description logic motivation for description logic (dl) dl and first order logic (fol) what is...

Description Logic

•Motivation for Description Logic (DL)

•DL and First Order Logic (FOL)

•What is Description Logic?

•DL Architecture

•Reasoning Structures

•Applications of DL

1Bart Gajderowicz, Ryerson University, Dept. of Computer Science

Introduction

March-03-09

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Description LogicMotivation

•Any area with information overload.

•Where amount of declarative information to be processed manually exceeds that of human abilities.

•Even if a large number of resources could be found, it would take an impractical number of human hours to process.

Bart Gajderowicz, Ryerson University, Dept. of Computer ScienceMarch-03-09

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•Need for greater push for standardization of terminologies:

•Semantic Web

•Health Services

•Bio-informatics


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•We would like to model complex concepts using modern modeling structures.

•OWL, Web Ontology Language, is a computer processible syntax for expressive Description Logic.


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•Propositional Logic (PL) has well defined operators and axioms creating atomic formulas.

•LHS and RHS Equivalences, such as:Commutativity ϕ ∨ ψ Ξ Ψ ∨ϕ

Tautology ϕ ∨Τ Ξ Τ

Unsatisfiability ϕ ∧ Ξ

Implication ϕ -> ψ Ξ ϕ ∨ ψ


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•We can already do a lot with PL

But …

• No access to the structure of atomic statements.

Bart Gajderowicz, Ryerson University, Dept. of Computer Science

Consider Statements

•Mary is female

•John is male

•Mary and John are siblings

Consider Propositions

•Mary-is-female

•John-is-male

•Mary-and-John-are-siblings

March-03-09

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•First Order Logic (FOL)

•The language of Predicate Logic

•Atomic formulas are sets of predicates interpreted as relations between elements of a domain.


Consider Predicates

•Female (Mary)

•Male (John)

•Siblings (Mary, John)March-03-09

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•Quantifiers:Existential x. ϕ Ξ For some x, ϕ is true

Universal x. ϕ Ξ For all x, ϕ is true

•Predicate Logic and FOL allows for formulas, arguments, free and bounded variables, and constants.

•Allows for literals, atomic formulas (relations), and negation.

•Well formed formulas, truth connectives, quantifiers over bounded variables.

March-03-09

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•Represent relationships between objects.

•More complex relationships can be defined in terms of more basic relationships or objects.

Woman Ξ Person FemaleΠ

Man Ξ Person Π Woman


Woman-is-a-Person-who-is-a-Female

Man-is-a-Person-who-is-not-a-Woman

verses

March-03-09

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Description Logic… and First Order Logic

•Description Logic (DL) is derived from FOL.

•DL Relationships are limited to 1-argument predicates called concepts and 2-argument predicates called roles.

FOLPredicates

DL Relationships

Description

unary concepts Woman( x )

binary roles Woman( x ).hasChild( y , x )


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•Concept C and its translation π(C)(x) are equivalent in the interpretation I = (ΔI, . I ) and all a ΔI , where Δ is the domain, and I is the interpretation.

•Consider:

a CI iff I |= π(C)(a).

a is an instance of an interpretation of WOMAN iff

a FOL translation of this concept is true for "a"

in the interpretation.


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•Translation of roles in FOL takes the form of a predicate with two arguments.

•Roles relate two concepts to each other.

•Roles cannot occur by themselves; only as a part of a complex concept.

•This puts restrictions on syntax of FOL formulas that are allowed.

•Important to guarantee decidability and good computational properties of DLs.


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•Translations require mappings on two variables

: π(x) and π(y)

πx(R:C) = y:R(y; x) ∧ πy(C)

Mother(x) is true iff x truth values match those of the definition of Woman with some y a child.


•Unary Concept translations

•Binary Role translation

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Description Logic First Order Logic

πx(A) A(x)

πx(C Π D) πx(C) ∧ πx(D)

πx(C D) πx(C) ∨ πx(D)

πx(R.C) y.R(x; y) ∧ πy(C)

πx( R.C) y.R(x; y) πy(C)


πy(A) A(y)

πy(C Π D) πy(C) ∧ πy(D)

πy(C D) πy(C) ∨ πy(D)

πy(R.C) y.R(y, x) ∧ πx(C)

πy(R.C) y.R(y, x) πx(C)


•Counting Quantifier translations

•Some x with at least n y’s

•A simple existential quantifier is insufficient:

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πx(≥ n R) ≥n y.R(x, y)

πx(≤ n R) ≤n y.R(x, y)


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Description LogicWhat is Description Logic?

•Sub-language of First Order Logic

•Model hierarchical and complex relationships.

•Collection of these statements is called TBox.

Man Ξ Person Π Woman

Father Ξ (Person Π Woman) Π hasChild:Person

Father Ξ (Man) Π hasChild:Person


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•The TBox is known as an ONTOLOGY (from Greek: science of beings, or science of things that exist).

•Ontology used to be a sub-discipline of philosophy.

•It is now at forefront of real world applications.


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•Models systems which are complex through size and cyclic definitions.

•“Man who has Only Sons” (Mos)

Momo Ξ Man Π hasChild.Momo

•“Man who has Only Male Offspring” (Momo)

Mos Ξ Man Π hasChild.Man


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•DL handles cyclic definitions through fixpoints

•Least fixpoint is the lowest x value where f(x) = x

•Greatest fixpoint is the greatest x value where f(x) = x

•Halting depends on the domain of x

•Example: a binary tree’s least-point are the nodes with only terminal nodes at the branches.

BinaryTree Ξ Tree Π (≤ 2 has-branch) Π (has-branch.BinaryTree)

BinaryEndTree Ξ Tree Π (≤ 2 has-branch) Π (has-branch.BinaryTree)


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•DL has variable-free syntax

•variables are implicitly understood

•There can be only two of them: x or y, but they can be reused.

•DL concepts denote sets of individuals.

•For concept C, and translation I.

a CI iff I |= π(C)(a).


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Description LogicDL Architecture

•Basic description language AL

•AL = attribute language

•A and B are atomic concepts

•C and D are concept descriptions

•R represents atomic roles

•Other Languages in this family are extensions of AL.


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•Basic description language AL

C, D A atomic concept

C, D Τ universal concept

C, D bottom concept

C, D A atomic negation

C, D C Π D intersection

C, D R. value restriction

C, D R.Τ Limited existential quantification


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•Atomic Concepts and Roles

Person Π Female Person who is not a female

Person Π hasChild.Τ Person with at least 1 child

Person Π hasChild. Person with no children

Person Π hasChild.Female Person with at least one daughter

Person Π hasChild.Female Person with only daughters

Mother Father Person who is a Parent


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•Limited to two-variable structures.

•Handles acyclic, as well as cyclic statements.

•Acyclic: LHS has no terms which appear in RHS

Momo Ξ Man Π hasChild.Momo

•Cyclic: LHS has a term which appears in RHS

Mos Ξ Man Π hasChild.Man


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•DL defines statements by stating assertions about a subject using its vocabulary.

•Type of statements which make up a DL Ontology are the ABox and the TBox formalisms.

•TBox = terminology used in a domain.

•ABox = assertions which make statements about particular object instances in that domain, using the terminology in TBox.


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TBoxTBox

ABoxABox

Description Language

Description Language ReasoningReasoning

KnowledgebaseKnowledgebase

ApplicationPrograms

Rules


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TBox

ABox

MotherWithoutDaughter(MARY) Father(PETER)

hasChild(MARY, PETER) hasChild(PETER, HARRY)

hasChild(MARY, PAUL)

Father Ξ Man Π hasChild.Person

Mother Ξ Woman Π hasChild.Person

MotherWithoutDaughter Ξ Mother Π hasChild.¬Woman


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Description LogicReasoning Structures

•DL Knowledgebase stores more then definitions and assertions.

•DL contains implicit knowledge which can be made explicit through inferences.

•Inferences are made through a set of rules of inference.

•TBox contains these statement.


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•Modeling a domain requires defining a terminology, say T, which satisfies all previously defined statements about that domain.

•Any new interpretation must satisfy the axioms of T, and result in a non-empty set of concepts.

•Otherwise it is unsatisfiable.


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Satisfiability: A concept C is satisfiable with respect to T if there exists a model I of T such that CI is nonempty:

I |= C

Subsumption: A concept C is subsumed by a concept D with respect to T if CI DI for every model I of T:

C T D T |= (C D)|Π|Π |Π|Π


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•Subsumption con’t

•LHS is subsumed by the more general terms in RHS:

Mother Woman

Father Man

Brunch Meal

Sport Car Vehicle|Π|Π

|Π|Π|Π|Π

|Π|Π


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Equivalence: Two concepts C and D are equivalent if C subsumes D and D subsumes C, with respect to T if CI = DI for every model I of T:

[ C ΞT D and T |= C Ξ D ] [ C D and D C ]

Disjointness: Two concepts C and D are disjoint with respect to T if:

CI DI = |Π|Π |Π|Π


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Description LogicApplications of DL

•Different extensions to AL-languages.

ALU Union (C D) I

ALε Full existential quantification

R.C

ALN Number restrictions ≥ n R and ≤ n R

ALFL- Non-atomic negation A

•AL in combination with U, ε, N, and FL- form 8 languages which are pairwise non-equivalent.


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•The special extension ALUε , represented by ALC, describe a commonly used language with axioms for union and existential quantification.

•The ALC extension is the basis for SHOIN, which in turn is the basis for the Web Ontology Language OWL

• OWL is formally recommended by the World Wide Web Consortium, W3C.


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•OWL is a markup language given the tasks of providing syntax used to model ontologies on the web.

•The Semantic Web is an attempt to organize the plethora of information available on the internet


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•On the web, instead of showing all available links and infinite levels of sub-topics, a high level “description” is given.

•Description identifies the subject of the content on the web.


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•Cyclic self-referenced nodes are contained within a higher level concept.

•Bound by fixpoints.

•This abstraction inherited from DL ensures satisfiability in the system.


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<owl:Class rdf:ID="Daugther"> <owl:equivalentClass> <owl:Class> <owl:intersectionOf rdf:parseType="Collection"> <owl:Class rdf:ID="Child"/> <owl:Class rdf:ID="Woman"/> </owl:intersectionOf> </owl:Class> </owl:equivalentClass> <owl:disjointWith> <owl:Class rdf:ID="Son"/> </owl:disjointWith></owl:Class>

•OWL Example: “Daugther” (misspelling preserved)

- http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl


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<owl:Class rdf:ID="Daugther"> <owl:equivalentClass> <owl:Class> <owl:intersectionOf rdf:parseType="Collection"> <owl:Class rdf:ID="Child"/> <owl:Class rdf:ID="Woman"/> </owl:intersectionOf> </owl:Class> </owl:equivalentClass> <owl:disjointWith> <owl:Class rdf:ID="Son"/> </owl:disjointWith></owl:Class>

•OWL Example: “Daugther” (misspelling preserved)

- http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl

Daugther Ξ Child Π Woman

Daugther Π Son Ξ


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•Description Logic can be analyzed for satisfiability with Theorem Provers (Prover9 / Vampire / Otter)

•Tableau Calculus

•Tableaux Calculus is a decision procedure solving the problem of satisafiability.

•If a formula is satisfiable, the procedure will constructively exhibit a model of the formula.


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•Tableau Calculus con’t

•The basic idea is to incrementally build the model by looking at the formula, by decomposing it in a top/down fashion.

•The procedure exhaustively looks at all the possibilities, so that it can eventually prove that no model could be found for unsatisfiable formulas.

•Unsatifiable formula contains a contradiction such as: p(a) Π p(a)


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•Closed vs Open World Semantics

•Connections to external databases require database schema mappings.

•Communicating TBoxes need to be mapped in order for the data in ABoxes to be meaningful.

•Closed systems represent internal databases where no mappings are required.

•A complete theory in a closed system may not be complete in an open world semantic


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Description LogicReferences

Enrico Franconi, DESCRIPTION LOGICS - Tutorial Course Information, Faculty of Computer Science, Free University of Bozen-Bolzano, Italy; http://www.inf.unibz.it/~franconi/dl/course/

Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, Peter F. Patel-Schneider (Eds.): The Description Logic Handbook: Theory,

Implementation, and Applications. Cambridge University Press 2003

Christine Golbreich, A SWRL/OWL Demo Ontology About Family Relationships, OWL Ontologies, ProtegeOntologiesLibrary;http://protege.cim3.net/file/pub/ontologies/family.swrl.owl/family.swrl.owl


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Description LogicThe End

Thank you

Q & A


Bart [email protected]/~bgajdero

March-03-09

description logic motivation for description logic (dl) dl and first order logic (fol) what is...

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