In this paper, we present a method for extracting subontologies from ELH ontologies for a set of symbols. The approach is focused on the generation of upwardly abstracted de nitions, which is a technique for computing de nitions expressed using closest primitive ancestors. The subontologies returned by the method are evaluated for quality and compared to extracts computed with locality-based modularisation and uniform interpolation methods. Our subontology generation method produces promising results in terms of size and relevance to the needs of domain experts.
Ontologies are formalised representations of domain knowledge. SNOMED CT,
the Gene ontology (GO) and the NCIt ontology are just a few of the major
ontologies used in the biomedical domain [
There are various methods and approaches for extracting subontologies in the
literature, including graph-based ontology partitioning [
tool, and their help in using it.
and reuse. The method computes a subset of the stated axioms of an ontology
that covers information related to the input signature. Modules can be large and
can contain information outside the meaning of the input signature [
Another method for producing compact representations is uniform
interpolation (UI), a logic-based method for computing restricted views of an ontology
which faithfully captures information about the set of speci ed concepts and role
names. This is accomplished by forgetting symbols that are not contained in the
input signature [
Locality-based modularisation and uniform interpolation are useful for a
variety of applications such as ontology summarisation, reuse, analysis, logical
difference, and information hiding [
In this paper, we introduce a notion of subontology based on the idea of
abstracted de nitions, because SNOMED CT users are already familiar with
a variety of normal forms [
The main contributions of the paper are threefold: { An investigation of computing upwardly abstracted de nitions for E LH ontologies that satisfy common modelling guidelines used in SNOMED CT, one of the most widely used biomedical ontologies. { A method for extracting subontologies based on the principle of abstracted de nitions. { An evaluation of the subontology extraction method and comparison with two existing subontology extraction methods, namely the bottom variant of SLBM, and the UI method, in order to comprehend bene ts of subontologies based on the abstracted de nitions.
The paper is organised as follows. Section 2 gives preliminary de nitions and background information. Section 3 goes into detail about the upwardly abstracted de nitions. The aim, requirements, algorithm and an example of the subontology generation method are presented in Section 4. Section 5 discusses related notions including SLBM and uniform interpolation. In Section 6, we evaluate the quality of the returned subontologies, and compare the results of our subontology generation method with SLBM and uniform interpolation in terms of precision and extract size. Finally, we conclude in Section 7. 2
Let NC and NR be disjoint sets of concept and role names respectively. The union
of such sets form the signature of an ontology O. The signature sig( ) is a set
of concept and role names that occur in , where is any syntactic object or
ontology. The set of E L-concepts C, and the sets of E LH-axioms are built
according to the grammar rules: C ::= A j C u C j 9r:C and ::= C v C j C
C j r v s where A 2 NC and r; s 2 NR.1 An E LH-TBox is a nite set of E
LHaxioms. The semantics, including the notions of model, satisfaction of concepts,
axioms and TBoxes as well as the logical consequence relation (entailment), are
de ned in the usual manner; see, for example [
A terminology is a TBox that contains only axioms of the form A C or A v C, with A appearing not more than once on the left-hand side of an axiom. If A does not depend on itself, i.e., does not occur in the set of symbols required to de ne itself for any A 2 NC, the terminology is acyclic.
Classi cation is a standard reasoning task that computes a hierarchy H for O. A hierarchy H is a nite set of subsumption axioms A v B such that O j= A v B, where A and B are concept names in sig(O).
Figure 1 shows an example of an E LH biomedical terminology, adapted
from [
Our study on generating upwardly abstracted de nitions is limited to axioms of the form A C or A v C in an E LH ontology or terminology, as the majority of biomedical ontologies lack GCIs of the form C v A, C v D, or C D, where C and D denote complex concepts and A denotes a concept name.
5 : EntireLiver v Liver
We start by de ning the key concepts used in this article.
De nition 1 (De ned (Primitive) Concept). Let O be an ontology, and C an E L-concept other than A. A concept name A is a de ned concept in O if there is an axiom of the form A C in O. Otherwise, it is called a primitive concept.
Regardless as to whether a named concept A is de ned or primitive, its abstracted de nition is computed by inferring its closest primitive ancestor(s) in the subsumption hierarchy. Closest primitive ancestors are de ned as follows. De nition 2 (Closest Primitive Ancestor). Let O be an ontology, A; P 2 sig(O) where A is a de ned or a primitive concept name and P is a primitive concept name. We say that P is a closest primitive ancestor to A in O if O j= A v P and there does not exist a primitive concept name Z 2 sig(O) (other than P or A) such that O j= A v Z and O j= Z v P . The set of closest primitive ancestors to A will be denoted by PA.
We de ne abstracted de nitions as follows.
De nition 3 (Upwardly Abstracted De nition). Let O be an ontology, and A a de ned (primitive) concept name in O. The abstracted de nition of A is A PA u EA (A v PA u EA), where PA is a conjunction of the closest primitive ancestors to A, while EA is a conjunction of existential restrictions (of the form 9r:C) required to complete the abstracted de nition of A such that O j= A PA u EA (or O j= A v PA u EA), and sig(PA u EA) sig(O). Example 1. Consider the ontology O = fA D u 9r:C1; D P u 9r:C2; P v 9r:C3g. An upwardly abstracted de nition of A is A P u 9r:C1 u 9r:C2 u 9r:C3.
Di erent equivalent logical forms of SNOMED CT concept de nitions are
discussed in [
The following example illustrates that abstracted de nitions may also be weaker than the original de nitions when O is an ontology rather than a terminology. Example 2. Let O = f 1; 2; 3g, where 1: A Du9r:C, 2: D P u9r:C and 3: A v P2. We notice that both ab1: A P u 9r:C and ab2: A P u P2 u 9r:C are entailed by O and are in fact abstracted de nitions of A. Let's consider the ontologies O1 = fab1; 2; 3g and O2 = fab2; 2; 3g in which the original de nition 1 of A is respectively replaced by ab1 and ab2. We observe that O O1 but O 6 O2 because O2 6j= O as O2 6j= 1 since ab2 is weaker than 1.
The abstracted de nition of A in O computed by our algorithm is the second case (ab2) as part of its search for all of the closest primitive ancestors. As seen in the example, this de nition is weaker than the original de nition. To de ne the logical strength of abstracted de nitions, we use the following de nition: De nition 4 (Logical Strength). An ontology O0 is weaker than another ontology O if O j= O0 but O0 6j= O. An axiom 0 is weaker than another axiom in O if O j= 0 but Onf g [ f 0g 6j= . 4
Our aim is to compute for a given set of symbols a domain-speci c subontology from a source ontology that satis es the following requirements: 1. The subontology must capture the meaning of the concepts in the focus set, using whenever possible abstracted de nitions in long canonical form. 2. The transitive closure of concept name subsumption of the subontology is a restriction of the original ontology's transitive closure of concept name subsumption over the signature of the subontology.
These requirements were established with a leading terminologist at SNOMED international.
Our method to compute subontologies is presented in Algorithm 1. The
algorithm takes as input an ontology O and a focus set F of concept and role
names to generate a subontology S. The rst step of the algorithm initialises
the output S, and the set of existential restrictions E , and + is set to F .
The second step classi es O using the ELK reasoner [
Computing an abstracted de nition for a focus concept A 2 F calls the method AbstractedDe nitionExtraction presented in Algorithm 2. The method starts with computing the Ancestors of A using H. The Ancestors set consists of all concept names in sig(O) that subsume A. In Line 2, the function ComputePrimitiveAncestors lters the set of ancestors by determining their status in O as de ned or primitive in order to obtain just the primitive concept ancestors. We use O to obtain the existential restrictions EA that occur in the right-hand side of A's de nition and all of A's ancestors' de nitions, which is computed by the function ComputeExistentialRestrictions in Line 3.
Output: Subontology S
+ := 1: S := ;, F , E := ;. 2: H := Classify(O) 3: for A 2 F do 4: if A is a de ned concept name in O then 5: A PA0 u EA0 := AbstractedDe nitionExtraction(A, O, H, TRUE) 6: else A v PA0 u EA0 := AbstractedDe nitionExtraction(A, O, H, FALSE) 7: + := +[ sig(A (v) PA0 u EA0) 8: E := E [ GetExistentialRestrictions(A (v) PA0 u EA0) 9: S := S [ A (v) PA0 u EA0 10: S := S [ ComputeAdditionalAxioms( +,E, H, O)
checker isDe nedConcept Output: The abstracted de nition of A 1: Ancestors := ComputeAncestorsOfA(A, H) 2: PA := ComputePrimitiveAncestors(Ancestors, O) 3: EA := ComputeExistentialRestrictions(A, Ancestors, O) 4: PA0 u EA0 := RemoveRedundantConcepts(PA, EA, H, O) 5: if isDe nedConcept then 6: return A PA0 u EA0 7: else return A v PA0 u EA0
The method RemoveRedundantConcepts in Line 4 returns the set (PA0 u E A0), which is a conjunction of the closest primitive ancestors and the most speci c existential restrictions, after the removal of possible redundant concepts from the set of primitive concepts PA and existential restrictions EA. A concept D is regarded as redundant if it occurs in another concept C where D is equivalent to or subsumes C. Redundant concepts in the sets PA and EA are removed according to the general rule C u D C , C v D. The removal of redundant concepts from the set PA results in the set of closest primitive ancestors. To remove redundant existential restrictions from the set EA, we follow the rules in Figure 2. For example, if the set E has two existential restrictions, E := 9t:(9r1:A1 u ::: u 9rn:An) and G := 9u:(9s1:B1 u ::: u 9sm:Bm), we check whether E v G. We do this by using the rst rule to check if the outer role t in E is equivalent to or subsumed by the outer role u in G. If this is the case, then we continue with the rst rule to check if an existential restriction 9ri:Ai is subsumed by or equivalent to an existential restriction 9sj :Bj under the nested roles t and u, respectively. Then, following rule 2, we determine if the subsumption checking performed using rule 1 is su cient for all 9ri:Ai to be subsumed by all 9sj :Bj . If this is the case, then it means that E v G, and G can be removed from the set E .
1.r v s; C v D ) 9r:C v 9s:D 2.8i = 1::n 9j = 1::m 9ri:Ci v 9sj:Dj ) 9t:(9r1:C1 u :: u 9rn:Cn) v 9t:(9s1:D1 u :: u 9sm:Dm)
Algorithm 2 concludes by returning the abstracted de nition of A as either A PA0 u E A0 or A v PA0 u E A0 depending on whether A is a de ned or a primitive concept in O. Lines 7 and 8 of Algorithm 1 add the signature, and the existential restrictions of the generated abstracted de nition to the sets + and E , respectively. Line 10 returns the subontology S after using the function ComputeAdditionalAxioms to add extra axioms to complete the hierarchy of the subontology. This is performed by looking for possible subsumption relations between concept and role names in the set +, as well as between concept names in + and the existential restrictions in E . For example, the axiom 3 in Figure 3 is an additional axiom that is required for the subontology hierarchy to be complete.
Our method abstracts the de nitions based on the closest primitive ancestors in order to include only what is truly necessary in the resulting subontology. For example, if a user is interested in computing a subontology using the ontology in Figure 1 for the focus set concepts A1:Hepatitis2 and A2:LargeLiver, then applying Algorithm 1 for A1 and A2 results in the subontology shown in Figure 3. It includes abstracted de nitions of A1 and A2, where the concept A3:LiverDisease occurring in the original de nitions of A1 and A2 has been abstracted away. The use of A3 would require the inclusion of its de nition and increase the size of the desired subontology without adding additional meaning. This is because A3 carries the same information that A1 and A2 inherit, and thus, the de nition of A3 is super uous with respect to the focus set fA1; A2g of interest to the user. We refer to the extra symbols occurring in the signature of the generated focus set de nitions as the supporting set, e.g., the concept EntireLiver occurring in the de nition of LargeLiver is a supporting concept.
1 : Hepatitis2 Disease u 9involves.In ammation u 9location.Liver 2 : LargeLiver v Disease u 9location.EntireLiver 3 : EntireLiver v Liver We de ne a subontology for a given set of focus symbols F as follows.
De nition 5 (Focus Set Subontology). Let O be an E LH ontology and F a focus set. S is a focus set subontology of O w.r.t. F if the following conditions are satis ed: (i) F sig(S); (ii) for every E LH-axiom where sig( ) sig(S) we have: (a) S j= =) O j= and (b) if is of the form A v B, then O j= =) S j= , where A and B are concept names.
Depending on the input ontology, our algorithm generates two types of subontologies: focus set subontologies, which may contain weaker abstracted de nitions (cf. ab2 in O2 in Example 2), and equivalent focus set subontologies, which contain abstracted de nitions that are equivalent to their original de nitions in O for all concepts in the focus set F . We de ne a subontology that contain equivalent focus set abstracted de nitions for all focus concepts in F as follows. De nition 6 (Equivalent Focus Set Subontology). Let O be an E LHacyclic terminology, and S a focus set subontology of O w.r.t. F . We say that S is an equivalent focus set subontology when O j= , S j= for all abstracted de nitions of focus concepts in F in S.
In order to obtain an equivalent focus set subontology, the input ontology to our method has to be E LH-acyclic terminology. 5 5.1
Syntactic Locality Based Modularisation
SLBM is widely used in ontology engineering to facilitate ontology reuse and
import [
SLBM retains the original forms of axioms in the resulting modules. However,
for the purpose of extracting de nitions of concepts in , modules computed
contain a signi cant number of supporting symbols. For instance, as demonstrated
in [
Uniform Interpolation
Uniform interpolation is a task that allows potentially undesirable symbols to
be eliminated from an ontology without a ecting the meaning of the remaining
symbols in the ontology [
Computing UIs for focus set concepts produces ontologies too small to include
de nitions of the focus concepts. For example, computing a UI for A1: Hepatitis2
and A2:LargeLiver for the ontology in Figure 1, requires forgetting the rest of the
ontology's symbols. This results in the UI fA1 v >, A2 v >g, because the RHSs
of the de nitions of A1 and A2 are eliminated. More informative UIs can be
generated by rst extending the focus set using a signature extension algorithm
as in [
The goal of the evaluation is to test empirically the following three hypotheses about our method: (1) The abstracted de nitions in the generated subontologies are equivalent to their original de nitions when the input ontology is a terminology (Section 6.1). (2) Abstracted de nitions aid in the reduction of axioms deemed redundant in relation to the input focus set. The study of the signature of the bottom modules demonstrates this (Section 6.2). (3) Subontologies are smaller in size than bottom modules, and smaller or equal in size to the UIs (Sections 6.2 and 6.3). To do this, we developed a Java prototype implementing our method using the OWL API. 2 We employed two distinct ontologies, SNOMED CT (July 2017) and the Gene ontology (GO) (February 2021). GO does not meet the requirements of a terminology, as it may contain more than one axiom for a concept name A. Both SNOMED CT and GO lack GCI axioms of the forms C v A and C v D where C and D are E L-concepts.
We compared the produced subontologies to the two extract types outlined in
Section 5: the ?-modules and the UIs. We computed UIs using a newly developed
UI tool for E LH ontologies [
SNOMED CT had a total of 335 245 logical axioms, 335 225 concept names and 97 role names. The Gene ontology had a total of 102 203 logical axioms, 44 085 concept names,3 and 8 role names. Both ontologies are in the scope of E LH, with the exception of one role chain axiom (r s v r) in SNOMED CT and one inverse role, 29 disjoint classes, four transitive roles, and two role chain axioms in the Gene ontology. Non E LH-axioms were omitted.
As focus sets, we used ve sets of human and animal medical conditions for
computing SNOMED CT extracts used in the experiments of [
Table 1. Results of logical strength test of the abstracted de nitions of medical conditions and gene slim focus sets in Snomed CT (SCT) and Gene Ontology (GO), respectively.
Quality criteria
Focus set
Anaemia (SCT)
Arthritis (SCT)
Diabetes (SCT)
Hypertension (SCT)
Obesity (SCT)
goslim mouse (GO)
goslim pir (GO)
goslim plant (GO)
goslim pombe (GO)
goslim yeast (GO)
names. For the Gene ontology, we used ve focus sets of the GO slim sets
provided in [
Given how the three methods work, we used the following input sets to generate the di erent extracts: (1) The focus set was used as the input signature for computing the bottom modules and subontologies, because both SLBM and our subontology generation method extend the signature as needed. (2) As input to the UI method, we used the signatures of the computed subontologies. As mentioned, UIs for focus sets would not adequately capture their de nitions. 6.1
Logical Strength of the Upwardly Abstracted De nitions To determine the logical strength of the generated abstracted de nitions, the test of De nition 4 was used. We found that when using SNOMED CT to compute subontologies, the abstracted de nitions generated have the same logical strength as the original de nitions. This is because SNOMED CT is a terminology that includes no more than one concept de nition for a concept name A. On the other hand, when the Gene ontology is used, the abstracted de nitions derived may be weaker than the original de nitions. The count numbers of the logical strength of the abstracted de nitions in the subontologies of SNOMED CT and the Gene ontology are shown in Table 1.
Subontology Results Against Bottom SLBM Table 2 shows that the number of symbols (concepts and roles) in the bottom modules were signi cantly larger than those in the subontologies. The average numbers of logical axioms, concepts, and roles in SNOMED CT's subontologies were 36.04%, 36.35%, and 39.79% less than the average numbers of those in the bottom modules, respectively.
Table 3 shows the computed values for Precision and Region for each bottom
module, as well as the number of signatures for subontologies denoted by jsig(S)j
and for bottom modules denoted by jsig(M)j for each SNOMED CT and Gene
ontology focus set. We use the formula given in [
To quantify the bene ts of abstracting focus concepts' de nitions in terms of reducing the size of an extract, we examined the set of symbols in the bottom modules that occur outside the set of their corresponding subontologies' signatures. Speci cally, we calculated the number of concepts that exist in the region between the concepts in the focus set F and their closest primitive ancestors, denoted by P F using the formula:
Region( F , P F ) := (Ancestors of F \ Descendants of P F ).
The de nitions of concepts found in the Region set can be viewed as redundant information in relation to the input focus set de nitions. This is because de nitions of concepts in the Region set should carry the same information that the abstracted de nitions of focus concepts inherit, which is especially true when the input ontology is a terminology. Thus, incorporating de nitions of focus concepts following abstraction is su cient for including information that is genuinely important in the extract, as the abstraction process aids in the inclusion of all of the focus concept's de ning characteristics. As can be observed from Table 3, 99.9% of concepts that exist outside the subontology's signature in the Diabetes bottom module occur in the Region set, demonstrating that the size of such an extract can be greatly reduced by abstracting the de nitions of focus concepts to their closest primitive ancestors. 6.3
Subontology Results Against UI
The UI tool was unable to generate views for the SNOMED CT medical
conditions focus sets (Table 2). This is due to the fact that forgetting becomes more
di cult when the input signature is small in comparison to the source ontology's
signature, particularly when forgetting from a very complex, large ontology such
as SNOMED CT [
As shown in Table 2, the number of concepts and roles for subontologies and UIs for the Gene ontology focus sets coincide, as the UIs were computed for the subontologies' signatures, con rming the expected 100% precision of UI. On the other hand, the results show that the number of logical axioms in UIs is greater than that in subontologies. There are two main explanations for this observation. The rst is that the UI method has incorporated inferred axioms of the form C v A, where A denotes a focus or a supporting concept name. For example, forgetting B1 and B2 from B1 C, B1 v B2, B2 v A, where all symbols in C are in the signature of the subontology, derives C v A. The second is that when a focus concept A is described by multiple axioms, our method generated an abstracted de nition that condenses these multiple axioms into a single axiom. For instance, the UI might contain two axioms for A, A C1 and A v C2, while our method infers the axiom A C1 u C2, which results in fewer axioms in the subontology, but can also result in a weaker de nition for A. 7
We presented a method of extracting subontologies, which is based on abstracted de nitions for given sets of focus symbols. Such abstracted de nitions are motivated by normal forms used in proximal primitive modelling in the SNOMED CT community. We empirically demonstrated that when the input ontology is a terminology, the method generates equivalent focus set subontologies; however, when the input ontology is not a terminology, the method may yield subontologies with weaker de nitions.
In comparison to bottom modules, our abstracted de nition-based subontologies contain signi cantly fewer supporting set symbols and contained fewer axioms. The sizes of the Region set in the bottom modules give additional insights into why the notion of abstracted de nitions in subontologies produce signi cantly smaller extracts than bottom modules while retaining all of the de ning characteristics of the focus set concepts.