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
- and role membership for pairs
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
_:a owl:someValuesFrom :Undergraduate
_: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 $(\text{funct} P^-)$
See Also
Sources
- Web Data Management, Manolescu, Ioana, et al. [1]