# Orchestrating 28 logical theories of mereo(topo)logy

Parts and wholes, again. This time it’s about the logic-aspects of theories of parthood (cf. aligning different hierarchies of (part-whole) relations and make them compatible with foundational ontologies). I intended to write this post before the Ninth Conference on Knowledge Capture (K-CAP 2017), where the paper describing the new material would be presented by my co-author, Oliver Kutz. Now, afterwards, I can add that “Orchestrating a Network of Mereo(topo) logical Theories” [1] even won the Best Paper Award. The novelties, in broad strokes, are that we figured out and structured some hitherto messy and confusing state of affairs, showed that one can do more than generally assumed especially with a new logics orchestration framework, and we proposed first steps toward conflict resolution to sort out expressivity and logic limitations trade-offs. Constructing a tweet-size “tl;dr” version of the contents is not easy, and as I have as much space here on my blog as I like, it ended up to be three paragraphs here: scene-setting, solution, and a few examples to illustrate some of it.

Problems

As ontologists know, parthood is used widely in ontologies across most subject domains, such as biomedicine, geographic information systems, architecture, and so on. Ontology (the philosophers) offer a parthood relation that has a bunch of computationally unpleasant properties that are structured in a plethora of mereologicial and meretopological theories such that it has become hard to see the forest for the trees. This is then complicated in practice because there are multiple logics of varying expressivity (support more or less language features), with the result that only certain fragments of the mereo(topo)logical theories can be represented. However, it’s mostly not clear what can be used when, during the ontology authoring stage one may want to have all those features so as to check correctness, and it’s not easy to predict what will happen when one aligns ontologies with different fragments of mereo(topo)logy.

Solution

We solved these problems by specifying a structured network of theories formulated in multiple logics that are glued together by the various linking constructs of the Distributed Ontology, Model, and Specification Language (DOL). The ‘structured network of theories’-part concerns all the maximal expressible fragments of the KGEMT mereotopological theory and five of its most well-recognised sub-theories (like GEM and MT) in the seven Description Logics-based OWL species, first-order logic, and higher order logic. The ‘glued together’-part refers to relating the resultant 28 theories within DOL (in Ontohub), which is a non-trivial (understatement, unfortunately) metalanguage that has the constructors for the glue, such as enabling one to declare to merge two theories/modules represented in different logics, extending a theory (ontology) with axioms that go beyond that language without messing up the original (expressivity-restricted) ontology, and more. Further, because the annoying thing of merging two ontologies/modules can be that the merged ontology may be in a different language than the two original ones, which is very hard to predict, we have a cute proof-of-concept tool so that it assists with steps toward resolution of language feature conflicts by pinpointing profile violations.

Examples

The paper describes nine mechanisms with DOL and the mereotopological theories. Here I’ll start with a simple one: we have Minimal Topology (MT) partially represented in OWL 2 EL/QL in “theory8” where the connection relation (C) is just reflexive (among other axioms; see table in the paper for details). Now what if we add connection’s symmetry, which results in “theory4”? First, we do this by not harming theory8, in DOL syntax (see also the ESSLI’16 tutorial):

logic OWL2.QL
ontology theory4 =
theory8
then
ObjectProperty: C Characteristics: Symmetric %(t7)

What is the logic of theory4? Still in OWL, and if so, which species? The Owl classifier shows the result:

Another case is that OWL does not let one define an object property; at best, one can add domain and range axioms and the occasional ‘characteristic’ (like aforementioned symmetry), for allowing arbitrary full definitions pushes it out of the decidable fragment. One can add them, though, in a system that can handle first order logic, such as the Heterogeneous toolset (Hets); for instance, where in OWL one can add only “overlap” as a primitive relation (vocabulary element without definition), we can take such a theory and declare that definition:

logic CASL.FOL
ontology theory20 =
theory6_plus_antisym_and_WS
then %wdef
. forall x,y:Thing . O(x,y) <=> exists z:Thing (P(z,x) /\ P(z,y)) %(t21)
. forall x,y:Thing . EQ(x,y) <=> P(x,y) /\ P(y,x) %(t22)

As last example, let me illustrate the notion of the conflict resolution. Consider theory19—ground mereology, partially—that is within OWL 2 EL expressivity and theory18—also ground mereology, partially—that is within OWL 2 DL expressivity. So, they can’t be the same; the difference is that theory18 has parthood reflexive and transitive and proper parthood asymmetric and irreflexive, whereas theory19 has both parthood and proper parthood transitive. What happens if one aligns the ontologies that contain these theories, say, O1 (with theory18) and O2 (with theory19)? The Owl classifier provides easy pinpointing and tells you the profile: OWL 2 full (or: first order logic, or: beyond OWL 2 DL—top row) and why (bottom section):

Now, what can one do? The conflict resolution cannot be fully automated, because it depends on what the modeller wants or needs, but there’s enough data generated already and there are known trade-offs so that it is possible to describe the consequences:

• Choose the O1 axioms (with irreflexivity and asymmetry on proper part of), which will make the ontology interoperable with other ontologies in OWL 2 DL, FOL or HOL.
• Choose O2’s axioms (with transitivity on part of and proper part of), which will facilitate linking to ontologies in OWL 2 RL, 2 EL, 2 DL, FOL, and HOL.
• Choose to keep both sets will result in an OWL 2 Full ontology that is undecidable, and it is then compatible only with FOL and HOL ontologies.

As serious final note: there’s still fun to be had on the logic side of things with countermodels and sub-networks and such, and with refining the conflict resolution to assist ontology engineers better. (or: TBC)

As less serious final note: the working title of early drafts of the paper was “DOLifying mereo(topo)logy”, but at some point we chickened out and let go of that frivolity.

References

[1] Keet, C.M., Kutz, O. Orchestrating a Network of Mereo(topo)logical Theories. Ninth International Conference on Knowledge Capture (K-CAP’17), Austin, Texas, USA, December 4-6, 2017. ACM Proceedings.

# New OWL files for the (extended) taxonomy of part-whole relations

Once upon a time (surely >6 years ago) I made an OWL file of the taxonomy of part-whole relations [1], which contains several parthood relations and a few meronyic-only ones that in natural language are considered ‘part’ but are not so according to mereology (like participation, membership). Some of these relations were defined with a specific domain and range that was a DOLCE category (it could just as well have been, say, GFO). Looking at it recently, I noticed it was actually a bit scruffy (but I’ll leave it here nonetheless), and more has happened in this area over the years. So, it was time for an update on contents and on design.

For the record on how it’s done and to serve, perhaps, as a comparison exercise on modeling, here’s what I did. First of all, I started over, so as to properly type the relations to DOLCE categories, with the DOLCE IRIs rather than duplicated as DOLCE-category-with-my-IRI. As DOLCE is way too big and slows down reasoning, I made a module of DOLCE, called DOLCEmini, mainly by removing the irrelevant object properties, though re-adding the SOB, APO and NAPO that’s in D18 but not in DOLCE-lite from DLP3791. This reduced the file from DOLCE-lite’s 534 axioms, 37 classes, 70 OPs, in SHI to DOLCEmini’s 388 axioms, 40 classes, 43 OPs, also in SHI, and I changed the ontology IRI to where DOLCEmini will be put online.

Then I created a new ontology, PW.owl, imported DOLCEmini, added the taxonomy of part-whole relations from [1] right under owl:topObjectProperty, with domain and range axioms using the DOLCE categories as in the definitions, under part-whole. This was then extended with the respective inverses under whole-part, all the relevant proper part versions of them (with inverses), transitivity added for all (as the reasoner isn’t doing it [2]) annotations added, and then aligned to some DOLCE properties with equivalences. This makes it to 524 axioms and 79 object properties.

I deprecated subquantityOf (annotated with ‘deprecated’ and subsumed by a new property ‘deprecated’). Several new stuff relations and their inverses were added (such as portions), and annotated them. This made it to the PW ontology of 574 axioms (356 logical axioms) and 92 object properties (effectively, for part-whole relations: 92 – 40 from dolce – 3 for deprecated = 49).

As we made an extension with mereotopology [3] (and also that file wasn’t great, though did the job nevertheless [4]), but one that not everybody may want to put up with, yet a new file was created, PWMT. PWMT imports PW (and thus also DOLCEmini) and was extended with the main mereotopological relations from [3], and relevant annotations were added. I skipped property disjointness axioms, because they don’t go well with transitivity, which I assumed to be more important. This makes PWMT into one of 605 (380 logical) axioms and 103 object properties, with, effectively, for parts: 103 – 40 from dolce – 3 for deprecated – 1 connection = 59 object properties.

That’s a lot of part-whole relations, but fear not. The ‘Foundational Ontology and Reasoner enhanced axiomatiZAtion’ (FORZA) and its tool that incorporates with the Guided ENtity reuse and class Expression geneRATOR (GENERATOR) method [4] describes a usable approach how that can work out well and has a tool for the earlier version of the owl file. FORZA uses an optional decision diagram for the DOLCE categories as well as the automated reasoner so that it can select and propose to you those relations that, if used in an axiom, is guaranteed not to lead to an inconsistency that would be due to the object property hierarchy or its domain and range axioms. (I’ll write more about it in the next post.)

Ah well, even if the OWL files are not used, it was still a useful exercise in design, and at least I’ll have a sample case for next year’s ontology engineering course on ‘before’ and ‘after’ about questionable implementation and (relatively) good implementation without the need to resorting to criticizing other owl files… (hey, even the good and widely used ontologies have a bunch of pitfalls, whose amount is not statistically significantly different from ontologies made by novices [5]).

References

[1] Keet, C.M., Artale, A. Representing and Reasoning over a Taxonomy of Part-Whole Relations. Applied Ontology, 2008, 3(1-2):91-110.

[2] Keet, C.M. Detecting and Revising Flaws in OWL Object Property Expressions. 18th International Conference on Knowledge Engineering and Knowledge Management (EKAW’12), Oct 8-12, Galway, Ireland. Springer, LNAI 7603, 252-266.

[3] Keet, C.M., Fernandez-Reyes, F.C., Morales-Gonzalez, A. Representing mereotopological relations in OWL ontologies with OntoPartS. 9th Extended Semantic Web Conference (ESWC’12), Simperl et al. (eds.), 27-31 May 2012, Heraklion, Crete, Greece. Springer, LNCS 7295, 240-254.

[4] Keet, C.M., Khan, M.T., Ghidini, C. Ontology Authoring with FORZA. 22nd ACM International Conference on Information and Knowledge Management (CIKM’13). ACM proceedings, pp569-578. Oct. 27 – Nov. 1, 2013, San Francisco, USA.

[5] Keet, C.M., Suarez-Figueroa, M.C., Poveda-Villalon, M. Pitfalls in Ontologies and TIPS to Prevent Them. In: Knowledge Discovery, Knowledge Engineering and Knowledge Management: IC3K 2013 Selected papers. Fred, A., Dietz, J.L.G., Liu, K., Filipe, J. (Eds.). Springer, CCIS 454, pp. 115-131. 2015.

# Part-whole relations, mereotopology and the OntoPartS tool

Part-whole relations are considered essential in knowledge representation and reasoning and, more practically, in ontology development and conceptual data modelling, especially in the subject domains of biology, medicine, geographic information systems, and manufacturing. In contrast to Ontology that sticks to one type of part-of, the modellers and subject domain experts have come up with a plethora of part-whole relations, some of which are considered real parthood relations and others only meronymic (or: due to imprecise natural language use). For instance, the Foundational Model of Anatomy has 8 basic locative part-whole relations [1], GALEN has come up with 26 part-whole relations [2], and in cognitive science and conceptual data modelling, it hovers around about 6 types [3,4]. They have been structured in a taxonomy of part-whole relations that makes a distinction between mereology and meronomy, transitivity and in- or non-transitivity, and the domain and range of the relationship [5], and some initial usage guidelines were proposed in [6].

But that’s not enough for the complex subject domains and demands on the representation and reasoning over the ontologies. This holds in particular when one has to represent that some things are contained in or located in something else. For instance, the way how Paris and France relate is somehow different from how the euro coin in your wallet relate to each other—the latter being an example of  (spatial) containment, but not structural part of—whereas in other case, the spatial containment of regions of space and the structural parthood of the objects occupying those regions do coincide, e.g., your heart in your body. Or consider representing that Alto Adige/Südtirol is a border province of Italy (bordering Austria), where we have to handle both the notion of administrative entities and connecting geographical regions. That is, handling regions and ‘things’ that occupy those regions (mereotopology).

Being more precise about how the things relate provides nice inferences. Take, e.g., NTPLI as ‘non-tangential proper located in’—a part is located in the whole but not at the boundary of it—and $EnclosedCountry \equiv Country \sqcap \exists NTPLI.Country$, with the following instances in our knowledge base $NTPLI(Lesotho, South Africa)$, $Country(Lesotho)$, and $Country(South Africa)$, then it deduces correctly that $EnclosedCountry(Lesotho)$, whereas with a mere ‘part-of’, we would not have been able to obtain this result.

Besides these examples, there are actual system requirements for, among others, annotating and querying multimedia documents and cartographic maps, such as annotating a photo of a beach where the area of the photo that depicts the sand touches the area that depicts the seawater so that, together with the knowledge that Varadero is a tangential proper part of Cuba, the semantically enhanced system can infer possible locations where the photo has been taken, or, vv., it can propose that the photo may depict a beach scene.

But how to cater for such things?

Let me summarise the three main basic problems that have to be resolved first:

1. There is lack of oversight on plethora of part-whole relations, that include real parthood (mereology) parts with their locations (mereotopology), and other part-whole relations (from meronymy);
2. The challenge to figure out which one to use when;
3. The underspecified representation and reasoning consequences when one has to put up with less expressive languages for which technological infrastructure exists.

We propose to solve that in the following way, which is described in detail in [7] that recently got accepted at the 9th Extended Semantic Web Conference (ESWC’12).

The short answer for the reader who is not interested in all the theory, design, and evaluation, but just wants to model quickly: the OntoPartS tool guides you to choose the most appropriate relation and saves the selection into your OWL file.

Now for a slightly longer answer. First, we extend the taxonomy of part-whole relations of [5] with the novel addition of a taxonomy of formally defined mereotopological relations, which is driven by the KGEMT mereotoplogical theory of Varzi [8], resulting in a taxonomy of 23 part-whole relations—mereological, mereotopological, and meronymic ones—therewith ensuring a solid ontological and logic-based foundation.

Second, some things have to be simplified from the KGEMT theory to make it implementable in OWL, and we describe the design rationale and trade-offs so that OntoPartS can load OWL/OWL2-formalised ontologies, and, if desired, modify the OWL file with the chosen relation. Which OWL species is best suited obviously depends on your individual requirements, but from a representation & reasoning and mereotopology viewpoint, OWL 2 DL and OWL 2 RL seem to fit better than the other ones. (Note: there are papers on DL and representing spatial relations and on DL and parthood, and alternative representation choices are discussed in the paper, yet, as far as we are aware of, none deals with mereotopological relations in OWL or, more generally, in DL.)

Third, there is the ‘how to select’ from the 23 relations. To enable a quick selection of the appropriate relation, we avail of a simplified OWL-ized DOLCE ontology—well, just the taxonomy of categories—for the domain and range restrictions imposed on the part-whole relations and with that, we can let the user take shortcuts compared to a lengthy decision procedure. In this way, we reduced the selection procedure to 0-4 options based on just 2-3 inputs. All of this has been structured neatly in implementation-independent activity diagrams, and subsequently has been implemented; see also the demos, the tool, and the OWL version of the taxonomy of the 23 relations.

Last, we have tested OntoPartS with modellers in controlled experiments and it was shown to improve efficiency and accuracy in modeling of part-whole relations.

As mentioned, further details can be found in [7], Representing mereotopological relations in OWL ontologies with OntoPartS, which I co-authored with Francis Fernández-Reyes, with the Instituto Superior Politécnico “José Antonio Echeverría” (CUJAE), and Annette Morales-González, with the Advanced Technologies Application Center (CENATAV), both located in Cuba (the example on semantic annotation of multimedia with spatial relations comes straight from the image processing research being done at CENATAV). A tidbit of non-scientific information: the first version of the OntoPartS tool was developed as part of the mini-project that Francis, Annette (and Alexis, who is into fish fulltime now) had chosen to carry out for the ontology engineering course I taught at the University of Havana in 2010 (mentioned earlier here and here). For the paper, we added some more theory, minor refinements to the tool, and a user evaluation with several CUJAE and UKZN students and a few FUB colleagues (thanks again for their cooperation and interest). We’ve started work on additional features, so if you have any particular request, drop me a line.

References

1. Mejino, J.L.V., Agoncillo, A.V., Rickard, K.L., Rosse, C.: Representing complexity in part-whole relationships within the foundational model of anatomy. In: Proc. of the AMIA Fall Symposium. pp. 450–454 (2003)
2. http://www.opengalen.org/tutorials/crm/tutorial9.html up to http://www.opengalen.org/tutorials/crm/tutorial16.html/.
3. Winston, M., Chaffin, R., Herrmann, D.: A taxonomy of part-whole relations. Cognitive Science 11(4), 417–444 (1987)
4. Odell, J.: Advanced Object-Oriented Analysis & Design using UML. Cambridge: Cambridge University Press (1998)
5. Keet, C.M., Artale, A.: Representing and reasoning over a taxonomy of part-whole relations. Applied Ontology 3(1-2), 91–110 (2008)
6. Keet, C.M.: Part-whole relations in object-role models. In: Proc. of ORM’06, OTM Workshops 2006. LNCS, vol. 4278, pp. 1116–1127. Springer (2006)
7. Keet, C.M., Fernández Reyes, F.C., Morales-González, A.: Representing mereotopological relations in OWL ontologies with OntoPartS. In Simperl, et al., eds.: Proc. of ESWC’12. LNCS, Springer (2012) 27-31 May 2012, Heraklion, Greece.
8. Varzi, A.: Handbook of Spatial Logics, chap. Spatial reasoning and ontology: parts, wholes, and locations, pp. 945–1038. Berlin Heidelberg: Springer Verlag (2007)