Representing and reasoning over a taxonomy of …Reasoning over a hierarchy of relations Orthogonal...

A taxonomyof part-whole

relations

Maria Keet

Background

Questions andproblems

Mereology

Computationalaspects

Types ofpart-wholerelations

Preliminaries

The taxonomy

Using thetaxonomy ofpart-wholerelations

Guidelines tochoose theappropriate type

Reasoning overa hierarchy ofrelations

Orthogonalsubtopics

Representing and reasoning over ataxonomy of part-whole relations

C. Maria (Marijke) Keet

KRDB Research Centre, Faculty of Computer ScienceFree University of Bozen-Bolzano, Italy

[email protected]

Seminar at DERI Galway, Ireland, 10 September 2009

1 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

1 BackgroundQuestions and problemsMereologyComputational aspects

2 Types of part-whole relationsPreliminariesThe taxonomy

3 Using the taxonomy of part-whole relationsGuidelines to choose the appropriate typeReasoning over a hierarchy of relations

4 Orthogonal subtopics

2 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

A few general questions and problems

Is a tunnel part of the mountain?

What is the difference, if any, between how Cellnucleus and Cell are related and how Receptor andCell wall are related?

And w.r.t. Brain part of Human and/versus Hand partof Boxer? (assuming boxers must have their own hands)

A classical example: hand is part of musician, musicianpart of orchestra, but clearly, the musician’s hands are notpart of the orchestra. Is part-of then not transitive, or isthere a problem with the example?

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics






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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics






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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics






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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Linguistic use of part-whole relations (meronymy)

Part of?? Centimeter part of Decimeter? Decimeter part of Meter— therefore Centimeter part of Meter? Meter part of SI— but not Centimeter part of SI

Transitivity?? Person part of Organisation? Organisation located in Bolzano— therefore Person located in Bolzano?— but not Person part of Bolzano

7 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Linguistic use of part-whole relations (meronymy)


Transitivity?? Person part of Organisation? Organisation located in Bolzano— therefore Person located in Bolzano?— but not Person part of Bolzano

8 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Linguistic use of part-whole relations


Transitivity?? Person member of Organisation? Organisation located in Bolzano— therefore Person located in Bolzano?— but not Person member of Bolzano

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


Which part of?? CellMembrane structural part of RedBloodCell? RedBloodCell part of Blood— but not CellMembrane structural part of Blood? Receptor structural part of CellMembrane— therefore Receptor structural part of RedBloodCell

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


Which part of?? CellMembrane structural part of RedBloodCell? RedBloodCell contained in? Blood— but not CellMembrane structural part of Blood? Receptor structural part of CellMembrane— therefore Receptor structural part of RedBloodCell

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Part-whole relations in realism-based ontologies

Instance-level relations in the OBO Relations Ontology[Setal05]

c part of c1 at t - a primitive relation between twocontinuant instances and a time at which the one is part ofthe otherp part of p1, r part of r1 - a primitive relation ofparthood, holding independently of time, either betweenprocess instances (one a subprocess of the other), orbetween spatial regions (one a subregion of the other)c contained in c1 at t , c located in c1 at t and not coverlap c1 at tc located in r at t - a primitive relation between acontinuant instance, a spatial region which it occupies, anda time

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Definitions in OBO Relations Ontology [Setal05]

Class-level relations

C part of C1 , for all c , t, if Cct then there is some c1

such that C1c1t and c part of c1 at t.P part of P1 , for all p, if Pp then there is some p1 suchthat: P1p1 and p part of p1.C contained in C1 , for all c , t, if Cct then there is somec1 such that: C1c1t and c contained in c1 at t

Need to commit to a foundational ontology. Recently,linked to BFO http://obofoundry.org/ro/#mappings (testrelease)

Same labels, different relata and only a textual constraint:Label the relations differently

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Definitions in OBO Relations Ontology [Setal05]

Class-level relations

C part of C1 , for all c , t, if Cct then there is some c1

such that C1c1t and c part of c1 at t.P part of P1 , for all p, if Pp then there is some p1 suchthat: P1p1 and p part of p1.C contained in C1 , for all c , t, if Cct then there is somec1 such that: C1c1t and c contained in c1 at t

Need to commit to a foundational ontology. Recently,linked to BFO http://obofoundry.org/ro/#mappings (testrelease)

Same labels, different relata and only a textual constraint:Label the relations differently

14 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Analysis of the issues from diverse angles

Mereological theories [Varzi04], usage & extensions (e.g.mereotopology, relation with granularity, set theory)

Early attempts with direct parthood [Sattler95], SEP triples[SH00], and other outstanding issues [AFGP96], some stillremaining [KA08].

Cognitive & linguistic issues from meronymy ([WCH87] and soforth)

Usage in conceptual modelling and ontology engineering

Subject domains: thus far, mainly geo, bio, medicine

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics







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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics







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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Ground Mereology

Reflexivity (everything is part of itself)

∀x(part of (x , x)) (1)

Antisymmetry (two distinct things cannot be part of each other, or: if they are, then they are the

same thing)

∀x , y((part of (x , y) ∧ part of (y , x))→ x = y) (2)

Transitivity (if x is part of y and y is part of z, then x is part of z)

∀x , y , z((part of (x , y) ∧ part of (y , z))→ part of (x , z)) (3)

Proper parthood

∀x , y(proper part of (x , y) ≡ part of (x , y) ∧ ¬part of (y , x))(4)18 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Ground Mereology

Proper parthood

∀x , y(proper part of (x , y) ≡ part of (x , y) ∧ ¬part of (y , x))(5)

Asymmetry (if x is part of y then y is not part of x)

∀x , y(part of (x , y)→ ¬part of (y , x)) (6)

Irreflexivity (x is not part of itself)

∀x¬(part of (x , x)) (7)

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Defining other relations with part of

Overlap (x and y share a piece z)

∀x , y(overlap(x , y) ≡ ∃z(part of (z , x) ∧ part of (z , y))) (8)

Underlap (x and y are both part of some z)

∀x , y(underlap(x , y) ≡ ∃z(part of (x , z) ∧ part of (y , z))) (9)

Over- & undercross (over/underlap but not part of)

∀x , y(overcross(x , y) ≡ overlap(x , y) ∧ ¬part of (x , y)) (10)

∀x , y(undercross(x , y) ≡ underlap(x , y) ∧ ¬part of (y , x))(11)

Proper overlap & Proper underlap

∀x , y(p overlap(x , y) ≡ overcross(x , y)∧overcross(y , x)) (12)

∀x , y(p underlap(x , y) ≡ undercross(x , y) ∧ undercross(y , x))(13)

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

With x as part, what to do with the remainder that makesup y?

Weak supplementation: every proper part must besupplemented by another, disjoint, part. MMStrong supplementation: if an object fails to includeanother among its parts, then there must be a remainder.EM

Problem with EM: non-atomic objects with the sameproper parts are identical, because of this (extensionalityprinciple), but sameness of parts may not be sufficient foridentity E.g.: two objects can be distinct purely based on arrangement of its parts, differences

statue and its marble (multiplicative approach)

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

With x as part, what to do with the remainder that makesup y?

Weak supplementation: every proper part must besupplemented by another, disjoint, part. MMStrong supplementation: if an object fails to includeanother among its parts, then there must be a remainder.EM

Problem with EM: non-atomic objects with the sameproper parts are identical, because of this (extensionalityprinciple), but sameness of parts may not be sufficient foridentity E.g.: two objects can be distinct purely based on arrangement of its parts, differences

statue and its marble (multiplicative approach)

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

General Extensional Mereology

Strong supplementation [EM]

¬part of (y , x)→ ∃z(part of (z , y)∧¬overlap(z , x)) (14)

And add unrestricted fusion [GEM]. Let φ be a property orcondition, then for every satisfied φ there is an entityconsisting of all entities that satisfy φ. 1 Then:

∃xφ→ ∃z∀y(overlap(y , z)↔ ∃x(φ∧overlap(y , x))) (15)

Note that in EM and upward we have identity, from whichone can prove acyclicity for ppo

There are more mereological theories, and the above is notuncontested (more about that later)

1Need to refer to classes, but desire to stay within FOL. Solution:axiom schema with only predicates or open formulas

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Relations between common mereological theories5

Ground Mereology M

Minimal Mereology MM

Extensional Mereology EM

Closure Mereology CM

Extensional Closure Mereology CEM = CMM

General Mereology GM

General Extensional Mereology GEM = GMM

Fig. 1: Hasse diagram of mereological theories; fromweaker to stronger, going uphill (after [44]).

We can define the sum σ and product π in GEM, which enables one to succinctlyrewrite sum (20), product (21), remainder (22), complement (23), and universal indi-vidual (24). See [44] sections 4.2 and 4.3 for further detail and discussion.

x + y = σz(part of(z, x) ∨ part of(z, y)) (20)

x× y = σz(part of(z, x) ∧ part of(z, y)) (21)

x− y = σz(part of(z, x) ∧ ¬overlap(z, y)) (22)

∼ x = σz(¬overlap(z, x)) (23)

U = σz(part of(z, z)) (24)

Given these basics, we can proceed to its mathematical analysis and some interestingproperties, which are described in the next section.

2.2 GEM and set theory

Set theory provides structural relations to abstract mathematical entities called setsby using the is element of relation (see [19] for a brief online introduction, amongmany sources and books). However, its grounding in reality is debatable due to themany abstract ingredients, which mereology may overcome at least to some extent (seee.g. the introduction of [6] for arguments and §5.2 below). Since mereological theoriesare formulated in predicate logic (see above in §2.1), one can assess how they relateto set theory from a mathematical perspective, comprehensively assessed by Pontowand Schubert [30].

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Can any of this be represented in a decidable fragment offirst order logic for use in ontologies and (scalable)software implementations?

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Things are improving...

Early days (90s) and simplest options: DL-role R aspartof, or has-part added as primitive role as �, modelit as the transitive closure of a parthood relation (16) anddefine e.g. Car as having wheels that in turn have tires[AFGP96] (17):

� .= (primitive-part) ∗ (16)

Car.

= ∃ � .(Wheel u ∃ � .Tire) (17)

Then Car v ∃ �.TireSEP triples with ALC [SH00]

What SHIQ fixes cf. ALC: Transitive roles, Inverse roles(to have both part-of and has-part), Role hierarchies (e.g.for subtypes of part-of), qualified Number restrictions (e.g.to represent that a bycicle has-part 2 wheels)

Build-your-own DL-language [BD05]

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics




Car.

= ∃ � .(Wheel u ∃ � .Tire) (17)




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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics




Car.

= ∃ � .(Wheel u ∃ � .Tire) (17)




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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

What we can(not) implement now with DL-basedontology languages

Table: Properties of parthood and proper parthood compared to theirsupport in DLRµ, SHOIN and SROIQ. ∗: properties of theparthood relation (in M); ‡: properties of the proper parthoodrelation (in M).

Language ⇒ DLRµ SHOIN SROIQ DL-LiteA

Feature ⇓ (∼ OWL-DL) (∼ OWL 2)

Reflexivity ∗ + – + –Antisymmetry ∗ – – – –Transitivity ∗ ‡ + + + –Asymmetry ‡ + + + +Irreflexivity ‡ + – + –

Acyclicity + – – –

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Taxonomy of types of part-whole relations

Clarifying and cleaning up ontologically the informalusages of “part of”

Sorting out (in)transitivity and why some part-wholerelations are non- or in-transitive, but parthood istransitive

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Addressing the issues

Efforts to disambiguate the confusion on part-wholerelations; e.g. an informal taxonomy [WCH87], list of 6 typesmotivated by conceptual modelling [Odell98] [GP95], andontology-inspired conceptual modelling [G05]

Location, containment, membership of a collective,quantities of a mass

Relatively well-settled debate on transitivity, or not [Varzi06]

and related papers in AO

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Overview

Mereological part of (and subtypes) versus ‘other’part-whole relations

Categories of object types of the part-whole relationchanges

Structure these relations by (non/in)transitivity and kindsof relata

Simplest mereological theory, M.

Commit to a foundational ontology: DOLCE [MBGGO03]

(though one also could choose, a.o., BFO, OCHRE, GFO,...)

32 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Overview

Mereological part of (and subtypes) versus ‘other’part-whole relations

Categories of object types of the part-whole relationchanges

Structure these relations by (non/in)transitivity and kindsof relata

Simplest mereological theory, M.

Commit to a foundational ontology: DOLCE [MBGGO03]

(though one also could choose, a.o., BFO, OCHRE, GFO,...)

33 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

DOLCE categories

PTParticular

EDEndurant

PDPerdurant

PEDPhysicalEndurant

NPEDNon-physical

Endurant

ASArbitrary

Sum

EVEvent

STStative

ACHAchievement

ACCAccomplishment

STState

PROProcess

NPOBNon-physical

object

MOBMental object

SOBSocial object

POBPhysicalobject

FFeature

MAmountof matter

NAPONon-agentive

physical object

APOAgentive

physical object

…

…

… … … …

… …

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Part-whole relations

Part-whole relation

mpart_of ((Meronymic) part-whole relation)

part_of (Mereological part-of relation)

member-of constitutes sub-quantity-of participates-in involved-in spatial-part-of

f-part-of

s-part-of

located-in contained-in member-of’

… … … …

… …

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


“member-bunch”, collective nouns (e.g. Herd, Orchestra)with their members (Sheep, Musician)

Definition 1 (member of)

∀x , y(member ofn(x , y) ,mpart of (x , y) ∧ (POB(x) ∨ SOB(x)) ∧ SOB(y))

“material-object”, that what something is made of (e.g., Vaseand Clay)

Definition 2 (constituted of)

∀x , y(constitutesit(x , y) ≡ constituted ofit(y , x) ,mpart of (x , y) ∧ POB(y) ∧M(x))

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


“quantity-mass”, “portion-object”, relating a smaller (or sub)part of an amount of matter to the whole. Two issues (glass ofwine & bottle of wine vs. Salt as subquantity of SeaWater)

Definition 3 (sub quantity of)

∀x , y(sub quantity ofn(x , y) , mpart of (x , y) ∧M(x) ∧M(y))

“noun-feature/activity”, entity participates in a process, likeEnzyme that participates in CatalyticReaction

Definition 4 (participates in)

∀x , y(participates init(x , y) , mpart of (x , y)∧ED(x)∧PD(y))

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


Object and its 2D or 3D region, such ascontained in(John’s address book, John’s bag) andlocated in(Galway, Ireland)

Definition 5 (contained in)

∀x , y(contained in(x , y) , part of (x , y) ∧ R(x) ∧ R(y) ∧∃z ,w(has 3D(z , x) ∧ has 3D(w , y) ∧ ED(z) ∧ ED(w)))

Definition 6 (located in)

∀x , y(located in(x , y) , part of (x , y) ∧ R(x) ∧ R(y) ∧∃z ,w(has 2D(z , x) ∧ has 2D(w , y) ∧ ED(z) ∧ ED(w)))

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics


processes and sub-processes (e.g. Chewing is involved in thegrander process of Eating)

Definition 7 (involved in)

∀x , y(involved in(x , y) , part of (x , y) ∧ PD(x) ∧ PD(y))

extensibility, e.g.

Definition 8 (s part of)

∀x , y(s part of (x , y) , part of (x , y) ∧ ED(x) ∧ ED(y))

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Using the taxonomy of part-whole relations

Representing it correctly in ontologies and conceptual datamodels

Reasoning with a taxonomy of relations

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Decision diagram [K06a]

X part-of Y → X f-part-of Y (functional part-of)

Does the part-of role relate roles?

X part-of Y → X involved-in Y

Is X a member of Y? (like player-team)

X part-of Y → X member-of Y

Is X made of Y? (like bike-steel,

vase-clay)

X part-of Y → Y constituted-of X

Is X a portion or subquantity of Y? (like slice-pie, wine or

other mass noun)

X part-of Y → X sub-quantity-of Y

Is X a spatial part of Y? (like oasis-desert,

nucleus-cell)

Are X and Y geographical object types? (as in place-area, like Massif

Central in France)

X part-of Y → X located-in Y

Then X part-of Y → X contained-in Y

(like a book in the bag)

Is X part of Y and X is also functionally dependent on Y (or vv)?

(like heart-body, handle-cup)

No

Is X part-of an event Y? (like bachelor-party, enzyme-reaction)

X part-of Y → X participates-in Y

Then X part-of Y → X s-part-of Y

(structural part-of, like shelf-cupboard)

Yes

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Decision diagram







vase-clay)



other mass noun)



nucleus-cell)


Central in France)






No





Yes

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Decision diagram







vase-clay)



other mass noun)



nucleus-cell)


Central in France)






No





Yes

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - before

has part part of

ConferenceBag Flap

ShoulderHandle

ConfProceedings

Compartment

Linen

has part part of

has part part of

has part part of

part ofhas part

part of

Env elope

part of

/has part

RegistrationReceipt

WineSample

WineTastingTicketpart ofhas part

WineTastingEv ent

allow s entry to

Winepart ofhas part

part of

/has part

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - after

Envelope is not involved-in, not a member-of, does notconstitute, is not a sub-quantity of, does notparticipate-in, is not a geographical object, but instead iscontained-in the ConferenceBag.

Transitivity holds for the mereological relations: derivedfacts are automatically correct, like RegistrationReceipt

contained-in ConferenceBag.

Intransitivity of Linen and ConferenceBag, because aconference bag is not wholly constituted of linen (themodel does not say what the Flap is made of).

Completeness, i.e. that all parts make up the whole, isimplied thanks to the closed-world assumption.ConferenceBag directly contains the ConfProceedings and Envelope

only, and does not contain, say, the Flap.

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - after







46 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - after







47 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - after







48 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Example - after

ConferenceBag Flap

ShoulderHandle

ConfProceedings

Compartment

Linen

has s-part s-part of

has f-part f-part of


contained incontains

constitutesconstituted of

Env elope

contained in

/contains

contains contained in

WineTastingTicket

RegistrationReceipt


WineTastingEv ent

allow s entry to

Wine

participates in

WineSample

sub-quantity -of

ConferenceBag Flap

ShoulderHandle

ConfProceedings

Compartment

Linen


has f-part f-part of



constitutesconstituted of

Env elope

contained in

/contains

contains contained in RegistrationReceipt

WineTastingTicketcontained incontains

WineTastingEv ent

allow s entry to

Wine

participates in

WineSample

sub-quantity -of

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Requirements

Represent at least Ground Mereology,

Express ontological categories and their taxonomicrelations,

Having the option to represent transitive and intransitiverelations, and

Specify the domain and range restrictions (/relata/entitytypes) for the classes participating in a relation.

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Current behaviour of reasoners

A1. Class hierarchy with asserted conditions

B. Correct role box (object properties) C. Wrong role box (object properties)

A2. Other class hierarchy with

the same asserted

conditions

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Current behaviour of reasoners

3. A1+C+racer: class hierarchy is inconsistent 4. A2+C+racer: Chassis reclassified as PD

1. A1+B+racer: ontology OK 2. A2+B+racer: ontology OK

5: Required inference result A1/A2+C+reasoner:

role hierarchy is inconsistent, with inconsistent roles “domain & range involved-in and part-of are inconsistent”, which can be fixed by the user, else the reasoner suggests:

Computing superroles reasoner log: “involved-in Moved to pwrelation“ and “part-of Moved to involved-in”

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

The RBox Compatibility service – definitions

Definition 9 (Domain and Range Concepts)

Let R be a role and R v C1 × C2 its associated Domain &Range axiom. Then, with the symbol DR we indicate theUser-defined Domain of R—i.e., DR = C1—while with thesymbol RR we indicate the User-defined Range of R—i.e.,RR = C2.

Definition 10 (RBox Compatibility)

For each pair of roles, R, S , such that 〈T ,R〉 |= R v S , check:

Test 1. 〈T ,R〉 |= DR v DS and 〈T ,R〉 |= RR v RS ;

Test 2. 〈T ,R〉 6|= DS v DR ;

Test 3. 〈T ,R〉 6|= RS v RR .

An RBox is said to be compatible iff Test 1 and (2 or 3) holdfor all pairs of role-subrole in the RBox.

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

The RBox Compatibility service – behaviour

If Test 1 does not hold: warning that domain & rangerestrictions of either R or S are in conflict with the rolehierarchy proposing either

(i) To change the role hierarchy or(ii) To change domain & range restrictions or(iii) If the test on the domains fails, then propose a new

axiom R v D ′R × RR , where D ′R ≡ DR u DS2, which

subsequently has to go through the RBoxcompatibility service (and similarly when Test 1 failson range restrictions).

2The axiom C1 ≡ C2 is a shortcut for the axioms: C1 v C2 and C2 v C1.54 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

The RBox Compatibility service – behaviour

If Test 2 and Test 3 fail: warn that R cannot be aproper subrole of S but that the two roles can beequivalent. Then, either:(a) Accept the possible equivalence between the two roles

or(b) Change domain & range restrictions.

Ignoring all warnings is allowed, too

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Extensions in various directions

Mereotopology, with location, GIS, Region ConnectionCalculus (http://www.comp.leeds.ac.uk/qsr/rcc.html)

Mereogeometry [BM07]

Mereology and/vs granularity (a.o. [BS03], [K08], [RRB06])

Temporalising the part-whole relations (a.o. [BD07] [AGK08])

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Some more issues in parthood relations

What to do with other (meta-)properties, such as:

Inseparability (x is inseparable of y iff x is existentiallydependent on y and x is necessarily a part of y , see [G05]

p169)Degrees of shareability (‘total’, part of more than wholeof the same type or of different types, etc. [MK99])Essential part & essential whole (like member-partnership[Odell98], brain-human)Immutable part & immutable whole (the boxer with hishands, an ecofarm with a piece of farmland)

De dicto/de re distinction (possible worlds) [G07], where– de re with ∀...→ �..., e.g. “every boxer necessarily has a hand”

– de dicto with �(∀...→ ...) e.g. “necessarily, every boxer has a hand”

Total/complete on the parts

Direct part to distinguish from part-by-transitivity

57 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Some more issues in parthood relations

What to do with other (meta-)properties, such as:

Inseparability (x is inseparable of y iff x is existentiallydependent on y and x is necessarily a part of y , see [G05]

p169)Degrees of shareability (‘total’, part of more than wholeof the same type or of different types, etc. [MK99])Essential part & essential whole (like member-partnership[Odell98], brain-human)Immutable part & immutable whole (the boxer with hishands, an ecofarm with a piece of farmland)

De dicto/de re distinction (possible worlds) [G07], where– de re with ∀...→ �..., e.g. “every boxer necessarily has a hand”

– de dicto with �(∀...→ ...) e.g. “necessarily, every boxer has a hand”

Total/complete on the parts

Direct part to distinguish from part-by-transitivity

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

(Non-)Sharable parts and wholes

Essential and mandatory parts; informally, distinguishbetween:

“must be part of some whole” and“must be part of the same whole” during its lifetime.

Sequentially or concurrently being part of > 1 whole ofsame or different type; distinguish between, a.o.:

p is part of w1 ∈W at time t1 and of w2 ∈W at time t2;p is part of w1 ∈W and w2 ∈W at time t1;p is part of w1 ∈W at time t1 and of wa ∈W ′ at time t2;p is part of w1 ∈W and wa ∈W ′ at time t1;

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Solution sketch

Temporalizing part-whole relations, and parts and wholes

DLRUS and ERVT [APS07], extended with status relations

Details in [AGK08]

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Examples

Scheduled: a relation is scheduled if its instantiation isknown but its membership will only become effective sometime later. For instance, a pillar for finishing the interior ofthe Sagrada Familia in Barcelona is scheduled to becomepart of that church.

Active: the status of a relation is active if the particularrelation fully instantiates the type-level relation. Forinstance, the Mont Blanc mountain is part of the Alpsmountain range, and the country Republic of Ireland ispart of the European Union.

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Examples

Suspended: to capture a temporarily inactive relation.For example, an instance of a CarEngine is removed fromthe instance of a Car it is part of, for purpose ofmaintenance at the car mechanic.

Disabled: to model expired relations that never again canbe used. For instance, to represent the donor of an organwho has donated that organ and one wants to keep trackof who donated what to whom: say, the heart p1 of donorw1 used to be a structural part of w1 but it will never beagain a part of it. The heart, p1, then may have becomeparticipant in a new part-of relation with a new whole, w2

where w1 6= w2, but the original part-of between p1 andw1 remains disabled.

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Effect of temporalising on the types of part-wholerelations

Not fully clear—yet.

Very preliminary results in [AK08].

If something is physically a proper part of a whole itcannot be directly part of another whole at the same time(idem proper containment and location)proper subprocess can participate in different granderprocesses concurrently, idem memberssubquantities: (i) for measured amounts of matter of thesame type (before/after), and (ii) amounts of differenttype (concurrently or sequentially)

63 / 68


relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Summary

Disambiguation of types of part-whole relations by meansof a taxonomy of types of part-whole relations

Guidelines for manual modelling

Reasoning over a relational hierarchy with the RBoxCompatibility service

A list of other part-whole topics, such as temporal,mereotopological, modality

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Discussion

Part-whole relation as first-class citizen?

Differences set theory and mereology

MathematicalPhilosophical (ontological)

Ease of representation vs accuracy

‘cannot’ include all mereological facets preciselyComputational support vs just representing domainknowledge or reality

“we did just fine without it for decades”

Better no constructor than a badly defined one?

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

References

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International Conference on Principles of Knowledge Representation and Reasoning (KR’08). GerhardBrewka, Jerome Lang (Eds.) AAAI Press. Sydney, Australia, September 16-19, 2008.

Artale, A. and Keet, C.M. Essential, mandatory, and shared parts in conceptual data models. In:

Innovations in Information Systems modeling: Methods and Best Practices. IGI Global, Advances inDatabase Research Series, Halpin, T.A., Proper, H.A., Krogstie, J. (Eds.).

Artale, A., Parent, C. & Spaccapietra, S. (2007b). Evolving objects in temporal information systems.

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Borgo, S., Masolo, C. Full mereogeometries. Journal of Philosophical Logic. (to appear).

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relations

Maria Keet

Background


Mereology



Preliminaries

The taxonomy




Orthogonalsubtopics

Gerstl, P., Pribbenow, S. Midwinters, end games, and body parts: a classification of part-whole

relations. Intl. Journal of Human-Computer Studies, 1995, 43:865-889.

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Masolo, C., Borgo, S., Gangemi, A., Guarino, N. and Oltramari, A. (2003). Ontology Library.

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Background


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Preliminaries

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Orthogonalsubtopics

Motschnig-Pitrik, R., Kaasbøll, J. Part-Whole Relationship Categories and Their Application in

Object-Oriented Analysis. IEEE Transactions on Knowledge and Data Engineering, 1999,11(5):779-797.

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