An Embedding-based Approach to Inconsistency-tolerant
Reasoning with Inconsistent Ontologies
Keyu Wang

Site Li

Jiaye Li

School of Computer Science andEngineering, Southeast University
Nanjing, China

School of Mathematics, Southeast
University
Nanjing, China

School of Mathematics, Southeast
University
Nanjing, China

Guilin Qi

Qiu JiSchool of Computer Science andEngineering, Southeast University
Nanjing, China

School of Modern Posts & Institute ofModern Posts, Nanjing University ofPosts and Telecommunications
Nanjing, China

ABSTRACT
Inconsistency handling is an important issue in knowledge management. Especially in ontology engineering, logical inconsistencies
may occur during ontology construction. A natural way to reason
with an inconsistent ontology is to utilize the maximal consistentsubsets of the ontology. However, previous studies on selecting
maximal consistent subsets have rarely considered the semanticsof the axioms, which may result in irrational inference. In this paper, we present a novel approach to reasoning with inconsistent
ontologies in description logics based on the embeddings of axioms. We first propose a sentence-embedding-based method and aknowledge-graph-embedding-based method for translating axioms
into semantic vectors to calculate semantic similarities among axioms. We then define an embedding-based approach for selecting
the maximal consistent subsets of the inconsistent ontology and use
it to define an inconsistency-tolerant inference relation. We show
the rationality of our inference relation by considering some logicalproperties. Finally, we conduct experiments on several ontologies
to evaluate the effectiveness and efficiency of our proposed method.
The experimental results show that our embedding-based method
can outperform existing inconsistency-tolerant reasoning methods
based on maximal consistent subsets.

1

INTRODUCTION

Ontologies are widely used in knowledge management and are
critical for the success of the Semantic Web because they provide
formal representation of knowledge shared within the Semantic
Web applications. The development of the Semantic Web is further
accelerated with the proposal of Knowledge Graph, which provides
users with more intelligent services, such as more accurate recommendation and search [10]. Ontologies also have a critical impact
on the performance of Knowledge Graph reasoning [28]. However,conflicting knowledge in ontologies is unavoidable. For example, ontology fusion [15], ontology evolution [14] and ontology migration[39] may result in inconsistent ontologies. Therefore, inconsistency
handling is an essential issue in ontology engineering.
A natural way to reason with inconsistent ontologies is to selectmaximal consistent subsets of an inconsistent ontology [36]. Anelementary method is called skeptical inference [27], i.e., an axiom
can be inferred if it can be inferred from every maximal consistentsubset of the inconsistent ontology. A well-known refinement ofskeptical inference is to utilize the cardinality-maximal consistentsubsets of the ontology for inference [3]. However, in [26], theshortcomings of these two methods are pointed out. They fail togive fine consideration to the difference in the reliability of axioms,
which results in weak reasoning power and poor answering quality. [26] gives a general class of monotonic selection relations forcomparing maximal consistent subsets. Each monotonic selection
relation corresponds to a rational inference relation. However, theapproaches given in [26] are limited to propositional logic and it is
not trivial to apply them to description logics. Axioms in an ontology contain semantic information, which can be used to define arational inconsistency-tolerant reasoning method. We use an example to illustrate this.
Example. We consider an example of an inconsistent ontologythat contains four axioms selected from the widely used ontology
OpenCyc1 :𝜑 1 : 𝐴𝑟𝑡𝑖 𝑓 𝑎𝑐𝑡𝑢𝑎𝑙𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑇𝑦𝑝𝑒 (𝑀𝑜𝑛𝑢𝑚𝑒𝑛𝑡)
𝜑 2 : 𝐸𝑥𝑖𝑠𝑡𝑖𝑛𝑔𝑆𝑡𝑢 𝑓 𝑓 𝑇𝑦𝑝𝑒 (𝑀𝑜𝑛𝑢𝑚𝑒𝑛𝑡)
𝜑 3 : 𝐷𝑖𝑠 𝑗𝑜𝑖𝑛𝑡𝑊 𝑖𝑡ℎ(𝐸𝑥𝑖𝑠𝑡𝑖𝑛𝑔𝑂𝑏 𝑗𝑒𝑐𝑡𝑇𝑦𝑝𝑒, 𝐸𝑥𝑖𝑠𝑡𝑖𝑛𝑔𝑆𝑡𝑢 𝑓 𝑓 𝑇𝑦𝑝𝑒)𝜑 4 : 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂 𝑓 (𝐴𝑟𝑡𝑖 𝑓 𝑎𝑐𝑡𝑢𝑎𝑙𝐹𝑒𝑎𝑡𝑢𝑟𝑒𝑇𝑦𝑝𝑒, 𝐸𝑥𝑖𝑠𝑡𝑖𝑛𝑔𝑂𝑏 𝑗𝑒𝑐𝑡𝑇𝑦𝑝𝑒)The existing methods based on propositional logic would assume
that these axioms have the same status as shown in the example(continue) in Section 2, regardless of the semantics of axioms. However, these axioms contain different semantic information, which
can be exploited to compute the reliability of axioms and select the
maximal consistent subset of the ontology.
Pre-trained transformer-based language models for embedding
such as BERT [11] and Knowledge Graph Embedding models such
as TransE [9] have been successfully applied in many natural language processing tasks and achieved good performance. Recently,
they have been applied to ontology matching [18] and instancematching [19] to encode the semantics of instances or concepts inan ontology. They were also used to learn ontologies from knowledge graphs [37]. This motivates us to use embedding-based models
to encode the semantic information of an axiom in a description
logics ontology and apply them to inconsistency-tolerant reasoning.
In this paper, we propose a novel approach to reasoning with inconsistent OWL ontologies based on the embeddings of axioms. We
1 https://sourceforge.net/projects/opencyc/

IJCKG 2023, December 8-9, 2023, Tokyo, Japanfirst introduce two methods, a sentence-embedding-based method
and a knowledge-graph-embedding-based method, for translating axioms into semantic vectors to compute semantic similarities
among them. By considering the number of occurrences in the maximal consistent subsets and the degree of semantic relationships
with other axioms, each axiom could be associated with a degreeof reliability. We then propose an embedding-based approach toscoring the maximal consistent subsets and select the maximal
consistent subset with the highest score. The selected maximal
consistent subset of the ontology can be used to infer consistentand reasonable results. We show the rationality of our proposed
reasoning method by considering some logical properties. Finally,
we conduct experiments on several ontologies to evaluate the effectiveness and efficiency of our proposed method. The experimental
results show that our embedding-based method can outperform
existing inconsistency-tolerant reasoning methods based on maximal consistent subsets. The data, source codes and technical report
including proofs are provided in the link: https://github.com/skyfish23/Embedding-based-infer.

2

RELATED WORK

There are mainly two classes of methods for inconsistency handlingin DL-based ontologies. One deals with inconsistencies by repairing them and the other tolerates inconsistencies and changes thesemantics of description logics. Our work falls into the latter class.
In this part, we mainly discuss existing approaches to reasoning
with inconsistent OWL DL ontologies based on consistent subsets.
[4, 5] provide surveys on this topic. To find consistent subsets, [23]proposes a linear extension strategy for checking whether an entailment could be inferred or not by defining syntactic relevancefunctions. This work is extended in [20] by defining semantic relevance functions with Google distances. However, such an approach
may not find maximal consistent subsets and it may result in many
unknown answers to queries.
To achieve some kind of minimal change in the calculation of
the consistent subsets, researchers have proposed various methods
to retain information as much as possible. The work in [12] focuseson SH IQ ontologies, and assumes that an ontology is composedof a consistent TBox, and the instance assertions in the ABox areassociated with weights. It answers a conjunctive query upon any
subset of the set including the TBox and a weight-maximally consistent subset from the ABox. Similarly, [7] computes four kinds ofmaximal subsets by exploiting additional information in the ABox,
which are maximal with respect to cardinality, weights, prioritized set inclusion or prioritized cardinality. They focus on DL-Lite,
which is a sub-language of OWL DL. Furthermore, [40] presents
a sound and complete method for DL-Lite ontologies and an approximation method for more expressive DLs, so that the queryanswering systems can scale up to billions of data. [8] provides
a practical approach by considering three well-known semantics
for DL-Lite ontologies and defines explanations for answers. Recently, [6] proposes propositional encoding of maximality and thendevelops several SAT-based algorithms to calculate answers.
There are also some other methods to perform reasoning over aninconsistent ontology which do not rely on (maximal) consistentsubsets of the ontology. For instance, various non-classical semantics could be adopted. The works in [30, 31] adopt four-valued andKeyu Wang, Site Li, Jiaye Li, Guilin Qi, and Qiu Ji

three-valued semantics for weakening an interpretation in DL fromtwo truth values to four and three values respectively. To infer more
useful information from an inconsistent ontology, [45] defines anovel description logics based on the quasi-classical logic.
Different from all these existing methods, our approach does notrely on weight information to select maximal consistent subsets
of an ontology as many real ontologies do not have weight information. Instead, we propose a new approach for selecting maximal
consistent subsets of an ontology by considering axiom embedding.

3

PRELIMINARIES

An ontology is a formal representation of knowledge that defines a
set of concepts and relationships between them, and OWL (Web
Ontology Language) is a standard language recommended by W3C2
used to represent ontologies. Description logics (DLs), as the logic
foundation of OWL, provide reasoning support for OWL ontologies
and we refer to the DL Handbook [2] for a detailed introduction.
A DL-based ontology describes the characteristics of some properties and the relationships between entities through various axioms. Such axioms can be categorized into TBox (terminologicalaxioms) and ABox (assertional axioms). A TBox establishes a conceptualization of a knowledge domain such as concept inclusion in
the form 𝐶 ⊑ 𝐷 , where 𝐶 and 𝐷 are concepts. An ABox describes
particular individuals, such as concept assertions in the form of
𝐶 (𝑎), where 𝐶 is a concept and 𝑎 is an individual.
An ontology is inconsistent if it has no model. To reason withan inconsistent ontology, we consider its maximal consistent subontologies, which are maximal subsets of the ontology that areconsistent.
Definition 1 (MCS). Given an ontology Σ, a maximal consistentsub-ontology(MCS) Σ′ of Σ satisfies:• Σ′ ⊆ Σ• Σ′ is consistent• If Σ′ ⊂ Σ′′ ⊆ Σ, then Σ′′ is not consistent.
We use mcs(Σ) to represent the set composed of all the maximal
consistent sub-ontologies of a given ontology Σ.
[26] first defines mappings that attach a score to each axiom ofan ontology and then aggregate those scores to rank each MCS.
The MCS with the highest score is selected and used to define theinconsistency-tolerant reasoning method. Since only axioms areused when computing scores for axioms or MCSs, we use the term
ontology to refer to a (finite) set of axioms for simplification in
the following definitions. In this paper, when we mention 𝛼 ∈ Σgiven an ontology Σ, we assume 𝛼 is an axiom and |Σ| denotes
the number of axioms in the ontology Σ. We adapt the notion of ascoring function given in [26] to description logics.
Definition 2 (scoring function). A scoring function s associateswith an ontology Σ and an axiom 𝛼 ∈ Σ a non-negative real number𝑠 (Σ, 𝛼) which is equal to 0 if and only if 𝛼 is trivial (i.e., such that 𝛼
is a tautology or a contradiction).
Different from [26], in DL ontologies in this paper, we exclude
the cases where the axioms are trivial. [26] designs several scoring
functions, including the #mc function mentioned in the following
Definition 3.
Definition 3 (#mc). Let Σ be an ontology and an axiom 𝛼 ∈ Σ.
2 https://www.w3.org/TR/owl-overview/An Embedding-based Approach to Inconsistency-tolerant Reasoning with Inconsistent Ontologies

IJCKG 2023, December 8-9, 2023, Tokyo, Japan

Figure 1: An embedding-based approach to reasoning with an inconsistent ontology.
Define:#𝑚𝑐 (Σ, 𝛼) = |{Σ𝑖 ∈ 𝑚𝑐𝑠 (Σ) | 𝛼 ∈ Σ𝑖 }|#𝑚𝑐 ). Let Σ be an ontology and Σ ∈ 𝑚𝑐𝑠 (Σ).
Definition 4 (𝑠𝑐𝑜𝑟𝑒 Σ,𝑠𝑢𝑚𝑖
Define:∑︁#𝑚𝑐(Σ𝑖 ) =#𝑚𝑐 (Σ, 𝛼)𝑠𝑐𝑜𝑟𝑒 Σ,𝑠𝑢𝑚
𝛼 ∈Σ𝑖

Example(continue) Ontology Σ = {𝜑 1, 𝜑 2, 𝜑 3, 𝜑 4 }. Σ has 4 MCSs
namely: {𝜑 2, 𝜑 3, 𝜑 4 }, {𝜑 1, 𝜑 3, 𝜑 4 }, {𝜑 1, 𝜑 2, 𝜑 4 }, {𝜑 1, 𝜑 2, 𝜑 3 }. We have#𝑚𝑐 (Σ, 𝜑 1 ) = #𝑚𝑐 (Σ, 𝜑 2 ) = #𝑚𝑐 (Σ, 𝜑 3 ) = #𝑚𝑐 (Σ, 𝜑 4 ) = 3, so the#𝑚𝑐 is equal to 9 (resp. 9, 9,score of Σ1 (𝑟𝑒𝑠𝑝.Σ2, Σ3, Σ4 ) w.r.t 𝑠𝑐𝑜𝑟𝑒 Σ,𝑠𝑢𝑚
9). These four MCSs all have the same value and are selected forreasoning, i.e., an axiom can be inferred if it can be inferred fromeach of these four MCSs.
Through the definitions above, the MCS Σ′ of the original inconsistent Σ with the highest score is selected, and the selected MCS
can be used for reasoning. In the example above, all the four MCSs
are selected because they all have the highest score, though thissituation is rare in our proposed approach. In this work, we also
define a reasoning method by attaching a score to each axiom in an
ontology and scoring the MCSs. But different from [26], we propose
an embedding-based novel method to score each axiom considering
not only the syntax but also the semantics of the axioms.
In this work, a sentence-embedding-based method and a knowledgegraph-embedding-based method are proposed for translating axioms into semantic vectors to calculate semantic similarities amongaxioms. We introduce some background knowledge about embedding models as follows.
Sentence Embedding. A sentence embedding is a mathematicalrepresentation of a sentence in a fixed-length vector space that
encodes the semantic meaning and contextual information of thesentence[33]. Generally, the more similar the semantics of two
sentences are, the closer their corresponding vectors are in thesemantic space. In this work, we leverage Sentence-BERT[35] and
CoSENT[43] to get the semantic vectors of axioms. Sentence-BERT
trains the upper classification function by supervised learning andmakes balances between performance and efficiency, while CoSENT
takes advantage of a ranking loss function, which makes the training process closer to prediction.
Knowledge Graph Embedding. Knowledge Graph Embedding(KGE) is a task for learning low-dimensional representation, typically called embeddings, of a knowledge graph’s entities and relations while preserving their semantics. A knowledge graph canbe assumed as the ABox of an ontology while KGE can also be applied to embedding ontologies. For example, [17] uses KGE models
such as TransE[9], DisMult[44] and HolE[34] to learn ontologies
by jointly embedding instances and concepts. There are various
KGE models and we refer to [42] for a comprehensive survey. In
this work, we use TransE [9] and RDF2Vec[38]. TransE is basedon the assumption that vs + vr is close to vo , where vs, vr, vo are
the embeddings of subject entity, relation and object entity respectively, while RDF2Vec utilizes random walks on the RDF graph tocreate sequences of RDF nodes, which are then used as input for
the word2vec[32] algorithm.

4

AN EMBEDDING-BASED METHOD TOREASON WITH INCONSISTENTONTOLOGIESThe overall framework of our method is shown in Figure 1. Our
method can be divided into six parts. The first five parts can beassumed as offline preparation, including translating axioms into
sentences or triples, using a sentence-embedding-based method ora knowledge-graph-embedding-based method to map the axioms
into semantic vectors in a continuous space, calculating similarity
between axioms based on the embeddings, generating and scoring
the MCSs. Finally, we select the MCS with the highest score forreasoning.

4.1

Semantic Representation & Embedding

We introduce two methods, a sentence-embedding-based method
and a knowledge-graph-embedding-based method in this part, andeither of the two methods can be used to convert axioms intosemantic vectors.
Sentence-embedding-based method. To represent the semantics of classes, individuals and properties in an ontology, we firstuse NaturalOWL [1] to translate axioms from OWL statements
to natural language sentences. For example, the axiom “ClassAssertion(ObjectMaxCardinality(1 madeFromGrape) product145)” istranslated to “product145 is a made from at most one Grape”. Table
1 shows most of the rules to translate OWL concepts or axioms into

IJCKG 2023, December 8-9, 2023, Tokyo, Japan

Ontology Elements

Keyu Wang, Site Li, Jiaye Li, Guilin Qi, and Qiu Ji

OWL Representation

Phrases or Sentences

Triples

Concpets

ObjectSomeValuesFrom(op A)
ObjectAllValuesFrom(op A)
ObjectHasValue(op a)
ObjectIntersectionOf(A B)
ObjectUnionOf(A B)
ObjectExactCardinality(n op A)
ObjectMinCardinality(n op A)
ObjectMaxCardinality(n op A)

op at least one A
op only A
op a
A and B
A or Bop exactly n A
op at least n A
op at most n A-

Axioms

SubClassOf(A B)
DisjointClasses(A B)
EquivalentClasses(A B)
ClassAssertion(A a)
ObjectPropertyAssertion(op a b)
DataPropertyAssertion(dp a v)

A is a kind of B
A isn’t a kind of B
A is a kind of B
a is a A
a op b
a dp v<A, SubClassOf, B><A, Disjointness, B><A, EquivalentClasses, B><a, isInstanceOf, C><a, op, b><a, dp, v>

Table 1: Rules to translate OWL concepts or axioms into sentences or triples, respectively, where A and B are concepts, a and b
are individual names, n indicates an integer, op and dp indicate an object property and a data property separately, and v is avalue.

natural language sentences implemented by NaturalOWL. Despite
the fact that NaturalOWL does not translate all kinds of axioms orentities such as property inclusion axioms and transitive properties, the system can handle OWL statements in most ontologies,
including various ontologies in our experiment.
Each axiom is transformed into a sentence in natural language
form by NaturalOWL[1], and then the sentence is input into acertain sentence embedding model to obtain the corresponding
semantic vector representation. Considering that BERT-based pretrained language models have achieved good performance in text
representation and text matching tasks, we use Sentence-BERT[35] and CoSERT [43] as sentence embedding models to computethe embeddings of the axioms in natural language sentence form.
We refer to the official website of PyPi3 for more details about theimplementation and experimental performance of these models.
Knowledge-graph-embedding-based method. The method
based on sentence embedding can make use of the textual information of ontology, but ignores other semantic information of ontology, such as logical expression and class hierarchies. We propose
another method using KG embedding, which is used to learn therelational information in the ontology. We implement the system“TripleOWL” of transforming axioms into triples and its transforming mechanism is similar to NaturalOWL. For example, an axiom“SubClassOf(IceCream Food)” is translated into “<IceCream, SubClassOf, Food>”. The last column of Table 1 shows most of the rules
to translate axioms to triples. Nonetheless, different from translating axioms to sentences, we fail to deal with complex concepts whentranslating axioms to triples, that is, we regard the complex concept
in an axiom as a whole, which may cause worse performances forwhen the ontology has many axioms with complex concepts. We
leave how to deal with complex concepts in KGE-based method
to future work. The three parts of a triple are called subject entity,
3 https://pypi.org/project/text2vec/

relation and object entity respectively for simplicity. Then the threecomponents of the triples can be embedded into vectors using KG
embedding models, and the vectors corresponding to the subject
entity, relation and object entity are concatenated together to obtainthe semantic vector representation of this axiom. We use TransE[9], RDF2Vec [38] in our experiments, which are implemented by
OpenKE platform [16] and pyRDF2Vec[41] respectively.

4.2

Semantic Similarity of Embedding

We compute the semantic similarities among axioms. We denotethe similarity of the axioms 𝛼, 𝛽 as 𝑆𝑖𝑚(𝛼, 𝛽) and the embedding of
the axiom 𝛼 as 𝐸𝑚𝑏 (𝛼). The similarity calculation method is given
as follows:
𝑆𝑖𝑚(𝛼, 𝛽) = 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 (𝐸𝑚𝑏 (𝛼), 𝐸𝑚𝑏 (𝛽))𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 represents certain similarity calculation method. The
most common ones are based on Cosine Distance and Euclidean
Distance:
𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐶𝑜𝑠 (v1, v2 ) = 12 (1 + | |v v| |1×·v| 2|v | | )1

𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐸𝑢𝑐 (v1, v2 ) =√ 1

21+ | |v1 −v2 | |where v1 and v2 are vectors of the same arity.
𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐶𝑜𝑠 and 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐸𝑢𝑐 satisfy the following three properties. For simplicity, we use 𝑆𝑖𝑚(𝜙, 𝜑) to denote either 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐶𝑜𝑠(𝐸𝑚𝑏 (𝜙), 𝐸𝑚𝑏 (𝜑)) or 𝑆𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦𝐸𝑢𝑐 (𝐸𝑚𝑏 (𝜙), 𝐸𝑚𝑏 (𝜑)).
Range. The semantic similarity is a real number between 0 and
1: 0 ≤ 𝑆𝑖𝑚(𝜙, 𝜑) ≤ 1 for any 𝜙 and 𝜑. The higher the similarity
between 𝜙 and 𝜑, the closer the 𝑆𝑖𝑚(𝜙, 𝜑) is to 1, and the opposite
is to 0.
Grammatical Reflexivity. Any axiom is always semantically closest to itself: 𝑆𝑖𝑚(𝜙, 𝜙) = 1 for any 𝜙.
Symmetry. The semantic similarity between two axioms is symmetric: 𝑆𝑖𝑚(𝜙, 𝜑) = 𝑆𝑖𝑚(𝜑, 𝜙) for any 𝜑 and 𝜙.

An Embedding-based Approach to Inconsistency-tolerant Reasoning with Inconsistent Ontologies

However, the similarity functions may not satisfy the semantic
reflexivity defined below, because two semantically equivalent axioms may be converted into different sentences. This does not affectthat our proposed method satisfies the properties in Section 5 so
our method still has the logical rationality. And in our work, we use
a unique set of transformation rules proposed by NaturalOWL[1],and the experimental part shows the significant performance of
our embedding-based method under this set of rules.
Semantic Reflexivity. Two semantically identical axioms should
be closest to each other: If |= 𝜙 ↔ 𝜑, then 𝑆𝑖𝑚(𝜙, 𝜑) = 1.
Example (continue) By applying the cosine similarity metric andSentence-BERT, we obtain 𝑆𝑖𝑚𝐶𝑜𝑠 (𝜑 1, 𝜑 2 ) = 0.41 and 𝑆𝑖𝑚𝐶𝑜𝑠 (𝜑 1, 𝜑 4 )= 0.62. This may be because 𝜑 4 has a same concept ArtifactualFeatureType with 𝜑 1 while 𝜑 3 doesn’t, so 𝜑 4 has higher semantic association with 𝜑 1 .

4.3

Semantic Selection Functions

We first define the degree of aggregation of each axiom in the MCS.
Then we score each axiom to represent the reliability of the axioms.
Finally we aggregate the scores of the axioms to get the score ofeach MCS.
We use the degree of aggregation to express how closely an axiom relates to other axioms in a MCS. The greater the aggregation
degree of an axiom in a MCS, the closer the axiom is to the semantics of other axioms in this MCS, which indicates more semantic
information it contains about this MCS.
Definition 5 (agg). Given an ontology Σ and Σ𝑖 ∈ 𝑚𝑐𝑠 (Σ), we define
the aggregation of an axiom 𝛼 ∈ Σ𝑖 as follows:
1 ∑︁𝑆𝑖𝑚(𝛼, 𝛽)𝑎𝑔𝑔(Σ𝑖 , 𝛼) =
|Σ𝑖 |
𝛽 ∈Σ𝑖

We then calculate the score for each axiom. If an axiom exists
in more MCSs and it has a higher degree of aggregation in the
MCSs it appears in which means it is more likely a true axiom inthese MCSs, this axiom is considered to be more reliable. Below, we
define scoring function for axioms.
Definition 6 (mc). Given an ontology Σ and an axiom 𝛼 ∈ Σ, we
define the score of an axiom 𝛼 ∈ Σ as follows:∑︁
𝑚𝑐 (Σ, 𝛼) =𝑎𝑔𝑔(Σ𝑖 , 𝛼){Σ𝑖 ∈𝑚𝑐𝑠 (Σ) |𝛼 ∈Σ𝑖 }Finally we accumulate the scores of axioms in each MCS to obtainthe scores of the MCSs.
Definition 7 (scoring function) Given an ontology Σ and Σ𝑖 ∈𝑚𝑐𝑠 (Σ), the scoring function for the maximal consistent sub-ontologies
are defined as follows:∑︁𝑠𝑐𝑜𝑟𝑒 (Σ𝑖 ) =𝑚𝑐 (Σ, 𝛼)𝛼 ∈Σ𝑖Each MCS is assigned a score by the scoring function in Definition 7. Then we select the MCS with the highest score for reasoning.
Example (continue). For the example mentioned above, we summarize the above calculations: MCS Σ3 = {𝜑 1, 𝜑 2, 𝜑 4 } gets the highest score. In our method, we conclude 𝜑 1, 𝜑 2, 𝜑 4 and use these axioms for reasoning.

5

IJCKG 2023, December 8-9, 2023, Tokyo, Japan

LOGICAL PROPERTIES

We consider the logical properties of the inference relation between
the axioms in Σ and reasoning results using our method. An inference relation is rational when it satisfies the six properties in
the minimal set of expected properties of preferential inference
relations (also called system P) and one of the rational inference
relations (also called system R) [27].
𝛼 ∧ 𝛽 |∼ 𝛾, 𝛼 |∼ 𝛽
Ref 𝛼 |∼ 𝛼Cut𝛼 |∼ 𝛾⊨ 𝛼 ↔ 𝛽, 𝛼 |∼ 𝛾𝛼 | ∼ 𝛾, 𝛽 |∼ 𝛾LLE
Or𝛽 |∼ 𝛾𝛼 ∨ 𝛽 |∼ 𝛾
⊨ 𝛼 → 𝛽, 𝛾 |∼ 𝛼𝛼 |∼ 𝛽, 𝛼 |∼ 𝛾RWCM𝛾 |∼ 𝛽𝛼 ∧ 𝛽 |∼ 𝛾𝛼 |≁ ¬𝛽, 𝛼 |∼ 𝛾RM
𝛼 ∧ 𝛽 |∼ 𝛾
For example, Cut expresses the fact that one may, in his way
towards a plausible conclusion, first add a hypothesis to the facts
he knows to be true and prove the plausibility of his conclusionfrom this enlarged set of facts, and then deduce (plausibly) thisadded hypothesis from the facts. We refer to the technical report4
for detailed explanations about these properties.
To investigate whether our inference relation satisfies the logicalproperties above, we present some significant definitions in [27]and theorems as follows:
Definition 8 (Aggregation function). ⊕ is an aggregation function
if for every positive integer n, for every non-negative real number
𝑥 1, ...𝑥𝑛 , ⊕(𝑥 1, ...𝑥𝑛 ) is a non-negative real number.
For example, this paper chooses sum as the aggregation function.
Definition 9 (𝑠𝑐𝑜𝑟𝑒 𝑠Σ,⊕ ). Let s be a scoring function defined in Definition 2 and let ⊕ be an aggregation function. Let Σ be an ontology𝑠 (Σ ) = ⊕and Σ𝑖 = {𝛼 1 ...𝛼𝑛 } ⊆ Σ . We define 𝑠𝑐𝑜𝑟𝑒 Σ,⊕𝑖
𝛼 ∈Σ𝑖 𝑠 (Σ, 𝛼).
In Definition 7, we denote 𝑠𝑐𝑜𝑟𝑒𝑚𝑐
(Σ
)
as
𝑠𝑐𝑜𝑟𝑒
(Σ𝑖 ) for simΣ,𝑠𝑢𝑚 𝑖
plicity. On the foundation of Definition 9, we can define the monotonic selection relation according to our method. Before that, we
propose a more general one.
Definition 10 (Monotonic selection relation). Given an ontology
Σ, let ⪰Σ ⊆ 2Σ × 2Σ be a reflexive, transitive and total relation
over the powerset of Σ. ⪰Σ is said to be a monotonic selection relation if for every consistent set Σ𝑖 ⊆ Σ , for every non-trivial axiom
𝛼 ∈ Σ\Σ𝑖 , Σ𝑖 ∪ {𝛼 } ≻Σ Σ𝑖 .
For instance, the relation ⪰𝑐𝑎𝑟𝑑 defined over 𝑃 (Σ) is a monotonic
selection relation, in which Σ𝑖 ⪰𝑐𝑎𝑟𝑑 Σ 𝑗 if and only if |Σ𝑖 | ≥ |Σ 𝑗 |.
To be specific, a selection relation and an inference relation in [26]are given as follows:Definition 11 ⪰𝑠Σ,⊕ . Let s be a scoring function and ⊕ an aggregation function. Let Σ be an ontology, Σ𝑖 , Σ 𝑗 ⊆ Σ. We state thatΣ𝑖 ⪰𝑠Σ,⊕ Σ 𝑗 if and only if score𝑠Σ,⊕ ( Σ𝑖 ) ≥ score𝑠Σ,⊕ Σ 𝑗 .
Based on Definition 11, we can compare the subsets of an ontology by the scores of them. We adopt the following notation forconvenience: mcs(Σ, 𝛼) = {Σ𝑖 ⊆ Σ | Σ𝑖 ∪ {𝛼 } ∈ mcs(Σ ∪ {𝛼 })}.
Definition 12 (𝑚𝑐𝑠 ⪰Σ ). Given an ontology Σ, an axiom 𝛼 ∈ Σ, anda monotonic selection relation ⪰Σ , we define 𝑚𝑐𝑠 ⪰Σ (Σ, 𝛼) = {Σ𝑖 ∈𝑚𝑐𝑠 (Σ, 𝛼)|there existsno Σ𝑖′ ∈ 𝑚𝑐𝑠 (Σ, 𝛼) such that Σ𝑖′ ≻Σ Σ𝑖 }.

4 https://github.com/sky-fish23/Embedding-based-infer/blob/main/technical%20report.pdf

IJCKG 2023, December 8-9, 2023, Tokyo, Japan

Ontology Name #class #prop. #indi. #axiom #MCS Expressivity
AUTO.-cocus-eda 158
86
114
907
6
ALCOIN
bioportal-metadata 98
183
61
822
14 ALUH INUOBM-lite-10-35 52
40
43
162
16
SH OINUOBM-lite-10-36 52
40
43
163
40
SH OIN
Table 2: Inconsistent ontologies used in the evaluation.

With respect to a monotonic selection relation, we define a selection mechanism consistingin keeping only the best subsets.

𝑚𝑐𝑠

Definition 13 [26] |∼Σ ⪰Σ , a binary inference relation based onbest subsets w.r.t ⪰Σ ). Given an ontology Σ, two axioms 𝛼 and 𝛽, and𝑚𝑐𝑠
a monotonic selection relation ⪰Σ , we state that 𝛼 |∼Σ ⪰Σ 𝛽 if andonly if for every Σ𝑖 ∈ 𝑚𝑐𝑠 ⪰Σ (Σ, 𝛼), we have Σ𝑖 ∪ {𝛼 } |= 𝛽.
We propose the inference relation to show how conclusions areinferred from an ontology.
Theorem 1. Let Σ be an ontology and let ⪰Σ be a selection relation
over Σ. Then ⪰Σ is monotonic if and only if there exists a scoring
function s for Σ such that 𝑠 (Σ, 𝛼) > 0 for every axiom 𝛼 ∈ Σ.
Proof sketch of Theorem 1. From the definition of monotonic
selection relation and ⪰ ⊕,Σ , this result can be easily induced. Especially, for necessity, to ensure the arbitrariness of 𝛼, we have toshow that for every axiom 𝛼 ∈ Σ, there exists a consistent subset
Σ𝑖 ⊆ Σ, such that 𝛼 ∉ Σ𝑖 .
Based on Theorem 1, we show that the given selection relation
is monotonic. Furthermore, Theorem 2 shows the equivalence ofthe rationality of an inference relation and the monotonicity of its
corresponding selection relation.
Theorem 2. A relation |∼ is rational if and only if there exists an𝑚𝑐𝑠
ontology Σ and a monotonic selection relation ⪰Σ such that |∼Σ ⪰Σ= |∼.
Proof sketch of Theorem 2. Theorem 2 is a rewriting of Theorem
5.18 in [27], so the proof is the same as that.
Theorem 3. The relation |∼ defined by the selection relation whichtakes our proposed scoring function is rational.
Proof sketch of Theorem 3. According to Theorem 2, we knowthat the rationality of the inference relation is equivalent to themonotonicity of corresponding selection relation. So we have toverify the positivity of our scoring functions to ensure every axiom
𝛼 ∈ Σ, 𝑠 (Σ, 𝛼) > 0 (We exclude the trivial axioms).
In conclusion, the selection relation based on our proposed scoring function is monotonic relation according to Theorem 1. And
Theorem 2 shows the rationality of the corresponding inference
relation. Due to the rationality of our proposed method, the reasoning satisfies all the seven logical properties mentioned above.
Detailed proofs of these theorems and explanation of the properties
can be found in the technical report 5 .

6EXPERIMENT AND EVALUATIONIn this section, we first introduce the experimental dataset and settings. Then we conduct some experiments to show the performanceof our proposed method.

Keyu Wang, Site Li, Jiaye Li, Guilin Qi, and Qiu Ji

6.1

Data

In order to fairly evaluate the quality of our proposed method in
answering queries, the test ontology dataset should be diverse. We
select four inconsistent ontologies, which vary in sources, size,
expressiveness, number of axioms and MCSs. Table 2 provides
details of the dataset, where #class, #prop., #indi., #axiom and #MCS
represent the number of classes, properties, individuals, axioms
and MCSs for the ontology respectively. Half of the ontologies areexisting inconsistent ontologies crafted by others, and others are
created by us based on existing ontologies.
AUT.-cocus-eda is constructed in [24] by merging two source
ontologies cocus and edas with their mapping generated by themapping system AUTOMSv2 which has participated in the famouscontest of ontology alignment evaluation initiative6 . Its inconsistency is caused by multiple sources, lying on two similar concepts‘cocus#person’ and ‘edas#person’. Bioportal-metadata7 is a reallife ontology from the world’s most comprehensive repository ofbiomedical ontologies [24]. Some inconsistencies come from multiple sources of certain individual. For instance, these four axioms
are inconsistent:
𝜑 1 : 𝑆𝑢𝑏𝐶𝑙𝑎𝑠𝑠𝑂 𝑓 (𝐾𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑃𝑎𝑟𝑎𝑑𝑖𝑔𝑚,
𝑑𝑜𝑐𝑢𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 𝑚𝑎𝑥 1 𝑟𝑑 𝑓 𝑠 : 𝐿𝑖𝑡𝑒𝑟𝑎𝑙 )𝜑 2 : 𝐷𝑜𝑚𝑎𝑖𝑛 (𝑑𝑜𝑐𝑢𝑚𝑒𝑛𝑡𝑖𝑜𝑛, 𝐾𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒𝑅𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑃𝑎𝑟𝑎𝑑𝑖𝑔𝑚)
𝜑 3 : 𝑑𝑜𝑐𝑢𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 (𝐽 𝑒𝑛𝑎-𝐴𝑅𝑄, “ℎ𝑡𝑡𝑝 : //𝑗𝑒𝑛𝑎.𝑠𝑜𝑢𝑟𝑐𝑒 𝑓 𝑜𝑟𝑔𝑒.𝑛𝑒𝑡 /𝐴𝑅𝑄/”)
𝜑 4 : 𝑑𝑜𝑐𝑢𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛 (𝐽 𝑒𝑛𝑎-𝐴𝑅𝑄, “ℎ𝑡𝑡𝑝 : //𝑗𝑒𝑛𝑎.𝑠𝑜𝑢𝑟𝑐𝑒 𝑓 𝑜𝑟𝑔𝑒.𝑛𝑒𝑡 /”)The two UOBM ontologies are constructed by us based on UOBMlite-10 with an increasing number of axioms. UOBM [13] indicates
University Benchmark and is enriched from the famous Lehigh
University Benchmark (LUMB) [29]. UOBM-lite-10 means that itis enriched from LUBM by adding OWL Lite constructors andcontains individuals from 10 universities. To construct inconsistent
ontologies, we borrowed the idea from [12] and develop the tool
Injector 8 to insert different numbers of conflicts. The mechanism
to generate conflicts in this work is the same as [12] and details
about the implementation of Injector are shown in section 5 of
[12]. A conflict is a set of instance assertions violating a functionalrole restriction or a disjointness constraint. UOBM-lite-10-35 and
UOBM-lite-10-36 are obtained by inserting 35 and 36 conflicts into
ontology UOBM-lite-10 respectively.

6.2

Experimental Settings

The experiment includes four parts: generating the MCSs, transforming axioms with NaturalOWL or TripleOWL, scoring the MCSs
and testing the queries. We perform the MCS generation system,
NaturalOWL system and TripleOWL system in Java 1.8.0 environment, on a laptop with 1.80 GHz Intel core CPU, 16 GB RAM and a
64-bit Windows 10 operating system. The MCS generation system
can be applied to any DLs. We implement the scoring function for
selecting MCSs using Python 3.9.7 and PyTorch 1.11.0 with thesame computer requirements as MCS generation system. For thequery test experiment, we used the widely used ontology editor
Protégé9 with version 5.5.0 and max memory set to 444MB. All
6 http://oaei.ontologymatching.org/2012/conference/
7 https://bioportal.bioontology.org/
8 https://anonymous.4open.science/r/Embedding-based-infer/data/Injector/

5 https://github.com/sky-fish23/Embedding-based-infer/blob/main/technical%20report.pdf

9 https://protege.stanford.edu/

An Embedding-based Approach to Inconsistency-tolerant Reasoning with Inconsistent Ontologies

MethodSimilarity

Skeptical inference
CMCS#mc
Sentence-BERT
CosineEuclidConSERT
CosineEuclidTransE
CosineEuclidRDF2Vec
CosineEuclidSkeptical inference
CMCS#mc
Sentence-BERT
CosineEuclidConSERT
CosineEuclidTransE
CosineEuclidRDF2Vec
CosineEuclidSkeptical inference
CMCS#mc
Sentence-BERT
CosineEuclidConSERT
CosineEuclidTransE
CosineEuclidRDF2Vec
CosineEuclid
Skep.
CMCS
#mc
Sentence-BERTConSERT
TransERDF2Vec

CosineEuclidCosineEuclidCosineEuclidCosineEuclidQueriesIA
CA
AUT.-cocus-edas
124
6
118
124
8
116
124
8
116
124
122
0
124
122
0
124
119
3
124
122
0
124
8
116
124
8
116
124
122
0
124
8
116
Bioportal metadata
41
22
19
41
22
19
41
22
19
41
36
0
41
38
0
41
38
0
41
38
0
41
38
0
41
38
0
41
36
2
41
22
19
UOBM-lite-10-35
90
74
16
90
74
16
90
74
16
90
80
0
90
89
0
90
84
0
90
89
0
90
83
1
90
89
0
90
78
3
90
74
16
UOBM-lite-10-36
92
61
31
92
71
12
92
71
12
92
82
0
92
86
0
92
82
0
92
86
0
92
82
0
92
86
0
92
86
0
92
71
12

IJCKG 2023, December 8-9, 2023, Tokyo, Japan

RA

CIA

IA RateICR Rate

0
0
0
2
2
1
3
0
0
2
0

0
0
0
0
0
1
0
0
0
0
0

4.84%
6.45%
6.45%
98.39%
98.39%
95.97%
98.39%
6.45%
6.45%
98.39%
6.45%

100%
100%
100%
100%
100%
99.21%
100%
100%
100%
100%
100%

0
0
0
3
3
3
3
3
3
2
0

0
0
0
2
0
0
0
0
0
1
0

53.66%
53.66%
53.66%
87.80%
92.68%
92.68%
92.68%
92.68%
92.68%
87.80%
53.66%

100%
100%
100%
95.12%
100%
100%
100%
100%
100%
97.56%
100%

0
0
0
4
0
4
0
4
0
4
0

0
0
0
6
1
2
1
2
1
5
1

82.22%
82.22%
82.22%
88.89%
98.89%
93.33%
98.89%
92.22%
98.89%
87.67%
82.22%

100%
100%
100%
93.33%
98.89%
97.78%
98.89%
97.78%
98.89%
94.44%
100.00%

0
5
5
5
5
5
5
5
5
5
5

0
4
4
5
1
5
1
5
1
1
4

63.30%
77.17%
77.17%
89.13%
93.48%
89.13%
93.48%
89.13%
93.48%
93.48%
77.17%

100%
95.65%
95.65%
94.57%
98.91%
94.57%
98.91%
94.57%
98.91%
98.91%
95.65%

Table 3: IA = Intended Answers, CA = Cautious Answers, RA = Reckless Answers, CIA = Counter-Intuitive Answers, IA Rate
=Intended Answers(%), ICR Rate = IA+CA+RA(%)

IJCKG 2023, December 8-9, 2023, Tokyo, Japanthe data and codes including MCS generator, NaturalOWL system,
TripleOWL system, Injector and scoring function are available athttps://anonymous.4open.science/r/Embedding-based-infer/.
We selected three existing methods based on MCSs as baselines
as follows.
• Skeptical inference [27].
• CMCS: It is based on selecting those cardinality-maximal consistent subsets, known as a refinement on skeptical inference [3].
• #mc: It is proposed by [26] considering the reliability of the information carried by the axioms as shown in Definitions 3, 4.
Our proposed method is according to Definition 5, 6, 7 and wehave implemented different variances corresponding to difference
embedding models.
[26] proposed a series of reasonable reasoning approaches basedon MCS which satisfy a set of logical properties, but there is a lackof experiment to evaluate the performance of the reasoning methods. In this section, each algorithm is evaluated with respect to its
effectiveness and efficiency. When reasoning with an inconsistent
ontology, the effectiveness of one algorithm indicates the quality of
the answers to the test queries. The efficiency of the algorithm isthe time to generate answers to the test queries. We use the sameevaluation method as [21, 22]. We compare answers generated by
the method to be evaluated with hand-crafted Gold Standard that
contains the humanly-judged correct answer which is supposedby human,including ontology experts and postgraduate students.
For a query, there might exist the following differences between an
answer generated by the method to be evaluated and its intuitive
answer.
• Intended Answer: the method’s answer is the same as the intuitive
answer.
•Counter-intuitive Answer: the method’s answer is opposite to theintuitive answer. Namely, the intuitive answer is ‘accepted’ whereas
the method’s answer is ‘rejected’, or vice versa.
•Cautious Answer: the intuitive answer is ‘accepted’ or ‘rejected’,but the method’s answer is ‘undetermined’.
•Reckless Answer: the method’s answer is ‘accepted’ or ‘rejected’
whereas the intuitive answer is ‘undetermined’. Under this situation
the method to be evaluated returns just one of the possible answers
without seeking other possibly opposite answers, which may leadto ‘undetermined’.
We randomly select classes, individuals and properties to form
the queries such as ‘Denmark(individual) Type country(concept) ?’,‘Administrator(concept) SubclassOf Person(concept) ?’. Especially,
we give greater weight to classes, individuals and properties thatare more likely to cause inconsistency when randomly forming
queries. We compared these answers against a hand-crafted Gold
Standard that contained the humanly-judged correct answer for allof these queries.

6.3

Evaluation Results

Table 3 presents the evaluation results. Baselines cannot give an effective MCS selection method because the semantic information of
the ontology is ignored, which leads to poor reasoning quality. Compared with baselines, our method has a significant improvement in
the intended answer (IA) rate. For AUT.cocus-edas, the IA rate of
the baselines is lower than 7%, but our proposed method based on

Keyu Wang, Site Li, Jiaye Li, Guilin Qi, and Qiu Ji

sentence embedding can be above 96%. The method based on knowledge graph embedding doesn’t perform well here, because there isrich and important textual information in this ontology, which is
not utilized in the knowledge-graph-embedding-based method, andsimply concatenating parts of the triple can’t represent the meaningof an axiom well especially for axioms with complex concepts. For
bioportal metadata and the two UOBM ontologies, our method has
improved the IA rate by more than 10% and 20% respectively withcomparison to the baselines. The baselines achieve high ICR rate at
the cost of low intended answer (IA) rate. However, our proposed
method can achieve both high ICR rate and IA rate. For AUT.cocusedas and bioportal metadata, most results for the ICR rate of our
method are 100%. For the two UOBM ontologies, our method canachieve more than 93% and the majority of the results are above
98% . These reflect the fact that the proposed method could provide
promising results when reasoning with inconsistent ontologies. It
owes to the semantic information of the axioms considered in ourembedding-based method.
We also evaluate the efficiency of our method with respect toselecting MCSs. For the four ontologies in our experiment, the consumed time of the proposed method is within 10 minutes. Although
the time consumed by the baselines is within several seconds andour proposed method spends more time than the baselines to scoreand select MCSs, it is efficient enough in practice as the procedureof scoring and selection only needs to be performed once for each
ontology and this process can be done offline. Suppose the selection is done, we also evaluate the time to execute each query. Our
proposed approach is very efficient and a query can be answered
within about half a second.

7

CONCLUSION AND DISCUSSION

In this paper, we introduced a new approach to reasoning withinconsistent ontologies based on maximal consistent subsets. As far
as we know, this is the first work that applies embedding techniques
to inconsistency-tolerant reasoning. In our work, we first proposed
two methods for turning axioms into semantic vectors and computed the semantic similarity between axioms. We then proposed
an approach for selecting the maximal consistent subsets and defined an inconsistency-tolerant inference relation. We showed thelogical properties of proposed inconsistency-tolerant inference relation. We have proved that the inference relation we proposedsatisfies the logic properties, which showed the rationality of theinference relation. We conducted extensive experiments on four
ontologies and the experimental results show that our embeddingbased method can outperform existing methods based on maximal
consistent subsets.
As for future work, we will give more careful consideration to
dealing with complex axioms. We also plan to extend our method
to inconsistency-tolerant inference with weighted ontologies andconsider applying ChatGPT for translating axiom into sentences.
Finally, we can explore applying embedding techniques to ontology
repair by defining some relevant relations like [25].

8

ACKNOWLEDGMENTS

We thank the reviewers for their valuable comments.This work
is partially supported by National Nature Science Foundation ofChina under No. U21A20488.

An Embedding-based Approach to Inconsistency-tolerant Reasoning with Inconsistent Ontologies

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