some comments on granularity scale & collectivity by rector & rogers thomas bittner ifomis...

Some comments on

Granularity Scale & Collectivity

by Rector & Rogers

Thomas Bittner

IFOMIS Saarbruecken

Overview

• Problems with doing ontology using DLs

• Problems with collectives

• Problems with indeterminacy

• Problems with transitivity

• Conclusions

Problems with doing ontology using Description Logics

Ontologies constrain intended meaning

The biomedical world

What you could say in L= Models of the language L

Language L(symbols+meaning)

We chose a language such that we canexpress the important aspects of theBio-medical world

This is what you actually say in your your ontology

The biomedical domain isamong the intended models= What you want to talk about


The biomedical world

Language L

Ontology

Models of the language L

Intended models

Guarino, 1998


GoodOntology

Guarino, 1998


BadOntology

Very badOntology

Guarino, 1998


BadOntology

Inappropriate tools which do not allow you to write good ontologies

• Mistakes when writing axioms• Too few axioms

Kinds of Ontology Languages


Different degrees of expressive power for the specification of the intended meaning

A shared vocabulary plus a specification of its intended meaning

Meaning specified implicitly and informally in natural language

Two extremes


Different degrees of rigor of the specification of the intended meaning

A shared vocabulary plus a specification of its intended meaning

Meaning specified implicitly andinformally in natural language

meaning specifiedexplicitly as a logical theory

Two extremes

In between a continuum of degree of expressive power


Terms

General Logic

Thesauri

formalTaxonomies

Frames(Protege)

Data Models(UML, STEP)

Description Logics

(DAML+OIL)

Principled, informal

hierarchies

ad hoc Hierarchies

(Yahoo!)structured Glossaries

XML DTDs

Data Dictionaries

(EDI)

‘ordinary’Glossaries

XML Schema

DB Schema

Glossaries & Data Dictionaries

MetaData,XML Schemas, & Data Models

Formal Ontologies & Inference

Thesauri, Taxonomies

Michael Gruninger, [email protected]


Terms

General Logic

Thesauri

formalTaxonomies

FramesProtege

Data Models(UML, STEP)

Description Logics

(DAML+OIL)

Principled, informal

hierarchies

ad hoc Hierarchies

(Yahoo!)structured Glossaries

XML DTDs

Data Dictionaries

(EDI)

‘ordinary’Glossaries

XML Schema

DB Schema

Glossaries & Data Dictionaries

MetaData,XML Schemas, & Data Models

Formal Ontologies & Inference

Thesauri, Taxonomies

Michael Gruninger, [email protected]

Why do we need formulate ontologies in very expressive languages?

Why do we need formulate ontologies in expressive languages?

It is the only way to produce good ontologies!!

GoodOntology


General Logic

Description Logics

(DAML+OIL)

Tradeoff betweenexpressive power and computability

How well canwe specify intendedmeaning

What can we compute automatically

We need BOTH kinds of languages

General Logic

Description Logics

(DAML+OIL)

Tradeoff betweenexpressive power and computability

How well canwe specify intendedmeaning

What can we compute automatically

Ontologies

Top Level Ontologies for arbitrary domains

• Endurant vs. perdurant (process)

• Parthood• Constitution

Ontologies


• Parthood• Containment• Constitution

Computational ontologies and for specific domains

• GALEN• FMA• SNOMED

Ontologies


• Parthood• Containment• Constitution


• GALEN• FMA• SNOMED

Focus on RELATIONS andproperties of relations

Focus on Class hierarchies

Ontologies



Requires high expressive power

Requires limitedExpressive power

Focus on RELATIONS andproperties of relations

Focus on Class hierarchies

Ontologies



Focus on high expressive power

Focus oncomputation

First order logic is the right language

Description logicsare the right tools

Ontologies



Alan and Jeremyuse Description Logicsto as tools to specifya top level ontology

Problems with collectives

Skin

The skin (an organ)

Object-like parts

Skin tissue

Skin tissue = collective of cells

Individual cell

Collective of cells/tissue

The organ ‘skin’

Levels of granularity

Individual cell

Collective of cells

The organ ‘skin’Entities of scale X

Entities of Scale Y

Collectives of Entities of scale Y

Level of granularity X

Level of granularity Y


Entities of scale X

Entities of Scale Y



Level of granularity Y

Entities are treatedas individuals

Members of theCollection areNOT treated asindividuals


Entities of scale X



Entities are treatedas individuals

Members of theCollection areNOT treated asindividuals

Collectives must have MANY members•Cell/molecules/atoms/

We are interested in BIG collectives

• In SMALL collectives we can individuate the members.

• Problem:– The sum/union of two BIG collectives IS a BIG

collection– The INTERSECTION of two BIG collectives is

NOT necessarily a BIG collection– Parthood relation between BIG collectives

CANNOT be modeled using the subset/subcollective relation

The INTERSECTION of two BIG collectives is NOT

necessarily a BIG collection

BIG BIG

small

Parthood relation between masses/collectives

• is DIFFERENT from parthood between individual entities

Weak supplementation principle does NOT hold

Weak supplementation principle

x proper-part-of y


x proper-part-of y (z)(z proper-part-of y AND overlap zx)


x proper-part-of y (z)(z proper-part-of y AND overlap zx)

Size of z doesNOT matter

Weak supplementation principle for big collectives

x p-mass-part-of y (z)(z p-mass-part-of y AND overlap zx)

BIG collective

BIG collective

small collective

The weak supplementation principle

relation Partial

order

WSP NPO

is-p-part-of yes yes

is-p-mass-of yes no

Contained-in yes noYou cannot make this distinctionin a Description Logic


BadOntology

Ontology does not make enough distinctionsDoes NOT constrain meaning well enough

Empty collectives

• Empty collectives do not have grains/members

• ‘Empty collectives are allowed. This is convenient …’ (Rector & Rogers)

This is always a bad justification!!

Empty collectives

• Empty collectives do not have grains/members

• ‘Empty collectives are allowed. This is convenient …’ (Rector & Rogers)

If we allow empty collectives then collectives areABSTRACT entities

Empty collectives are abstract!

• Abstract entities can be parts of concrete entities– Collective-of-blood-cells part-of blood

concreteabstract

Blood cell grain-of Collective-of-blood-cells

abstractconcrete


Blood cell grain-of Collective-of-blood-cells

abstractconcrete

Blood cell part-of Collective-of-blood-cells

abstractconcrete


Blood cell part-of Collective-of-blood-cells

abstractconcrete

Abstract entities are immaterial and immaterial entities cannot have material parts

–E.g., a hole CANNOT have a material part

So how can a blood cell be part of an ABSTRACT collective of blood cells?


BadOntology

Collectives are concrete

Collectives are abstract

So how can a blood cell be part of a collective of blood cells?

Give up emptycollectives

Give up that is-grain-ofis a parthood relation

I suggest: Do BOTH!!

Problems with indeterminacy

Grains, collections, and indeterminacy

“Granular parts are parts by way of being members of a collective that is part of the whole and of indeterminate in number: removing one does not (normally) diminish the whole.” (Rector & Rogers)

Grains, collections, and indeterminacy

“Granular parts are parts by way of being members of a collective that is part of the whole and of indeterminate in number: removing one does not (normally) diminish the whole.” (Rector & Rogers)

There are some grains (e.g., cells) and it is indeterminatewhether they are members/parts of some collection.

Parthood, granularity, indeterminacy

Individual cell

Collective of cells


ConstitutesGross-part-of

is_grain_of

Determinate parthood

INdeterminacy

Individual cell

Collective of cells

is_grain_of INdeterminacy

At a given time tit is indeterminate (vague) whether a cell is member ofa collective

Collectives have differentmembers at different times and it is hardto keep track of thosechanges


At a given time tit is indeterminatewhether a cell ismember of the collective

Collectives have differentmembers at different times and it is hardto keep track of thosechanges

Only true for some cells•At the boundary of the skin•For most cells it is pretty clear whether they are parts of a collective

This is neither indeterminacy norvagueness


Time-indexed is_grain_of

So, what does indeterminacy mean???

BadOntology

Ontology does not make enough distinctionsDoes NOT constrain meaning well enough

More problems with indeterminacy

Individual cell

Collective of cells


Constitutes(determinate)

is_grain_of(indeterminate)

Collective of cells


Individual cell

Part-ofimplies

Part-ofimplies

Determinate??????????

INdeterminate??????????

Problems with transitivity


Two questions are confused

Does the relation Xhave the property Y?

e.g., is parthood transitive

Can we exploit the fact thatrelation X has property Yfor reasoning purposes

e.g., can we exploit transitivityfor reasoning

Ontology Knowledge rep. & reasoning

What does it mean ‘relation R is transitive’

For ALL x,y,z [IF R(x,y) AND R(y,z) THEN R(x,z)]

IF is_grain_of(x,y) AND is_grain_of(y,z) THEN is_grain_of(x,z)

If this formula is true in the bio-medical domain then is_grain_of is transitive in this domain

Is is_grain_of transitive ?

IF is_grain_of(x,y) AND is_grain_of(y,z) THEN is_grain_of(x,z)

The premise is false or The conclusionis true

is_grain_of(x,y)

individual collective

is_grain_of(y, z)

is_grain_of(x,y) AND is_grain_of(y,z)

is_grain_of is (trivially) transitive


Two questions are confused

Does the relation Xhave the property Y?

e.g., is grain_of transitive

Can we exploit the fact thatrelation X has property Yfor reasoning purposes

e.g., can we exploit transitivityfor reasoning

Formal Ontology Computational ontologies

Conclusions

The short version• It is WRONG to consider description logics

as tools for formal ontology, i.e., as formal languages in order to represent top-level ontologies

• DLs are VERY valuable and capable tools for computational ontologies that support reasoning

The short version• It is WRONG to consider description logics

as tools for formal ontology, i.e., as formal languages in order to represent top-level ontologies

• DLs are VERY valuable and capable tools for computational ontologies that support reasoning

• Computational ontologies should be derived (built in compliance with) a formal ontology in First Order Logic

The GOOD and the BAD

• Using SEP-triples in a formal (top-level) ontology IS BAD

• Using SEP-triples in a computational ontology to provide computationally efficient transitivity reasoning is GOOD (assuming that you have an underlying formal ontology that tells you what you are reasoning about)

The GOOD and the BAD

• To ignore properties that you cannot express in your language is BAD in a formal ontology

• To ignore properties that you cannot express in your (computable) language is all one can do in a computational ontology

Ontologies for biomedicine

• Formal top-level ontology expressed in first order logic

Ontologies for biomedicine

• Formal top-level ontology expressed in first order logic

• Computational ontologies in DLs based on a formal ontology

some comments on granularity scale & collectivity by rector & rogers thomas bittner ifomis...

Documents

ontology models

intended meaning

intended models guarino

natural language meaning

bad ontology guarino

good ontology guarino

xml schemas

expressive power slide