7.3: Part-Whole Relations
- Page ID
- 6437
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A, if not the, essential relation in Ontology and ontologies is the part-whole relation, which is deemed as essential as subsumption by the most active adopters of ontologies—i.e., bio- and medical scientists—while its full potential is yet to be discovered by, among others, manufacturing to manage components of devices. Let’s start with a few modelling questions to get an idea of the direction we are heading at:
- Is City a subclass of or a part of Province?
- Is a tunnel part of the mountain? If so, is it a ‘part’ in the same way as the sand of your sandcastle on the beach?
- What is the difference, if any, between how Cell nucleus and Cell are related and how Cell Receptor and Cell wall are related? Or between the circuit on the ethernet card embedded on the motherboard and the motherboard in the computer?
- Assuming boxers must have their own hands and boxers are humans, is Hand part of Boxer in the same way as Brain is part of Human?
- Consider that “Hand is part of Musician” and “Musician part of Orchestra”. Clearly, the musician’s hands are not part of the orchestra. Is part-of then not transitive, or is there a problem with the example?
To shed light on part-whole relations in its broadest sense and sort out such modeling problems, we will look first at mereology, which is the Ontology take on part-whole relations, and to a lesser extent meronymy, which is more popular in linguistics. Subsequently, the different terms that are perceived to have something to do with part-whole relations are structured into a taxonomy of part-whole relations, based on [KA08], which has been adopted elsewhere, such as in NLP.
Mereology
The most ‘simple’ mereological theory is commonly considered to be Ground Mereology. We take the one where parthood is primitive^{12}, i.e., part-of
is not defined but only characterized with some properties. In particular, the three characterizing properties are that parthood is reflexive (everything is part of itself, Eq. 6.2.1), antisymmetric (two distinct things cannot be part of each other, or: if they are, then they are the same thing, Eq. 6.2.2), and transitive (if \(x\) is part of \(y\) and \(y\) is part of \(z\), then \(x\) is part of \(z\), Eq. 6.2.3):
\[\forall \texttt{x(part_of(x, x))}\]
\[\forall \texttt{x, y((part_of(x, y)}\wedge\texttt{part_of(y, x))}\to\texttt{x = y)}\]
\[\forall\texttt{x, y, z((part_of(x, y)}\wedge\texttt{part_of(y, z))}\to\texttt{part_of(x, z))}\]
With parthood, on can define proper parthood:
\[\forall\texttt{x, y(proper_part_of(x, y)}\equiv\texttt{part_of(x, y)}\wedge\neg\texttt{part_of(y, x))}\]
and its characteristics are that it is transitive (Eq. 6.2.5), asymmetric (if \(x\) is part of \(y\) then \(y\) is not part of \(x\), Eq. 6.2.6) and irreflexive (\(x\) is not part of itself, Eq. 6.2.7). Irreflexivity follows from the definition of proper parthood and then, together with antisymmetry, one can prove asymmetry of proper parthood (proofs omitted).
\[\forall\texttt{x, y, z((proper_part_of(x, y)}\wedge\texttt{proper_part_of(y, z))}\to\texttt{proper_part_of(x, z))}\]
\[\forall\texttt{x, y(proper_part_of(x, y)}\to\neg\texttt{proper_part_of(y, x))}\]
\[\forall\texttt{x}\neg\texttt{(proper_part_of(x, x))}\]
These basic axioms already enables us to define several other common relations. Notably, overlap (\(x\) and \(y\) share a piece \(z\)):
\[\forall\texttt{x, y(overlap(x, y)}\equiv\exists\texttt{z(part_of(z, x)}\wedge\texttt{part_of(z, y)))}\]
and underlap (\(x\) and \(y\) are both part of some \(z\)):
\[\forall\texttt{x, y(underlap(x, y)}\equiv\exists\texttt{z(part_of(x, z)}\wedge\texttt{part_of(y, z)))}\]
The respective definitions of proper overlap & proper underlap are similar.
But there are ‘gaps’ in Ground Mereology, some would say; put differently: there’s more to parthood than this. For instance: what to do—if anything—with the ‘remainder’ that makes up the whole? There are two options:
- Weak supplementation: every proper part must be supplemented by another, disjoint, part, resulting in Minimal Mereology (MM).
- Strong supplementation: if an object fails to include another among its parts, then there must be a remainder, resulting in Extensional Mereology (EM).
There is a problem with EM, however: non-atomic objects with the same proper parts are identical (extensionality principle), but sameness of parts may not be sufficient for identity. For instance, two objects can be distinct purely based on arrangement of its parts, like there is a difference between statue and its marble and between several flowers bound together and a bouquet of flowers. This is addressed in General Extensional Mereology (GEM); see also Figure 6.2.1.
One can wonder about parts some more: does it go on infinitely down to even smaller than the smallest, or must it stop at some point? If one is convinced it stops with a smallest part, this means a ‘basic element’ exists, which is called Atom in mereology. The alternative—going on infinitely down into parts of parts—is that at the very basis there is so-called atomless ‘gunk’. These different commitments generate additional mereological theories. If that is not enough for extensions: one could, e.g., temporalize each mereological theory, so that one can assert that something used to be part of something else; this solves the boxer, hand, and brain example mentioned in the introduction (we’ll look at the solution in Section 10.2: Time and Temporal Ontologies "Temporal DLs"). Another option is to also consider space or topology, which should solve the tunnel/mountain question, above; see also, e.g., [Var07]. These extensions do not yet solve the cell and the musician questions. This will be addressed in the next section.
Figure 6.2.1: Hasse diagram of mereological theories; from weaker to stronger, going uphill (after [Var04]). Atomicity can be added to each one.
Modeling and Reasoning in the Context of Ontologies
Mereology is not enough for ontology engineering. This is partially due to the ‘spillover’ from conceptual data modeling and cognitive science, where a whole range of relations are sometimes referred to as a parthood relation, but which are not upon closer inspection. In addition, if one has only part-of
in one’s ontology with no domain or range axiom, the reasoner will not complain when one adds, say, \(\texttt{Hand}\sqsubseteq\exists\texttt{part-of.Musician}\) and \(\texttt{Musician}\sqsubseteq\exists\texttt{part-of.Performance}\), even though ontologically this is not quite right. A philosopher might say “yeah, well, then don’t do this!”, but it would be more useful for an ontology developer to have relations at one’s disposal that are more precise, both for avoiding modeling mistakes and for increasing precision to obtain a better quality ontology.
This issue has been investigated by relatively many researchers. We shall take a closer look at a taxonomy of part-whole relations [KA08] that combines, extends, and formalizes them. The basic version of the informal graphical rendering is depicted in Figure 6.2.2.
Figure 6.2.2: Taxonomy of basic mereological (left-hand branch) and meronymic (righthand branch) part-whole relations, with an informal summary of how the relations are constrained by their domain and range; s-parthood = structural parthood. (Source: based on [KA08])
The relations have been formalized in [KA08]. It uses DOLCE in order to be precise in the domain and range axioms; one could have taken another foundational ontology, but at the time it was a reasonable choice (for an assessment of alternatives, see [Kee17a]). The more precise characterizations (cf. the figure) and some illustrative examples are as follows.
- involvement for processes and sub-processes; e.g.
Chewing
(a pedurant, PD) is involved in the grander process ofEating
(also a perdurant), or vv.:
\[\forall\texttt{x,y(involved_in (x,y)}\equiv\texttt{part_of(x,y)}\wedge\texttt{PD(x)}\wedge\texttt{PD(y))}\]
- containment and location for object and its 2D or 3D region; e.g.,
contained_in(John’s address book, John’s bag)
andlocated_in(Tshwane, South Africa)
. They are formalized as Eqs. 6.2.11 and 6.2.12, respectively, where has_2D and has_3D are shorthand relations standing for DOLCE’s qualities and qualia:
\[\forall\texttt{x, y(contained_in(x, y)}\equiv\texttt{part_of(x, y)}\wedge\texttt{R(x)}\wedge\texttt{R(y)}\wedge\exists\texttt{z, w(has_3D(z, x)}\wedge\texttt{has_3D(w, y)}\wedge\texttt{ED(z)}\wedge\texttt{ED(w)))}\]
\[\forall\texttt{x, y(located_in(x, y)}\equiv\texttt{part_of(x, y)}\wedge\texttt{R(x)}\wedge\texttt{R(y)}\wedge\exists\texttt{z, w(has_2D(z, x)}\wedge\texttt{has_2D(w, y)}\wedge\texttt{ED(z)}\wedge\texttt{ED(w)))}\]
Observe that the domain and range is Region (R), which has an object occupying it, i.e., this does not imply that those objects are related also by structural parthood. Also, the 2D vs 3D distinction is not strictly necessary, but prior research showed that modelers like to make that difference explicit.
- structural parthood between endurants (ED) specifically:
\[\forall\texttt{x, y(s_part_of(x, y)}\equiv\texttt{part_of(x, y)}\wedge\texttt{ED(x)}\wedge\texttt{ED(y))}\]
Practically, this is probably better constrained by PED, physical endurant, such as a wall being a structural part of a house.
- stuff part or “quantity-mass”, e.g.,
Salt
as a stuff part ofSeaWater
relating different types of amounts of matter (M) or stuffs, which are typically indicated with mass nouns and cannot be counted other than in quantities. A partial formalization is as follows (there is a more elaborate one [Kee16]):
\[\forall\texttt{x, y(stuff_part(x, y)}\equiv\texttt{part_of(x, y)}\wedge\texttt{M(x)}\wedge\texttt{M(y))}\]
- portion, elsewhere also called “portion-object”, relating a smaller (or sub) part of an amount of matter to the whole, where both are of the same type of stuff; e.g., the wine in the glass of wine & wine in the bottle of wine. A partial formalization is as follows (there is a more elaborate one [Kee16]):
\[\forall\texttt{x, y(portion_of(x, y)}\equiv\texttt{part_of(x, y)}\wedge\texttt{M(x)}\wedge\texttt{M(y))}\]
- membership for so-called “member-bunch”: collective nouns (e.g.,
Herd
,Orchestra
) with their members (Sheep
,Musician
, respectively), where the subscript “n” denotes non-transitive and POB physical object and SOB social object:
\[\forall\texttt{x, y(member_of}_{\texttt{n}}\texttt{(x, y)}\equiv\texttt{mpart_of(x, y)}\wedge\texttt{(POB(x)}\vee\texttt{SOB(x))}\wedge\texttt{SOB(y))}\]
That is, sometimes transitivity might hold in a chain of memberships, but as soon as POB and SOB are mixed, that stops working, like with the hand in the example at the start of the section, for it is a POB.
- participation where an entity participates in a process (also called “nounfeature/ activity”), like
Enzyme
that participates inCatalyticReaction
or aMusician
participating in aPerformance
, where the subscript “it” denotes intransitive:
\[\forall\texttt{x, y(participates_in}_{\texttt{it}}\texttt{(x, y)}\equiv\texttt{mpart_of(x, y)}\wedge\texttt{ED(x)}\wedge\texttt{PD(y))}\]
From this definition, it becomes obvious why a ‘musician is part of a performance’ does not work: the domain and range are disjoint categories, so they never can line up in a transitivity chain.
- constitution or “material-object”, to relate that what something is made of to the object, such as the
Vase
and the (amount of)Clay
it is constituted of, where the subscript “it” denotes intransitive:
\[\forall\texttt{x, y(constitutes}_{\texttt{it}}\texttt{(x, y)}\equiv\texttt{constituted_of}_{\texttt{it}}\texttt{(y, x)}\equiv\texttt{mpart_of(x, y)}\wedge\texttt{POB(y)}\wedge\texttt{M(x))}\]
This can be put to use with manual or software-supported guidelines, such as OntoPartS [KFRMG12], to choose the most appropriate part-whole relation for the modeling problem at hand. Several OWL files with taxonomies of part-whole relations, including aligned to other foundational ontologies are also available^{13}.
Note that the mereological theories from philosophy are, as of yet, not feasible to implement in OWL: there is no DL that actually allows one to represent all of even the most basic mereological theory (Ground Mereology), as shown in Table 6.2.1, let alone add definitions for relations. This is possible within the DOL framework (recall Section 4.3: OWL in Context "The Distributed Ontology, Model, and Specification Language DOL"). More precisely with respect to the table’s languages beyond OWL: \(\mathcal{DLR}_{\mu}\) is a peculiar DL [CDGL99] and HOL stands for higher order logic (like, second order, beyond first order). Acyclicity means that an object \(x\) does not have a path to itself through one or more relations R on which acyclicity is declared. The reason why acyclicity is included in the table is because one actually can prove acyclicity with the axioms of proper parthood. It needs second order logic, though; formally, acyclicity is \(\forall x(\neg\varphi (x, x))\) where \(\varphi\) ranges over one or more relations (of proper parthood, in this case).
Language \(\Rightarrow\) Feature \(\Downarrow\) |
DL | Lite | 2DL | 2QL | 2RL | 2EL | \(\mathcal{DLR}\mu\) | FOL | HOL |
---|---|---|---|---|---|---|---|---|---|
Reflexivity^{\(P\)} | \(-\) | \(-\) | \(+\) | \(+\) | \(-\) | \(+\) | \(+\) | \(+\) | \(+\) |
Antisymmetry^{\(P\)} | \(-\) | \(-\) | \(-\) | \(-\) | \(-\) | \(-\) | \(-\) | \(+\) | \(+\) |
Transitivity^{\(P,P P\)} | \(+\) | \(+\) | \(+\) | \(-\) | \(+\) | \(+\) | \(+\) | \(+\) | \(+\) |
Asymmetry^{\(P P\)} | \(-\) | \(-\) | \(+\) | \(+\) | \(+\) | \(-\) | \(+\) | \(+\) | \(+\) |
Irreflexivity^{\(P P\)} | \(-\) | \(-\) | \(+\) | \(+\) | \(+\) | \(-\) | \(+\) | \(+\) | \(+\) |
Acyclicity | \(-\) | \(-\) | \(-\) | \(-\) | \(-\) | \(-\) | \(+\) | \(-\) | \(+\) |
Table 6.2.1: Properties of parthood (\(.^{P}\) ) and proper parthood (\(.^{P P}\) ) in Ground Mereology and their inclusion in the OWL family, FOL, \(\mathcal{DLR}_{\mu}\), and HOL.
Notwithstanding this, what sort of things can be derived with the part-whole relations, and what use may it have? The following example provides a few of the myriad of illustrations.
Example \(\PageIndex{1}\):
Informally, e.g., when it is possible to deduce which part of the device is broken, then only that part has to be replaced instead of the whole it is part of (saving a company money), and one may want to deduce that when a soccer player has injured her ankle, she has an injury in her limb, but not deduce that if she has an amputation of her toe, she also has an amputation of her foot that the toe is (well, was) part of. If a toddler swallowed a Lego brick, it is spatially contained in his stomach, but one does not deduce it is structurally part of his stomach (normally it will leave the body unchanged through the usual channel). A consequence of asserting reflexivity of parthood in the ontology is that then for a domain axiom like \(\texttt{Twig}\sqsubseteq\exists\texttt{s-part-of.Plant}\), one deduces that each Twig
is a part-of
some Twig
as well, which is an uninteresting deduction, and, in fact, points to a defect: it should have been asserted to be a proper part—which is irreflexive—of Plant
.
A separate issue that the solution proposed in [KA08] brought afore, is that it requires one to declare the taxonomy of relations correctly. This can be done by availing of the RBox Compatibility service that we have seen in Section 5.2: Methods to Improve an Ontology’s Quality "Combining Logic and Philosophy: Role Hierarchies". While the part-whole taxonomy, the RBox Compatibility service, and the OntoPartS tool’s functionalities do not solve all modeling problems of part-whole relations, at least they provide an ontologist with a sound basis and some guidelines.
As noted before, various extensions to mereology are being investigated, such as mereotopology and mereogeometry, the notion of essential parthood, and portions and stuffs. For mereotopology, the interested reader may want to consult, among others, ontological foundations [Var07] and its applicability and modeling aspects in the Semantic Web setting with OWL ontologies [KFRMG12] and DOL [KK17b], the introduction of the RCC8 spatial relations [RCC92], and exploration toward integrating RCC8 with OWL [GBM07, SS09]. Useful starting points for portions and stuff parts from the viewpoint of ontology and formalizations are [BD07, DB09, Kee16].
Other foundational ontology aspects, such as philosophy of language, modal logic, change in time, properties, the ontology of of relations, and dependence, will not be addressed in this course. The free online Stanford Encyclopedia of Philosophy^{14} contains comprehensive, entry-level readable, overviews of such foundational issues.