Introduction to Description Logic and Ontology

Languages

Jidi (Judy) ZhaoApril 21, 2023

CS6999 Presentation

Talk Outline

• Introduction to Ontologies• Introduction to Description Logic

(DL)• Reasoning in DL• Introduction to Ontology

Languages: OWL• Extensions of DL and Research

Challenges2

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What is an ontology? Many definitions have been given:

from Philosophy: “a systematic explanation of being”

Neches gives some guidelines: “…defines the basic terms and relations including the vocabulary of a topic area as well as the rules for combining terms and relations to define extensions to the vocabulary.”

Gruber, the most quoted: “…an explicit specification of a conceptualization”

An ontology defines the concepts used to describe and represent an area of knowledge, as well as relations among them.

Types of Ontologies Top-level Ontologies

The Standard Upper Ontology (SUO): http://suo.ieee.org/

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Types of Ontologies Top-level Ontologies

The Standard Upper Ontology (SUO): http://suo.ieee.org/

WordNet:http://wordnet.princeton.edu/

Sowa’s top-level ontology Cyc’s upper ontology

Domain Ontologies E-commerce Medicine Engineering Enterprise Chemistry ….

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Thing

Living Nonliving

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Methodologies for Ontology Engineering

Building domain ontologies from huge ontologies (SENSUS, Cyc, AKT,…)

OTK (On-To-Knowledge) Methodology Univ. of Karlsruhe

Methontology Univ. Politecnica de Madrid

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Methontology: A Methodology for Building Ontologies

Methontology Ontology Development Process Life Cycle (Fernández-López et al., 1997;1999)

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Tools for Ontology Engineering

OilEd from University of Manchester http://oiled.man.ac.uk/

Ontolingua from KSL (Stanford University) http://www-ksl.stanford.edu

OntoSaurus from ISI (USA) http://www.isi.edu/isd/ontosaurus.html

OntoEdit from Karlsrhue Univ. http://ontoserver.aifb.unikarlsruhe.de/ontoedit/

Protégé from SMI (Stanford University) http://protege.stanford.edu/

WebOnto from KMI (Open University) http://kmi.open.ac.uk/projects/webonto/

WebODE from UPM http://webode.dia.fi.upm.es/webODE/

KAON from AIFB and FZI at the University of Karlsruhe http://kaon.semanticweb.org/

Talk Outline

• Introduction to Ontologies• Introduction to Description Logic

(DL)• Reasoning in DL• Introduction to Ontology

Languages: OWL• Extensions of DL and Research

Challenges9

Description Logic Brachman and Levesque [1984] “there is a

tradeoff between the expressiveness of a representation language and the difficulty of reasoning over the representations built using that language”.

The more expressive the language, the harder the reasoning.

Description Logics overcome the ambiguities of early semantic networks

and frames first realized in the system KL-One [Brachman and

Schmolze, 1985] Well-studied and decidable (most DL languages) Tight coupling between theory and practice

Architecture of a DL System

from DL Handbook

DL Basics Concepts (unary predicates/formulae with one free

variable) E.g., Person, Female

Roles (binary predicates/formulae with two free variables)

E.g., hasChild Individuals (constants)

E.g., Mary, John Constructors

Uniont, Intersectionu Exists restriction9: 9hasChild.Doctor Value restriction8: 8hasChild.Doctor Complement /negation:: :Mother Number restriction ≥n, ≤n Inverse role (-): isChildOf ≡ hasChild–

transitive role (+): hasSister Role hierarchy : hasDaughter v hasChild

Axioms Subsumptionv: MothervParent Assertion: Mary: Mother, Mary hasChild John12

What does 8 R.C and 9 R.C mean?

hasPet

A Fido

A Fluffy

B Tabby

C Rover

C Flip

Dog

Fido

Rover

Flip

Cat

Fluffy

Tabby

A DogLover is someone whose pets are all dogs, in this case {C}DogLover = 8 hasPet.Dog{p | 8 a, (p, a) 2 hasPet ! a 2 Dog}Also writen more simply as {p | hasPet(p, a) ! Dog(a) }

A DogLiker is someone who owns a dog , in this case {A, C}DogLiker = 9 hasPet.Dog{p | hasPet(p, a) Æ Dog(a) }This slide is from Dr. Bruce Spencer’s slides (2007).

The DL Family Smallest propositionally closed DL is ALC

Concepts constructed using boolean operatorst , u , :

plus restricted quantifiers 9 , 8 Only atomic roles

E.g.,Person u 8hasChild.(Doctor t 9hasChild.Doctor)

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The DL Family (cont.) S often used for ALC extended with transitive

roles (R+) Additional letters indicate other extensions, e.g.:

H for role hierarchy O for nominals (e.g., {Mary, John}) I for inverse roles N for number restrictions Q for qualified number restrictions (e.g.,

≥2hasChild.Doctor) R for limited complex role inclusion axioms, role

disjointness ALC+ transitive role (R+)+role hierarchy (H) +O

+ I + Q = SHOIQ

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DL Semantics

Semantics given by standard FO model theory The vocabulary is the set of names (consist of concepts

and roles ) we use in our model of (part of) the world {Daisy, Cow, Animal, Person, Car, drives, …}

An interpretation I is a tuple (I, •I) I is the domain (a set) •I is a mapping that maps:

Names of objects (individuals/constants) to elements of I

Names of unary predicates (classes/concepts) to subsets of I

Names of binary predicates (properties/roles) to subsets of I ×I

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DL Semantics(adapted from Horrocks 2006)

Interpretation domain IInterpretation function •I

Individuals iI 2 I

Mary

John

Concepts CI µ I

Teacher

Student

Car

Roles RI µ I × I

hasChild

owns

(Teacher u Student)17

A Knowledge Base (KB) <T,A>= a Tbox + an Abox

A TBox (terminology) is a set of inclusion axioms and equivalence axioms

the vocabulary of an application domain e.g.: { Mother v Person, GrandMother ≡ Person u

9hasChild.Parent }

An ABox (Assertion) is a set of assertions about individuals

about named individuals in terms of this vocabulary e.g.: {Mary: Mother, Anita hasChild Mary}

DL Knowledge Bases

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Talk Outline

• Introduction to Ontologies• Introduction to Description Logic

(DL)• Reasoning in DL• Introduction to Ontology

Languages: OWL• Extensions of DL and Research

Challenges19

Tableau Reasoning (1) Key reasoning tasks

Satisfiability: asat(A), whether the assertions in a KB have a model Instance checking: C(a)? Concept satisfiability: C? Retrieval: retrieve a set of individuals that instantiate C Subsumption: B v A ?

A subsumes B if every individual of concept B is also of concept A. Equivalence: A≡B? , B v A ? And A v B?

Reasoning tasks reducible to KB (un)satisfiability: asat(A) Instance checking: instance(a, C, A) , :asat (A [ {a: :C}) Concept satisfiability: sat(C) , asat(A [ {a:C}) Concept subsumption:

C v D w.r.t. KB A , A [ {:D u C} is not satisfiable , :asat(A [ {a::D u C})

Retrieval:check each individual in the Abox, reducible to instance checking

DL systems typically use tableau algorithms to decide the satisfiability (consistency) of KB

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CS6795 Semantic Web Techniques21

Tableau Reasoning (2) Tableau algorithms work by trying to construct a concrete

example (model) consistent with KB. A KB A is satisfiable iff a fully expanded clash-free graph is

constructed. Tableau reasoning contains a set of completion rules

operating on constraint sets or tableau Clash: a clash is an obvious contradiction, e.g., A(x), :A(x) Proof procedure:

start from assertions about individuals (ABox axioms) unfold the TBox so that atomic concepts only appear on the right

side of axioms transform all concepts into negation normal form (i.e. negation

only occurs in front of atomic concept names): : (C u D) ! :C t :D : 9R.C ! 8 R.:C

apply completion rules in arbitrary order as long as possible stops when a clash is found terminates if no completion rule is applicable A KB is satisfiable iff a clash-free tableau can be derived

completion rules

Tableau Reasoning (3)

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Tableau Reasoning (5): Concept SubsumptionKB:

Reasoning task: mother v woman?

Exercise: Is the concept : woman u mother satisfiable?

Tableau Reasoning (4): asat(A) E.g., KB:

{HappyParent≡Person u ∀hasChild.(Doctor t 9hasChild.Doctor), John:HappyParent, John hasChild Mary, Mary: :Doctor, Wendy hasChild Mary, Wendy marriedTo John}

from Harrock, 2006

Person∀hasChild.(Doctor t ∃hasChild.Doctor)

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Tableau Reasoning (6) Some completion rules are nondeterministic (e.g.,

9 , ≤ ). Blocking Strategies are often needed to ensure

termination. E.g., KB:

{Person v 9hasParent.Person, John:Person}

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Tableau Reasoning (7)

In general, (representation of) model consists of: Named individuals

forming arbitrary directed graph

Trees of anonymous individuals rooted in named individuals

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Similar tableaux expansions can be designed for more expressive DL languages.

A tableau algorithm has to meet three requirements:

Soundness: if a complete and clash-free graph is found by the algorithm, we can construct a model.

Completeness: Given a model, the algorithm can always find an complete and clash-free graph

Termination: the algorithm can terminate in finite steps with specific result.

Tableau Reasoning (8)

Software for DL Reasoning

Pellet

KAON2 CEL

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Efficiency of Tableau Reasoning

I can’t find an efficient algorithm, but neither can all these famous people.

NP-Complete Cartoons, http://max.cs.kzoo.edu/~kschultz/CS510/ClassPresentations/NPCartoons.html

Talk Outline

• Introduction to Ontologies• Introduction to Description Logic

(DL)• Reasoning in DL• Introduction to Ontology

Languages: OWL• Extensions of DL and Research

Challenges30

Traditional Ontology Languages Ontolingua and KIF LOOM OKBC F-logic

Ontology Markup Languages SHOE RDF and RDF Schema OIL DAML+OIL OWL

Ontology Languages

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Semantic Web led to requirement for a “web ontology language”

set up Web-Ontology (WebOnt) Working Group

WebOnt developed OWL language OWL based on earlier languages OIL and

DAML+OIL OWL now a W3C recommendation

OIL, DAML+OIL and OWL based on Description Logic

The Web Ontology Language OWL

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OWL Three species of OWL

OWL full is the union of OWL syntax and RDF

OWL DL restricted to FOL fragment (is equivalent to SHOIN(Dn) DL)

OWL Lite is an “easier to implement” subset of OWL DL

OWL DL Benefits from many years of DL research Well defined semantics Formal properties well understood

(complexity, decidability) Known reasoning algorithms Implemented systems (highly optimised)

Adapted from ENC 2004 Tutorial by Peter F. Patel-Schneider

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OWL RDF/XML Exchange Syntax

<owl:Class> <owl:intersectionOf rdf:parseType=“collection"> <owl:Class rdf:about="#Person"/> <owl:Restriction> <owl:onProperty rdf:resource="#hasChild"/> <owl:allValuesFrom> <owl:unionOf rdf:parseType=“collection"> <owl:Class rdf:about="#Doctor"/> <owl:Restriction> <owl:onProperty

rdf:resource="#hasChild"/> <owl:someValuesFrom

rdf:resource="#Doctor"/> </owl:Restriction> </owl:unionOf> </owl:allValuesFrom> </owl:Restriction> </owl:intersectionOf></owl:Class>

E.g., Person u ∀hasChild.(Doctor t ∃hasChild.Doctor):

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Class/Concept Constructors

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Ontology Axioms

OWL ontology equivalent to DL KB (Tbox + Abox)

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Talk Outline

• Introduction to Ontologies• Introduction to Description Logic

(DL)• Reasoning in DL• Introduction to Ontology

Languages: OWL• Extensions of DL and Research

Challenges37

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Extensions of DL

Combinations of DL and Logic Programs (LP) Uncertainty extension of DL Concrete domain constraints Modal, epistemic, and temporal operators Open world vs. close world …..

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Venn Diagram of DL, LP, and FOC

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Motivation(1)

DL cannot represent “more than one free variable at a time”.

(1) A rule involving multiple variables. E.g.,Man(?X) ∧ Woman(?Y) !PotentialFriendshipBetween(?X,?Y).

(2) Chaining to derive values of Properties. E.g., Father(?X,?Y) ∧ Father(?Y,?Z)! Grandfather(?X,?Z). (not allowed in SHOIN)

Work(?X, ?Y) ∧ Live(?X, ?Z) ∧ Loc(?Y,?W) ∧ Loc(?Z,?W)!HomeWorker(?X).

X

YWork

Z

Live W

Loc

Loc

Motivation(2)

•Horn Logic cannot represent a (1) disjunction or (2) existential in the head.•(1) State a subclass of a complex class expression which is a disjunction. E.g.,

(Human u Adult) v (Man t Woman)•(2) State a subclass of a complex class expression which is an existential. E.g.,

Radio v 9hasPart.Tuner41

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Different approaches1. Approaches reducing description logics to logic programs

A. DLPB. OWL 2 RL

2. Homogeneous approachesA. OWL RulesB. SWRL

3. Hybrid approaches accessing description logic through queries in logic programsA. AL-Log

Uncertainty extension of DL Handling uncertain knowledge is becoming a critical

research direction for the (Semantic) Web. knowledge on the Web is often uncertain and imprecise. E.g., many concepts needed in business domain

ontology modeling lack well-defined boundaries or, precisely defined criteria of relationship between concepts

Domain modeling and Ontology reasoning Quantify degree of an individual belonging to a class Quantify degree of subsumption between a class and its

subclasses Concept mapping between ontologies

Quantify degree of alignment between classes of two ontologies

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URW3 Situation Report: uncertainty ontology

URW3

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Probability, Possibility and Fuzzy logic

Probabilistic Description Logic: Statistical information e.g. John is a student with the probability 0.6

and a teacher with the probability 0.4 Fuzzy Description Logic:

Express vagueness and imprecision e.g. John is tall with the degree of truth 0.9

Possibilistic Description Logic: Particular rankings and preferences e.g. John prefers an ice cream to a beer

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Research Challenges Syntax and Semantics Decidability Reasoning algorithms for

possible extensions Soundness and completeness Complexity/efficiency Effective methods for

reasoning under uncertainty

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Questions?