meghyn slides-hse-2014

Ontologies in Computer Science:The Description Logic Approach

Meghyn Bienvenu

Laboratoire de Recherche en Informatique,CNRS & Universite Paris-Sud, France

What is an ontology?

Depends on who you ask...

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In philosophy:the study of the nature of being, becoming, existence, or reality, as well as thebasic categories of being and their relations (source: wikipedia)

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In computer science:the formal specification of the knowledge of a particular domain,thereby making it amenable to machine processing

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In computer science:the formal specification of the knowledge of a particular domain,thereby making it amenable to machine processing

Such a specification consists of:

• terminology (or vocabulary) of the domain

• semantic relationships between terms

– relations of specificity or generality, equivalence, disjointness, ...

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What are ontologies good for?

To standardize the terminology of an application domain

• meaning of terms is constrained, so less misunderstandings

• by adopting a common vocabulary, easy to share information

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To present an intuitive and unified view of data sources

• ontology can be used to enrich the data vocabulary, making it easier forusers to formulate their queries

• especially useful when integrating multiple data sources

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To present an intuitive and unified view of data sources

• ontology can be used to enrich the data vocabulary, making it easier forusers to formulate their queries

• especially useful when integrating multiple data sources

To support automated reasoning

• uncover implicit connections between terms, or errors in modelling

• exploit knowledge in the ontology during query answering, to get back amore complete set of answers to queries

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Ontologies in Medecine

Comprehensive medical ontologies: SNOMED CT (∼ 400,000 terms!), GALEN

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Specialized ontologies: FMA (anatomy), NCI (cancer), ...

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• querying & exchanging medical records (e.g. find patients eligible for medical trials)

– myocardial infarction vs. MI vs. heart attack vs. 410.0

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• querying & exchanging medical records (e.g. find patients eligible for medical trials)

– myocardial infarction vs. MI vs. heart attack vs. 410.0

• supports tools for annotating and visualizing patient data (scans, x-rays)

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Ontologies in Biology

Hundreds of ontologies at BioPortal (http://bioportal.bioontology.org/):Gene Ontology (GO), Cell Ontology, Pathway Ontology, Plant Anatomy, ...

• help scientists to share, query, compare, and visualize experimental data

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Description Logics in a Nutshell

Description logics (DLs) are a family of knowledge representation languages.

• originated from work on semantic networks, frame languages

• nowadays, popular means for specifying ontologies

• basis of the web ontology language OWL (W3C standard)

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Formally: fragments of first-order logic.

• inherit well-defined semantics

• succinct, variable-free syntax

• related to modal logics, guarded fragment, ...

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Computational properties well-understood (decidability, complexity)

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Computational properties well-understood (decidability, complexity)

Many implemented reasoners and tools available for use

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Description Logic Basics

Building blocks of DLs:

• atomic concepts (unary predicates, classes)

Animal, Dog, Cow, Giraffe, Plant, Herbivore, Carnivore, Domesticated...

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Building blocks of DLs:

• atomic concepts (unary predicates, classes)

Animal, Dog, Cow, Giraffe, Plant, Herbivore, Carnivore, Domesticated...

• atomic roles (binary predicates, properties)

eats, parentOf, hasPart, hasHabitat, ...

• individuals (constants)

fluffy, giraffe0354, specimen32091, ...

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Can build complex concepts and roles using constructors:

• conjunction (u), disjunction (t), negation (¬)

Dog t Cat Animal u ¬Carnivore Plant u Carnivore

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• restricted forms of existential and universal quantification (∃, ∀)

∃parentOf.> ∃eats.Animal Animal u ∀eats.Plant

( > acts as a “wildcard”, denotes set of all things)

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• inverse (−) and composition (◦) of roles

parentOf− parentOf ◦ parentOf

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• inverse (−) and composition (◦) of roles

parentOf− parentOf ◦ parentOf

NB: available constructors depends on the particular DL

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Description Logic Knowledge Bases

DL knowledge bases (KBs) consist of two parts.

TBox (‘T’ for “terminology”)Contains general knowledge about the domain of interest (axioms)

• Plants and animals are disjoint classes of living things

• Giraffes are animals that only eat leaves

• Herbivores are animals that eat only plants

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Description Logic Knowledge Bases

DL knowledge bases (KBs) consist of two parts.

TBox (‘T’ for “terminology”)Contains general knowledge about the domain of interest (axioms)

• Plants and animals are disjoint classes of living things

• Giraffes are animals that only eat leaves

• Herbivores are animals that eat only plants

ABox (‘A’ for “assertion”)Contains facts about specific individuals (assertions)

• Fluffy is a dog. Fluffy is the parent of Sparky.

• Specimen0305 is a Venus fly trap.

NB: open world assumption (unlike databases)

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Examples of TBox Axioms

Plants and animals are disjoint classes of living things

Plant v LivingThing Animal v LivingThing Plant v ¬Animal

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Giraffes are animals that only eat leaves

Giraffe v Animal u ∀eats.Leaf

FOL-translation: ∀x (Giraffe(x)→ (∀y eats(x, y)→ Leaf(y)))

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Giraffes are animals that only eat leaves


FOL-translation: ∀x (Giraffe(x)→ (∀y eats(x, y)→ Leaf(y)))

Herbivores are animals that eat only plants

Herbivore v Animal u ∀eats.Plant Animal u ∀eats.Plant v Herbivore

Abbreviation: Herbivore ≡ Animal u ∀eats.Plant

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A Few More Examples

Defining grandparents

Grandparent ≡ ∃parentOf.∃parentOf.>

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A Few More Examples



Defining the childOf and grandchildOf relations

childOf ≡ parentOf− grandchildOf ≡ childOf ◦ childOf

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A Few More Examples



Defining the childOf and grandchildOf relations

childOf ≡ parentOf− grandchildOf ≡ childOf ◦ childOf

Asserting transitivity of binary relations

partOf ◦ partOf v partOf

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Description Logic Semantics (What does it all mean?)

Interpretation I (“possible world”)

• (possibly infinite) domain of objects ∆I

• atomic concept A ; set of objects AI ⊆ ∆I (which objects are A’s?)

• atomic role r ; pair of objects rI (which objects are in relation r?)

• individual a ; object aI (which object is a?)

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• complex concepts / roles ; sets / pairs of objectse.g. (C uD)I = CI ∩DI (¬C)I = ∆I \ CI

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Satisfaction in an interpretation

• I satisfies C v D ⇔ CI ⊆ DI

• I satisfies C(a) ⇔ aI ∈ CI

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Satisfaction in an interpretation

• I satisfies C v D ⇔ CI ⊆ DI

• I satisfies C(a) ⇔ aI ∈ CI

Model of a KB K = interpretation that satisfies all axioms & assertions in K

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Example: Interpretations

∆I

fluffyI

AnimalI

HerbivoreI

PlantI

Which axioms and assertions are satisfied in I?

Animal v ¬Plant Herbivore v Animal Herbivore v ∀eats.Plant

Animal(fluffy) Herbivore(fluffy)

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Reasoning Tasks

SatisfiabilityDoes the KB have a model?

• want to test whether the KB is internally coherent (e.g. no contradictions)

• essential tool for debugging ontologies

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Example: Satisfiability

Suppose our KB contains the following statements:

Plant v ¬Animal

VenusFlyTrap v CarnivorousPlant

VenusFlyTrap(speciment0305)

CarnivorousPlant ≡ Plant u Carnivore

Carnivore ≡ Animal u ∃eats.Animal

Question: Is this KB satisfiable?

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Plant v ¬Animal






No!

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Plant v ¬Animal






No!

Can infer Plant(speciment0305) and Animal(speciment0305)

This contradicts the axiom Plant v ¬Animal

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Reasoning Tasks

SatisfiabilityDoes the KB have a model?

SubsumptionIs it the case that concept C is a subset of D in all models?

• want to understand the relationship between different concepts

• classification: organize atomic concepts into hierarchy

• useful both for visualizing, exploring, and debugging ontologies

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Example: Subsumption


Tree v Plant

Leaf v ∃partOf.Tree

Herbivore ≡ Animal u (∀eats.¬Animal) u (∀eats.¬(∃partOf.Animal))

Animal t ∃partOf.Animal v ¬(Plant t ∃partOf.Plant)

Claim: the above implies Giraffe v Herbivore

Why?

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Reasoning Tasks

SatisfiabilityIs there an interpretation that satisfies all axioms and assertions in the KB?

SubsumptionIs it the case that concept C is always interpreted as a subset of D?

Instance CheckingFind all individuals a that are always interpreted as belonging to C.

• want to infer to which classes the individuals belong

• simple way of querying the ABox (“data”)

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Reasoning Tasks

SatisfiabilityIs there an interpretation that satisfies all axioms and assertions in the KB?

SubsumptionIs it the case that concept C is always interpreted as a subset of D?

Instance CheckingFind all individuals a that are always interpreted as belonging to C.

• want to infer to which classes the individuals belong

• simple way of querying the ABox (“data”)

Conjunctive Query AnsweringFind all tuple of individuals ~a that always satisfy q.

• want to perform database-style querying over DL KBsconjunctive queries = select-project-join queries in SQL

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Example: Conjunctive Queries


MountainGorilla v Gorilla

Gorilla v Primate

MountainGorilla v CriticallyEndangered

keptAt(molly, berlin zoo), MountainGorilla(molly), Zoo(berlin zoo)

Example conjunctive query

q(x, y) = Primate(x) ∧ CriticallyEndangered(x) ∧ keptAt(x, y) ∧ Zoo(y)

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Complexity vs. Expressivity

Extensive investigations of computational complexity of reasoning

• wide range of DLs (from very simple to highly expressive)

• many reasoning tasks, different complexity measures

Nowadays, complexity landscape is well-understood

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• for most DLs, reasoning is provably intractable (EXPTIME-hard or worse)

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• for most DLs, reasoning is provably intractable (EXPTIME-hard or worse)

• useful tractable DLs have been identified

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Example: Complexity vs. Expressivity

“Standard” description logic ALC

C := > | A | ¬C | C u C | C t C | ∃r.C | ∀r.C

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C := > | A | ¬C | C u C | C t C | ∃r.C | ∀r.C EXPTIME-complete

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Same holds for ALCI (= ALC + inverse roles r−)

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“Lightweight” description logic EL

C := > | A | C u C | ∃r.C

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C := > | A | C u C | ∃r.C PTIME-complete

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What about ELI (= EL + inverse roles r−) ?

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What about ELI (= EL + inverse roles r−) ? EXPTIME-complete

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Main Reasoning Paradigms

Tableau method (used for expressive DLs)

• Idea: to show that T |= C v D, try to build a countermodel, i.e.a model of T that contains an individual in C u ¬D

• Shows good performance on reasonably-sized ontologies

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Forward chaining (aka saturation, consequence-based reasoning)

• Works for Horn DLs (e.g. EL), with no disjunction

• Apply the axioms as rules to generate the consequence of the KB,check if produce desired subsumption / assertion

• Can classify SNOMED in a few seconds!

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Forward chaining (aka saturation, consequence-based reasoning)

• Works for Horn DLs (e.g. EL), with no disjunction

• Apply the axioms as rules to generate the consequence of the KB,check if produce desired subsumption / assertion

• Can classify SNOMED in a few seconds!

Backward chaining (aka query rewriting)

• Used for conjunctive query answering in DL-Lite family of lightweight DLs

• Allows us to reduce query answering in DLs to database querying

Primate(x) ; Gorilla(x) ∨ Chimpanzee(x) ∨ . . . ∨ Lemur(x)

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Current Topics in DL Research

Improving scalability of DL reasoners

• need to be able to handle very large KBs (esp. ABoxes ∼ datasets)

• much work focuses on conjunctive queries, lightweight DLs (EL, DL-Lite)

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Reasoning support for building and maintaining ontologies

• explanation (locating source of inconsistencies / undesired subsumptions)

• modularity (how to extract relevant subset of an ontology)

• smart versioning tools (syntactic vs. semantic difference)

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Reasoning support for building and maintaining ontologies

• explanation (locating source of inconsistencies / undesired subsumptions)

• modularity (how to extract relevant subset of an ontology)

• smart versioning tools (syntactic vs. semantic difference)

Capturing and reasoning with “imperfect” knowledge

• vagueness (what does it mean to be “tall”?)

• contextual information (“tall for an elephant” or “tall for a building”?)

• defeasible reasoning (“birds usually fly, but penguins don’t”)

• inconsistency-tolerance (draw sensible inferences despite contradictions)

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Thanks for your attention!

Questions?

meghyn slides-hse-2014

Science

ontologies basis

ontologies good

galen hse

galen specialized ontologies

queries hse

information hse

wikipedia hse

experimental data hse