# An Embedding-Based Approach to Repairing OWL Ontologies
## 5 Experiments
### Results of Comparing Repair Algorithms
#### 5.4.1 Results about Effectiveness  
Table 4 presents the number of the axioms removed by each repair algorithm to restore the coherence of an incoherent ontology.  
Obviously, LogicAlg cannot deal with ontology km1500-3500 because the reasoning process of computing entailments for each axiom MIPS cannot be finished within the limited time (i.e., 1000 milliseconds). Similar to the work in LogicAlg [27], we mainly consider those entailments that are subsumption. In ontology km1500-3500, too many such entailments can be inferred for some axioms, and thus, the process is quite time-consuming. Take the following axiom in a MIPS in km1500-3500 as an example:  
SubClassOf(information_c service_c)  
Figure 2: Time in milliseconds (y-axis) for each ontology to compute similarities for all axiom pairs in each ontology by using Cosine Distance.  
LogicAlg needs to compute all descendants (denoted as \(S_{\textit{des}}\)) of the concept information_c and all ancestors (denoted as \(S_{\textit{anc}}\)) of the concept service_c by invoking the reasoner first, then create axioms in the form of \(\texttt{SubClassOf}(C_{1},C_{2})\) (\(C_{1}\in S_{\textit{des}}\) and \(C_{2}\in S_{\textit{anc}}\)) as this axiom's entailments. Since service_c is an unsatisfiable concept and all concept names except unsatisfiable concepts are regarded as this concept's ancestors, we obtain 2937 ancestors in total (i.e., 3671 concept names minus 734 unsatisfiable concepts) for service_c. Moreover, the concept information_c has 18 descendants obtained by reasoning. Therefore, more than 50 thousand entailments (i.e., 18 \(\times\) 2937 = 52,866) were obtained for the axiom. This is the main reason why LogicAlg failed to compute a diagnosis within limited resources.  
Comparing various repair algorithms, the following observations can be obtained:  
1. The baseline algorithm BaseAlg is able to find a minimal diagnosis as expected. This can be explained by the fact that it applies the ILP-based approach to all MIPS directly, and the ILP-based approach has been proven to find minimal diagnoses [26].
2. The score-based algorithm ScoreAlg has similar performances as BaseAlg since it selects the axioms with the highest score from each MIPS. Thus, it can find a minimal diagnosis in most cases. In addition, the ranking strategy in ShapAlg is similar to that in ScoreAlg. The more times an axiom appears in MIPS, the higher the rank of the axiom is. One main difference between them is that the former is also dependent on the size of a MIPS. Thus, both algorithms perform similarly.
3. For those ontologies that BaseAlg removed more than five axioms, the original signature-based algorithm SigAlg often removed much more axioms than others, and it removed 193 axioms in total for all selected ontologies. For example, SigAlg removed 47 axioms for the ontology Economy while no more than 36 axioms for others.
4. For each ontology that BaseAlg removed less than five axioms, LogicAlg can always find a minimal diagnosis as BaseAlg. It is because most of the axioms in MIPS do not have any entailments such that nearly all axioms in a MIPS have the same rank. For instance, although the ontology Wiktionary-confof-edas has 82 MIPS and 24 distinct axioms in these MIPS, only 3 axioms have entailments.
5. Two embedding-based algorithms by considering the signature of axioms performed similarly. Namely, two distance measures do not make any big difference. Furthermore, both of them outperformed the original signature-based algorithm SigAlg. It shows that SigCosAlg and SigEucAlg can reduce the number of removed axioms.  
Figure 3: Time in milliseconds (y-axis) for each ontology to compute similarities for axiom pairs in each ontology by using two distance metrics.  
As a whole, two threshold-based algorithms performed better than the three signature-based ones, especially ThrEucAlg. Each of them removed less than 170 axioms in total (143 axioms for ThrEucAlg)while more than 175 axioms for a signature-based algorithm.