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 can express the important aspects of the Bio-medical world This is what you actually say in your your ontology The biomedical domain is among the intended models = What you want to talk about

Ontologies constrain intended meaning The biomedical world Language L Ontology Models of the language L Intended models Guarino, 1998

Ontologies constrain intended meaning Good Ontology Guarino, 1998

Ontologies constrain intended meaning Bad Ontology Very bad Ontology Guarino, 1998

Ontologies constrain intended meaning Bad Ontology 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

Kinds of Ontology Languages Different degrees of rigor of the specification of the intended meaning A shared vocabulary plus a specification of its intended meaning Meaning specified implicitly and informally in natural language meaning specified explicitly as a logical theory Two extremes In between a continuum of degree of expressive power

Kinds of Ontology Languages Terms General Logic Thesauri formal Taxonomies 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,

Kinds of Ontology Languages Terms General Logic Thesauri formal Taxonomies 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,

Kinds of Ontology Languages Terms General Logic Thesauri formal Taxonomies 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,

Kinds of Ontology Languages Terms General Logic Thesauri formal Taxonomies 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,

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!! Good Ontology

Kinds of Ontology Languages General Logic Description Logics (DAML+OIL) Tradeoff between expressive power and computability How well can we specify intended meaning What can we compute automatically

Kinds of Ontology Languages General Logic Description Logics (DAML+OIL) Tradeoff between expressive power and computability How well can we specify intended meaning What can we compute automatically

Kinds of Ontology Languages General Logic Description Logics (DAML+OIL) Tradeoff between expressive power and computability How well can we specify intended meaning What can we compute automatically

We need BOTH kinds of languages General Logic Description Logics (DAML+OIL) Tradeoff between expressive power and computability How well can we specify intended meaning What can we compute automatically

Ontologies Top Level Ontologies for arbitrary domains Endurant vs. perdurant (process) Parthood Constitution

Ontologies Top Level Ontologies for arbitrary domains Parthood Containment Constitution Computational ontologies and for specific domains GALEN FMA SNOMED

Ontologies Top Level Ontologies for arbitrary domains Parthood Containment Constitution Computational ontologies and for specific domains GALEN FMA SNOMED Focus on RELATIONS and properties of relations Focus on Class hierarchies

Ontologies Top Level Ontologies for arbitrary domains Computational ontologies and for specific domains Requires high expressive power Requires limited Expressive power Focus on RELATIONS and properties of relations Focus on Class hierarchies

Ontologies Top Level Ontologies for arbitrary domains Computational ontologies and for specific domains Focus on high expressive power Focus on computation First order logic is the right language Description logics are the right tools

Ontologies Top Level Ontologies for arbitrary domains Computational ontologies and for specific domains Alan and Jeremy use Description Logics to as tools to specify a 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

Levels of granularity Entities of scale X Entities of Scale Y Collectives of Entities of scale Y Level of granularity X Level of granularity Y Entities are treated as individuals Members of the Collection are NOT treated as individuals

Levels of granularity Entities of scale X Collectives of Entities of scale Y Level of granularity X Entities are treated as individuals Members of the Collection are NOT treated as individuals 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 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 

Weak supplementation principle x proper-part-of y  (  z)(z proper-part-of y AND  overlap zx)

Weak supplementation principle x proper-part-of y  (  z)(z proper-part-of y AND  overlap zx) Size of z does NOT 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 small collective

The weak supplementation principle relationPartial order WSPNPO is-p-part-ofyes is-p-mass-ofyesno Contained-inyesno You cannot make this distinction in a Description Logic

Ontologies constrain intended meaning Bad Ontology Ontology does not make enough distinctions Does 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 are ABSTRACT entities

Empty collectives are abstract! Abstract entities can be parts of concrete entities –Collective-of-blood-cells part-of blood concrete abstract Blood cell grain-of Collective-of-blood-cells abstract concrete

Empty collectives are abstract! Blood cell grain-of Collective-of-blood-cells abstract concrete Blood cell part-of Collective-of-blood-cells abstract concrete

Empty collectives are abstract! Blood cell part-of Collective-of-blood-cells abstract concrete 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?

Ontologies constrain intended meaning Bad Ontology Collectives are concrete Collectives are abstract

So how can a blood cell be part of a collective of blood cells? Give up empty collectives Give up that is-grain-of is 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 indeterminate whether they are members/parts of some collection.

Parthood, granularity, indeterminacy Individual cell Collective of cells The organ ‘skin’ Constitutes Gross-part-of is_grain_of Determinate parthood INdeterminacy

Individual cell Collective of cells is_grain_of INdeterminacy At a given time t it is indeterminate (vague) whether a cell is member of a collective Collectives have different members at different times and it is hard to keep track of those changes Parthood, granularity, indeterminacy

At a given time t it is indeterminate whether a cell is member of the collective Collectives have different members at different times and it is hard to keep track of those changes 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 nor vagueness Parthood, granularity, indeterminacy Time-indexed is_grain_of

So, what does indeterminacy mean??? Bad Ontology Ontology does not make enough distinctions Does NOT constrain meaning well enough

More problems with indeterminacy Individual cell Collective of cells The organ ‘skin’ Constitutes (determinate) is_grain_of (indeterminate) Collective of cells The organ ‘skin’ Individual cell Part-of implies Part-of implies Determinate ?????????? INdeterminate ??????????

Problems with transitivity

Two questions are confused Does the relation X have the property Y? e.g., is parthood transitive Can we exploit the fact that relation X has property Y for reasoning purposes e.g., can we exploit transitivity for 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 conclusion is 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

Problems with transitivity Two questions are confused Does the relation X have the property Y? e.g., is grain_of transitive Can we exploit the fact that relation X has property Y for reasoning purposes e.g., can we exploit transitivity for 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