Description logic

DL has a self-describing naming system [Baader et al, 2007, Appendix: DL Terminology]. Unfortunately, it is not entirely consistent across the literature.

My attempt at a summary is below. See also Evgeny Zolin’s encyclopedic page on complexity of reasoning in description logics .

• $\mathcal{A}\mathcal{L}$ [Attributive Concept Language, the most basic DL]
  • Atomic concept ($A$), universal concept ($\top$), bottom concept ($\bot$)
  • Atomic negation ($\neg A$)
  • Concept intersection ($C \cap D$)
  • Value restriction ($\forall R.C$)
  • Limited existential quantification ($\exists R.\top$)
  • Concept hierarchy ($C \subseteq D$, $C \equiv D$)
• $\mathcal{A}\mathcal{L}\mathcal{C}$ (equivalently $\mathcal{A}\mathcal{L}\mathcal{U}\mathcal{E}$)
  • $\mathcal{C}$: Concept negation ($\neg C$)
  • $\mathcal{U}$: Concept union ($C \cup D$)
  • $\mathcal{E}$: “Full” existential quantification ($\exists R.C$)
• $\mathcal{H}$: Role hierarchies ($R \subseteq S$, $R \equiv S$)
• $\mathcal{I}$: Inverse roles ($R^{-}$)
• $\mathcal{N}$: (Unqualified) number restriction ($\geq n~R$, $\leq n~R$, $= n~R$)
• $\mathcal{Q}$: Qualified number restriction ($\geq n~R.C$, $\leq n~R.C$, $= n~R.C$)
• $\mathcal{O}$: Nominals [class literals] ($\{x_1,x_2,\ldots,x_n\}$)
• $\mathcal{F}$: Depending on author, EITHER
  • Functional roles ($\leq 1~R$), a restricted form of $\mathcal{Q}$, OR
  • Agreement and disagreement [Baader et al, 2007, Table A.1]
• $\mathcal{R}$: Depending on author, EITHER
  • Role intersection ($R \cap S$), OR
  • Regular role inclusion axioms [regular RIAs] ($R_1 \circ R_2 \circ \cdots \circ R_n \subseteq S$)
    • Regular means acyclic, which ensures decidability
    • May include other features: disjoint roles, reflexive, irreflexive, etc.
• $(\mathcal{D})$: Concrete domains, e.g., natural numbers with $<,\leq,=,>,\geq$

• $\mathcal{S}$ = $\mathcal{A}\mathcal{L}\mathcal{C}$ + transitive relations [as declaration, not as transitive closure]
• OWL 1 Lite = $\mathcal{SHIF}$ (ref )
• OWL 1 DL = $\mathcal{SHION}$ (ref )
• OWL 2 DL = $\mathcal{SRIOQ}$ (ref )

Books

• Baader et al, 2007: The Description Logic Handbook, 2nd ed.
  • Most comprehensive reference on DL
  • Covers OWL 1 (Ch. 14) but not OWL 2
• Robinson & Bauer, 2011: Introduction to Bio-ontologies
  • DL from a bioinformatics perspective
• Hitzler et al, 2010: Foundations of Semantic Web Technologies
  • Coverage of RDFS and OWL 2 with more mathematical formality than usual

Surveys and tutorials

• Rudolf, 2011: Foundations of description logics (doi, pdf)
• Krotzsch, Simancik, Horrocks, 2012: A description logic primer (arxiv)

DL in Semantic Web

• Horrocks, Kutz, Sattler, 2005: The irresistible SRIQ (pdf)
• Horrocks, Kutz, Sattler, 2006: The even more irresistible SROIQ (pdf)
• Knorr, Alferes, Hitzler, 2011: Local closed world reasoning with description logics under the well-founded semantics (doi, pdf)
• Sengupta, Krisnadhi, Hitzler: 2011: Local closed world semantics: Grounded circumscription for OWL (doi, pdf)

Critical perspectives on DL

• Doyle & Patil, 1991: Two theses of KR: Language restrictions, taxonomic classification, and the utility of representation services (doi, pdf)
  • Cogent criticisms of the restricted language thesis and restricted classification thesis
• Hitzler, 2009: Towards reasoning pragmatics (doi)