ML Wiki

Descriptive Logic

Descriptive Logic (DL)

• Formal basis for OWL
• FOL give formal definitions of RDFS and OWL statements
• Classes - unary predicates
• Properties - binary predicates
• but inference in FOL is not decidable
• DL is a subset of FOL where many interesting properties are decidable
• so it allows reasoning
• exactly what needed for Ontologies

Mapping between OWL, FOL and DL:

DL Knowledge Base

A DL knowledge base consists of:

• intentional part (TBox, $T$) - ontology: classes and concepts
• assertional part (ABox, $A$) - data
• So a DL KB is a tuple $\langle T, A\rangle$

In Semantic Web knowledge base is

TBox

TBox defines the ontology that serves as conceptual view over the data in the ABox

Terminology

• Classes = Concepts ($B, C, ...$)
• Properties = Roles ($R, E, ...$)

A TBox $T$ is a set of terminological axioms

• in form of inclusion and equivalences between
• concepts: $B \sqsubseteq C$ or $B \equiv C$
• roles: $R \sqsubseteq E$ or $R \equiv E$

ABox

an ABox - set of assertions that

• state membership of individuals to concepts
• $C(a)$
• and role membership for pairs
• $R(a, b)$

Constructing a DL Knowledge Base

"Ingridients":

• a vocabulary $\langle C, R, O \rangle$:
• set $C$ of atomic concepts ($A, B, ...$)
• set $R$ of atomic roles ($P, Q, ...$)
• and set $O$ of individuals ($a, b, c, ...$)
• a set of constructs for building complex concept and roles from atomic ones
• a language of axioms for stating the constraints on the vocabulary
• used to express domain constraints

Constructs and Axioms

Conjunction construct $\sqcap$

• $\text{Student} \sqcap \text{Researcher}$
• this is a complex concept build from atomic concepts $\text{Student}$ and $\text{Researcher}$

Inclusion $\sqsubseteq$ and equivalence $\equiv$ axioms

• We can relate any concepts (atomic and complex) to atomic concepts
• e.g.
• $\text{PhDStudent} \sqsubseteq \text{Student} \sqcap \text{Researcher}$
• $\text{PhDStudent} \equiv \text{Student} \sqcap \text{Researcher}$

Restriction constructs

• value restriction: $\forall \ R.C$ (owl:allValuesFrom)
• existential restriction: $\exists \ R.C$ (owl:someValuesFrom)

Examples

• $\text{MathStudent} \equiv \text{Student} \ \sqcap \ \forall \text{RegisteredTo} . \text{MathCourse}$
• a math student is a student is he's a student and registered to math courses only
• $\text{MathStudent} \equiv \text{Student} \ \sqcap \ \exists \text{RegisteredTo} . \text{MathCourse}$
• a math student is a student is he's a student and registered to at least one math course

Inclusion axiom $\sqsubseteq$

• expresses relation between concepts / roles
• left side: more specific, right side: more general
• e.g.
• $\text{MathCourse} \sqsubseteq \text{Course}$ (concepts)
• $\text{LateRegisteredTo} \sqsubseteq \text{RegisteredTo}$ (roles)

General Inclusion Axioms

General Inclusion Axioms (CGIs)

• inclusions between complex concepts

Example

• $\exists \text{TeachesTo} . \text{UndergraduateStudent} \sqsubseteq \text{Professor} \sqcup \text{Lecturer}$
• only professor or lecturer may teach undergraduate students
• in OWL it will be the following
_:a rdfs:subClassOf owl:Restriction
_:a owl:onProperty :TeachesTo
_:b owl:unionOf (:Professor :Lecturer)
_:a rdfs:subClassOf _:b

_:a and _:b are just blank no-name nodes

DL-Lite

In DL, reasoning is not always tractable

• but there's a trade-off between expressiveness and tracability
• DL-Lite $\subset$ DL
• Conjunctive Queries can be run over such DL-Lite KBs

Allowed:

• Constructs:
• unqualified existential restriction on roles, inverse of roles ($\exists R$ and $\exists R^-$)
• negation on roles
• Axioms in TBox
• $B \sqsubseteq C$ and $B \sqsubseteq \lnot C$
• where $B$ and $C$ are atomic concepts or existential restrictions
• negation allowed only in the right side of inclusion statements

Also, there are two families of DL-Lite:

DL-Lite${}_R$

• allow role inclusion statements
• $P \sqsubseteq Q$ or $P \sqsubseteq \lnot Q$
• where $P$ and $Q$ are atomic or inversion of atomic roles

DL-Lite${}_F$

• allow functional statements on roles
• $(\text{funct} P)$ or $(\text{funct} P^-)$