on logical foundation of the semantic web...national university of singapore, 2005 d. lucanu: on...

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On Logical Foundation of theSemantic Web

An institution-based approach

Dorel Lucanu

{dlucanu}@info.uaic.ro

“Alexandru Ioan Cuza” University, Romania

National university of Singapore, 2005 D. Lucanu: On Logical Foundation of the Semantic Web – p. 1/39

Outline

motivation

institutions

Semantic Web (SW) languagesRDF and RDF SchemaOWLSWRLSWRL FOL

relationships between SW languages

conclusion


Motivation

an integrating mathematical structure for Semantic Weblanguages

soundness of the reasoners for Web ontologies

translating Web ontologies into other formalisms

“NOTE: There is a strong correspondence between thesemantics for OWL DL defined in this section(RDF-based) and the Direct Model-Theoretic Semanticsdefined in . . . . If, however, any conflict should ever arisebetween these two forms, then the DirectModel-Theoretic Semantics takes precedence.” (OWLWeb Ontology Language Semantics and Abstract Syntax Section 5.RDF-Compatible Model-Theoretic Semantics,http://www.w3.org/TR/owl-semantics/rdfs.html)


Institutions

formalize the notion of "a logic"

study the properties of a logicrepresentationimplementationtranslation of logics


Institutions: Ingredients

signatures Σ

formalize vocabulariesExample: (Many Sorted) First Order Logic withEquality (FOLEQ)

Booleans Integers

sorts Bool Int

constants false true : Bool 0 : Int

operations and : Bool Bool→Bool succ pred : Int→Int

× : Int Int→Int



signatures (continued)

are organized as categories, where the signaturemorphisms formalizes the translations betweenvocabularies

φ : Booleans → Integers

Bool 7→ Intfalse 7→ 0, true 7→ succ(0)and 7→ ×



sentencesabstracts the notion of formula

is formalized as a functorsen : Sign → Set

sen(Σ) = the set of well-formed first-order formulas builtover Σ(∀ y : Int) y × 0 = 0(∀ y : Bool) y and false = false

if φ : Σ → Σ′ then sen(φ) : sen(Σ) → sen(Σ′)

sen(φ)(y and false = false) = (y × 0 = 0)

Notation: sen(φ)(s)not= φ(s)



modelsare interpretations of the syntactical constructs

are parameterized over signatures: Mod(Σ) = thecategory of the Σ-models (interpretations of thevovabulary Σ)

are formalized as a functor:Mod : Signop → Cat

if φ : Σ → Σ′ then Mod(φop) : Mod(Σ′) → Mod(Σ)

Z is a model for IntegersMod(φop)(Z) is Z viewed as a model for Booleans

Notation: Mod(φop)(M)not= M�φ



satisfaction relationrelates the models and the sentences: M |=Σ s

where M is Σ-model and s is a Σ-sentence

it is the subject of the satisfaction condition whichexpresses the invariance of truth under change ofnotation

M ′ |=Σ′ φ(s) iff M ′�φ|=Σ s

where φ : Σ → Σ′, M ′ is a Σ′-model, and s is aΣ-sentence

Z |=Integers y × 0 = 0 iffZ�φ|=Booleans y and false = false


Institutions: Specifications and Theories

a specification is a pair (Σ, S), where Σ is a signatureand S is a set of sentences

semantical consequences: (Σ, S) |= s iff(∀M)(M |=Σ S ⇒ M |=Σ s)

a theory is a specfication (Σ, S) s.t.(∀ s)((Σ, S) |= s ⇒ s ∈ S

the inclusion Th → Spec is an equivalence of categories

theoroidal institutions:signatures are theoriesa (Σ, S)-sentence is a Σ-sentence(Σ, S)-models are Σ-models satisfying S

M |=(Σ,S) s iff M |=Σ s


Institutions: Properties of interest

theory colimitsthe module expressions are evaluated as colimits oftheories

model amalgamation

expresses the possibility of amalgamation of consistentmodels for different specification modules

liberalityexpresses the possibility of free constructionsgeneralizing the principle of “initial semantics”


Relating Institutions

morphism: capture the way in which a “richer”institution is built over a “simpler” one

comorphism: capture the way in which a “simpler”institution is embedded (encoded) into a “richer” one

both are the subject of a corresponding satisfactioncondition

there exist a variety of definitions for morphisms andvariety of definitions for comorphisms in literature

a prover from the target logic can be used to proveproperties from the source logic only if certainconditions are fulfilled


Institutions: Main references

Introducing Institutions, by J. Goguen and R. Burstall,1984Institutions: Abstract model theory for specification andprogramming, by J. Goguen and R. Burstall, 1992Structuring theories on consequence, by J. Fiadeiroand A.Sernadas - 1988May I Borrow Your Logic?, by M. Cerioli and J.Meseguer,1993Moving Between Logical Systems, Andrzej Tarlecki,1995Institution Morphisms, by J. Goguen and Gr. Rosu,2002Grothendieck Institutions, by R. Diaconescu


Semantic Web

From Semantic Web talk by Tim Berners-Lee at XML 2000


RDF

proposed in October 1997

in February 1999 becomes a W3C recommendation

it is a standard for representing information in the Web

a expression in RDF is a collection of triples, eachconsisting of a subject, a property (predicate), and anobject

propertysubject object


RDF - example<rdf:Description rdf:about=

"http://www-cs-faculty.stanford.edu/˜knuth/"><hasName rdf:resource="Donald Knuth" />

</rdf:Description>

<rdf:Description rdf:about="http://www.amazon.com/exec/.../104-3442396-7552717">

<hasAuthor rdf:resource="http://www-cs-faculty.stanford.edu/˜knuth/" />

</rdf:Description>

http://.../~knuth/hasAuthor

hasName

http://.../...152717

’Donald Knuth’


The institution RDF++

We consider given a datatype D

signatures: Σ = (RR, BN)

sentences: F ::= (s, p, o) | u ≡ v | F ∧ F | ¬ F | (∀ y)F

models: A = (ResA, P ropA, resA, [[ ]]A), whereResA a set of resourcesPropA a set of properties (assume that PropA ⊆ ResA)resA : RR → ResA

[[ ]]A : PropA → P(ResA × (ResA ∪ [[D]]))

satisfaction:A |= (s, p, o) iff resA(p) ∈ PropA and

(resA(s), resA(o)) ∈ [[resA(p)]]AA |= u ≡ v iff resA(u) = resA(v)


Interpretation of the blank nodes

Σ = (RR, BN)a Σ-model A and a Σ-sentence s

φ : Σ → Σ′ = (RR ∪ BN, ∅)A |=Σ s iff there is a Σ′-model A′ s.t.

A′�φ= A and A′ |=Σ′ s

The satisfaction of the RDF graphs:

a RDF graf is a set S of triples

A |=Σ S iff A |=Σ ∧s∈S s

which is not always the same with saying that A satisfies allthe sentences in S


A specification in RDF++

(Σ, S) ={

({bk , dkhp, hasAuthor , hasName}, ∅),

(bk , hasAuthor , dkhp),

(dkhp, hasName,"Donald Knuth")

}


RDF++: properties

SignRDF++ has colimits

e.g., merge:

(RR1 ∩ RR2, ∅) −−−→ (RR1, BN1)yy

(RR2, BN2) −−−→ (RR1 ∪ RR2, BN1∐

BN2)

RDF++ is liberal (free constr. is a generalized Herbrandconstr.)

RDF++ has amalgamation property (Mod(RDF++) preservesfinite limits)


The specification RDFV(RDF Vocabulary)

RR(RDF) ={rdf : type, rdf : Property, rdf : list, rdf : nil, . . .}

BN(RDF) = ∅

S(RDF) = {

(rdf : type, rdf : type, rdf : Property),

(rdf : nil, rdf : type, rdf : List),

(∀ s, p, o)(s, p, o) → (p, rdf : type, rdf : Property),

. . .

}


RDF Schema

proposed in March 1999

is a standard which describes how to use RDF todescribe RDF vocabularies

it is claimed that it is a semantical extension of RDF

introduces the basic primitives for ontology modeling:classes, subclassessubpropertiesdomain, range. . .


RDF Schema: Example

<rdfs:Class rdf:about="Book" />

<rdfs:Class rdf:about="Person" />

<rdfs:Class rdf:about="Author"><rdfs:subClassOf rdf:resource="#Person" />

</rdfs:Class>

<rdf:Property rdf:about="hasAuthor"><rdfs:domain rdf:resource="Book" /><rdfs:range rdf:resource="Author" />

</rdf:Property>


RDFS is a theory!RR(RDFS) = RR(RDF) ∪ {rdfs : Class, rdfs : subClassOf,

rdfs : subPropertyOf, rdfs : domain, . . .}BN(RDFS) = those used in sentences

S(RDFS) = S(RDF) ∪

{

(rdf : type, rdfs : domain, rdfs : Resource),

(rdfs : domain, rdfs : domain, rdf : Property),

(∀ u, v , x, y)(x, rdf : domain, y) ∧ (u, x, v) → (u, rdf : type, y)

(∀ x, y)(x, rdfs : subClassOf, y) → (x, rdf : type, rdfs : Class),

(∀ x, y)(x, rdfs : subClassOf, y) → (y , rdf : type, rdfs : Class),

(∀ u, x, y)(x, rdfs : subClassOf, y) ∧ (u, rdf : type, x) → (u, rdf : type, y),

. . .

}


The institution RDFS

signatures: theory morphisms RDFS → (Σ, S)

(Sign(RDFS) is a comma category)RDFS → (Σ, S) how use RDF to describe RDF vocabularies

sentences: Σ-sentences

models: (Σ, S)-modelsMod(Σ, S) → Mod(RDFS) → ModRDF semantical extension

satisfaction: A |=RDFS→(Σ,S) s iff A |=Σ s

semantics of a class:[[C]]A = {x | A |= (x, rdf : type, C)}

There is a simple theoroidal comorphism from RDFS toRDF++.


OWL

proposed in March 2002a language used to describe Web ontologieshas three levels: OWL LITE, OWL DL, OWL Fullincludes RDF Schemanew items:

makes distinction between individual-valuedproperties and data-valued propertiescardinality restrictionsoperations with classesrestrictions on propertiesontology imports. . .


OWL: Example

each book has at least one author

<owl:Class rdf:ID="Author"><rdfs:subClassOf>

<owl:Restriction><owl:onProperty rdf:resource=

"#hasAuthor" /><owl:minCardinality rdf:datatype=

"#&xsd;nonNegativeInteger">1</owl:minCardinality>

</owl:Restriction></rdfs:subClassOf>

</owl:Class>


OWL is also a theory!RR(OWL) = RR(RDFS)∪{owl : Thing, owl : Class, owl : subClassOf,

owl : ObjectPropertyO, owl : DatatypeProperty, . . .}BN(OWL) = those used in sentences

S(OWL) = S(RDFS) ∪

{

(owl : Nothing, rdf : type, owl : Class),

(owl : Thing, rdf : type, owl : Class),

¬(∃ x)(x, rdf : type, owl : Nothing),

(∀ x, C)(x, rdf : type, C) ∧ (C, rdf : type, owl : Class) →

(x, rdf : type, owl : Thing), . . .

}

There is a forgetful morphism from OWL to RDFS.There is a simple theoroidal comorphism from OWL to RDF++.


OWL DL hides some vocabulary itemsΣ(OWLDL) = hide

rdf:type, rdf: Property,...,owl: TransitiveProperty,...

inΣ(OWL)

. . . and adds some new constraints:

DLCONSTR = {

(∀ x, C)(x, rdf : type, owl : Thing) ∧ (C, rdf : type, owl : Class) →

¬(x ≡ C),

. . .

}

φ : Σ(OWLDL) ↪→ Σ(OWL)Mod(OWLDL) = {A�φ | A ∈ Mod(OWL) ∧ A |= DLCONSTR}


The institution OWL

signatures: theory morphisms OWL → (Σ, S)

(Sign(OWL) is a comma category)

sentences: Σ-sentences

models: (Σ, S)-models

satisfaction: A |=(Σ,S) s iff A |=Σ s

There is a forgetful morphism from OWL to RDFS.

There is a simple theoroidal comorphism from OWL to RDF++.


OWL: problems with amalgamation

PETS + BOOKS

PETS

-

BOOKS

�

can we amalgamate a PETS-model A1 andBOOKS-model A2 in a PETS+BOOKS model?

NO if [[owl : Thing]]A16= [[owl : Thing]]A2


OWL: problems with amalgamation

solution: transform such a diagram into a pushout

PETS + BOOKS

PETS

-

BOOKS

�

∅

φ2

-

�

φ1

we have to consider a ∅-model A0

A1 and A2 are consistent iff A1�φ1= A = A2�φ2


SWRL

proposed in November 2004extends OWL with Horn rulesexample: citation implies not self-citation

<ruleml:imp><ruleml:_body>

<swrlx:individualPropertyAtom swrlx:property="writtenBy"><ruleml:var>x1</ruleml:var><ruleml:var>x2</ruleml:var>

</swrlx:individualPropertyAtom>...

</ruleml:_body><ruleml:_head>

...</ruleml:_head>

</ruleml:imp>


The institution SWRL

signatures: OWL signatures

sentences:

writtenBy(x1, x2) ∧ citedBy(x1, x3) → x2 6= x3.

models: OWL models

satisfaction: as in OWL and HornLog

There is a forgetful morphism from SWRL to OWL.

There is a simple theoroidal comorphism from SWRL toRDF++.


SWRL FOLproposed in November 2004extends OWL with first-order formulasexample: any cited author has written a paper which iscited by someone else

<Assertion owlx:name="Example"><Forall>

<ruleml:var>x1</ruleml:var><Implies><swrlx:classAtom owlx:name="CitedAuthor"><owlx:Class owlx:name="CitedAuthor" /><ruleml:var>x1</ruleml:var>

</swrlx:classAtom><Exists>...

</Exists></Implies>

</Forall></Assertion


The institution SWRLFOL

signatures: OWL signatures

sentences:

(∀ x1)CitedAuthor(x1) → (∃ x2, x3)writtenBy(x2, x1) ∧

citedBy(x2, x3)

models: OWL models

satisfaction: as in OWL and FOL

There is a forgetful morphism from SWRLFOL to SWRL.

There is a simple theoroidal comorphism from SWRLFOL toRDF++.


Relationships between SW logics

SWRLFOL −−−→ RDF++y =

SWRL −−−→ RDF++y =

OWL −−−→ RDF++y =

RDFS −−−→ RDF++

−−−→ morphism

−−−→ comorphism


Conclusioncontributions

the institution RDF++RDFS, OWL, SWRL, and SWRLFOL are in facttheories in RDF++the institutions RDFS, OWL, SWRL, and SWRLFOL definedas particular theoroidal institutionsthe relationships between these institutions

advantages:a rigurous and systematic approach of the logicsunderlying SW languagesan important step towards structuring and re-usingontology partsa solid framework for relating SW languages with otherformalisms and for proving the soundness of thereasoners


Questions?

Thank you!


on logical foundation of the semantic web...national university of singapore, 2005 d. lucanu: on...

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