Extracting a subset of a given OWL ontology that captures all the ontology's knowledge about a specified set of terms is a wellunderstood task. This task can be based, for instance, on locality-based modules (LBMs). These come in two flavours, syntactic and semantic, and a syntactic LBM is known to contain the corresponding semantic LBM. For syntactic LBMs, polynomial extraction algorithms are known, implemented in the OWL API, and being used. In contrast, extracting semantic LBMs involves reasoning, which is intractable for OWL 2 DL, and these algorithms had not been implemented yet for expressive ontology languages. We present the first implementation of semantic LBMs and report on experiments that compare them with syntactic LBMs extracted from real-life ontologies. Our study reveals whether semantic LBMs are worth the additional extraction effort, compared with syntactic LBMs.
Extracting a subset of a given OWL ontology that captures all the ontology’s
knowledge about a specified set of concept and role names is an interesting task
for various applications, and it is by now well-understood [
We present the first implementation of semantic LBMs and report on experiments that compare them with syntactic LBMs extracted from real-life ontologies. The contributions of this paper are as follows: we show with statistical significance that, for almost all members of a large corpus of existing ontologies, there is no difference between any syntactic LBM and its corresponding semantic LBM. In the few cases where differences occur, these differences are modest and not worth the increased computation time needed to compute semantic LBMs. In addition, we isolate two types of axioms that lead to differences, where one is a simple tautology that can, in principle, be detected by a straightforward addition to the syntactic locality checker. Furthermore, our results show that the extraction of semantic LBMs, which is in principle hard, seems feasible in practice. The lesson we learn from these results is that “Cheap is Great”! 2
We assume the reader to be familiar with OWL and the underlying description
logic SROIQ [
Let NC be a set of concept names, and NR a set of role names. A signature is a set of terms, i.e., a set NC [ NR of concept and role names. We can think of a signature as specifying a topic of interest. Axioms that only use terms from can be thought of as “on-topic”, and all other axioms as “off-topic”. For instance, if = fAnimal; Duck; Grass; eatsg, then Duck v 9eats:Grass is on-topic, while Duck v Bird is off-topic.
Any concept, role, or axiom that uses only terms from is called a -concept, -role, or -axiom. Given any such object X, we call the set of terms in X the signature of X and denote it with Xe .
Given an interpretation I, we denote its restriction to the terms in a signature with Ij . Two interpretations I and J are said to coincide on a signature , in symbols Ij = J j , if I = J and XI = XJ for all X 2 .
There are a number of variants of the notion of conservative extensions, which capture the desired preservation of knowledge to different degrees. We focus on the deductive variant.
Definition 1. Let M
O be SROIQ-ontologies and a signature. (1) O is a deductive -conservative extension ( -dCE ) of M if, for all SROIQaxioms with e , it holds that M j= if and only if O j= . (2) M is a dCE-based module for of O if O is a -dCE of M.
Unfortunately, deciding in general if a set of axioms is a module in this sense
is hard or even impossible for expressive DLs [
It has been shown in [
Furthermore, locality can be tested using available DL-reasoners [
Definition 3. An axiom is called syntactically ?-local (>-local ) w.r.t. signature if it is of the form C? v C, C v C>, C? C?, C> C>, R? v R (R v R>), or Trans(R?) (Trans(R>)), where C is an arbitrary concept, R is an arbitrary role name, R? 2= (R> 2= ), and C? and C> are from Bot( ) and Top( ) as defined in Part (a) (resp. (b)) of the table below. (a) ?-Locality
Let A?; R? 2= ; C? 2 Bot( ), C(>i) 2 Top( ); n 2 N n f0g Bot( ) ::= A? j ? j :C> j C u C? j C? u C j 9R:C? j >n R:C? j 9R?:C j >n R?:C Top( ) ::= > j :C? j C1> u C2> j >0 R:C (b) >-Locality
Let A>; R> 2= ; C? 2 Bot( ), C(>i) 2 Top( ); n 2 N n f0g
Bot( ) ::= ? j :C> j C u C? j C? u C j 9R:C? j >n R:C?
Top( ) ::= A> j > j :C? j C1> u C2> j 9R>:C> j >n R>:C> j >0 R:C
It has been shown in [
For each of the four locality notions, modules of O are obtained by starting with an empty set of axioms and subsequently adding axioms from O that are non-local. In order for this procedure to be correct, the signature against which
locality is checked has to be extended with the terms in the axioms that are
added in each step, so that the resulting module M consists of all the non-local
axioms with respect to [ Mf. Definition 4 (1) introduces locality-based
modules, which are always dCE-based modules [
Definition 4. Let x 2 f;; ; ?; >g, yz 2 f>?; signature. ;g, O an ontology and a (1) An ontology M is the x-module of O w.r.t. if it is the output of Algorithm 1. We write M = x-mod( ; O). (2) An ontology M is the yz-module of O w.r.t. , written M = yz-mod( ; O), if M = y-mod( ; z-mod( ; O)). (3) Let (Mi)i>0 be a sequence of ontologies such that M0 = O and Mi+1 = yz-mod( ; Mi) for every i > 0. For the smallest n > 0 with Mn = Mn+1, we call Mn the yz -module of O w.r.t. , written M = yz -mod( ; O). Algorithm 1 Extract a locality-based module
Input: Ont. O, sig. , x 2 f;; ; ?; >g
O M ;; O0 repeat changed for all if
false 2 O0 do not x-local w.r.t.
M M [ f g; O0 until changed = false return M [ Mf then
true
As for (1), it has been shown in [
Roughly speaking, a - or >-module for gives a view from above because it contains all subclasses of class names in , while a ;- or ?-module for gives a view from below since it contains all superconcepts of concept names in .
Modulo the locality check, Algorithm 1 runs in time cubic in jOj + j j [
but, as said above, the ;/ -locality check is more complex. A module extractor is implemented in the OWL API6 and SSWAP7. To summarize: 1. Given an ontology O, the semantic module Msem for a signature is contained in the corresponding syntactic module Msyn for the same seed signature.8 This means that in principle more unnecessary axioms for preserving entailments over can end up in syntactic modules rather than in semantic modules. 2. The extraction of a syntactic module can be done in polynomial time w.r.t. the size of the ontology O. In contrast, the extraction of a semantic module is as hard as reasoning. 3
The main aim of this paper is to investigate how well syntactic locality approximates semantic locality. In particular, we want to see how (un)likely it is that syntactic locality-based modules are larger than semantic locality-based ones and how large these differences are. We also want to understand empirically how much more costly semantic locality is in terms of performance.
Selection of the Corpus. For our experiments, we have built a corpus containing:
(1) from the TONES repository,9 those ontologies that have already been studied
in a previous work on modularity [
We then filter out all those the ontologies for which at least one of the following problems occurs: the ontology is impossible to download; the .owl file is corrupted when downloaded; the file is not parseable; the ontology is inconsistent. Furthermore, due to time constraints, we exclude from this preliminary investigation all ontologies whose size exceeds 10; 000 axioms.
This selection results in a corpus of 156 ontologies, which greatly differ in
size and expressivity [
Range of expressivity ALCN -SHIN (D)=SOIN (D) AL-SROIF (D)=SHOIQ(D)
Range #axs. Range sig. size 38–4,735 21–3,161 13–9,629 14–9,221 6 http://owlapi.sourceforge.net 7 http://sswap.info 8 Recall that ?-syntactic modules approximate ;-semantic modules, while >-syntactic modules approximate -semantic modules. 9 http://owl.cs.manchester.ac.uk/repository/ 10 http://bioportal.bioontology.org Comparing Syntactic and Semantic Locality. In order to compare syntactic and semantic locality, we want to understand: 1. whether, for a given seed signature , the semantic -module is likely to be smaller than the syntactic -module, and if so by how much,11 2. how feasible the extraction of semantic modules is.
Here, we focus on the two corresponding notions of ;-semantic locality and
?-syntactic locality. In particular, ?-syntactic locality has been throughly
investigated in previous work [
Due to the recursive nature of the locality-based module extraction algorithm, we want to investigate locality both on a – per-axiom basis: given an axiom and a signature , is it likely that is semantically ;-local w.r.t. but not syntactically ?- local w.r.t. ? – per-module basis: given a signature , is it likely that ?-mod( ; O) 6= ;-mod( ; O)? If yes, is it likely that the difference is large?
Hence we need to pick, for each ontology in our corpus, a suitable set of signatures, and this poses a significant problem. First, we do not yet have enough insight into what typical seed signatures are for module extraction. One could assume that large ones are rarely relevant for module extraction—why bother with extracting a large module—but this still leaves a large, i.e., exponential space of possible seed signatures. If m = #Oe, there are 2m possible seed signatures for which axioms can be tested for locality and for which modules can be extracted. Hence a full investigation is infeasible.
One could assume that the comparison between semantic and syntactic
modules could be easier since many signatures can lead to the same module. In other
words, the statistically significant number of modules w.r.t. the total number
of modules is not larger than that of seed signatures needed w.r.t. the total
number of seed signatures. In previous work [
As a consequence, we compare the two kinds of locality of axioms—both
on a per-axiom basis and a per-module basis—w.r.t. random signatures. To
avoid any bias, we select a random signature as follows: we set each named
entity E in the ontology to have probability p = 1=2 of being included in the
signature. Thus each seed signature has the same probability to be chosen. For
ontologies whose signature exceeds 9 entities, in order to get results where the
11 Recall that the semantic -module is always a subset of the syntactic -module.
true proportion of differences between the two notions of locality lies in the
confidence interval ( 5%) with confidence level 95%, we have to select only 400
random signatures [
Non-random seed signatures. A module, in general, does not necessarily show any
internal coherence: intuitively, if we had an ontology describing some knowledge
from both the domains of Geology and of Philosophy, we could still extract the
module for the signature = fEpistemology; Mineralg. This module is likely
to be the union of the two disjoint modules for 1 = fEpistemologyg and
2 = fMineralg. This combinatorial behaviour can lead to exponentially many
modules in the size of the signature of the ontology and indeed, as mentioned
above, the number of modules in ontologies seems to be exponential [
In contrast to general modules, genuine modules can be called coherent: they are defined as those modules that cannot be decomposed into the union of two different modules. Notably, there are only linearly many genuine modules in the size of the ontology O, and the set of genuine modules is a base for all general modules: any module is either genuine or the union of genuine modules. The linear bound on the number of genuine modules is due to the fact that, for each genuine x-module M, there is an axiom such that M = x-mod( ~; O).
Thus genuine modules can be said to be interesting modules that we can fully investigate. Hence in addition to the above mentioned investigation of ?and ;-modules for random signatures, we also look at all axiom signatures.
In summary, we test: (T1) for random seed signatures , (a) for each axiom in our corpus, is semantically ;-local w.r.t. but not syntactically ?- local w.r.t. ? (b) is ?-mod( ; O) 6= ;-mod( ; O)? If yes, we determine the difference and its size. (T2) for each axiom signature from our corpus, is ?-mod( ~; O) 6= ;-mod( ~; O)?
If yes, we determine the difference and its size. 4
No differences. The main result of the experiment is that, for 151 of the 156 ontologies we tested, no difference between ?- and ;-locality can be observed. These 151 ontologies exclude the two NCBO BioPortal ontologies EFO (Experimental Factor Ontology) and SWO (Software Ontology), as well as Koala, miniTambis, and Tambis. More specifically, for every generated seed signature, the corresponding ?- and ;-module agree, and every axiom is either ?- and ;-local, or neither. This statement applies to all randomly generated seed signatures as well as for all axiom signatures – which are seed signatures for all genuine modules. We can therefore draw the following conclusions for the 151 ontologies with respect to (T1) and (T2) above. (T1) Given an arbitrary seed signature , there is no difference (a) between ?- and ;-locality of any given axiom w.r.t. and (b) between the ?- and ;-modules for , both times at a significance level of 0:05. (T2) Given any axiom signature , there is no difference between the ?- and ;-modules for .
In the case of the 151 ontologies, the extraction of a ;-module (with tautology
tests performed by FaCT++) often took considerably longer than the extraction
of the corresponding ?-module. For example, for MoleculeRole, the largest of
the 151 ontologies, times to extract a ?-module (test all axioms for ?-locality,
respectively) ranged between 27 and 169ms (21 and 77ms, respectively), while
the extraction of a ;-module (test of all axioms for ;-locality, resp.) took up
to 6 as long, on average 2.7 (2.0 , resp.). It is also worth noting that the
ontologies Galen and People, which are renowned for having particularly large
?-modules [
Type-1 axioms occur in the “inferred view” of the ontologies EFO and SWO. Table 2 shows the relatively modest differences caused by these axioms. In all cases, there are no other axioms in the differences. This means that no differences occur for the non-inferred original versions of EFO and SWO.
Type-2 culprits are complex definitions A C of a concept name A where C is a disjunction that contains both a universal and an existential (or minimum cardinality) restriction on the same role. This affects the ontologies Koala, miniTambis, and Tambis. The effect is best illustrated for Koala, which contains exactly one such axiom, namely M S u 8c:F u 8g:fmg u =3 c:>; where we have abbreviated the concept names MaleStudentWith3Daughters, Student, Female, the roles hasChildren, hasGender, and the nominal male. Now if the signature against which the axiom is tested for locality contains fS; c; gg but Ontology
#axs SWO EFO Koala miniTambis Tambis time culprit ratio type and avg. frequency neither M nor F, then this axiom is not ?-local because none of the conjuncts on the right-hand side is in Bot( ). On the other hand, this axiom is a tautology when M and F are replaced by ?: the conjunction 8c:? u =3 c:> cannot have any instances, regardless of how c is interpreted.
For Koala, this effect only causes two singleton differences between sets of local axioms for the randomly generated seed signatures, as shown in Table 2. For axiom signatures, there is no difference. Interestingly, this effect does not propagate to modules: for all signatures, ?- and ;-modules are the same. The reason might be that (a) g is used in many axioms and is thus very likely to contribute to the extended signature during module extraction, and (b) then the axiom defining F is no longer local, which “pulls” F into the extended signature, preventing the observed effect.
In miniTambis and Tambis, this effect is much stronger and affects a large proportion of modules, as shown in Table 2. The differences in these cases do not only consist of culprit axioms, but also of axioms that become non-local after the signature has been extended by the terms in the culprit axioms. Still, the size of the differences is mostly modest while, for Tambis, the ;-locality test (;-module extraction) takes on average over three times (five times) as long as the ?-locality test (?-module extraction).
Summary. We obtain two main observations from the experiments carried out. – In practice, there is no or little difference between semantic and syntactic locality. That is, the computationally cheaper syntactic locality is a good approximation of semantic locality. – Though in principle hard to compute, semantic modules can be extracted rather fast in practice.
These results suggest that it is questionable to conclude that semantic locality
should be preferred to syntactic locality. In terms of computation time, there is
often a benefit in using syntactic locality: the average speed-up compared to the
extraction of a semantic-locality based module is by a factor of up to 6. For
some particular module pairs, it is higher by an order of magnitude. The gain
in module size is zero or so small that it is hard to justify the extra time spent.
In particular, there is no gain in size for the ontologies Galen and People, which
are “renowned” for having disproportionately large modules [
Our results are interesting not only because they provide an evaluation of how good the cheap syntactic locality approximates semantic locality, but also because they enabled us to fix bugs in the implementation of syntactic modularity. For example, earlier data from the experiment have shown that reflexivity axioms had been treated incorrectly by the syntactic locality checker. Future Work. It is evident that this work is preliminary. It investigates only the differences between the related notions of ?- and ;-locality. We plan to extend the same study to other notions of locality, in particular, nested modules (>? - vs. ; -modules) – these notions are the most economical in terms of module size. Moreover, we want to extend the investigation to the remaining larger ontologies in the BioPortal repository and further large ontologies, e.g., some versions of the NCI Thesaurus12. Preliminary results with a version that is not among the regular releases show differences due to type-2 culprits, but we have not included them here because the differences disappear after removing axioms that were introduced due a problem with object and annotation properties when the ontology file is parsed by the OWL API. This behaviour is yet to be investigated and explained.
Another interesting extension is to modify the seed signature sampling. Currently, the random variable “size of the seed signature generated” follows the binomial distribution with expected value m=2 and variance m=4. Hence, most signatures in the sample have size around m=2; small and large signatures are underrepresented. For example, for one ontology with 915 terms, all signature sizes lay between 422 and 509. One might argue that, for big ontologies, the typical module extraction scenario does not require large seed signatures – but it does sometimes require relatively small seed signatures, for example, when a module is extracted to efficiently answer a given entailment query of typically small size. 12 Downloadable from http://evs.nci.nih.gov/ftp1/NCI_Thesaurus On the other hand, large modules resulting from larger seed signatures may be more likely to differ. We therefore plan an alternative seed signature sampling via bins for average signature sizes: repeat the current sampling procedure scaled to several subintervals of the range of possible signature sizes.
Our current results answer the question whether there is a significant difference between the two locality notions with respect to a given signature. It is also interesting to ask the same question relative to a given module. To answer it, the sampling of modules instead of seed signatures requires further investigation. Acknowledgment. We thank Rafael Gonçalves and the anonymous reviewers for helpful comments.