they
lack important features such as explainability and
trustworthiness. This led to the research area of Neuro-Symbolic
AI [
Those approaches are based on the idea of embedding
concepts as convex sets in a vector space and logical
operations between concepts as geometrical operations between
those sets, e.g., representing concept conjunction as set
intersection. An instance belonging to a concept is then
modeled as a point in the convex set representing this
concept. For those instances, e.g., relations between them or
their concept membership can be predicted. The approaches
are, e.g., able to model the description logic (DL) ℰ ℒ++, e.g.,
by representing concepts as boxes [
However, concepts are normally not static but evolve
over time. A cured person is, for example, a person who has
recovered from an illness. Therefore, to model the concept
of cure, it is necessary to model both the concepts of
healthiness and of illness and a temporal combination thereof. To
model such ideas, temporal logic can be used and, in the
context of ontologies, this leads to the area of several diferent
temporal description logics (see, e.g., [
This directly leads to the question whether existing approaches for embedding ℒ can be extended to model also temporal aspects and especially whether such a model has the same expressivity as the DL interpretation, thus, whether a temporal DL-ontology is satisfiable if and only if the corresponding geometric model is satisfiable.
Though, temporal knowledge graph embeddings are a
widely studied topic (see, e.g., [
The basic idea is to extend the cone-based embedding of
Özçep et al. [
In contrast to classical KGE-approaches, the focus lies here on representing concepts and their temporal aspects and not on relations. Thus, a Boolean temporal ℒontology is considered. The main result is that it is actually possible to model temporal DL-ontologies via those models based on cones, namely that a temporal DL-ontology is satisfiable if and only if it is satisfiable in such a geometric model.
After a short introduction to the description logic ℒ
and to the temporal description logic ℒ in Section 2,
in Section 3, the cone embedding of Özçep et al. [
In the following, the description logic ℒ and the temporal description logic ℒ is shortly introduced.
We are going to work with the description logic of Boolean
ℒ, i.e., ℒ without considering roles [
| ⊥ | ⊤ | ¬ | ⊓ | ⊔ where ∈ is an atomic concept and stands for arbitrary concepts. A classical ℒ interpretation is a pair (∆ , (· )ℐ ) consisting of a set ∆ , called the domain, and an interpretation function (· )ℐ which maps constants to elements in ∆ , concept names to subsets of ∆ , and role names to subsets of ∆ × ∆ . The semantics of arbitrary concept descriptions for a given interpretation ℐ is as follows: • ⊤ℐ = ∆ • ⊥ℐ = ∅ • ( ⊓ )ℐ = ℐ ∩ ℐ • ( ⊔ )ℐ = ℐ ∪ ℐ • (¬)ℐ = ∆ ∖ ℐ An ontology is defined as a pair = ( , ) of a terminological box (TBox) and an assertional box (ABox) . A TBox consists of general inclusion axioms (GCIs) ⊑ (“ is subsumed by ”) with concept descriptions , . An ABox consists of a finite set of assertions, i.e., facts of the form () for ∈ . An interpretation ℐ models a GCI ⊑ , for short ℐ |= ⊑ , if ℐ ⊆ ℐ . An interpretation ℐ models an ABox axiom (), for short ℐ |= (), if ℐ ∈ ℐ . An interpretation is a model of an ontology ( , ) if it models all axioms appearing in ∪ . An ontology entails a (TBox or ABox) axiom , for short |= , if all models of are also models of .
Each TBox generates a Boolean algebra, the so-called Lindenbaum-Tarski algebra, as follows: For concepts , let ∼ be the relation defined by ∼ if |= ⊑ and |= ⊑ . Relation ∼ is an equivalence relation inducing for each concept an equivalence class []. Deifne operations ⊓, ⊔, ¬ on the equivalence classes by setting []⊓[] = [ ⊓], []⊔[] = [ ⊔] and ¬[] = [¬] which can be shown to fulfill the axioms of a Boolean algebra.
There are several diferent temporal description logics, for
a survey, see, e.g., [
| ◇ | □ | Note that ◇ is used as shorthand for ⊤ and □ as shorthand for ¬ ◇ ¬ .
A temporal interpretation = (∆ , (· ) ) is based on a non-empty domain ∆ and an interpretation function that maps every concept name ∈ to a subset ⊆ N× ∆ and every individual name ∈ to an element ∈ ∆ . (, ) ∈ describes that is an instance of at time point . Constants are considered as rigid, meaning they are interpreted the same way at every time point. Additionally, the constant domain assumption is assumed to be valid, meaning that constants are not destroyed or created over time. Though, there are diefrent interpretations possible, here the standard assumption of a bounded past and an unbounded future is used, where the time flow is discrete. Thus, the time is represented as (N, <). Alternatively, it is possible to interpret a temporal interpretation as an infinite sequence (0), (1), . . . of (non-temporal) interpretations based on the same domain ∆ .
The semantic is defined based on the semantic for ℒ with the extension of • (∘ ) = {(, ) | ( + 1, ) ∈ } • ( ) = {(, ) | ∃ ≥ ((, ) ∈ ∧ (, ) ∈ for ≤ < )}.
A temporal interpretation is a model of a concept if is satisfied at time point 0, i.e., (0, ) ∈ for some ∈ ∆ . An interpretation is a model of a TBox if and only if ⊆ for all ⊑ in .
The geometric interpretation that is used as basis for the
embedding approach introduced in this paper was presented
by Özçep et al [
∘ = { | for all ∈ : ⟨, ⟩ ≤ 0},
where ⟨· , ·⟩ denotes the usual scalar product. Now, the
DLinterpretation can be interpreted geometrically: The domain
∆ is interpreted as R and a concept interpretation ℐ is
interpreted as cone , the negation of this concept (¬)ℐ
as its polar ∘ , the conjunction of two concepts as set
intersection between the respective cones and disjunction via
de Morgan. An instance belonging to a concept is then
represented as a point in the respective cone. Based on this
interpretation, it can be shown that each set of cones in
R closed under set-intersection and polarity leads to an
orthologic [
, ∈ {R, R+, R− , {0}}, where R+ = { ∈ R | ≥ 0} and R− = { ∈ R | ≤ 0}. Thus, al-cones can be considered as unions of neighboring hyperoctants. So, in dimensions we have 4 possible al-cones. Conjunction, disjunction and negation are defined analogously to the case of closed convex cones, as al-cones are a specialization.
Example 1. An example for a cone-model can be seen in Figure 1 on the left, an example for an al-cone model in Figure 1 in the middle. In both of them, the TBox { ⊑ } is modeled. First consider the cone model: The dashed cone on the upper left is the interpretation of the concept , fulfilling the TBoxaxiom, as it is a subset of the gray cone, the interpretation of . The negation of concept is interpreted as the polar cone of cone ℐ , thus the cone containing all rays having an angle of 90∘ or more to all rays in in ℐ . Conjunction is interpreted as set intersection, thus ( ⊓ ¬)ℐ is the gray, dashed cone in the top right. Disjunction can be determined via de Morgan, thus, e.g., ( ⊔ ¬)ℐ would be (¬(¬ ⊓ ))ℐ , thus the polar of the gray, dashed cone in the upper right, therefore the convex hull of the cones of ℐ and (¬)ℐ . ABox-instances are interpreted as points in the space, therefore an instance with () would be placed in the cone ℐ as ℐ .
The al-cone example in Figure 1 in the middle represents the same TBox, however, based on a restricted cone model not based on arbitrary but axis-aligned cones. There, is interpreted as positive -axis and is interpreted as upper right quadrant. The polar al-cones are defined the same as for the case of classical cones, thus, for example, (¬)ℐ is the lower left quadrant.
In both cases, ⊥ is interpreted as point of origin, as there all cones intersect, leading to a contradiction, thus to ⊥ℐ .
An al-cone interpretation can be defined formally as follows: Definition 1. [6, Definition 1] A Boolean al-cone interpretation ℐ is a structure (∆ , (· )ℐ ) where ∆ is R for some ∈ N, and where (· )ℐ maps each concept symbol to some al-cone and each constant to some element in ∆ ∖ {⃗0}. An al-cone interpretation for arbitrary Boolean ℒ concepts is deifned recursively as (⊤)ℐ = ∆ , (⊥)ℐ = {⃗0}, ( ⊓ )ℐ = ℐ ∩ ℐ , (¬)ℐ = ∘ , and ( ⊔ )ℐ = (¬(¬ ⊓ ¬))ℐ .
The notions of an al-cone being a model and that of entailment are defined as in the classical case (but using al-cone interpretations).
This then leads to the following proposition: Proposition 1. [6, Proposition 2] Boolean ℒ-ontologies are classically satisfiable if and only if they are by a geometric model on some finite R based on al-cones of the form 1 × · · · × with ∈ {{0}, R+, R− , R} for ∈ {1, . . . , }.
The aim of this paper is to show a similar result for temporal description logics.
When considering those cone models, it gets apparent that on each ray (a ray through a point ∈ R contains all for ≥ 0) only one concept is represented and that instances placed on such a ray are indistinguishable on a conceptual level. Therefore, without loss of expressivity it is possible to model an al-cone model instead of in R on the unit -sphere (an unit -sphere can be defined as = { ∈ R : ‖‖ = 1}) and thus gain the space to model temporal aspects.
Example 2. In Figure 1 on the right an example for a geometric model based on a sphere can be seen for a simple ontology with only one concept . This concept is represented as a sphere part with radius one in the upper right quadrant. It is based on the al-cone of the upper right quadrant where each vector in this al-cone is set to unit length. ℐ can easily be extended into an al-cone by considering the convex closure of it. The easiest way to determine the negation of is to consider the al-cone extension of ℐ (thus, the upper right quadrant), taking the polarity (the lower left quadrant) and then intersect it with a sphere of radius one, leading to the sphere part that can be seen in the figure. An instance with () can then be placed on the sphere part representing with ||ℐ || = 1.
It can be shown that both the al-cone model and the sphere model have in fact the same expressivity. Proposition 2. Boolean ℒ-ontologies are satisfiable by a geometric model on some finite R based on al-cones of the form 1 × · · · × with ∈ {{0}, R+, R− , R} for ∈ {1, . . . , } if and only if they are satisfiable by a geometric model on some finite (with ≤ 2) where concepts are represented as intersection between al-cones and the unit sphere.
→ The concept representation can be directly
transformed to a spherical representation without
loss of expressivity, as for a concept ℐ for each
∈ ℐ , |||| ∈ ℐ . Therefore, it is possible to
interpret a concept ℐ = { | ∈ ℐ,cone&|||| =
1}, where ℐ,cone represents the original concept
representation as al-cone. It can be trivially seen
that the represented TBox-axioms do not change.
The instances can then be placed on the unit sphere
in the same manner as it is done for the construction
of the al-cone model in the proof of Proposition 1
(as can be seen in [
A diference to the cone-based model is that ⊥ℐ is no longer represented as {0} but as ∅ which, however, does not influence the satisfiability.
Assume a geometric model based on a sphere for > 0 is given. As each concept ℐ is based on the intersection of the unit sphere with an al-cone, it is trivially possible to extend the representation on the sphere to an al-cone again such that for ℐ, = { | ∈ ℐ & ≥ 0} (thus by considering the convex hull).
The interpretation of instances does not change.
Note that the increase of dimensions depends on the structure of the cone-based model and thus, the doubling of dimensions is only an upper bound.
When reducing an al-cone model to a sphere model, this opens up the opportunity to model spheres of diferent radii in a vector space, thus modeling diferent geometric models in the same vector space, as Proposition 2 is trivially adaptable to a sphere of a diferent radius. This directly leads to the possibility of modeling temporal description logics: As each temporal interpretation can be modeled as a sequence (0), (1), . . . of non-temporal interpretations, it is possible to model each () on a sphere with radius + 1 in the vector space ( + 1 is considered, as (0) is interpreted as model on the unit sphere). To follow the assumption that instances are rigid, each instance is represented as a ray, interfering with all spheres, thus with concepts of all time points. Then, an instance has an interpretation for each time point, thus for each distance 1, 2, . . . to the point of origin.
Now, the classical operators need to be adjusted to model the temporal case and the temporal operators need to be defined. The domain ∆ is, in contrast to the ℒ-case not represented as R but as the set of unit vectors in R, 28–34 thus ∆ = { | |||| = 1& ∈ R}. The tuple (, ), thus an instance at a specific time point is then represented as ( + 1) · for ∈ ∆ and a concept is represented as = {( + 1) · | ∈ ()}. Then, the interpretation of ∘ and can be straightforwardly adapted.
(∘ ) ={( + 1) · | ( + 2) · ∈ } (1) ( ) ={( + 1) · | ∃ ≥ : ( + 1) · ∈ &( + 1) · ∈ for ≤ < )}
The temporal interpretation of each time step leads to the representation of concepts as sphere parts. Now the question arises how the concepts look like when considering them in a combination of all time steps. The main observation is that the resulting sets are neither closed convex cones nor al-cones anymore, as for a point in a concept not necessarily the ray through this point is contained in that concept. It needs to be examined in a practical setting whether there are concepts which are rigid or at least constant in several successive time steps to be able to use the advantages of convexity as much as possible. However, if only one time step is considered, then it is again possible to consider closed convex cones, resp. al-cones by using the construction mentioned in the proof of Proposition 2.
This directly leads to the definition of a temporal cone interpretation, following and extending the ideas of the classical cone interpretation of Definition 1.
Definition 2. A temporal al-cone interpretation is a structure (∆ , (· ) ) where ∆ is the set of unit vectors in R for some ∈ N and where (· ) maps each concept symbol for each radius to a union of intersections of al-cones and the sphere with radius for ∈ {1, . . . , } for ∈ N or for ∈ N and each constant to an element of ∆ .
• ( ⊓ ) = ∩ • (¬) defined based on the polarity of intersected with a sphere of radius |||| for ∈ • ∘ , as denoted in Equation (1) and • ◇ , □ and ⊔ interpreted via the other operators as stated above
The construction is in the following illustrated by an
example taken from [
Example 3. As an example, a TBox is modeled which contains the statement that any non-EU country has to be first an EU-member candidate before it can be an EU-member: ¬ _ ⊓ ◇ _ ⊑ ◇ ( _ _) One exemplary model can be seen in Figure 2. The first model for time step 0 is on the sphere with radius 1. At this time point, there aren’t any EU-candidates but some EU-members. Then, to fulfill the axiom, there must be a point in time where a non-EU-member is included into the EU, this is modeled here in the second time step (in the al-cone model concatenated with a sphere with radius three) in the upper right quadrant. To model the subsumption mentioned in the axiom, it is necessary to incorporate the concept of a EU-candidate. Whereas at time step 0, there weren’t any EU-candidates, in the first time step, the instances in the upper right quadrant became EU-candidates. Those instances are in the second time step EU-members, thus, the axiom is fulfilled for the upper right quadrant. The other three quadrants do not fulfill the premise of the subsumption, as either the instances are not a non-member of the EU or they aren’t becoming a member eventually. In this low dimensional example, there are concepts represented by a point at a time step, meaning that only one instance can be placed in this concept. This can be solved by increasing the dimension of the model, as described in the proof of Proposition 2 and is omitted here for readability reasons.
One basic property of cone-based models is their ability
to model unknown or indeterminable information: the
socalled faithfulness [
Example 4 (Example 3 continued). As stated before, the upper right quadrant of the model in Figure 2 fulfills the premise and the conclusion of the stated axiom. The axes represent some atomic concepts. The other three quadrants, however, allow for the incorporation of faithfulness: The most obvious case is the lower left quadrant, there it is at no time point known whether a state is member of the EU or not, however, it is known that the state is definitely not a candidate (assume for example that someone has created the ABox knowledgeable about EU-candidates and the plans which states will be candidates, but not as knowledgeable about the history of the EU and the actual members). The lower right quadrant denotes states which are at the moment neither member nor candidate but for which it is at least thinkable of that they are maybe later on a candidate and a member. The upper left quadrant denotes states where it is known for some point in the future that they will be EU-members, however, it is not known whether they are members already or need to be candidates ifrst.
This example illustrates the wide possibilities of using faithfulness in the context of temporal models. Due to the restricted size of the model, it is not possible to model each possibility for unknown information, it is however, possible to focus on specific possibilities which should be modeled.
These temporal cone models can be used as basis for a learning approach. One option would be to extend the approach to the handling of relations (and thus to leave the Boolean case) and to do temporal knowledge graph embedding. The cone models in the form presented here also enable for embedding approaches by themselves, e.g., having a training set with instances, their attributes and their concept memberships and a test set only with instances and their attributes to predict the concepts. Additionally, possibly a temporal DL-ontology is given. Then, an embedding can be learned, e.g., via a neural network, where the optimizing function is on the one hand based on the axioms of the ontology and on the other hand based on the correct placement of instances. By choosing a suficient dimension of the cone model, it is possible to find a trade of between faithfulness on the one site and recall on the other. 1 0 (¬)0 (¬)1 (¬)2 1
2 1 (¬)0 2
3 (¬)1 28–34
The construction principle mentioned in the last section is
usable for modeling background knowledge in form of an
ontology in a learning approach, as there it is mostly
appropriate to model an approximation of the ontology. However,
when there is a need for modeling the ontology exactly, it is
necessary to prove that each classically satisfiable ontology
is also satisfiable when considering the proposed
embedding. In the following, a theorem proves this statement for
ℒ ontologies and temporal embeddings as proposed
in Definition 2. The proof for the case of Boolean ℒ
in [
With the help of this theorem, it is possible to prove the following main result of this paper, stating the strong connection between temporal cone embeddings and ℒ ontologies.
Theorem 2. A Boolean ℒ -ontology with constant domains is satisfiable if it has a geometric model on some finite R based on a temporal al-cone-interpretation as introduced in Definition 2.
Proof.
→ The ultimately periodic model property of Theorem 1 can be used as a basis for the geometric model based on cones. Based on this theorem, it is enough to model a finite set of interpretations (0), . . . , () up to the starting index and additionally the first period, thus, ( + 1), . . . , ( + ). After the time point + , the information on the period can be used to define all following time points based on the preceding ones, thus a further modeling is not necessary. Therefore, there are only ifnite many atomic concepts possible. Then it is possible to create an intermediate al-cone model where analogously to the proof of Proposition 1 each concept is placed on one half-axis. The instances can then be interpreted as points in this space as done in Proposition 1. This al-cone model can then be modified to lead to a temporal coneinterpretation as follows: First, for each atomic concept, a temporal representation (thus based on a representation for each time step) is created following the rules of Equation (1). Then, the point representing an instance in the intermediate model is changed to a ray through the point of origin, meaning ℐ = { | ≥ 0}.
Proposition 2 in combination with Proposition 1 shows that each sub-model () for ≥ 0 represents a satisfiable ℒ-ontology. As a temporal DLinterpretation can be interpreted as combination of classical interpretations () for ≥ 0 and the operators modeled can be straightforwardly interpreted as classical operators, the classical interpretation is also satisfiable.
Analogously to the case of cone models introduced in Section 3, for the temporal case it is also possible to use instead of al-cones cones as basis for the concept representation and thus increase the expressivity beyond ℒ .
The embedding of temporal description logic is an important
research topic in KGE, as the triples stored in knowledge
graphs, e.g., (, , ) can be extended with
temporal information, as not all triples are valid at every time
point. If and are separated today, the triple can
be extended to (, , , 2023). There are several
KGE-approaches handling the embedding of this temporal
information (see, e.g., an extension of the classical
KGEapproach TransE, TTransE [
In this paper, I have presented a model showing that for
temporal description logics similar approaches as for classical
description logics can be used to embed them into a vector
space, here demonstrated based on an adaptation of a model
based on closed convex cones. This is a first step towards a
learning approach which is not only able to model temporal
aspects regarding instances but also modeling their
conceptual behavior, thus it enables designing a learning approach
respecting background knowledge information. This model
opens up the opportunity to be extended to metric temporal
description logics (see, e.g., the work of Gutiérrez-Basulto
et al [
I acknowledge the financial support through the ‘Abstractron’ project funded by the Autonome Provinz Bozen Südtirol (Autonomous Province of Bolzano-Bozen) through the Research Südtirol/Alto Adige 2022 Call.