DL related work

Networks, unary/binary predicates, SROIQ/OWL 2 DL, Lloyd–Topor, ABox↔TBox

[NaBr2003] summarises DL terminology: in the representation system, a concept denotes a set of individuals; a terminology is the intensional structure over the domain. Non-logical, GUI-oriented approaches (semantic networks, frames) used ad hoc structures; they lacked precise semantics until DL-style accounts. Unary predicates denote sets of individuals; binary predicates denote relationships. Reasoning in fragments of first-order logic yields different complexities.

The example network (generalised from p. 9 of [NaBr2003]) uses ◊is / subclass links, role ◊Child with domain/range and cardinality intuitions, and class ^Female with a .Gender default — compare Gender partitioning on taoke.de. The canonical page notes a modelling caveat: if ^ANM (animals) is a superclass of ^Human, “every mother is a woman” may fail unless “woman” is restricted to humans.

An atomic concept corresponds to a unary predicate; an atomic role to a binary predicate, mapped to triples as on the main DL page. Concept expressions are variable-free; intersection C ⊓ D aligns with C(x) ∧ D(x) in FOL over the domain. Open-world assumption and possibly infinite domains distinguish DL from many database-oriented modelling languages [NaBr2003].

Examples such as ≡Human∧¬Female vs Human ⊓ ¬Female and ^Male, ^Female linked via ◊EQ to concept expressions illustrate relator-style diagrams (Fig. DL-CM00 on taoke.de). Role intersection (e.g. “has-daughter” as ◊Child ⊓ ◊Female) and cardinality constraints appear in shorthand such as Woman ⊓ (≤ 2 ◊Child ⊓ ◊Female).

Following [Rudo2011], the expressive DL SROIQ underpins OWL 2 DL (decidable reasoning). Illustrative axioms on taoke.de include: subsumption Actor ⊑ Artist; role quantification mixing ∃knows.Actor and ∀hasfriend.Envious; role inverses; cardinality (Polygamist ⊑ >2 Married.⊤); nominals; role chains hasChild⁻ ◦ hasChild ⊑ hasSibling; and regular role hierarchies with formal conditions on role inclusion axioms.

Concrete OWL syntax and profiles are covered in [AlHe2020].

Axioms can be normalised using equivalences such as {A ⊓ B ⊑ C} ⇔ {A ⊑ C, B ⊑ C} and {A ⊑ B ⊓ C} ⇔ {A ⊑ B, A ⊑ C}. A common normalisation uses C ⊑ D ⇔ ⊤ ⊑ ¬C ⊔ D (canonical “Normalization axiom” on taoke.de).

With nominals, ABox assertions can be rewritten as TBox inclusions, e.g. C(a) ⇔ {a} ⊑ C, r(a,b) ⇔ {a} ⊑ ∃r.{b}, negated role assertions and equality/inequality similarly (canonical list on taoke.de).

Role chain axioms relate to their inverses by reversing the chain and inverting each role. Self restrictions appear in examples such as PersonCommittingSuicide ≡ ∃kills.Self, Narcissist ≡ ∃loves.Self. The page contrasts open-world vs. closed-world perspectives and mentions that nominals help express “nothing but” statements but do not fully encode CWA.

RDF’s triple structure requires encoding complex axioms with blank nodes and list structures [AlHe2020]; the canonical text points to examples in [Rudo2011] for graph-shaped encodings.