Machine learning with Semantic Web ontologies follows several
strategies, one of which involves projecting ontologies into graph
structures and applying graph embeddings or graph-based machine
learning methods to the resulting graphs. Several methods have been
developed that project ontology axioms into graphs. However, these
methods are limited in the type of axioms they can project
(totality), whether they are invertible (injectivity), and how they
exploit semantic information. These limitations restrict the kind of
tasks to which they can be applied.
Category-theoretical semantics of logic languages formalizes
interpretations using categories instead of sets, and categories
have a graph-like structure.  We developed CatE, which uses the
category-theoretical formulation of the semantics of the Description
Logic $\mathcal{ALC}$ to generate a graph representation for
ontology axioms. The CatE projection is total and injective, and
therefore overcomes limitations of other graph-based ontology
embedding methods which are generally not invertible.  We apply CatE
to a number of different tasks, including deductive and inductive
reasoning, and we demonstrate that CatE improves over state of the
art ontology embedding methods.  Furthermore, we show that CatE can
also outperform model-theoretic ontology embedding methods in
machine learning tasks in the biomedical domain.

§ INTRODUCTION

Description logic (DL) theories are used to structure knowledge in a
particular domain. Description Logics are useful as they usually
enable decidable reasoning procedures. However, performing such
operations might become computationally expensive, especially for
large theories such as those found in bioinformatics ontologies like
the Gene Ontology (GO) that contains around 50,000 concepts and over
100,000 axioms. Furthermore, while reasoning is useful in many cases,
there are scenarios and applications where incomplete or inconsistent
information is present and forms of approximate or paraconsistent
reasoning would instead be more useful to be applied.

Leveraging different forms of knowledge in machine learning algorithms
can enhance their capabilities. For instance, zero-shot learning
models use an underlying knowledge source to generate predictions
about unseen data samples. This approach is useful in tasks such as
image classification [Lampert et al., 2009]. Additionally, background
knowledge can be utilized to impose inductive bias on machine learning
models before they start learning [Cozman and Munhoz, 2021]. It can also act as constraints to
control the learning procedure. Leveraging background
knowledge from ontologies has proven to be effective in
addressing problems in bioinformatics.

Transforming (or projecting) an ontology into a graph has been a
widely explored approach that enables the use of graph embedding methods to generate vector representation of ontology
entities (concepts, individuals and roles) [Rodríguez-García and Hoehndorf, 2018, Chen et al., 2020, Chen et al., 2021]. Different
projection methods have been proposed such as
Onto2Graph [Rodríguez-García and Hoehndorf, 2018], DL2Vec [Chen et al., 2020],
or OWL2Vec* [Chen et al., 2021]. The RDF syntax tree of DL axioms can
also be considered as a graph projection.

All these methods implement a function $p$ that takes one or more
axioms and generates a set of graph edges [Zhapa-Camacho and Hoehndorf, 2023].

An embedding is a structure-preserving map between two mathematical
structures. In order to generate embeddings from the ontology
structure to the graph structure, the function $p$ must be invertible
in the sense that an axiom should be mapped to a unique set of edges
such that it is possible to retrieve the axiom information from the
graph. Not all
of the current graph projection methods implement invertible

embeddings. For
example, DL2Vec, OWL2Vec* and some implementations of Onto2Graph might
map more than one axiom into the same set of edges, which makes the
projection $p$ not injective; therefore, translating back from the
graph to the axioms is not a well defined operation since the same set
of edges might represent more than one axiom. Using the syntax tree as
a graph representation solves the invertibility problem; however,
other disadvantages appear. For example, using the RDF syntax of DL
axioms generate graphs that are noisy since they include a high number
of blank nodes. This has an effect on the embedding step from the
graph structure into $\mathbb{R}^n$ [Chen et al., 2021, Zhapa-Camacho and Hoehndorf, 2023], affecting the predictive performance.

Another property of function $p$ is totality, which means that
$p$ should be able to  use all the possible information in the theory
(total function), which is not the case in some projections
methods. Totality can be defined on the set of asserted axioms and
also in the deductive closure of the DL theory. Methods such as
Onto2Graph explore the information in the deductive closure to
generate edges, whereas OWL2Vec* or RDF projections define a fixed set
of graph edges to associate to particular axioms.

Furthermore, concept descriptions of DL languages have an inherent
ordered structure that fulfills the properties of a
lattice [Borgwardt and Peñaloza, 2011, Borgwardt and Peñaloza, 2012]. Current graph representations do not consider
this aspect and the generated graphs might violate the lattice

We propose the use of a new projection method that fulfills the
totality, invertibility and lattice-preserving properties. Our method
is based on the diagrammatic representation of logical formulas under
the lens of category theory.  Category theory characterizes the common
properties across different mathematical structures by using a single
structure called category.  A category contains objects and
morphisms, which have a graph-like structure with nodes and
edges, respectively. Categorical logic [Awodey, 2010] is the study
of logic languages (i.e., propositional, first-order) under the lens
of category theory. In recent years, formalization under category
theory for description logics have been developed [Duc, 2022]
together with reasoning procedures such as unsatisfiability
computation in the category space [Brieulle et al., 2022].
Leveraging the categorical representation of a Description Logic
theory can be used to overcome previous limitations of graph
representation of ontologies. We show that this new projection
function is invertible, total, and preserves the lattice structure of
concepts in a Description Logic theory. Furthermore, we use the
generated graph structure to generate embeddings for ontology entities
in $R^{n}$ where we can formulate knowledge base completion tasks as axiom
prediction. We conduct several experiments
related to knowledge base completion such in a inductive and deductive
Furthermore, we show that we can take advantage of a category-theoretical reasoning procedures to augment the
generated graph and improve the axiom prediction performance.

We evaluate the performance of categorical projections (a) deductive
reasoning tasks such as prediction of axioms $\sub{C}{D}$ in the
deductive closure and prediction of concept satisfiability formulated as
prediction of axioms $\sub{C}{\bot}$ and (b) inductive reasoning tasks
such as predictions of axioms $\sub{C}{D}$ and $\sub{C}{\exists R.D}$
that are plausible but not necessarily can be entailed. For prediction
of axioms $\sub{C}{\exists R.D}$, we conduct the experiment on a task in
the biomedical domain.

In summary, our contributions are the following:

* We developed a novel graph-based embedding method for
Description Logic theories based on category-theoretical semantics.

* We demonstrate that our method improves results on several tasks
involving inductive and deductive reasoning.

* We apply our method to a use case from the biomedical domain and
predict protein–protein interactions based on axiom prediction, and
demonstrate that we can improve the state of the art in this task.

* We show that partial saturation procedures in the categorical space can
be leveraged to enhance the predictive performance in the inductive
reasoning task.

§ PRELIMINARIES

§.§ Description Logics

A Description Logic language consists of a signature
$\Sigma = (\mathbf{C}, \mathbf{R}, \mathbf{I})$, where $\mathbf{C}$ is
a set of concept names, $\mathbf{R}$ is a set of role names and
$\mathbf{I}$ is a set of individual names. In the language
$\mathcal{ALC}$, complex concept descriptions can be constructed
inductively, and if $A$ and $B$ are concept descriptions and $r$ a
role name, then $A \sqcap B$, $A \sqcup B$, $\neg A$, $\exists r.A$,
and $\forall r.A$ are concept descriptions. If $A$ and $B$ are concept
descriptions, $a$ and $b$ individual names, $r$ a role name, then
$A \sqsubseteq B$, $r(a,b)$, and $B(a)$ are axioms in
$\mathcal{ALC}$. A set of axioms is an $\mathcal{ALC}$ theory.

An interpretation of an $\mathcal{ALC}$ theory consists
of an interpretation domain $\Delta^{\mathcal{I}}$ and an
interpretation function $\cdot^{\mathcal{I}}$.  For every

name $C \in \mathbf{C}$, $\interp{C} \subseteq \interp{\Delta}$;
individual name $a \in \mathbf{I}$, $\interp{a} \in \interp{\Delta}$;
role name $r \in \mathbf{R}$, $\interp{r} \in \interp{\Delta} \times
\interp{\Delta}$; and, inductively:

\begin{equation*}
\begin{split}
{\bot}^{\mathcal{I}}            &=\emptyset \\
{\top}^{\mathcal{I}}             &=\Delta^{\mathcal{I}} \\
(\neg A)^{\mathcal{I}}           &=\Delta^{\mathcal{I}} \backslash A^{\mathcal{I}} \\
(C \sqcap D)^{\mathcal{I}}       &=C^{\mathcal{I}} \cap D^{\mathcal{I}} \\
\end{split}
\quad \quad
\begin{split}
(C \sqcup D)^{\mathcal{I}}     &=C^{\mathcal{I}} \cup D^{\mathcal{I}}\\
(\exists R . C)^{\mathcal{I}} &=\left\{a \in \Delta^{\mathcal{I}}
\mid \exists b . ((a, b) \in R^{\mathcal{I}} \land b \in \interp{C})\right\}\\
(\forall R . C)^{\mathcal{I}}    &=\left\{a \in \Delta^{\mathcal{I}}
\mid \forall b . ((a, b) \in R^{\mathcal{I}} \rightarrow b \in
\end{split}
\end{equation*}

An interpretation $\mathcal{I}$ is a model for an axiom $\sub{C}{D}$
if and only if $\interp{C} \subseteq \interp{D}$ [Baader et al., 2003];
for an axiom $B(a)$ if and only if $\interp{a} \in \interp{B}$; and
for an axiom $r(a,b)$ if and only if
$(\interp{a},\interp{b}) \in \interp{r}$.  Given an $\mathcal{ALC}$
theory $T$, a concept description $C$ is called unsatisfiable if
$\interp{C} = \emptyset$ in all models of $T$.

A preorder is a set $P$ equipped with a reflexive and
transitive binary relation $\leq$. A partial order is a preorder that
is also antisymmetric. A lattice is a partially ordered set where each
two-element subset has a least upper and greatest lower bound. If a
lattice has a greatest element, it is denoted $\top$, and if it has a
least element it is denoted $\bot$.  The inclusion operation
$\subseteq$ induces a lattice structure over the set of

concept descriptions. In this
lattice, bottom and top elements are $\bot$ and $\top$, respectively
and for any concept description $X$ fulfills
$\interp{\bot} \subseteq \interp{X} \subseteq \interp{\top}$ (Figure <ref>).

§.§ Category Theory and ALC description logic

Category theory is a field in mathematics that studies the properties
of mathematical structures

and the
relationships among them. A category $\mathcal{C}$ is a
structure with two components: a collection of objects called
$Obj(\mathcal{C})$ and a collection of morphisms called
$Mor(\mathcal{C})$. These collections must satisfy two properties: (i)
identity: for each $e \in Obj(\mathcal{C})$, there is
$id_e \in Mor(\mathcal{C})$ of the form $e \xrightarrow{id_e} e$ and
(ii) composition: given $a, b, c \in \obj{C}$ and $f, g \in \mor{C}$
of the form $a \xrightarrow{f} b $ and $b \xrightarrow{g} c$, then
$h \equiv g \circ f \in \mor{C}$ of the form $a \xrightarrow{h} c$
(Figure <ref>).

Given two categories C and D, a functor
$F: \cat{C}\rightarrow \cat{D}$ is a mapping that maps
every object $o \in \obj{C}$ to an object $F(o) \in
\obj{D}$. Furthermore, $F$ maps
every morphism
$f \in \mor{C}$ to a morphism $F(f) \in \mor{D}$
(Figure <ref>).

Category with identity and composition morphisms.

Functor $F$ between categories C and D

Lattice representation. $\bot$ is
the bottom element and $\top$ is to top element.

Representation of (a) identity and composition morphisms, (b)
functor between two categories and (c) a lattice.

Based on the categorical formulation of properties that are common in
different mathematical structures (preorders, groups, topological
spaces, logical systems) some categorical structures have been defined
in the literature. Those structures describe operations that are
equivalent across different mathematical structures.

Categorical logic consists on formalizing logics under the perspective of
category theory. For example, propositional and first-order logics have been characterized under
in this way, where logical formulas become
objects and the entailment relation ($\vdash$) is a morphism [Awodey, 2010].
Furthermore, operations such as conjunction and disjunction in logic correspond to
products and coproducts in categories.

Existential and universal quantifiers are formulated using
functors [Awodey, 2010, Lawvere, 1969].

Recent developments have focused on formalizing description logics,
especially the $\mathcal{ALC}$ language under category
theory [Duc, 2022]. In this formulation, two categories are
defined, one for concept descriptions $\mathbf{Con}$ and one for roles
$\mathbf{Rol}$. The morphisms are given by the subsumption relation
($\sqsubseteq$) (i.e., $A \sqsubseteq B$ corresponds to morphism
$A \rightarrow B$).  Sound and complete reasoning procedures for

$\mathcal{ALC}$ under the categorical formulation have also been
developed [Brieulle et al., 2022].

§.§.§ Products, coproducts, intersection and union.

Given concept descriptions $\phi$ and $\gamma$ in a category, the
morphism $\phi \rightarrow \gamma$ exists if
$\phi \sqsubseteq \gamma$. Operations such as intersection and union
correspond to categorical structures called product and
coproduct, respectively. The intersection $\phi \sqcap \gamma$
is represented as the diagram in Figure <ref> where every
morphism corresponds to an subsumption operation. In a product,
morphisms $\pi_i$ are called projections and the object
$\phi \sqcap \gamma$ is a limit in the sense that if there are
objects $X$ such that $X \xrightarrow{p_1} \phi$ and
$X \xrightarrow{p_2} \gamma$, then there is a unique morphism
$X \xrightarrow{m} \phi \sqcap \gamma$ such that
$p_1 \equiv \pi_1 \circ m$ and $p_2 \equiv \pi_2 \circ m$.

The dual of a product is a coproduct. Coproducts are used to represent
unions $\phi \sqcup \gamma$ (Figure <ref>). Morphisms
$\iota_i$ are called injections and the object
$\phi \sqcup \gamma$ is a colimit because if there are objects
$X$ such that $\phi \xrightarrow{i_1} X$ and
$\gamma \xrightarrow{i_2} X$, then there is a unique morphism
$\phi \sqcup \gamma \xrightarrow{m} X$ such that
$i_1 \equiv m \circ \iota_1$ and $ i_2 \equiv m \circ \iota_2$.

\begin{equation*}
\bfig
\Atrianglepair/>`-->`>`<-`>/[X`\phi`\phi \sqcap \gamma`\gamma;p_1`m`p_2`\pi_1`\pi_2]
\efig
\end{equation*}

Intersection as product

\begin{equation*}
\bfig
\Vtrianglepair/>`<-`>`-->`>/[\phi`\phi \sqcup \gamma`\gamma`X;\iota_1`\iota_2`i_1`m`i_2]
\efig
\end{equation*}

Union as coproduct

Representation of union and intersection concepts as
categorical diagrams

§.§.§ Functors, existential and universal quantifications.

Based on Duc, 2022,

the  category is equipped with two functors
$domain: \rol{} \rightarrow \con{}$ and
$codomain: \rol{} \rightarrow \con{}$ mapping every role
$R \in \rol{}$ to objects $domain(R)$ and $codomain(R)$ in
, respectively.  A concept description $\exists R. C$
induces the existence of a role $R_{\exists R. C} \in \obj{\rol{}}$
together with objects
$domain(R_{\exists R.C})$ and $ codomain(R_{\exists R. C}) \in
\obj{\con{}}$. The following morphisms are also induced:

\begin{align*}
R_{\exists R.C} \to R & \\
codomain(R_{\exists R. C}) \to C & \\
domain(R_{\exists R. C}) \leftrightarrow \exists R. C & \\
\forall R', R' \to R, codomain(R') \to C & \To domain(R') \to domain(R_{\exists R.C})
\end{align*}

The representation of universal quantification is done by linking
the concept $\forall R. C$ to the concept $\neg \exists R. \neg C$
with some morphisms for subroles of role $R$:

\begin{align*}
\forall R.C \leftrightarrow \neg \exists R. \neg C & \\
\forall R', R' \to R, domain(R') \to \forall R .C & \To codomain(R') \to C
\end{align*}

Universal restriction requires the existence of negated concept
descriptions. Under category theory the negated concept $\neg C$ is
represented with the morphisms in Figure <ref>.

\begin{align*}
C \sqcap \neg C \rightarrow \bot &\\
\top  \rightarrow C \sqcup \neg C & \\
\forall X, C \sqcap X \rightarrow \bot & \To X \to \neg C\\
\forall X, \top \to C \sqcup X  & \To  \neg C \to X
\end{align*}

Categorical semantics of negation

\begin{equation*}
\bfig
\morphism/@{>}@<2pt>/<0,300>[\bot`C\sqcap \neg C;]
\morphism/@{<-}@<-2pt>/<0,300>[\bot`C\sqcap \neg C;]
\morphism(0,300)<-250,300>[C \sqcap \neg C`C;]
\morphism(0,300)<250,300>[C\sqcap \neg C`\neg C;]
\morphism(-250,600)<250,300>[C`C \sqcup \neg C;]
\morphism(250,600)<-250,300>[\neg C`C\sqcup \neg C;]
\morphism(0,900)/@{>}@<2pt>/<0,300>[C\sqcup \neg C`\top;]
\morphism(0,900)/@{<-}@<-2pt>/<0,300>[C \sqcup \neg C`\top;]

\morphism/<-/[\bot`C \sqcap X_1;]
\morphism(500,0)<0,300>[C \sqcap X_1`X_1;]
\morphism(500,300)<-250,300>[X_1`\neg C;]

\morphism(0,1200)[\top`C\sqcup X_2;]
\morphism(500,900)<0,300>[X_2`C\sqcup X_2;]
\morphism(250,600)<250,300>[\neg C`X_2;]
\efig
\end{equation*}

Diagrammatic representation of negation

Representation of negation of a concept ($\neg C$) with
objects and morphisms.

The subsumption between two concept descriptions is also a
morphism. Following [Duc, 2022], every time a subsumption
$\sub{C}{D}$ is introduced as morphism $C \to D$, the morphism
$\top \to \neg C \sqcup D$ is also introduced.  For a more detailed
description of the categorical representation of $\mathcal{ALC}$, we
refer to Duc, 2022 and
Brieulle et al., 2022.

§.§ Knowledge Graph Embeddings

We define a knowledge graph as a tuple $G=(V, E, L)$, where $V$ is a
set of nodes, $L$ is a set of edge labels and
$E \subseteq V \times L \times V$ is a set of labeled edges.

A knowledge graph embedding (KGE) embeds a graph $G$ into a
real-valued space $\mathbb{R}^n$, which means that nodes in $V$ and
labels in $L$ are represented as vectors in $\mathbb{R}^n$. In machine
learning, several KGE methods have developed such as
TransE [Bordes et al., 2013], DistMult [Yang et al., 2015], or
ConvKB [Nguyen et al., 2018] by defining the plausibility of a graph edge
$(h,r,t)$ with a scoring function. For example, for TransE, the score
function is $f_{TransE} = ||h + r -t||$ implying that the nodes $h$
and $t$ are separated by the edge label $l$. These KGE methods have
been utilized on tasks such as link prediction or query
answering [Wang et al., 2017, Huang et al., 2019].

Hierarchical representation learning focuses on generating
representation of entities under a hierarchical structure.
OrderE is a method that focuses on embeddings graphs with hierarchy
structure. In OrderE, the scoring function $f$ is denoted as
order-embedding since for entities $a,b \in X$,

\[
h \preceq_{X} t \mbox{ if and only if } f(h) \preceq_{\mathbb{R}^n} f(t)
\]

this formulation is lattice-compatible. And the scoring method for a
triple is:

\[
s(h,r,t) = - ||\max(0,t-h)||
\]

§.§ Graph projections

To use DL axioms in a machine learning model, several embedding
approaches have been developed [Kulmanov et al., 2020]. One type of
approach is to transform (project) ontology axioms into
graphs [Chen et al., 2020, Rodríguez-García and Hoehndorf, 2018, Chen et al., 2021, Hinnerichs and Hoehndorf, 2021]. The
resulting graph structure can then be used by a KGE method to predict
new edges (knowledge graph completion) which are then translated back
into axioms.
Different graph projections and their properties have been analyzed
in [Zhapa-Camacho and Hoehndorf, 2023], in particular injectivity, totality, and
use of domain knowledge.

The projection function found in OWL2Vec* [Chen et al., 2021] ($p_{owl2vec*}$)
defines a set of axioms patterns and associate one or more graph edges to
them; for example $p_{owl2vec*}(\{\sub{C}{ \exists R. (D\sqcap E)}\}) =
\{(C,R,D), (C, R, E)\}$.

is not injective since different axioms can produce the same set of
edges, for example $p_{owl2vec*}(\{\sub{C}{ \exists R. (D\sqcap E)}\})
= p_{owl2vec*}(\{\sub{C}{ \exists R. (D\sqcup E)}\})$,
therefore predicting new edges and trying to translate back into
axioms will trivially generate multiple axioms with the same score.

In Onto2Graph [Rodríguez-García and Hoehndorf, 2018],
edges are created using relational patterns
that act as query templates. For example, one can decide to associate
the  pattern
$?X \sqsubseteq \exists ?R . ?Y$ with edges $(?X,
?R,?Y)$. The pattern is used to query axioms $\sub{X_i}{\exists R_j. Y_k}$ and
generate edges $(X_i, R_j, Y_k)$ for the axioms that hold true in the
ontology. Depending on the patterns and edges that are defined,
$p_{onto2graph}$ can be total and injective or not. In our analysis,
we use the original implementation of Onto2Graph, that uses the
pattern $?X \sqsubseteq \exists ?R . ?Y$, which makes the projection
not total but injective.

Using syntax trees is another
approach to represent logic formulas[Crouse et al., 2020]. We use RDF rendering of OWL ontologies to create
graph representations of axioms. In this projection method, every
formula has a different syntax tree, which makes the projection
invertible. Also, totality is ensured because every axiom can be
represented using its syntax tree.
One limitation of this approach is that syntax trees might introduced a
large amount of blank nodes that introduces noise in the graph, affecting
the prediction performance [Chen et al., 2021, Zhapa-Camacho and Hoehndorf, 2023].

§ METHOD: CATE

§.§ Categorical graph projection

To construct the graph projection in CatE, we use categorical diagrams
defined in the literature [Awodey, 2010, Duc, 2022, Brieulle et al., 2022] and outlined in
Section <ref>. Every role and every concept
description together with potential sub-descriptions used in the theory

are represented as nodes in the graph and there is only one edge
label that we denote $arrow$. CatE projection only handles the TBox of
an ontology. Every axiom $\sub{C}{D}$ is converted into an
directed edge
$(\nd{C}, arrow, \nd{D})$ that we shorten to $(\nd{C}, \nd{D})$ given
that the edge label is always the same. $\nd{C}$ is the node
representation of concept $C$. Concepts $C, D$ might be
complex, in which case they would induce new edges to the graph, which
correspond to the semantics of the objects in the category. Thus,
the core of CatE is how to process each concept description, which we
do in a recursive manner following the diagrams
from Section <ref>. Although, the categorical formulation in [Duc, 2022] uses two
categories, one for concept descriptions and one for roles, we do not
differentiate between those two categories and map all the generated
diagrams into the same graph.

Let $\Sigma = (\mathbf{C},\mathbf{R},\mathbf{I})$ be the signature
of a $\mathcal{ALC}$ theory and be $\mathbf{CD} = \{C \mbox{ st. }  C = A \mid A \sqcap B \mid A \sqcup B
\mid \neg A \mid \exists R. B \mid \forall R .B \}$ be the set of
possible $\mathcal{ALC}$ concept
descriptions with signature $\Sigma$. Let $G = (V,E)$ be a directed graph with
$\mathbf{C} \cup \mathbf{R} \subseteq V$ and
$E\subseteq V \times V$. Let $domain: \mathbf{R} \to \mathbf{C}$ and
$codomain: \mathbf{R} \to \mathbf{C}$ be bijective functions mapping
every role in $\mathbf{R}$ to domain and codomain concepts
respectively. Let $\nd{\cdot}: \Sigma \to V$ be a bijective node
representation function of entities in $\Sigma$ in the graph
$G$. The projection function $p_{CatE}: \mathbf{CD} \to G$ is defined recursively:

\begin{equation*}
\begin{aligned}
\pcat{C} =            & \emptyset \\
\pcat{C \sqcap D} =   & \{(\nd{C\sqcap D}, \nd{C}), (\nd{C\sqcap D}, \nd{D})\} \cup \pcat{C} \cup \pcat{D} \\
\pcat{C \sqcup D} =   & \{(\nd{C}, \nd{C\sqcup D}), (\nd{D}, \nd{C\sqcup D})\} \cup \pcat{C} \cup \pcat{D} \\
\pcat{\exists R. D} = & \{(\nd{\rolesub{R}{D}}, \nd{R}), (\nd{codomain(\rolesub{R}{D})}, \nd{D}), \\
& (\nd{domain(\rolesub{R}{C})}, \nd{\exists R. D})  \} \cup \pcat{D} \\
\pcat{\forall R. D} = & \{(\nd{\forall R. D}, \nd{\neg \exists R. \neg D}), (\nd{\neg \exists R. \neg D}, \nd{\forall. R. D})\} \cup \pcat{\neg \exists R. \neg D} \\
\pcat{\neg C} =       & \{(\nd{C \sqcap \neg C}, \nd{\bot}), (\nd{\top}, \nd{C\sqcup \neg C})\}  \cup \pcat{C} \\
& \cup \pcat{C \sqcap \neg C} \cup \pcat{C \sqcup \neg C}  \\
\end{aligned}
\end{equation*}

Let $\Sigma = (\mathbf{C},\mathbf{R},\mathbf{I})$ be the signature
of a $\mathcal{ALC}$ theory and $Ax$ be the set of all possible axioms with
signature $\Sigma$. Let $G = (V,E)$ be a directed graph with
$\mathbf{C} \cup \mathbf{R} \subseteq V$ and
$E\subseteq V \times V$. Let $\nd{\cdot}: \Sigma \to V$ be a
bijective node representation function of entities in $\Sigma$ in
the graph $G$. The projection function $\tilde{p}_{CatE}: Ax \to G$
maps axioms into edges and is defined as:

\[
\tilde{p}_{CatE}(\sub{C}{D}) = \{(\nd{C}, \nd{D}) \} \cup \pcat{C} \cup \pcat{D}
\]

A function $f:X \to Y$ between sets $X$ and $Y$ is total if for every $x \in X$ there exist some $y =f(x) \in Y$

A function $f: X \to Y$ is injective if for every $a, b \in X$, if $f(a) = f(b)$, then $a=b$.

The projection function $\tilde{p}_{CatE}$ is total and injective.

Totality: Let us assume $\tilde{p}_{CatE}$ is not total. Therefore,
there exist one axiom $\sub{C}{D} \in Ax$ for which
$\pcatt{\sub{C}{D}}$ is not defined.
$\pcatt{\sub{C}{D}} = \{(\nd{C}, \nd{D}) \} \cup \pcat{C} \cup
\pcat{D}$. The function $\nd{\cdot}$ is bijective by definition,
therefore, the lack of totality must be in either concepts $C$ or
$D$. Without loss of generality, let us assume $C$ is not
defined. By definition of $p_{CatE}$, if $C$ is not defined then $C$ cannot belong to any of
the patterns
$ A \mid A \sqcap B \mid A \sqcup B \mid \neg A \mid \exists R. B
\mid \forall R .B$, therefore the axiom $\sub{C}{D} \nin Ax$, which is a contradiction.

Injectivity: By contraposition from the  definition of injectivity, if $a \neq b$, then $f(a)
\neq f(b)$. Thus, let us assume two different axioms $\sub{C}{D}$ and
$\sub{E}{F}$. Without loss of generality, we can take concepts $C, E$. Regardless the concept complexity, by the totality property
already defined $\pcatt{\sub{C}{D}}$ contains at least one edge involving the
concept $C$ and this edge has the form

\[
(\nd{C}, \nd{X}) \quad or \quad (\nd{X}, \nd{C})
\]

which, in any case, involves composing with the function
$\nd{\cdot}$, which is bijective and produces $\nd{C} \neq \nd{E}$,
making the graph representation different at least in one edge and
therefore $\pcatt{\sub{C}{D}}
\neq \pcatt{\sub{E}{F}}$.

§.§ Reasoning over the graph

It is possible to perform “saturation” steps within the category
representing the semantics of an $\mathcal{ALC}$ theory to materialize
some of the entailed statements
[Brieulle et al., 2022]. Saturation introduces new objects
and morphisms that correspond to new nodes and edges. The saturation
procedure consist on a set of rules that are applied to the
categorical diagrams until a fixed point is reached. Let $G = (V, E, L)$ be the graph generated by $\tilde{p}_{CatE}$, we denote the
rules as $precondition \To consequence$ where the precondition are
edges in $E$ or nodes in $V$ and the consequence are new edges to be
added to $E$.

\begin{align*}
\nd{\neg \neg C} \in V & \To \pcat{C} \\
\forall X\in \mathbf{C}, (\nd{C\sqcap X}, \nd{\bot}) \in E & \To \tilde{p}_{CatE}(\sub{X}{\neg C}) \\
\forall X, (\nd{\top}, \nd{C\sqcup X}) \in E & \To \tilde{p}_{CatE}(\sub{\neg C}{X}) \\
(\nd{C}, \nd{\neg D}) & \To \pcatt{\sub{D}{\neg C}}  \\
\forall R', (\nd{R'}, \nd{R}), (\nd{codomain(R'), \nd{C}}) \in E & \To \pcatt{\sub{domain(R')}{domain(\rolesub{R}{C})}} \\
(\nd{C}, \nd{\bot}) \in E & \To \pcatt{\sub{\exists R. C}{\bot}} \\
(\nd{C}, \nd{D}) \in E & \To \pcatt{\sub{\exists R. C}{\exists R. D}}
\end{align*}

§.§ Preserving the lattice structure

Since categories contain a lattice structure, using the categorical
representation of axioms will preserve the lattice structure between
concept descriptions that arises from a Description Logic theory
(Figure <ref>); therefore, every edge can be seen as a
subsumption relationship and new predicted edges can also be
interpreted as subsumptions.

§.§ Graph embedding strategies

We apply two different embedding strategies to the categorical
representation of description logic axioms. We first treat the
category as a graph with a single relation and apply the TransE
model. This strategy is called CatE-Graph. Secondly, considering that
there is only one possible relation label and that a graph embedding
method such as TransE cannot capture the inherent lattice structure of
category, we treat the category as a preorder and use the scoring  method introduced
in Vendrov et al., 2016 that we call CatE-Preorder.

In CatE-Graph, we treat every object in the category as a node
and the morphism as a edge label. Then, triples $(h,r,t)$ are scored
by the TransE scoring function $s(h,r,t) = -||h + r - t||_2$.

In the case of CatE-Preorder, we treat the whole category as a preorder and
apply the OrderE scoring function $s(h,r,t) = -||\max(0, t-h)||_2$.

For both strategies we use the same training procedure. We apply
negative sampling by corrupting the nodes $t$ randomly and we use BPR
loss [Rendle et al., 2012] as ranking loss method.

§ EVALUATION AND RESULTS

To evaluate the performance of the proposed method, we evaluate
prediction of axioms tasks involving deductive and inductive
reasoning. For deductive reasoning, we test the prediction of axioms
$\sub{C}{D}$ existing in the deductive closure of the DL theory. One
particular case that we evaluate is the prediction of axioms
$\sub{C}{\bot}$, which correspond to prediction of concept
unsatisfiability.  For inductive reasoning, we test the prediction of
axioms $\sub{C}{D}$ and $\sub{C}{\exists R.D}$ that are plausible but
not necessarily exist in the deductive closure. For the case of
prediction of axioms $\sub{C}{\exists R.D}$, we chose a task in the
biomedical domain and formulate protein–protein interaction (PPI)
prediction as axiom prediction.

For all tasks except PPI prediction, we compare our method with other
graph-projection methods such as Onto2Graph, the graph component of
OWL2Vec* and the RDF rendering of the ontology and used TransE as
graph embedding method. We optimized
hyperparameters embedding dimension, L2 regularization, margin and
batch size using grid search strategy in ranges [16, 32, 64, 128,
256], regularization [0.0001, 0.001, 0], and margin [0, 0.02, 0.04,
0.08], [512, 1024, 4096, 8192, 16384], respectively. Depending on the
task, we used different subsets of the hyperparameters search. For
example, in cases where ontologies are small such as the
unsatisfiability computation task, the smaller embedding dimension and
batch sized were selected. Furthermore, we used Cyclic Learning
Rate [Smith, 2017] with minimum learning rate $1e^{-5}$ and maximum
learning rate $1e^{-4}$.

For the PPI prediction task, we compare against methods found in
Kulmanov et al., 2020, Xiong et al., 2022 and Tang et al., 2022,
that include graph-based, syntactic, geometric, and fuzzy-logic based
models. In this task, we used the minimum learning rate to $1e^{-4}$
and maximum learning rate to $1e^{-3}$.

For all tasks, we report Mean Rank and ROC AUC and depending on the task we report metrics in the range
Hits@$\{1, 3, 10, 50, 100\}$. We report filtered metrics, which means that samples in the training set have been removed in the evaluation procedure.

§.§ Deductive reasoning

In the context of DL theories in ontologies, we define deductive
reasoning as predicting axioms that can be entailed by an automated
reasoner. Therefore, in this task we use ontology asserted axioms as
training set and we generate the testing set by computing the
ground-truth entailed axioms using automated reasoners such as
ELK [Kazakov et al., 2013] or Hermit [Glimm et al., 2014].

§.§.§ Unsatisfiability of named concepts

Given a model $\mathcal{I}$, a concept $C$ is unsatisfiable if  $\mathcal{I}$ satisfies the
axiom $\sub{C}{\bot}$. In category theory semantics, unsatisfiability
is translated to the morphism $C\to \bot$. Therefore, we formulate
satisfiability computation as prediction of edges ($\nd{C}, \nd{\bot}$)  in
the embedding space $\mathbb{R}^{n}$ and we rank the axiom of interest
over all edges ($\nd{C'},  \nd{\bot}$) with $C'$ being named
Baseline methods might not add the bottom and top concepts in their
projection procedure; therefore we manually add edges ($\nd{\bot},
subclassof, \nd{C}$) and ($\nd{C},subclassof, \nd{\top}$) for all named concepts
$C$. Due to the time and memory complexity to compute
ground-truth data for large ontologies, we selected small ontologies
available in the AberOWL[http://aber-owl.net/] repository, namely the
Food-Biomarker Ontology (FOBI) and the  Neural Reprogramming Ontology (NRO) (Table <ref>) and  used automated reasoners ELK [Kazakov et al., 2013] and
Hermit [Glimm et al., 2014] to generate the true unsatisfiable named concepts of the
ontologies.  Furthermore, in this case the number of
testing samples and ontology named concepts is small, in which case Hits@$k$
for small $k$ might change drastically between methods and therefore, we report Mean Rank, ROC AUC and Hits@$\{50, 100\}$.
We can see in Table <ref> that the graph representation
using categorical properties improves the prediction of unsatisfiable
named concepts in a DL theory.

Datasets information for unsatisfiability prediction task.

Ontology     Total named concepts     Unsatisfiable named concepts
FOBI     1334     148

NRO     563     45

Prediction of unsatisfiable named concepts on different

2*Projection function     4cFOBI     4cNRO

MR              H@50            H@100           1cAUC     MR     H@50     H@100     AUC

Onto2Graph        241.19          14.86           26.35           82.16            156.71     20.0     35.56     73.20

OWL2Vec*          325.69          25.00           41.89           75.95            189.93     11.11     33.33     67.39

RDF               148.29          75.68     76.35           89.20            200.11     20.0     26.67     65.45
CatE-Graph        64.13     75.68     79.73     95.51      108.29      2.22     62.22     81.58

CatE-Preorder     81.65     31.08    85.81     94.08      97.09     17.78     53.33     83.30

§.§.§ Prediction of axioms $\sub{C}{D}$ in the deductive closure

We also evaluated the prediction capability for predicting axioms
in the deductive closure of an ontology similar to an automated reasoner. We use the Gene Ontology (GO)
axioms in two ways and follow the evaluation procedure for
graph-projection methods [Zhapa-Camacho and Hoehndorf, 2023], where a reduced version of GO is generated
by removing 10% of axioms $\sub{C}{D}$ from GO
denoted as $\mbox{GO}_{sub}$ and a fragment of the deductive closure
is constructed as testing set  using axioms
$\sub{C}{D}$ that are in the deductive closure of GO but not in the
deductive closure of $\mbox{GO}_{sub}$.
When using  GO as training set the task involves  deductive
reasoning and using GO$_{sub}$ as training set involves inductive reasoning. We compare against Onto2Graph, the
projection found in OWL2Vec* and RDF rendering of the ontology. For
baselines, we used results previously reported on the same
task [Zhapa-Camacho and Hoehndorf, 2023].

The results are shown in Table <ref>. For most metrics, CatE outperforms other graph
projection methods. Furthermore, among different graph embedding
strategies, we can see that adjacency
information explored by Cat-Graph contributes to get higher scores in
H@k metrics, whereas lattice structure preserved by CatE-Preorder
helps in overall axiom ranking showed by lower Mean Rank and higher ROC

9pt10ptResults of prediction of axioms of the form
$C\sqsubseteq D$ using different
projection functions on different versions of the Gene
Ontology. Baseline results were taken from Zhapa-Camacho and Hoehndorf, 2023

2*Method     5p4cmPredictions of axioms $C\sqsubseteq D$ in $GO_{sub}$      5p4cmPredictions of axioms $C\sqsubseteq D$ in $GO$

MR     H@1     H@10     H@100     AUC     MR     H@1     H@10     H@100     AUC
Onto2Graph          4439.27         0.41           1.35            10.09           91.31            2947.81         0.41            0.85           12.32           94.23

OWL2Vec*            5374.64         1.79         9.55          34.34     89.48            4514.51           1.98          10.03        33.87     91.16

RDF                 4122.82       0.41           2.82            10.10           91.93          4158.77            0.36           2.85           10.03           91.86
CatE-Graph     2856.11     7.44     12.49     22.27            94.41      2704.70     7.44     12.53     21.54           94.71

CatE-Preorder     2455.12     1.99     12.40     23.49      95.19     2389.96      2.07     12.37     22.72     95.32

§.§ Task 2: Inductive reasoning

In this context, we define inductive reasoning as the task of
generating conclusions that are not entailed by the axioms but are
plausible to hold true. This task can also be denoted as ontology
completion. We evaluate two tasks involving inductive reasoning:
prediction of axioms $\sub{C}{D}$ and prediction of axioms
$\sub{C}{\exists R .D}$

§.§.§ Prediction of axioms $\sub{C}{D}$  inductively

Predicting axioms that
are plausible to be part of the ontology and cannot necessarily be
entailed by a reasoning procedure is a task involving inductive reasoning. Predicting such axioms
implies leveraging the remaining information on the ontology. We focus
on the prediction of subsumption axioms $\sub{C}{D}$. For this
task, we follow the same testing procedure as in
OWL2Vec* [Chen et al., 2021]. We use two datasets: Gene Ontology [Ashburner et al., 2000] and
Food Ontology [Dooley et al., 2018], where 10% of subsumption axioms $\sub{C}{D}$ have
been randomly removed for validation and 20% for testing. The
training set contains the 70% of subsumption axioms together
with the other axioms existing in the ontology initially. We evaluate the performance on
this task as a ranking problem and report mean rank, hits@10, hits@100 and
ROC AUC. Additionally, we noticed that some of the projection methods generated
edges where training and testing sets overlapped. We removed those
edges manually from the training set.

The results are shown in Table <ref>. We can see that
the categorical projection method outperforms the baseline
methods. Furthermore, the among the category embedding approaches,
CatE-Preorder performs better in all metrics (lower in Mean Rank and
higher in H@10,100 and AUC) than CatE-Graph, which is due to
lattice-preserving property of order embeddings that does not exists
in TransE, whose scoring method is distance based only. Additionally,
in CatE-Preorder, applying one saturation step improves the
axiom prediction performance in most of the metrics. When we
apply two saturation steps, the performance decreases. This phenomenon
is surprising since we are adding more knowledge to the
graph. However, it can be explained because saturation steps
introduces information over all the knowledge base and at some point
the large amount of information might become noisy for the particular
task we set up.

Inductive reasoning over axioms $C\sqsubseteq D$. S1 and S2
correspond to 1 and 2 saturation steps, respectively.

2*Projection Function     4c|GO     4cFoodOn

MR          H@10     H@100     AUC    MR          H@10     H@100     AUC

Onto2Graph      6804.09           3.31           12.91           84.63            3394.5       6.23           18.08           87.97

OWL2Vec*        6546.47           2.24           10.10           85.21            3472.31            6.68           18.95           87.69

RDF             6609.14           2.86           11.63           85.07            3489.36           7.18            19.22           87.63

CatE-Graph        6978.77            1.60           11.91           84.23           3861.53           6.87            19.09           86.31

CatE-Graph S1     7413.39            0.61           9.31            83.25           4785.31           6.87            18.85           83.03

CatE-Graph S2      7513.95            0.45           8.79            83.02           5143.72           6.87            18.68           81.76

CatE-Preorder     6083.40      7.95     27.12     86.26     3318.93     6.04            20.14           88.23

CatE-Preorder S1     5881.81      6.87     27.75     86.71     5539.79           10.14     23.43     80.35

CatE-Preorder S2     6854.35            4.31           19.03           84.51           5597.66           6.80            21.00            80.15

§.§.§ Biomedical task: PPI prediction

Protein-protein interaction (PPI) prediction is a widespread task in
bioinformatics. PPIs can be encoded in axioms
$\sub{p_1}{\exists interacts .p_2}$ where $p_1, p_2$ are proteins. We used the PPI
dataset provided in Kulmanov et al., 2020.
For this task, we compare our method against state-of-the-art methods
found in Kulmanov et al., 2020, Xiong et al., 2022 and
Tang et al., 2022. In terms of the categorical representation, this case is different
from previous tasks since it includes existential restrictions. In
this regard, CatE can only make predictions over concepts that exist
in the graph. Therefore, after generating the
category projection, we manually add edges $\pcatt{\sub{\bot}{\exists interacts. p_i}}$ and $\pcatt{\sub{\exists interacts. p_i}{\top}}$.

The results are shown in Table <ref>. We can see that
Cat-Preorder manages to rank PPI axioms higher than other methods
(mean rank and AUC), although in the case of Hits@$k$ metrics its
perfomance is below other methods. Nevertheless, in CatE-Preorder, the
performance is at least comparable with other methods even though the
underlying structure of the set of concepts is a preorder, which is
simpler than a geometric structures like the one formulated in
ELEm [Kulmanov et al., 2019] or BoxEL [Xiong et al., 2022], and fuzzy logic
based structures such as F$\mathcal{ALC}$ON [Tang et al., 2022].

[PPI on Yeast.]Protein-protein interaction prediction on
Yeast. Baseline results were taken from
Kulmanov et al., 2020, Xiong et al., 2022 and
Tang et al., 2022. R means raw metrics and F means filtered metrics where training sample scores have been removed.

2*Method     2cHits@10     2cHits@100     2cMean Rank     2cAUC
R     F     R     F     R     F     R     F
TransE                 0.06     0.13     0.32     0.40     1125      1075     0.82     0.83

BoxE                   0.08     0.14     0.36     0.43     633       620      0.85     0.85

SimResnik              0.09     0.17     0.38     0.48     758       707      0.86     0.87

ELEm                   0.08     0.17     0.44     0.62     451       394      0.92     0.93

EmEL++                 0.08     0.16     0.45     0.63     451       397      0.90     0.91

OPA2Vec                0.06     0.13     0.39     0.58     523       467      0.87     0.88

BoxEL                  0.09     0.20     0.52     0.73     423       379      0.93     0.94

F$\mathcal{ALC}ON$     0.09     0.23     0.51     0.75     –        –       –       –

CatE-Preorder          0.07     0.20     0.53     0.73     250       202      0.96     0.97

§ LIMITATIONS AND FUTURE WORK

Although the CatE projection enables representation of arbitrary
concepts in $\mathcal{ALC}$ language, inferring over concepts not
existing in the training graph is not possible. For example,
in the PPI prediction task all the $\exists interacts. protein_i$
concepts were introduced a priori to enable inferences of the type
$\sub{protein_i}{\exists interacts .protein_j}$. Therefore, all
concepts required for inference must exist in the graph and future
work will include the development of strategies to deal with unseen

Another limitation of the current  method is that it only handles TBox
axioms. Further work should analyze the extension to ABox and its
compatibility with category-theoretical semantics.

Furthermore, the categorical semantics include one category for
concepts and one for roles. We did not differentiate between the two
categories and projected both to the same graph. Further work can
investigate the impact of treating each category separately since that
formulation is closer to the categorical semantics.

§ CONCLUSION

We developed CatE, a novel graph representation for ontology axioms in the
$\mathcal{ALC}$ DL language. The CatE projection is based on category-theoretical
formulation of the $\mathcal{ALC}$ language, and therefore, the
transformation into graph benefits from properties such as preserving
the lattice structure.  We have shown that CatE projection
function is total (i.e., can use all the information of the theory)
and injective (i.e., two different axioms have different graph
representations). We applied CatE to different knowledge base completion
tasks demonstrating improvement over previous methodology.

*Source code availability: Source
code, data
and hyperparameter selection for each task and dataset is
available at <https://github.com/bio-ontology-research-group/catE>.

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