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


  • 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 $R \equiv E$


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


  • 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)


  • $\text{MathStudent} \equiv \text{Student} \ \sqcap \ \forall \text{RegisteredTo} . \text{MathCourse}$
    • 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


  • $\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


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


  • 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:


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


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

See Also


  • Web Data Management, Manolescu, Ioana, et al. [1]